Positions of Powder Diffraction Peaks

8.4 Positions of Powder Diffraction Peaks

As discussed earlier in general terms, diffraction peaks appear at specific angles due to scattering by periodic lattices. Further, as shown by the Braggs and Ewald (Sect. 7.2), these angles are a discontinuous function of Miller indices, the inter- planar distances (lengths of independent reciprocal lattice vectors) and the wave- length (radius of the Ewald’s sphere). Therefore, both the unit cell dimensions and

8.4 Positions of Powder Diffraction Peaks 163 the wavelength are the two major factors that determine Bragg angles for the same

combination of h, k, and l. As we will also see later, the observed peak positions can

be distorted by instrumental and specimen parameters.

8.4.1 Peak Positions as a Function of Unit Cell Dimensions

The interplanar distance is a function of the unit cell parameters and Miller in- dices, h, k, and l, which fully describe every set of crystallographic planes. The

corresponding formulae for the inverse square of the interplanar distance, 1 /d 2 , are usually given separately for each crystal system, 8 as shown in (8.2)–(8.7).

Cubic:

1 h 2 +k 2 +l 2

2 d (8.2) a Tetragonal:

1 h 2 +k 2 l 2

2 = a 2 d + c 2 (8.3) Hexagonal:

ac sin 2 β Triclinic:

1 h 2 2kl

(cos β cos γ − cosα)+

(cos α cos γ − cosβ)+

(cos α cos β − cosγ) /

ab

c 2 sin 2 γ

2 2 (1−cos 2 α − cos β − cos γ + 2 cos α cos β cos γ)

8 Primitive rhombohedral lattices, i.e., when a ◦ are nearly always treated in the hexagonal basis with rhombohedral (R) lattice centering. In a primitive rhombohedral

lattice: 1 (h 2 +k 2 +l 2 ) sin 2 α + 2(hk + kl + hl)(cos 2 α − cosα)

a 2 (1 − 3cos 2 α + 2 cos 3 α)

164 8 The Powder Diffraction Pattern

Fig. 8.10 The illustration of a reciprocal lattice vector, d ∗ hkl , as a vectorial sum of three basis unit vectors, a ∗ ,b ∗ and c ∗ multiplied by h, k and l, respectively.

The most complex formula is the one for the triclinic crystal system, in which a total of six independent parameters are required to describe the unit cell dimensions. On the other hand, (8.7) is the most general, since (8.2)–(8.6) are easily derived from it. For example, after introducing the corresponding relationships between the unit cell dimensions for the tetragonal crystal system (i.e., a

90 ◦ ) into (8.7), the latter is straightforwardly simplified to (8.3). Thus, the simplified formulae (8.2)–(8.6) are only useful in manual calculations, but when the list of possible d’s (or θ’s) is generated using a computer program, it makes better sense

to employ only the most general equation, since obviously the resultant 1 /d 2 values are correct upon the substitution of the appropriate numerical values for a, b, c, α, β, and γ into (8.7).

The usefulness of the reciprocal lattice concept may be once again demonstrated here by illustrating how easily (8.2)–(8.7) can be derived in the reciprocal space employing reciprocal lattice vectors. When the derivation is performed in the direct space, the geometrical considerations become quite complex.

Consider a reciprocal lattice as shown in Fig. 8.10. Any reciprocal lattice vector,

d ∗ hkl , is a sum of three non-coplanar vectors (a ∗ ,b ∗ and c ∗ are the unit vectors of the reciprocal lattice and h, k, and l are integers):

(8.8) For example, in the orthorhombic crystal system α ∗ =β ∗ =γ ∗ = 90 ◦ . Hence, (8.8)

d ∗ hkl = ha ∗ + kb ∗ + lc ∗

is transformed into:

(8.9) and (8.5) is obtained immediately because d ∗ = 1/d, a ∗ = 1/a, b ∗ = 1/b and

(d ∗ 2 ∗ 2 ∗ 2 ∗ hkl 2 ) = (ha ) + (kb ) + (lc )

c ∗ = 1/c.

