Shapes of Powder Diffraction Peaks

8.5 Shapes of Powder Diffraction Peaks

All but the simplest powder diffraction patterns are composed from more or less overlapped Bragg peaks due to the intrinsic one-dimensionality of the powder dif- fraction technique coupled with the usually large number of “visible” reciprocal lattice points, that is, those that have d ∗ hkl ≤ 2/λ and the limited resolution of the in- strument (e.g., see the model in Figs. 8.8d and 8.2). Thus, processing of the data by fitting peak shapes to a suitable function is required in order to obtain both the posi- tions and intensities of individual Bragg peaks. The same is also needed in structure refinement using the full profile fitting approach – the Rietveld method.

The observed peak shapes are best described by the so-called peak-shape func- tion (PSF), which is a convolution 11 of three different functions: instrumental broad-

11 A convolution ( ⊗) of two functions, f and g, is defined as an integral

f (t) ⊗ g(t) =

f (τ)g(t − τ)dτ =

g (τ) f (t − τ)dτ

which expresses the amount of overlap of one function g as it is shifted over another function f . It, therefore, “blends” one function with another. The convolution is also known as “folding” (e.g., see E.W. Weisstein, Convolution, Eric Weisstein’s world of mathematics, http://mathworld. wolfram.com/Convolution.html).

8.5 Shapes of Powder Diffraction Peaks 169 ening, Ω, wavelength dispersion, Λ, and specimen function, Ψ. Thus, PSF can be

represented as follows:

(8.18) where b is the background function.

PSF (θ) = Ω(θ) ⊗ Λ(θ) ⊗ Ψ(θ) + b(θ)

The instrumental function, Ω, depends on multiple geometrical parameters: the locations and geometry of the source, monochromator(s), slits, and specimen. The wavelength (spectral) dispersion function, Λ, accounts for the distribution of the wavelengths in the source and it varies depending on the nature of the source, and the monochromatization technique. Finally, the specimen function, Ψ, originates from several effects. First is the dynamic scattering, or deviations from the kinemat-

ical model. They yield a small but finite width (the so-called Darwin 12 width) of the Bragg peaks. The second effect is determined by the physical properties of the specimen: crystallite (grain) size and microstrains. For example, when the crystal- lites are small (usually smaller than ∼1µm) and/or they are strained, the resultant Bragg peak widths may increase substantially.

It is worth noting that unlike the instrumental and wavelength dispersion func- tions, the broadening effects introduced by the physical state of the specimen may be of interest in materials characterization. Thus, effects of the average crystallite size ( τ) and microstrain (ε) on Bragg peak broadening (β, in radians) can be described in the first approximation as follows:

(8.20) where k is a constant, that depends on the definition of a microstrain. It is important

β = k · ε · tanθ

to note that β in (8.19) and (8.20) is not the total width of a Bragg peak but it is an excess width, which is an addition to all instrumental contributions. The latter is usually established by measuring a standard material without microstrain and grain- size effects at the same experimental conditions.

In general, three different approaches to the description of peak shapes can be used. The first employs empirical peak-shape functions, which fit the profile with- out attempting to associate their parameters with physical quantities. The second is a semi-empirical approach that describes instrumental and wavelength dispersion functions using empirical functions, while specimen properties are modeled using

12 Sir Charles Galton Darwin (1887–1962) the British physicist, who begun working with Ernest Rutherford and Niels Bohr, later using his mathematical skills to help Henry Moseley with his

work on X-ray diffraction. A brief biography is available on WikipediA at http://en.wikipedia.org/ wiki/Charles Galton Darwin.

170 8 The Powder Diffraction Pattern realistic physical parameters. In the third, the so-called fundamental parameters ap-

proach, 13 all three components of the peak-shape function (8.18) are modeled using rational physical quantities.

