Intensity of Powder Diffraction Peaks

8.6 Intensity of Powder Diffraction Peaks

Any powder diffraction pattern is composed of multiple Bragg peaks, which have different intensities in addition to varying positions and shapes. Numerous factors have either central or secondary roles in determining peak intensities. As briefly mentioned in Sect. 8.1 (Table 8.2), these factors can be grouped as: (i) structural factors , which are determined by the crystal structure; (ii) specimen factors owing to its shape and size, preferred orientation, grain size and distribution, microstructure and other parameters of the sample, and (iii) instrumental factors, such as proper- ties of radiation, type of focusing geometry, properties of the detector, slit and/or monochromator geometry.

The two latter groups of factors may be viewed as secondary, so to say, they are less critical than the principal part defining the intensities of the individual diffrac- tion peaks, which is the structural part. 34 Structural factors depend on the internal (or atomic) structure of the crystal, which is described by relative positions of atoms in the unit cell, their types and other characteristics, such as thermal motion and population parameters. In this Chapter, we consider secondary factors in addition to introducing the concept of the integrated intensity, while Chap. 9 is devoted to the major component of Bragg peak intensity – the structure factor.

8.6.1 Integrated Intensity

Consider the Bragg peak, which is shown in Fig. 8.17, and let us try to answer the question: which quantity most adequately describes its intensity, that is, what is the combined result of scattering from a series of crystallographic planes (hkl) or, which is the same, from the corresponding point in the reciprocal lattice? Is it the height of the peak (i.e., the Y coordinate of the highest point)? Is it the area under the peak? Is it something else?

The value of the peak maximum (Y max ) is intuitively, and often termed as its intensity. It can be easily measured, and is indeed used in many applications where relative intensities are compared on a qualitative basis, for example, when searching for a similar pattern in a powder diffraction database. This approach to measuring intensity is, however, unacceptable when quantitative values are needed, because both the instrumental and specimen factors may cause peak broadening, which may

be different for identical Bragg peaks produced by the same crystalline material. On the other hand, the area under the peak remains unchanged in most cases, even when substantial broadening, especially anisotropic, is present (see (8.21) indicating that Bragg peak intensity is a multiplier applied to the corresponding peak-shape

34 Some of the external factors, e.g., preferred orientation (see Sect. 8.6.6), may have a tremendous effect on the diffracted intensity. However, all secondary factors have similar or identical effects

on the diffracted intensity, regardless of the crystal (atomic) structure of the material.

8.6 Intensity of Powder Diffraction Peaks 183

Y max

I hkl Y i

(arb. units) Y

Bragg angle, 2θ (deg.)

Fig. 8.17 The relationship between the measured shape (the open circles connected with the solid lines ) and the integrated intensity (shaded area) of the Bragg peak. The background is shown using the dash-dotted line. The maximum measured intensity is indicated as Y max . The measured intensities and the corresponding values of the background are indicated for one of the points as Y i and b i , respectively.

function, which has unit area). Thus, the shaded area in Fig. 8.17 is known as the integrated intensity, and it represents the true intensity of Bragg peaks in powder diffraction.

The intensity, I hkl , scattered by a reciprocal lattice point (hkl) corresponds to the integrated intensity of the matching Bragg peak. For simplicity, it is often called “intensity.” What is actually measured in a powder diffraction experiment is the intensity in different points of the powder pattern, and it is commonly known as profile intensity. Profile intensity is usually labeled Y i , where i is the sequential point number, normally beginning from the first measured data point (i = 1).

Assuming that powder diffraction data were collected with a constant step in 2 θ, the area of an individual peak may be calculated simply by adding the intensities (Y -coordinates) of all points measured within the range of the peak after the con- tribution from the background has been subtracted in every point. The background is shown as a nearly horizontal dash-dotted line in Fig. 8.17. The observed inte- grated intensity (I hkl ) of a Bragg peak (hkl) is, therefore, determined from numerical integration as:

I hkl obs = ∑ Y i −b i

i =1

where j is the total number of data points measured within the range of the peak. 35 35 Strictly speaking, each Bragg peak begins and ends when its contribution becomes indistin-

guishable from that of the background. The determination of peak range at its base presents a challenging numerical problem since (i) diffracted intensity is always measured with a finite error; (ii) it is nearly impossible to achieve zero background, and (iii) Bragg peaks often overlap with

184 8 The Powder Diffraction Pattern The integrated intensity is a function of the atomic structure, and it also depends

on multiple factors, such as certain specimen and instrumental parameters. Consid- ering (7.7) and after including necessary details, earlier grouped as “geometrical” effects, the calculated integrated intensity in powder diffraction is expressed as the following product:

(8.41) where

hkl =K×p hkl ×L θ ×P θ ×A θ ×T hkl ×E hkl × |F hkl |

– K is the scale factor, that is, it is a multiplier required to normalize experimen- tally observed integrated intensities with absolute calculated intensities. Absolute calculated intensity is the total intensity scattered by the content of one unit cell in the direction ( θ), defined by the length of the corresponding reciprocal lattice vector. Therefore, the scale factor is a constant for a given phase and it is de- termined by the number, spatial distribution, and states of the scattering centers (atoms) in the unit cell.