8.4 Positions of Powder Diffraction Peaks 165 In the triclinic crystal system, the equivalent of (8.9) is more complex

d ∗2 =h 2 a ∗2 +k 2 b ∗2 +l 2 c ∗2 +

(8.10) 2hka ∗ b ∗ cos γ ∗ + 2hla ∗ c ∗ cos β ∗ + 2klb ∗ c ∗ cos α ∗ but it becomes considerably more intuitive and easier to understand in terms of

reciprocal lattice parameters than (8.7), which is given in terms of direct space unit cell dimensions.

According to the Braggs’ law (7.9), the diffraction angle, θ hkl , of a reflection from a series of lattice planes (hkl) is determined from the interplanar distance, d hkl , and the wavelength, λ as:

8.4.2 Other Factors Affecting Peak Positions

Equations (8.7) and (8.10) are exact, assuming that both the powder diffractome- ter and the sample are ideal. In reality, various instrumental and specimen features may affect the observed positions of Bragg peaks. These factors are often known as systematic aberrations (distortions), and they are usually assembled into a single correction parameter, Δ2θ. The latter is applied to the idealized Bragg angle, 2θ calc , calculated from the unit cell dimensions and wavelengths using (8.7) or (8.10) and (8.11), so that the experimentally observed Bragg angle, 2 θ obs , is given as:

(8.12) For the most commonly used Bragg–Brentano focusing geometry (see

2 θ obs = 2θ calc + Δ2θ

Sect. 11.3), the overall correction is generally a sum of six factors:

+p 4 sin 2 θ+p 5 cos θ+p 6 (8.13) tan 2 θ sin 2 θ tan θ

The first two parameters, p 1 and p 2 , account for the axial divergence of the inci- dent beam (see Sect. 11.2) and they can be expressed as:

3R 2 ; p 2 =− 3R 2 (8.14) where h is the length of the specimen parallel to the goniometer axis, R is the go-

p 1 =−

niometer radius, K 1 and K 2 are constants established by the collimator. Soller slits (see Sect. 11.2.1) usually minimize the axial divergence and therefore, these two corrections are often neglected for practical purposes.

In addition to axial divergence, the first parameter (p 1 ) includes a shift that is due to peak asymmetry caused by other factors. One of these is the finite length of the receiving slit of the detector, which results in the measurement of a fixed length

166 8 The Powder Diffraction Pattern of an arc (see Fig. 8.4), rather than an infinitesimal point of the Debye ring. The

curvature of the Debye ring increases 9 with the decreasing Bragg angle, and the resultant increasing peak asymmetry cannot be corrected for by using Soller slits. This effect can be minimized by reducing the detector slit length, which however, considerably lowers the measured intensity.

The third parameter, p 3 , is given as: α 2

p 3 =−

where α is the in-plane divergence of the X-ray beam (see Sect. 11.2) and K 3 is

a constant. This factor accounts for the zero curvature of flat samples, typically used in Bragg–Brentano goniometers. This geometry of the sample distorts the ideal focusing in which the curvature of the sample surface should vary with Bragg angle. The aberrations are generally insignificant and they are usually neglected in routine powder diffraction experiments.

The fourth parameter is

2 µ eff R

where µ eff is the effective linear absorption coefficient (see Sect. 8.6.5 and (8.51)). This correction is known as the transparency-shift error, and it may play a role when examining thick (more than 50–100 µm) samples. The transparency-shift er- ror is caused by the penetration of the beam into the sample, and the penetration

depth is a function of Bragg angle. Usually p 4 is the refined parameter since µ eff is rarely known (both the porosity and the density of the powder sample are usu- ally unknown). The transparency-shift error could be substantial for low absorbing samples, for example, organic compounds, and it is usually negligible for highly absorbing specimens, that is, compounds containing heavy chemical elements. For low absorbing materials this shift can be reduced by using thin samples, however, doing so significantly decreases intensity at high Bragg angles. The latter is already small when a compound consists of light chemical elements due to their low X-ray scattering ability.