8.5.1 Peak-Shape Functions

Considering Figs. 8.8 and 8.9, and (8.18), the intensity, Y (i), of the ith point (1 ≤ i ≤ n , where n is the total number of measured points) of the powder diffraction pattern, in the most general form is the sum of the contributions, y k , from the m overlapped

individual Bragg peaks (1 ≤ k ≤ m) and the background, b(i). 14 Therefore, it can be described using the following expression:

Y (i) = b(i) + ∑ I k [y k (x k ) + 0.5y k (x k + Δx k )]

k =1

where: I k is the intensity of the kth Bragg reflection, x k = 2θ i –2 θ k and Δx k is the difference between the Bragg angles of the K α 2 and K α 1 components in the doublet (if present). The presence of Bragg intensity as a multiplier in (8.21) enables one to introduce and analyze the behavior of different normalized functions independently of peak intensity, that is, assuming that the definite integral of a peak-shape function, calculated from negative to positive infinity, is unity in each case.

The four most commonly used empirical peak-shape functions (y) are as follows: Gauss 15 :

1 C /2 y (x) = G(x) = G √ exp

y (x) = L(x) = L πH ′ 1 +C L x 2 −1

13 J. Bergmann, Contributions to evaluation and experimental design in the fields of X-ray pow- der diffractometry, Ph.D. thesis (in German), Dresden University for Technology (1984). See

http://www.bgmn.de/methods.html for more information and other references. 14 Several functions commonly used in approximating the background are discussed later, see

(13.1)–(13.6). 15 Johann Carl Friedrich Gauss (1777–1885) was the German mathematician. A brief biography is

available on WikipediA, http://en.wikipedia.org/wiki/Carl Friedrich Gauss. 16 Hendrik Antoon Lorentz (1853–1928) was a Dutch physicist best known for his contributions to

the theory of electromagnetic radiation. In 1902 he shared the Nobel Prize in physics with Pieter Zeeman “in recognition of the extraordinary service they rendered by their researches into the in- fluence of magnetism upon radiation phenomena.” See http://nobelprize.org/nobel prizes/physics/ laureates/1902/lorentz-bio.html for details.

8.5 Shapes of Powder Diffraction Peaks 171 Pseudo-Voigt 17 :

1 C /2 C

y (x) = PV (x) = η √ G exp

−C L

G x 2 + (1 − η )

(1 +C L x 2 ) −1 (8.24)

y (x) = PVII(x) =

(1 +C P x 2 ) −β (8.25)

Γ(β − / 2 ) πH

where

– H and H ′ , are the full widths at half maximum (often abbreviated as FWHM). –x = (2θ i − 2θ k )/H k , is essentially the Bragg angle of the ith point in the powder

diffraction pattern with its origin in the position of the kth peak divided by the peak’s FWHM.

–2 θ i , is the Bragg angle of the ith point of the powder diffraction pattern; –2 θ k , is the calculated (or ideal) Bragg angle of the kth Bragg reflection.

–C G = 4 ln 2, and C 1/2 √ G / πH is the normalization factor for the Gauss function

such that

G (x)dx = 1.

1 –C /2 L = 4, and C ∞ L / πH ′ is the normalization factor for the Lorentz function such that

L (x)dx = 1.

–C p = 4(2 1 /β − 1), and [Γ(β)/Γ(β − 1 1 / /2 2 )]C P

πH is the normalization factor

for the Pearson-VII function such that P VII (x)dx = 1.

–H = (U tan 2 θ +V tan θ +W ) 1 /2 , which is known as Caglioti formula, is the full width at half maximum as a function of θ for Gauss, pseudo-Voigt and Pearson-

VII functions, and U, V and W are free variables. 19 –H ′ = X/ cos θ +Y tan θ, is the full width at half maximum as a function of θ for the Lorentz function, and X and Y are free variables. – η=η 0 +η 1 2 θ+η 2 2 θ 2 , where 0 ≤ η ≤ 1, is the pseudo-Voigt function mix- ing parameter, i.e., the fractional contribution of the Gauss function into the lin- ear combination of Gauss and Lorentz functions, and η 0 , η 1 and η 2 are free variables.