–p hkl is the multiplicity factor, that is, it is a multiplier which accounts for the presence of multiple symmetrically equivalent points in the reciprocal lattice, or in other words, the number of symmetrically equivalent reflections.

–L θ is Lorentz multiplier, which is defined by the geometry of diffraction. –P θ is the polarization factor, that is, it is a multiplier, which accounts for a partial

polarization of the scattered electromagnetic wave (see the footnote and Thom- son’s equation in Sect. 7.1.1, p. 136).

–A θ is the absorption multiplier, which accounts for absorption of both the incident and diffracted beams and nonzero porosity of the powdered specimen. –T hkl is the preferred orientation factor, that is, it is a multiplier, which accounts for possible deviations from a complete randomness in the distribution of grain orientations.

–E hkl is the extinction multiplier, which accounts for deviations from the kinemat- ical diffraction model. In powders, these are quite small and the extinction factor is nearly always neglected.

–F hkl is the structure factor (or the structure amplitude), which is defined by the details of the crystal structure of the material: coordinates and types of atoms, their distribution among different lattice sites, and thermal motion.

The subscript hkl indicates that the multiplier depends on both the length and di- rection of the corresponding reciprocal lattice vector d ∗ hkl . Conversely, the subscript

one another. Thus, for all practical purposes, the beginning and the end of any Bragg peak (i.e., its width at the base) is usually assumed in terms of a certain number of full widths at half maxi- mum to the left and to the right from peak maximum. For Bragg peaks, which are well-represented by pure Gaussian distribution, the number of FWHM’s can be limited to 2–3 on each side, while in the case of nearly Lorentzian distribution this number should be increased substantially (see Fig. 8.12). In some instances, the number of FWHM’s can reach 10–20. It is also possible to de- fine peak limits in terms of maximum intensity, for example, a peak extends only as far as profile intensity (Y i ) remains greater or equal than a certain small predetermined fraction of the maximum intensity (Y max ).

8.6 Intensity of Powder Diffraction Peaks 185 θ indicates that the corresponding parameter is only a function of Bragg angle

and, thus it only depends upon the length of the corresponding reciprocal lattice vector, d ∗ hkl .

8.6.2 Scale Factor

As described earlier, the amplitude of the wave (and thus, the intensity, see (7.1)–(7.7) and relevant discussion in Sect. 7.1) scattered in a specific direction by a crystal lattice is usually calculated for its symmetrically independent minimum – one unit cell. In order to compare the experimentally observed and the calculated intensities directly, it is necessary to measure the absolute value of the scattered intensity. This necessarily involves

– Measuring the absolute intensity of the incident beam exiting through the slits and reaching the sample. – Precise account of inelastic and incoherent scattering, and absorption by the sam- ple, sample holder, air, and other components of the system, such as windows of

a sample attachment, if any. – Measuring the portion of the diffracted intensity that passes through receiving slits, monochromator and detector windows. – Correction for efficiency of the detector, number of events generated by a single photon, detector proportionality, etc., all of which must be precise and repro- ducible.

– Knowledge of many other factors, such as the volume of the specimen which par- ticipates in scattering of the incident beam, the fraction of the irradiated volume which is responsible for scattering precisely in the direction of the receiving slit, and so on.

Obviously, doing all this is impractical, and in reality the comparison of the ob- served and calculated intensities is nearly always done after the former are normal- ized with respect to the latter using the so-called scale factor. As long as all observed intensities are measured under nearly identical conditions (which is relatively easy to achieve), the scale factor is a constant for each phase, and is applicable to the entire diffraction pattern.

Thus, scattered intensity is conventionally measured using an arbitrary relative scale, and the normalization is usually performed by analyzing all experimental and calculated intensities using a least squares technique. 36 The scale factor is one of the variables in structure refinement and its correctness is critical in achieving the

36 In certain applications, e.g., when the normalized structure factors should be calculated (see Sect. 10.2.2), the knowledge of the approximate scale factor is required before the model of the

crystal structure is known. This can be done using various statistical approaches taking into ac- count that the structure factor for the 000 reflection is equal to the number of electrons in the unit cell [e.g., see A.J.C Wilson, Determination of absolute from relative X-ray intensity data, Nature (London) 150, 151 (1942)], consideration of which is beyond the scope of this book.

186 8 The Powder Diffraction Pattern best agreement between the calculated and observed intensities. 37 Its value is also

essential in quantitative analysis of multiple phase mixtures.