The fifth parameter characterizes specimen displacement, s, from the goniometer axis and it is expressed as

9 Strictly speaking, the curvature of the Debye ring increases both below and above 2 θ = 90 ◦ , e.g., see Fig. 8.2. Both the curvature and associated asymmetry become especially significant when

2 θ ≤ ∼20 ◦ and 2 θ ≥ ∼160 ◦ . At low Bragg angles, this contribution to asymmetry results in the enhancement of the low angle slopes of Bragg peaks, while at high Bragg angles the asymmetry effect is opposite. Asymmetry at high Bragg angles is often neglected because the intensity of Bragg peaks is usually low due to a variety of geometrical and structural factors, which is discussed

in Sects. 8.6 and 9.1. It is also worth mentioning that at 2 θ∼ = 90 ◦ the contribution from p 1 becomes negligible because tan 2 θ → ∞.

8.4 Positions of Powder Diffraction Peaks 167 where R is the radius of the goniometer. This correction may be substantial, es-

pecially when there is no good and easy way to control the exact position of the specimen surface.

The last parameter, p 6 , is constant over the whole range of Bragg angles and the corresponding aberration usually arises due to improper setting(s) of zero angles for one or more diffractometer axes: detector and/or X-ray source. Hence, this distor- tion is called the zero-shift error. The zero-shift error can be easily minimized by proper alignment of the goniometer. However, in some cases, for example, in neu- tron powder diffraction, zero shift is practically unavoidable and, therefore, should

be always accounted for. Equations (8.13)–(8.17) define the most important factors affecting peak posi- tions observed in a powder diffraction pattern, some of which combine several ef- fects that have the same 2 θ dependence. The latter however is not really impor- tant when parameters p i are refined rather than modeled. A comprehensive analysis of errors in peak positions for a general case of focusing geometry (not only the Bragg–Brentano focusing geometry) may be found in the International Tables for Crystallography in Tables 5.2.4.1, 5.2.7.1, and 5.2.8.1 on pp. 494–498. 10

In order to account for several different factors simultaneously, high accuracy of the experimental powder diffraction data is required, in addition to the availability of data in a broad range of Bragg angles. Even then, it may be difficult since p 4 and p 5 are strongly correlated, and so is the zero-shift parameter, p 6 . Generally, they cannot be distinguished from one another when only a small part of the diffraction pattern has been measured (e.g., below 60–70 ◦ 2 θ). Thus, refinement of any single parameter (p 4 ,p 5 or p 6 ) gives similar results, that is, the satisfactory fit between the observed and calculated 2 θ values. The problem is: how precise are the obtained unit cell parameters? If the wrong correction was taken into account, the resultant unit cell dimensions may be somewhat different from their true values. The best way to deal with the ambiguity of which correction to apply, is to use an internal standard, which unfortunately contaminates the powder diffraction pattern with Bragg peaks of the standard material.

An example of how important the sample displacement correction may become is shown in Fig. 8.11, where the differences between the observed and calculated Bragg angles are in the −0.03 to +0.04 ◦ range before correction (open circles). They fall within the −0.01 to +0.01 ◦ range (filled triangles) when the sample displacement parameter was refined, together with the unit cell dimensions. Even though the difference in the unit cell dimensions obtained with and without the sample displacement correction (Fig. 8.11) is not exceptionally large, it is still ten to twenty times the least squares standard deviations, that is, the differences in lattice parameters are statistically significant.

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

168 8 The Powder Diffraction Pattern 0.04

No correction: a = 10.4309(4) Å

Corrected for sample displacement: a = 10.4356(2) Å

c = 3.0217(1) Å −0.04 p 5 = − 0.00103(3)

Bragg angle, 2θ (deg.)

Fig. 8.11 The differences between the observed and calculated 2 θ values plotted as a function of 2 θ without (open circles) and with (filled triangles) specimen displacement correction. The corresponding values of the unit cell parameters and the specimen displacement parameter are indicated on the plot. The material belongs to the tetragonal crystal system.