17 Named after Woldemar Voigt (1850–1919), the German physicist best known for his work in crystal physics. A brief biography is available on WikipediA, http://en.wikipedia.org/wiki/

Woldemar Voigt. 18 Named after Karl Pearson (1857–1936) the British mathematician who derived several prob-

ability distribution functions known today as Pearson I to Pearson XII. A brief biography and a description of Pearson distributions are available at WikipediA: http://en.wikipedia.org/ wiki/Karl Pearson, and http://en.wikipedia.org/wiki/Pearson distribution, respectively.

19 G. Caglioti, A. Paoletti, and F.P. Ricci, Choice of collimators for a crystal spectrometer for neutron diffraction, Nucl. Instrum. Methods 3, 223 (1958).

172 8 The Powder Diffraction Pattern

(arb. units) Lorentz Gauss ); L(x)

G(x

x = 2θ i −2θ k (deg.)

Fig. 8.12 The illustration of Gauss (dash-dotted line) and Lorentz (solid line) peak-shape func- tions. Both functions have been normalized to result in identical definite integrals ( ∞ ∞ −∞ G (x)dx =

−∞ L (x)dx) and full widths at half maximum (FWHM). The corresponding FWHM’s are shown as thick horizontal arrows.

– Γ, is the gamma function. 20 – β=β 0 +β 1 / 2θ + β 2 / (2θ) 2 , is the exponent as a function of Bragg angle in the Pearson-VII function, and β 0 , β 1 and β 2 are free variables.

The two simplest peak-shape functions ((8.22) and (8.23)) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Fig. 8.12, from which it is easy to see that the Lorentz function is sharp near its maximum, but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base, but has a rounded maximum. Both functions are centrosymmetric, that is, G (x) = G(−x) and L(x) = L(−x).

The shapes of real Bragg peaks, which are the results of convoluting multiple instrumental and specimen functions (8.18), are rarely described well by simple Gaussian or Lorentzian distributions, especially in X-ray diffraction. Usually, real peak shapes are located somewhere between the Gauss and Lorentz distributions

20 Gamma function is defined as ∞ Γ(z) = t z −1 e t dt , or recursively for a real argument as

Γ(z) = (z − 1)Γ(z − 1). It is nonexistent when z = 0,−1,−2,..., and becomes (z − 1)! when z = 1, 2, 3, . . . Gamma function is an extension of the factorial to complex and real argu-

ments, (e.g., see E.W. Weisstein, Gamma function, Eric Weisstein’s world of mathematics, http://mathworld.wolfram.com/GammaFunction.html for more information).

8.5 Shapes of Powder Diffraction Peaks 173 and they can be better represented as the mixture of the two functions. 21 An ideal

way would be to convolute the Gauss and Lorentz functions in different proportions. This convolution, however, is a complex procedure, which requires numerical inte- gration every time one or several peak-shape function parameters change. Therefore,

a much simpler linear combination of Gauss and Lorentz functions is used instead of a convolution, and it is usually known as the pseudo-Voigt function (8.24). The Gaussian and Lorentzian are mixed in η to 1–η ratio, so that the value of the mixing parameter, η, varies from 0 (pure Lorentz) to 1 (pure Gauss). Obviously, η has no physical meaning outside this range. When during refinement η becomes negative, this is usually called super-Lorentzian, and such an outcome points to an incorrect choice of the peak-shape function. Usually, Pearson VII function should be used instead of pseudo-Voigt, see next paragraph.