8.6.3 Multiplicity Factor

As we established earlier, a powder diffraction pattern is one-dimensional, but the associated reciprocal lattice is three-dimensional. This translates into scatter- ing from multiple reciprocal lattice vectors at identical Bragg angles. Consider two points in a reciprocal lattice, 00l and 00 ¯l. By examining (8.2)–(8.7), it is easy to

see that in any crystal system 1 /d 2 (00l) = 1/d 2 (00¯l). Thus, Bragg reflections from these two reciprocal lattice points are observed at exactly the same Bragg angle. Now consider the orthorhombic crystal system. Simple analysis of (8.5) indicates that the following groups of reciprocal lattice points will have identical reciprocal lattice vector lengths and thus, are equivalent in terms of the corresponding Bragg angle:

h 00 and ¯h00 – 2 equivalent points 0k0 and 0¯k0

– 2 equivalent points 00l and 00 ¯l

– 2 equivalent points hk

0, ¯hk0, h¯k0 and ¯h¯k0 – 4 equivalent points

h 0l, ¯h0l, h0 ¯l and ¯h0 ¯l – 4 equivalent points 0kl, 0¯kl, 0k ¯l and 0¯k ¯l

– 4 equivalent points hkl , ¯hkl, h¯kl, hk ¯l, ¯hk ¯l, ¯h¯kl, ¯h¯k ¯l and hkl – 8 equivalent points

Assuming that the symmetry of the structure is mmm, these equivalent recipro- cal lattice points have the same intensity, in addition to the identical Bragg angles. Consequently, in general there is no need to calculate intensity separately for each reflection in a group of equivalents. It is enough to calculate it for one of the corre- sponding Bragg peaks, and then multiply the calculated intensity by the number of the equivalents in the group, that is, by the multiplicity factor. The multiplicity fac- tor is, therefore, a function of lattice symmetry and combination of Miller indices. In the example considered here (orthorhombic crystal system with point group sym- metry mmm), the following multiplicity factors could be assigned to the following types of reciprocal lattice points:

p hkl = 2 for h00, 0k0 and 00l p hkl = 4 for hk0, 0kl and h0l, and p hkl = 8 for hkl

Reciprocal lattices and therefore, diffraction patterns are generally centrosym- metric, regardless of whether the corresponding direct lattices are centrosymmetric

37 The correctness of the scale factor is dependent on many parameters. The most critical are: the photon flux in the incident beam remains identical during measurements at any Bragg angle;

the volume of the material producing scattered intensity is constant; the number of crystallites approaches infinity and their orientations are completely random; the background is accounted precisely; the absorption of X-rays (when relevant) is accounted.

8.6 Intensity of Powder Diffraction Peaks 187 or not. Thus, pairs of reflections with the opposite signs of indices, (hkl) and (¯h¯k ¯l)

– the so-called Friedel pairs – usually have equal intensity. Yet, they may be dif- ferent in the presence of atoms that scatter anomalously (see Sect. 9.1.3) and this phenomenon should be taken into account when multiplicity factors are evaluated comprehensively. Relevant details associated with the effects of anomalous scatter- ing on the multiplicity factor are considered in Sect. 9.2.2.

8.6.4 Lorentz-Polarization Factor

The Lorentz factor takes into account two different geometrical effects and it has two components. The first is owing to finite size of reciprocal lattice points and finite thickness of the Ewald’s sphere, and the second is due to variable radii of the Debye rings. Both components are functions of θ.

Usually, the first component is derived by considering a reciprocal lattice rotating at a constant angular velocity around its origin. Under these conditions, various reciprocal lattice points are in contact with the surface of the Ewald’s sphere for different periods of time. Shorter reciprocal lattice vectors are in contact with the sphere for longer periods when compared with longer vectors. In powder diffraction, this contribution arises from the varying density of the equivalent reciprocal lattice points resting on the surface of the Ewald’s sphere, which is a function of d ∗ . It can

be shown that the first component of the Lorentz factor is proportional to 1 / sin θ. The second component accounts for a constant length of the receiving slit. As a result, a fixed length of the Debye ring is always intercepted by the slit regardless of Bragg angle. The radius of the ring (r D ) is, however, proportional to sin 2 θ. 38 Because the scattered intensity is distributed evenly along the circumference of the ring, the intensity that reaches the detectors becomes inversely proportional to r D and, therefore, directly proportional to 1 / sin 2θ.

The two factors combined result in the following proportionality:

sin θ sin 2θ

which after recalling that sin 2 θ = 2 sin θ cos θ and ignoring all constants (which are absorbed by the scale factor), becomes

cos θ sin 2 θ

The polarization factor arises from partial polarization of the electromagnetic wave after scattering. Considering the orientation of the electric vector, the partially polarized beam can be represented by two components: one has its amplitude paral- lel (A || ) to the goniometer axis and another has the amplitude perpendicular (A ⊥ ) to

38 This proportionality holds as long as the distance between the specimen and the receiving slit of the detector remains constant at any Bragg angle.

