Section 2.3.1.1.6. Aberrations related to the specimen

The major displacement errors arising from the specimen are (1) displacement of the specimen surface from the axis of rotation, (2) use of a flat rather than a curved specimen, and (3) specimen transparency. These are illustrated schematically for the focusing plane in Fig. 2.3.1.10(a) . The rays from a highly absorbing or very thin specimen with the same curvature as the focusing circle converge at A without broadening and at the correct 2θ. The rays from the flat surface cause an asymmetric profile shifted to B. Penetration of the beam below the surface combined with the flat specimen causes additional broadening and a shift to C.

Figure 2.3.1.10| top | pdf | (a) Origin of specimen-related aberrations in focusing plane of conventional reflection specimen diffractometer (Fig. 2.3.1.3). A no aberration from curved specimen; B flat specimen; C specimen displacement from 0. (b) Computed angular shifts caused by specimen displacement, R = 185 mm. (c) Flat-specimen asymmetric aberration, Si(422), Cu Kα1, Kα2 peak intensities normalized. (d) Computed flat-specimen centroid shifts for various apertures; parabola for constant irradiated 2 cm specimen length. (e) Transparency asymmetric aberration, LiF(200) powder reflection, Cu Kα, peak intensities normalized, thin specimen (solid-line profile) 0.1 mm thick; thick specimen (dotted-line profile) 1.0 mm, αES 1°, αRS 0.046°. (f) Computed transparency centroid shifts for various values of linear absorption coefficient.

The most frequent and usually the largest source of angular errors arises from displacement of the specimen surface from the diffractometer axis of rotation. It is not easy to avoid and may arise from several sources. It is advisable to check the reproducibility of inserting the specimen in the diffractometer by recording an isolated peak at low 2θ for each insertion. If only a radial displacement s occurs, the reflection is shifted where R is the diffractometer radius. A plot of equation (2.3.1.10) is shown in Fig. 2.3.1.10(b). The shift is to larger or smaller angles depending on the direction of the displacement and there is no broadening if the displacement is only radial and relatively small. Even a small displacement causes a relatively large shift; for example, if s = +0.1 mm and R = 185 mm, = + 0.06° at 20°2θ. This gives rise to a systematic error in the recorded reflection angles, which increases with decreasing 2. It could be handled with a plot, providing it was the only source of error. There are other possible sources of displacement such as (a) if the bearing surface of the specimen post was not machined to lie exactly on the axis of rotation, (b) improper specimen preparation or insertion in which the surface was not exactly coincident with the bearing surface or (c) nonplanar specimen surface, irregularities, large particle sizes, and specimen transparency. Source (a) leads to a constant error in all measurements, and errors due to (b) and (c) vary with each specimen.

Ideally, the specimen should be in the form of a focusing torus because of the beam divergence in the equatorial and axial planes. The curvatures would have to vary continuously and differently during the scan and it is impracticable to make specimens in such forms. An approximation is to make the specimen in a flexible cylindrical form with the radius of curvature increasing with decreasing 2θ (Ogilvie, 1963). This requires a very thin specimen (thus reducing the intensity) to avoid cracking and surface irregularities, and also introduces background from the substrate. A compromise uses rigid curved specimens, which match the SFC (Fig. 2.3.1.3) at the smallest 2θ angle to be scanned, and this eliminates most of the aberration (Parrish, 1968). A major disadvantage of the curvature is that it is not possible to spin the specimen.

In practice, a flat specimen is almost always used. The specimen surface departs from the focusing circle by an amount h at a distance l/2 from the specimen centre: This causes a broadening of the low-2θ side of the profile and shifts the centroid to lower 2θ: For α = 1° and = 20°, = −0.016°. The peak shift is about one-third as large as the centroid shift in the forward-reflection region. This aberration can be interpreted as a continuous series of specimen-surface displacements, which increase from at the centre of the specimen to a maximum value at the ends. The effect increases with α and decreasing 2θ. The profile distortion is magnified in the small 2θ-angle region where the axial divergence also increases and causes similar effects. Typical flat-specimen profiles are shown in Fig. 2.3.1.10(c) and computed centroid shifts in Fig. 2.3.1.10(d).

The specimen-transparency aberration is caused by diffraction from below the surface of the specimen which asymmetrically broadens the profile (Langford & Wilson, 1962). The peak and centroid are shifted to smaller 2θ as shown in Fig. 2.3.1.10(e). For the case of a thick absorbing specimen, the centroid is shifted and for a thin low-absorbing specimen where μ is the effective linear absorption coefficient of the specimen used, t the thickness in cm, and R the diffractometer radius in cm. The intermediate absorption case is described by Wilson (1963). A plot of equation (2.3.1.13) for various values of μ is given in Fig. 2.3.1.10(f). The effect varies with sin2θ and is maximum at 90° and zero at 0° and 180°. For example, if μ = 50 cm⁻¹, the centroid shift is −0.033° at 90° and falls to −0.012° at 20°2θ.

The observed intensity is reduced by absorption of the incident and diffracted beams in the specimen. The intensity loss is , where μ is the linear absorption coefficient of the powder sample (it is almost always smaller than the solid material) and x_s is the distance below the surface, which may be equal to the thickness in the case of a thin film or low-absorbing material specimen. The thick (1 mm) specimen of LiF in Fig. 2.3.1.10(e) had twice the peak intensity of the thin (0.1 mm) specimen.

The aberration can be avoided by making the sample thin. However, the amount of incident-beam intensity contributing to the reflections could then vary with θ because different amounts are transmitted through the sample and this may require corrections of the experimental data. Because the effective reflecting volume of low-absorbing specimens lies below the surface, care must be taken to avoid blocking part of the diffracted beam with the antiscatter slits or the specimen holder, particularly at small 2θ.

There are additional problems related to the specimen such as preferred orientation, particle size, and other factors; these are discussed in Section 2.3.3.