The fourth commonly used peak-shape function is Pearson-VII (8.25). It is sim- ilar to Lorentz distribution, except that the exponent ( β) varies in the Pearson-VII, while it remains constant ( β = 1) in the Lorentz function. Pearson-VII provides an intensity distribution close to the pseudo-Voigt function: when the exponent, β = 1, it is identical to the Lorentz distribution, and when β∼ = 10, Pearson-VII becomes nearly pure Gaussian. Thus, when the exponent is in the range 0 .5 < β < 1 or β > 10, the peak shape extends beyond Lorentz or Gauss functions, respectively, but these values of β are rarely observed in practice. An example of the X-ray powder dif- fraction profile fitting using Pearson-VII function is shown in Fig. 8.13. Both the pseudo-Voigt and Pearson-VII functions are also centrosymmetric.

The argument, x, in each of the four empirical functions establishes the loca- tion of peak maximum, which is obviously observed when x = 0 and 2θ i = 2θ k .A second parameter, determining the value of the argument, is the full width at half maximum, H. The latter varies with 2 θ and its dependence on the Bragg angle is most commonly represented by an empirical peak-broadening function, which has three free parameters U, V , and W (except for the pure Lorentzian, which usually has only two free parameters). Peak-broadening parameters are refined during the profile fitting. Hence, in the most general case the peak full width at half maximum at a specific 2 θ angle is represented as

(8.26) As an example, the experimentally observed behavior of FWHM for a standard

H = U tan 2 θ +V tan θ +W

reference material SRM-660 (LaB 6 ) is shown in Fig. 8.14, together with the corre- sponding interpolation using (8.26), both as functions of the Bragg angle, 2 θ, rather than tan θ.

It is worth noting that the Lorentzian broadening function (H ′ ) parameters, X and Y , have the same dependence on Bragg angle as crystallite size- and microstrain- related broadening (compare (8.19) and (8.20) with (8.23) and following explana- tion of notations). Therefore, when Bragg peaks are well-represented by Lorentz

21 The most notable exception is the shape of peaks in neutron powder diffraction (apart from the time-of-flight data), which is typically close to the pure Gaussian distribution. Peak shapes in TOF

experiments are usually described by a convolution of exponential and pseudo-Voigt functions.

174 8 The Powder Diffraction Pattern 100

Kα 2 (arb. units) 60

Y Y obs -Y calc

Relative intensity,

Bragg angle, 2θ (deg.)

Fig. 8.13 The example of using Pearson-VII function to fit experimental data (open circles) repre-

senting a single Bragg peak containing K α 1 and K α 2 components.

0.16 LaB 6 , Cu Kα, Ge(Li) detector U = 0.004462

0.10 FWHM (deg. 2

Bragg angle, 2θ (deg.)

Fig. 8.14 Experimentally observed full width at half maximum of LaB 6 (open circles) as a function of 2 θ. The solid line represents a least squares fit using (8.26) with U = 0.004462, V = −0.001264, and W = 0.003410.

8.5 Shapes of Powder Diffraction Peaks 175 distribution, these physical characteristics of the specimen can be calculated from

FWHM parameters after the instrumental and wavelength dispersion parts are sub- tracted.

The mixing coefficient, η for pseudo-Voigt function and the exponent, β for Pearson-VII function, generally vary for a particular powder diffraction pattern. Their behavior is typically modeled with a different empirical parabolic function of tan θ and 2θ, respectively, as follows from (8.24) and (8.25). Peak shapes in the majority of routinely collected X-ray diffraction patterns are reasonably well- represented using pseudo-Voigt and/or Pearson-VII functions. On the other hand, noticeable improvements in the experimental powder diffraction techniques, which occurred in the last decade, resulted in the availability of exceptionally precise and high resolution data, especially when employing synchrotron radiation sources, where the use of these relatively simple functions is no longer justified. Furthermore, the ever-increasing computational power facilitates the development and utilization of advanced peak-shape functions, including those that extensively use numerical integration.