188 8 The Powder Diffraction Pattern the same axis. The diffracted intensity is proportional to the square of the amplitude

and the two projections of the partially polarized beam on the diffracted wavevector are proportional to 1 for (A || ) 2 and cos 2 2 θ for (A ⊥ ) 2 . Thus, partial polarization af- ter scattering yields the following overall factor (also see Thomson equation in the footnote on page 136):

1 + cos 2 2 θ

When a monochromator is employed, it introduces additional polarization, which is accounted as:

where 2 θ M is the Bragg angle of the reflection from a monochromator (it is a con- stant for a fixed wavelength), and K is the fractional polarization of the beam. For neutrons K = 0; for unpolarized and unmonochromatized characteristic X-ray radi- ation, K = 0.5 and cos 2θ M = 1, while for a monochromatic or synchrotron radiation K should be established experimentally (i.e., measured) or refined.

The Lorentz and polarization contributions to the scattered intensity are nearly always combined together in a single Lorentz-polarization factor, which in the case when no monochromator is employed is given as:

or assuming K = 0.5 with a crystal monochromator

Once again, all constant multipliers have been ignored in (8.46) and (8.47). The Lorentz-polarization factor is strongly dependent on the Bragg angle as shown in Fig. 8.18. It is near its minimum between ∼80 ◦ and ∼120 ◦ 2 θ, and increases sub- stantially both at low and high angles. The latter (above approximately 150 ◦ 2 θ) are usually out of range in most routine powder diffraction experiments. As is easy to see from Fig. 8.18, additional polarization caused by the presence of a monochro- mator results in a small change in the behavior of the Lorentz-polarization factor, but it must be properly accounted for, especially when precision of diffraction data is high.

8.6.5 Absorption Factor

Absorption effects in powder diffraction are dependent on both the geometry and properties of the sample and the focusing method. For example, when a flat sample is studied using the Bragg–Brentano technique, the scattered intensity is not affected

8.6 Intensity of Powder Diffraction Peaks 189 50

30 No monochromator LP

10 2θ M ≅ 26.5 o

Bragg angle, 2θ (deg.)

Fig. 8.18 Lorentz-polarization factor as a function of Bragg angle: the solid line represents calcu- lation using (8.46) (no monochromator), and the dash-dotted line is calculated assuming graphite monochromator and Cu K α radiation with K = 0.5 (8.47).

by absorption as long as the specimen is highly impermeable, homogeneous and thick enough so that the incident beam never penetrates all the way through the sample at any Bragg angle. On the contrary, absorption by a thick flat sample in the transmission geometry has considerable influence on the scattered intensity, much stronger than if a thin sample of the same kind is under examination.

When X-rays penetrate into the matter, they are partially transmitted, and par- tially absorbed. Thus, when an X-ray beam travels the infinitesimal distance, dx, its intensity is reduced by the infinitesimal fraction dI /I (Fig. 8.19a), which can be defined using the following differential equation:

where µ is the proportionality coefficient expressed in the units of the inverse dis- tance, usually in cm −1 . This coefficient is also known as the linear absorption coef- ficient of a material.

The linear absorption coefficient of any chemical element is a function of the wavelength (photon energy), and both µ(λ) and µ(E) dependencies of Fe and Gd are shown in Fig. 8.20. In the range of wavelengths, which are of interest to powder diffraction, the µ(λ) functions consist of several continuous branches separated by abrupt changes in the absorption properties at certain, element specific wavelengths. The points, at which the discontinuities of the absorption coefficient occur, are called

190 8 The Powder Diffraction Pattern dx

Total path: l = x I t +x S beam d x I Sca ttere

Inciden

beam I 0 I t

b dV x S

Fig. 8.19 Schematic explaining the phenomenon of absorption of X-rays by the matter (a) and the illustration of the derivation of (8.51) (b). The incident beam penetrates into the sample by the

distance x I before being scattered by the infinitesimal volume dV. The scattered beam traverses the distance x S before exiting the sample. In the Bragg–Brentano geometry x I =x S .

Energy, E (keV) Energy, E (keV) )

cient … ffi 1000

cient ffi 1000

b Wavelength, λ (Å) Fig. 8.20 The behavior of the linear absorption coefficients of the elemental iron (a) and gadolin-

a Wavelength, λ (Å)

ium (b) as functions of the wavelength of the X-rays (bottom scale) and photon energy (top scale). The numerical data used to prepare both plots have been taken from the National Institute of Stan- dards and Technology Physical Reference Data web page. 39

absorption edges. A single absorption edge – the K edge – is observed at the shortest wavelength, the next set of edges is called L, followed by the M edges, and so on. The number of absorption edges increases as the electronic structure of the element becomes more complex. For example, when 0 < λ ≤ 13 ˚A, iron has a single K absorption edge, but Gd in addition to the K edge has three L edges, followed by five M absorption edges. The linear absorption coefficient changes its value by a factor of 6–8 at the K edge; the relative changes become much smaller at the majority of L and M edges.

The continuous change of the liner absorption along each of the two branches is approximately defined as µ = kZ 3 λ 3 , where Z is the atomic number of the chemi- cal element and k is a constant, specific for each of the two continuous parts of the

39 http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html.

8.6 Intensity of Powder Diffraction Peaks 191 absorption function. The continuous branches correspond to the absorption occur-

ring due to random scattering of photons by electrons, which is observed in all direc- tions, thus reducing the number of photons in the transmitted beam in the direction of the propagation vector.

The appearance of the discontinuities is known as the true absorption, and it can

be understood by considering (6.3). As the wavelength increases, the energy of the X-ray photons decreases and at a certain λ it matches the energies required to excite K electrons from their ground states for the K edge, L electrons for the L edges, M electrons for the M edges, and so on. 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 vacant K (L, M, . . . ) levels in the atoms of the absorber – a photoelectric effect, during which a fluorescent X-ray photon can be emitted in any direction. Both scattered and true absorption result in the reduction of the transmitted intensity, as defined by (8.48).

Absorption coefficients for all chemical elements are usually tabulated (see Table 8.3) in the form of mass absorption coefficients 40 µ/ρ (the units are cm 2 /g), instead of the linear absorption coefficients, µ. The linear absorption coefficient of any material (solid, liquid, or gas) is then calculated as:

µ=ρ m ∑ w i

i =1

where w i is the mass fraction of the chemical element in the material, ( µ/ρ) i is elemental mass absorption coefficient, and ρ m is the density of the material. For example, the liner absorption coefficient of the stoichiometric mixture of gaseous

hydrogen and oxygen (2 mol of H 2 per 1 mol of O 2 ) is only ∼ 1/1,200 of the linear absorption coefficient of water because the density of water is just over 1,200 times greater than the density of the mixture of the two gases at atmospheric pressure and room temperature.

After integrating (8.48), the transmitted intensity (I t ) is easily calculated in terms

of a fraction of the initial intensity of the beam (I 0 ):

(8.50) Hence, the intensity of the X-ray beam or other type of radiation is reduced due

I t =I 0 exp (−µx)

to absorption after passing through a layer of a material with a finite thickness. Now, consider Fig. 8.19b, where the incident beam is scattered by the infinitesimal volume

dV of the flat sample in the reflection geometry. The total path of both the incident and diffracted beams through the sample is l =x I +x S . Thus, to calculate the ef- fect of absorption in this and in any other geometry, it is necessary to perform the integration over the entire volume of the specimen which contributes to scattering.

40 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 is directly proportional to the

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

192 8 The Powder Diffraction Pattern Table 8.3 Mass absorption coefficients (in cm 2 /g) of selected chemical elements for the com-

monly used anode materials. 41 The mass absorption coefficients of the best β-filter elements (see Sect. 11.2.2) for the corresponding anode materials are underlined.

Element \ Cr Fe Cu Mo Anode

K β K α K β H 0.412

Taking into account (8.50), the following integral equation expresses the reduc- tion of the diffracted intensity, A, as the result of absorption: 42

It is important to recognize that an effective linear absorption coefficient, µ eff , has been introduced into (8.51) to account for a lower density of dusted or packed powder when compared with the linear absorption coefficient, µ, of the bulk. The latter is usually used in diffraction from single crystals.

41 Taken from the International Tables for Crystallography, vol. C, Third edition, E. Prince, Ed., published jointly with the International Union of Crystallography (IUCr) by Springer (2004).

42 In single crystal diffraction, absorption correction is usually applied to the observed intensities and therefore, A is sometimes called the transmission factor, while the corresponding absorption

correction is A ∗ = 1/A.

8.6 Intensity of Powder Diffraction Peaks 193 Equation (8.51) can be solved analytically for all geometries usually employed

in powder diffraction. 43 For the most commonly used Bragg–Brentano focusing geometry, the two limiting cases are as follows:

– The material has very high linear absorption coefficient, or it is thick enough so that there is a negligible transmission of the incident beam through the sample at any Bragg angle. The resultant absorption factor in this case is a constant, and it is usually neglected in (8.41) because it becomes a part of the scale factor:

A = eff

– The material has low linear absorption, or the sample is thin so that the incident beam is capable of penetrating all the way through the sample. The absorption correction in this case is a function of Bragg angle as shown in (8.53). Once

again, the constant coefficient 1 / 2 µ eff is omitted since it becomes a part of the scale factor:

1 − exp(−2µ

eff t / sin θ)

∝ 1 − exp(2µ eff t / sin θ) (8.53)

eff

In (8.53), t is sample thickness. Ignoring the absorption correction, especially when µ and/or t are small, which means a weakly absorbing or thin sample, results in the underestimated calculated intensity at high Bragg angles. As a result, unphysical (negative) values of thermal displacement parameters are usually obtained.

The major difficulty in applying an absorption correction (8.53) arises from usu- ally unknown µ eff . Obviously, the linear absorption coefficient, µ, can be easily cal- culated when the dimensions of the unit cell and its content are known (see (8.49)), but it is applicable only for a fully dense sample. When a pulverized sample is used (and typically it is), µ eff cannot be determined easily without measuring sample den- sity. Often the combined parameter ( µ eff t ) can be refined or estimated and accounted in intensity calculations during Rietveld refinement (Sect. 15.7).