Most often, various modifications of the pseudo-Voigt function are employed to achieve improved precision, enhance the asymmetry approximation, account for the anisotropy of Bragg peak broadening, etc. For example, a total of four differ- ent functions (not counting those for the time-of-flight experiments) are employed

in GSAS. 22 The first function is the pure Gaussian (8.22), which is suitable for neutron powder diffraction data. 23 The second is a modified pseudo-Voigt (the so- called Thompson modified pseudo-Voigt), 24 where the function itself remains iden- tical to (8.24), but it employs a multi-term Simpson’s integration introduced by C.J. Howard. 25 Its FWHM (H ) and mixing (η) parameters are modeled as follows:

H = ∑ a i H 5 G −i H L i

i =0

η= L ∑ b

i =1

22 C.A. Larson and R.B. Von Dreele, GSAS: General structure analysis system. LAUR 86- 748 (2004). The cited user manual and software are freely available http://www.ccp14.ac.uk/

solution/gsas/. 23 We note that GSAS is continuously under development and new functions are often added.

Hence, the numbering of peak shape functions in this book may not correspond to the numbering scheme in GSAS.

24 P. Thompson, D.E. Cox, and J.B. Hastings, Rietveld refinement of Debye–Scherrer synchrotron X-ray data from A1 2 O 3 , J. Appl. Cryst. 20, 79 (1987).

25 C.J. Howard, The approximation of asymmetric neutron powder diffraction peaks by sums of Gaussians, J. Appl. Cryst. 15, 615 (1982).

176 8 The Powder Diffraction Pattern where a i and b i are tabulated coefficients. Further,

H G = 2σ

2 ln 2

(8.29) σ= U tan 2 θ +V tan θ +W + P/ sin 2 θ

H L = (X + X a cos φ)/ cos θ + (Y +Y a cos φ) tan θ (8.31) and H G is the Gaussian full width at half maximum, modified by an additional

broadening parameter, P; H L is Lorentzian full width at half maximum, which ac- counts for the anisotropic FWHM behavior by introducing two anisotropic broad-

ening parameters, X a (crystallite size) and Y a (strain), and φ is the angle between a common anisotropy axis and the corresponding reciprocal lattice vector. The major benefit achieved when using the modified pseudo-Voigt function is in the separation of FWHM’s due to Gaussian and Lorentzian contributions to the peak-shape function. They represent two different effects contributing to the com- bined peak width, which are due to the instrumental (Gauss) and specimen (Lorentz) broadening. The specimen-broadening parameters X and Y , being coefficients of

1 / cos θ and tan θ, could be directly associated with the crystallite size and micros- train, respectively. Anisotropic broadening can be refined using two additional pa- rameters, X a and Y a . The crystallite size (p ) in ˚ A can be obtained from these para- meters as follows:

180K λ p iso =p ⊥ =

π(X + X a ) and microstrain (s ) in percent as: π

πX

a −Y instr ) · 100% (8.33) where the subscript iso indicates isotropic parameters, ⊥ and || denote parame-

s iso =s ⊥ = 180 (Y −Y instr ) · 100% and s =

(Y +Y

ters that are perpendicular and parallel, respectively, to the anisotropy axis, K is the Scherrer constant, 26 and Y instr is the instrumental part in the case of strain broadening. The third function used in GSAS, is similar to the second function as described in (8.27)–(8.31). However, it fits real Bragg peak shapes better, due to improved han- dling of asymmetry, which is treated in terms of axial divergence. 27 This function is formed by a convolution of pseudo-Voigt with the intersection of the diffraction cone and a finite receiving slit length using two geometrical parameters, S/L and D/L, where S and D are the sample and the detector slit dimensions in the direction

26 K is known as the shape factor or Scherrer constant which varies in the range 0 .89 < K < 1, and usually K = 0.9 [H.P. Klug and L.E. Alexander, X-ray diffraction procedures for polycrystalline

and amorphous materials, Second edition, John Wiley, NY (1974) p. 656]. 27 L.W. Finger, D.E. Cox, A.P. Jephcoat, A correction for powder diffraction peak asymmetry due

to axial divergence, J. Appl. Cryst. 27, 892 (1994).