Another problem with pulverized samples is that their packing density varies as

a function of the depth. This is known as the porosity effect, and for the Bragg– Brentano geometry, it may be expressed using two different approaches: The first has been suggested by Pitschke et al. 44

43 Analytical integration necessarily assumes that µ eff remains constant, even though the irradiated area of the specimen surface changes as a function of Bragg angle, as discussed in Chap. 12,

Sects. 12.1.3 and 12.2.3. When diffraction from a single crystal is of concern, analytical solution of this equation is rarely possible and it is usually integrated numerically using the known dimensions of a single crystal and the orientations of both the incident and diffracted beams with respect to crystallographic axes for each individual reflection hkl.

44 W. Pitschke, N. Mattern, and H. Hermann, Incorporation of microabsorption corrections into Rietveld analysis, Powder Diffraction 8, 223 (1993).

194 8 The Powder Diffraction Pattern and the second by Suortti 45

a 1 + (1 − a 1 ) exp(−a 2 / sin θ)

a 1 + (1 − a 1 ) exp(−a 2 )

where a 1 and a 2 are two variables that can be refined. Both approximations also account for surface roughness as well as for absorption effects. They give practically identical results and the only difference is that the Suortti formula works better at low Bragg angles, according to Larson and Von Dreele. 46

Approximations given in (8.53)–(8.55) also account for some other effects that distort intensity, for example improper size of the incident beam, causing the beam to be broader than the sample at low Bragg angles. The refinement of the corre- sponding parameters may become unstable because of correlations with some struc- tural parameters (e.g., with the scale factor and/or thermal displacement parameters of atoms). Therefore, any of these corrections should be introduced and/or refined with care.

8.6.6 Preferred Orientation

Conventional theory of powder diffraction assumes completely random distribution of the orientations among the infinite amount of crystallites in a specimen used to produce a powder diffraction pattern. In other words, precisely the same fraction of the specimen volume should be in the reflecting position for each and every Bragg reflection. Strictly speaking, this is possible only when the specimen contains an in- finite number of crystallites. In practice, it can be only achieved when the number of

crystallites is very large (usually in excess of 10 6 –10 7 particles). Nonetheless, even when the number of crystallites approaches infinity, this does not necessarily mean that their orientations are completely random. The external shape of the crystallites plays an important role in achieving randomness of their orientations in addition to their number.

When the shapes of crystallites are isotropic, random distribution of their ori- entations is not a problem, and deviations from an ideal sample are usually neg- ligible. However, quite often the shapes are anisotropic, for example, platelet-like or needle-like and this results in the introduction of distinctly nonrandom crystal- lite orientations due to natural preferences in packing of the anisotropic particles. The nonrandom particle orientation is called preferred orientation, and it may cause considerable distortions of the scattered intensity.

Preferred orientation effects are addressed by introducing the preferred orien- tation factor in (8.41) and/or by proper care in the preparation of the powdered

45 P. Suortti, Effects of porosity and surface roughness on the X-ray intensity reflected from a powder specimen, J. Appl. Cryst. 5, 325 (1972).

46 C.A. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los Alamos National Laboratory Report, LAUR 86-748 (2000).

8.6 Intensity of Powder Diffraction Peaks 195 specimen. The former may be quite difficult and even impossible when preferred

orientation effects are severe. Therefore, every attempt should be made to physically increase randomness of particle distributions in the sample to be examined during

a powder diffraction experiment. The sample preparation is discussed in Sect. 12.1, and in this section we discuss the modeling of the preferred orientation by various functions approximating the radial distribution of the crystallite orientations.

Consider two limiting anisotropic particle shapes: platelet-like and needle-like. The platelets, when packed in a flat sample holder, tend to align parallel to one an- other and to the sample surface. 47 Then, the amount of plates that are parallel or nearly parallel to the surface is much greater than the amount of platelets that are perpendicular or nearly perpendicular to the surface. In this case, a specific direc- tion that is perpendicular to the flat sides of the crystallites is called the preferred orientation axis. It coincides with a reciprocal lattice vector d ∗T hkl that is normal to the flat side of each crystallite. Therefore, intensity of reflections from reciprocal lattice points with vectors parallel to d ∗T hkl is larger than intensity of reflections produced by any other point of the reciprocal lattice (minimum for those with reciprocal lattice vectors perpendicular to d ∗T hkl ) simply because the distribution of their orientations is highly anisotropic. The preferred orientation in cases like that is said to be uni- axial, and the preferred orientation axis is perpendicular to the surface of the flat specimen.

The needle-like crystallites, when packed into a flat sample, will also tend to align parallel to the surface. 47 However, the preferred orientation axis, which in this case coincides with the elongated axes of the needles, is parallel to the sample sur- face. In addition to the nearly unrestricted distribution of needles’ axes in the plane parallel to the sample surface (which becomes nearly ideally random when the sam- ple spins around an axis perpendicular to its surface), each needle may be freely rotated around its longest direction. Hence, if the axis of the needle coincides, for example, with the vector d ∗T hkl , then reflections from reciprocal lattice points with vectors parallel to d ∗T hkl are suppressed to a greater extent and reflections from recip- rocal lattice points with vectors perpendicular to d ∗T hkl are strongly increased. This example describes the so-called in-plane preferred orientation.

In both cases, the most affected is the intensity of Bragg peaks that correspond to reciprocal lattice points that have their corresponding reciprocal lattice vectors parallel or perpendicular to d ∗T hkl , while the effect on intensity of other Bragg peaks is intermediate. Hence the preferred orientation effect on the intensity of any reflection hkl can be described as a radial function of angle φ hkl between the corresponding vector d ∗ hkl and a specific d ∗T hkl , which is the preferred orientation direction. The angle φ hkl can be calculated from:

where d ∗ hkl is the reciprocal lattice vector corresponding to a Bragg peak hkl and

d ∗T hkl is the reciprocal lattice vector parallel to the preferred orientation axis. The 47 Also see the schematic shown in Fig. 12.3.

196 8 The Powder Diffraction Pattern

axis d *T hkl T || = 1/τ T || = 1/τ 3 T || =0.67

T ⊥ /T = 2.5

a ||

Fig. 8.21 Preferred orientation functions for needles represented by the ellipsoidal (a) and March– Dollase (b) functions with the magnitude T ⊥ /T || = 2.5, and the two functions overlapped when T ⊥ /T || = 1.5 (c). The two notations, T || and T ⊥ , refer to preferred orientation corrections in the directions parallel and perpendicular to the preferred orientation (PO) axis, respectively.

numerator is a scalar product of the two vectors and the denominator is a product of the lengths of two vectors.

The simplest radial function that describes the anisotropic distribution of the pre- ferred orientation factor as a function of angle φ hkl is an ellipse (Fig. 8.21a), and the corresponding values of T hkl , which are used in (8.41), can be calculated using the following expression: 48

1 N T −1/2

hkl =

∑ i 1 +( τ 2 − 1)cos 2 φ

In (8.57) the multiplier T hkl is calculated as a sum over all N symmetrically equiv- alent reciprocal lattice points, and τ is the preferred orientation parameter refined against experimental data. The magnitude of the preferred orientation parameter is defined as T ⊥ /T || , where T ⊥ is the factor for Bragg peaks with reciprocal lattice vectors perpendicular, and T || is the same for those which are parallel to the pre- ferred orientation axis, respectively. In the case of the ellipsoidal preferred orien- tation function, T ⊥ /T || is equal to τ for the needles (in-plane preferred orientation) and 1 /τ for platelets (axial preferred orientation).

48 V.K. Pecharsky, L.G. Akselrud, and P.Y. Zavalij, Method for taking into account the influence of preferred orientation (texture) in a powdered sample by investigating the atomic structure of a

substance, Kristallografiya 32, 874 (1987). Engl. transl.: Sov. Phys. Crystallogr. 32, 514 (1987).

8.6 Intensity of Powder Diffraction Peaks 197

A different approach has been suggested by Dollase, 49 where the preferred ori- entation factor is represented by a more complex March–Dollase function:

1 N 1 −3/2 T =

N ∑ τ cos φ hkl + sin φ

Here, the preferred orientation magnitude T ⊥ /T || is τ 41 /2 for needles and its in- verse ( τ −41/2 ) is for plates. An example of the March–Dollase preferred orientation

function for needles with magnitude T ⊥ /T || = 2.5 is shown in Fig. 8.21b. In both cases ((8.57) and (8.58)) the preferred orientation factor T hkl is propor- tional to the probability of the point of the reciprocal lattice, hkl, to be in the re- flecting position (i.e., the probability of being located on the surface of the Ewald’s sphere). In other words, this multiplier is proportional to the amount of crystallites with hkl planes parallel to the surface of the flat sample.

Both approaches work in a similar way. In the case of platelet-like particles, the function is stretched along T || (T || >T ⊥ ), while in case of needles, it is stretched along T ⊥ (T || <T ⊥ ). Therefore, in both cases τ < 1 describes preferred orientation of the platelets and τ > 1 describes preferred orientation of the needles. Obviously τ = 1 corresponds to a completely random distribution of reciprocal lattice vectors and the corresponding radial distribution functions become a circle with unit radius (both (8.57) and (8.58) result in T hkl = 1 for any φ hkl ).

Both functions give practically the same result at low and moderate degrees of nonrandomness (i.e., at low preferred orientation contribution). The example with T ⊥ /T || = 1.5 is shown in Fig. 8.21c, where the two functions ((8.57) and (8.58)) are nearly indistinguishable. Unfortunately, strong preferred orientation cannot be adequately approximated by either of these functions, and the best way around it is to reduce the preferred orientation by properly preparing the sample.