8.5 Shapes of Powder Diffraction Peaks 177 parallel to the goniometer axis, and L is the goniometer radius. These two para-

meters can be measured experimentally, or refined (after being suitably constrained because D and S are identical in a typical powder diffraction experiment) when low Bragg angle peaks are present. This peak-shape function also supports an empirical extension of microstrain anisotropy described by six parameters. The result is added

to Y in the second part of (8.31) as γ L d 2 , where:

γ L = γ 2 2 11 2 h + γ 22 k + γ 33 l + γ 12 hk + γ 13 hl + γ 23 kl (8.34) Thus, the total number of parameters for this peak-shape function is 19.

The fourth function is also a modified pseudo-Voigt, and it accounts for anisotropic microstrain broadening as suggested by P. Stephens: 28

H S = ∑ S HKL h H k K l L

HKL

where S HKL are coefficients and H, K, and L (these are not the same as Miller indices hkl), represent permutations of positive integers restricted to H + K + L = 4. These coefficients are further restricted by Laue symmetry, so that a total of 2 in the cubic crystal system to 15 coefficients in the triclinic crystal system may be used to describe strain broadening. The latter contributes to Gaussian and Lorentzian

broadening by adding σ 2 S d 4 and γ S d 2 to U and Y in (8.26) and (8.31), respectively. Here, σ S = (1 − η)H S , and γ S = ηH S , where η is the pseudo-Voigt function mixing parameter, as in (8.24).

Both the third and fourth functions describe asymmetric peaks much better than the first two and the simple pseudo-Voigt (8.24), especially at low Bragg angles. The fourth function is also an excellent approximation of Bragg peaks when significant anisotropic broadening caused by microstrains is present. When the anisotropy is low, this function is similar to the third one but with a noticeably reduced number of free variables. Thus, the number of fitting parameters for the fourth function depends on the Laue class, and it varies from 14 to 27. The number of free variables may be reduced further since the coefficients S HKL have physical meaning, and some of them may be set to known predetermined values (for further details and examples

see the original paper 28 ). The attractiveness of this model is that the anisotropy of microstrains can be visualized as the three-dimensional surface in reciprocal space with radial distances defined as:

D S (hkl) =

∑ S HKL h H k K l L

C HKL

In the modified pseudo-Voigt functions described earlier ((8.27)–(8.31)), both the Gaussian to Lorentzian mixing parameter ( η, (8.27)) and their individual contri-

28 P.W. Stephens, Phenomenological model of anisotropic peak broadening in powder diffraction, J. Appl. Cryst. 32, 281 (1999).

178 8 The Powder Diffraction Pattern butions to the total peak width (H, (8.32)) are tabulated. This feature may be used

to lower the number of free parameters and to obtain more realistic peak-shape pa- rameters that are due to the physical state of the specimen. Either or both may be achieved by using one of the following approaches:

– Employing a high-quality standard sample (e.g., LaB 6 , see the footnote on page 155) that has no measurable contributions from small crystallite size and microstrains, the peak-shape function parameters (V , W and P), responsible for the instrumental and wavelength dispersion broadening, can be determined ex- perimentally. These should remain constant during following experiments when using different materials and, thus should be kept fixed in future refinements. Obviously, the goniometer configuration must be identical in the experiments conducted using both the standard and real samples. This method requires mea- suring a standard every time when any change in the experimental settings oc- curs, including replacement of the X-ray tube, selection of different divergence or receiving slits, monochromator geometry, filter, and other optical components.

– Taking advantage of the fundamental parameters approach, which is based on

a comprehensive description of the experimental conditions and hardware con- figuration. It is developed quite well and as a result, the corresponding peak- shape parameters may be computed, and not necessarily refined. This technique requires realistic data about the experimental configuration, such as slit open- ings and heights, in-plane and axial divergences, monochromator characteristics, source and sample geometry and dimensions, and other data. Indeed, consider- able effort is involved in order to obtain all required physical characteristics of the powder diffractometer, the source, and the specimen. The resultant peak shape is then obtained as a convolution (8.18) of the modeled instrumental function, Ω, wavelength distribution in the incident spectrum, Λ, and sample function, Ψ,

with the pseudo-Voigt function. 29 For example, the instrumental function can be obtained by convolution of primitive (fundamental) functions describing effects of the corresponding instrumental characteristics on the peak shape, as shown in Fig. 8.15. The fundamental parameters approach is implemented in several soft-

ware products, including Koalariet/XFIT 30 and BGMN 31 , TOPAS 32 and others. More detailed information about both the technique and its implementation may

be found in the corresponding references. 33

29 From this point of view, some applications of the modified pseudo-Voigt function (e.g., third and fourth peak-shape functions employed in GSAS) are in a way similar to the fundamental

parameters approach as they use instrumental parameters to describe certain aspects of peak shape. 30 See http://www.ccp14.ac.uk/tutorial/xfit-95/xfit.htm.

31 See http://www.bgmn.de/. 32 Bruker AXS: TOPAS V3: General profile and structure analysis software for powder diffraction

data. User’s Manual, Bruker AXS, Karlsruhe, Germany (2005).

33 R.W. Cheary, A.A. Coelho, J.P. Cline, Fundamental parameters line profile fitting in labo- ratory diffractometers. J. Res. Natl. Inst. Stand. Technol. 109, 1 (2004) [http://nvl.nist.gov/pub/

nistpubs/jres/109/1/j91che.pdf]; R.W. Cheary and A. Coelho, A fundamental parameters approach to X-ray line-profile fitting, J. Appl. Cryst. 25, 109 (1992); R.W. Cheary and A.A. Coelho, Ax-

8.5 Shapes of Powder Diffraction Peaks 179

Axial width

Slit

X-ray

In plane

divergence Fig. 8.15 Graphical representation of typical fundamental functions defining convoluted instru-

focus

divergence

mental profile.

8.5.2 Peak Asymmetry

All peak-shape functions considered so far were centrosymmetric with respect to their arguments (x ), which implies that both the low and high angle slopes of Bragg peaks have mirror symmetry with respect to a vertical line intersecting the peak maximum (e.g., see Fig. 8.12). In reality, Bragg peaks are asymmetric due to vari- ous instrumental factors such as axial divergence and nonideal specimen geometry, and due to the nonzero curvature of the Debye rings (e.g., see Fig. 8.4), especially at low Bragg angles. The combined asymmetry effects usually result in the low an- gle sides of Bragg peaks being considerably broader than their high angle sides, as illustrated schematically in Fig. 8.16. Peak asymmetry is usually strongly de- pendent on the Bragg angle, and it is most prominently visible at low Bragg angles

(2θ below ∼20 ◦ –30 ◦ ). At high Bragg angles peak asymmetry may be barely visible, but it is still present.

A proper configuration of the instrument and its alignment can substantially re- duce peak asymmetry, but unfortunately, they cannot eliminate it completely. The major asymmetry contribution, which is caused by the axial divergence of the beam, can be successfully controlled by Soller slits, especially when they are used on both the incident and diffracted beam’s sides. The length of the Soller slits is critical in handling both the axial divergence and asymmetry; however, the reduction of the axial divergence is usually accomplished at a sizeable loss of intensity.

Since asymmetry cannot be completely eliminated, it should be addressed in the profile-fitting procedure. Generally, there are three ways of treating the asymmetry of Bragg peaks, all achieved by various modifications of the selected peak-shape function:

ial divergence in a conventional X-ray powder diffractometer. II. Realization and evaluation in a fundamental-parameter profile fitting procedure, J. Appl. Cryst. 31, 862 (1998); J. Bergmann, R. Kleeberg, A. Haase, and B. Breidenstein, Advanced fundamental parameters model for im- proved profile analysis, Mater. Sci. Forum 347, 303 (2002) and references therein.