The platelets and needles discussed here are the two limiting but still the simplest possible cases. Particles may (and often do) have shapes of ribbons. These particles will pack the same way needles do – parallel to the sample surface but the ribbons will not be randomly oriented around their longest axes – they will tend to align their flat sides parallel to the sample surface. This case should be treated using two different preferred orientation functions simultaneously: one along the needle and one perpendicular to its flat surface. Thus, both types of functions ((8.57) and (8.58)) can be modified as follows:

T total =k 0 + ∑ k i T i

i =1

where T total is the overall preferred orientation correction, N a is the number of dif- ferent preferred orientation axes, T i is the preferred orientation correction for the ith axis, and k i is the corresponding scale factor, which reflects the contribution of each

49 W.A. Dollase, Correction of intensities for preferred orientation in powder diffractometry: Ap- plication of the March model, J. Appl. Cryst. 19, 267 (1986).

198 8 The Powder Diffraction Pattern axis. Here, k 0 is the portion of the sample not affected by preferred orientation at

all. Equation (8.59) is sometimes used even when only one kind of the preferred orientation is present, thus giving the following very simple expression:

(8.60) Yet another approach, which is based on the algorithm described by Bunge, 50

T total = k + (1 − k)T hkl

uses spherical harmonics expansion to deal with preferred orientation in three di- mensions as a complex radial distribution:

T (h, y) = 1 +

mn m (h)k ∑ n 2l +1 ∑ ∑ C l k l l (y)

where – h represents reflection, and y sample orientations;

– L is the maximum order of a harmonic; –C mn l are harmonic coefficients; –k (h) and k(y) are harmonic factors as functions of reflection and sample orienta-

tions, respectively. The expression for harmonic factors is complex and is defined azimuthally by

means of a Lagrange function. Sample orientation in routine powder diffraction ex- periment is fixed, and so is the corresponding harmonic factor k (y), which simplifies (8.61) to:

T (h) = 1 + ∑ ∑ C m k m l l (h)

l =2 2l +1 m =−l

The magnitude of the preferred orientation can be evaluated using the following function:

J =1+

2l +1 ∑ |C l |

which is unity in the case of random orientation, otherwise J > 1. When all grains are perfectly aligned (single crystal) the function (8.63) becomes infinity.

Only even orders are taken into account in (8.62) and (8.63), due to the presence of the inversion center in the diffraction pattern. The number of harmonic coef- ficients C and terms k (h) varies, depending on lattice symmetry and desired har- monic orderL. The low symmetry results in multiple terms (triclinic has five terms for L = 2) and therefore, low orders 2 or 4 are usually sufficient. High symmetry requires fewer terms (e.g., cubic has only 1 term for L = 4), so higher orders may

be required to adequately describe preferred orientation. The spherical harmonics approach is realized in GSAS. 51

50 H.-J. Bunge, Texture analysis in materials science, Butterworth, London (1982). 51 R. B. Von Dreele, Quantitative texture analysis by Rietveld refinement, J. Appl. Cryst. 30, 517

8.6 Intensity of Powder Diffraction Peaks 199 Fig. 8.22 The illustration

Z of the complex distribution of reciprocal lattice vectors modeled using a spherical harmonic preferred orienta- tion function for the (100) reflection.

An example of the preferred orientation modeled using second- and fourth- order spherical harmonics in the orthorhombic crystal system [space group Cmc2 1 ,

C 0 = 0.17(1), C 2 = 1.65(1), C 0 = 0.17(2), C 2 2 2 4 4 = 0.04(1), C 4 4 = 0.56(2)] is shown in Fig. 8.22. Here the surface represents the probability of finding 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.

At present, 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 gradu- ally, and only as long as improvements are obvious, and the results make sense. An 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. In its full form, (8.61) may

be used in complex texture analysis, where powder diffraction data have been col- lected, not only as a function of Bragg angle 2 θ, but also at different orientations along Debye rings and with tilting the sample.

8.6.7 Extinction Factor

Extinction effects, which are dynamical in nature, may be noticeable in diffrac- tion from nearly perfect and/or large mosaic crystals. Two types of extinction are generally recognized: primary, which occurs within the same crystallite, and sec- ondary, which originates from multiple crystallites. Primary extinction is caused by back-reflection of the scattered wave into the crystal and it decreases the measured

200 8 The Powder Diffraction Pattern

ctor To dete

To detector hkl

hkl

Fig. 8.23 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.

scattered intensity (Fig. 8.23, left). Further, 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, pri- mary extinction effects are often smaller than experimental errors; however, when necessary, they may be included in (8.41) as: 52

(8.64) where E B and E L are Bragg (2 θ = π) and Laue (2θ = 0) components, both defined as

E hkl =E B sin 2 θ+E L cos 2 θ

various functions of the extinction parameter, x, which is normally a refined variable:

x = (KN λ F D ) c 2 hkl

(8.65) In (8.65), K is the shape factor (it is unity for a cube of edge D, K = 3 / 4 for a sphere

of diameter D, and K = 8 / 3 π for a cylinder of diameter D), λ is the wavelength, F hkl is the calculated structure amplitude and N c is the number of unit cells per unit volume. Secondary extinction (Fig. 8.23, 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 ex- tinction. It is not detected in diffraction from polycrystalline materials and therefore, is always neglected.

52 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).

8.8 Problems 201