180 8 The Powder Diffraction Pattern

. units)

G(x) (arb

x = 2θ i - 2θ k (deg.)

Fig. 8.16 The schematic illustrating the asymmetric Bragg peak (solid line) when compared with the symmetric peak composed of the dash-dotted line (left slope) and the solid line (right slope). Both peaks are modeled by the pure Gauss function (8.22) using two different FWHM’s on differ- ent sides of the peak maximum in the asymmetric case.

– In the first method, the symmetry of a function is broken by introducing a multi- plier, which increases the intensity on one side from the peak maximum (usually the low Bragg angle side), and decreases it on the opposite side. The same mod- ification of intensities can also be achieved by introducing different peak widths on the opposite sides of the peak-maximum, as has been done in Fig. 8.16. The following equation expresses the intensity correction, A, as a function of Bragg angle:

z i × |z i |

A (x i )=1−α

tan θ

In (8.37) α is a free variable, or the asymmetry parameter, which is refined during profile fitting and z i is the distance from the maximum of the symmetric peak to the corresponding point of the peak profile, or z i = 2θ k –2 θ i . This modification is applied

separately to every individual Bragg peak, including K α 1 and K α 2 components. Since (8.37) is a simple intensity multiplier, it may be easily incorporated into any of the peak-shape functions considered earlier. In addition, in the case of the Pearson-

VII function, asymmetry may be treated differently. It works nearly identical to

8.5 Shapes of Powder Diffraction Peaks 181 (8.37) and all variables have the same meaning as in this equation, but the expression

itself is different:

A (x i )=1+α

where C P = 4(2 1 /β − 1), see (8.25). – Equations (8.37) and (8.38) are quite simple, but they are also far from the best in

treating peak asymmetry, especially when high-quality powder diffraction data are available. Better results can be achieved by introducing the so-called split pseudo-Voigt or split Pearson-VII functions. Split functions employ two sets of peak-shape parameters (all or only some of them) separately to represent the op- posite sides of each peak. For example, in a split Pearson-VII function, a different exponent β and its dependence on the Bragg angle may be used to model the low (left) and high (right) angle sides of the peak, while keeping the same FWHM parameters U, V , and W . This results in a total of nine peak-shape function pa-

0 , β 1 and β 2 , where superscripts and refer to parameters of the left and right sides, respectively, of the peak (see (8.25) and

rameters: U, V , W , β L 0 , β L 1 , β L 2 , β R R

LR

the following explanation of notations). It is also possible to split the peak width (FWHM parameters), but then a total of twelve parameters should be refined, which is usually an overwhelming number of free variables for an average, or even good-quality powder diffraction experiment.

– In some advanced implementations of the modified pseudo-Voigt function, an asymmetric peak can be constructed as a convolution of a symmetric peak shape and a certain asymmetric function, which can be either empirical, or based on the real instrumental parameters. For example, as described in Sect. 8.5.1, and using the Simpson’s multi-term integration rule, this convolution can be approximated using a sum of several (usually 3 or 5) symmetric Bragg peak profiles:

y (x) asym = ∑ g i y (x) sym

i =1

where: n is the number of terms, n = 3 or 5; y sym and y asym are modeled sym- metric and the resulting asymmetric peak-shape functions, respectively, and g i are the coefficients describing Bragg angle dependence of the chosen asymmetry parameter. This approach is relatively complex, but in the case of high accuracy data (e.g., precision X-ray or synchrotron powder diffraction), it adequately de- scribes the observed asymmetry of Bragg peaks. An even more accurate method employs the modeling of asymmetry by using geometrical parameters responsi- ble for axial divergence (see Sect. 8.5.1; Finger, Cox, and Jephcoat reference on page 176). Nevertheless, lower quality routine powder diffraction patterns to a large extent can be treated using the simpler (8.37).

182 8 The Powder Diffraction Pattern