Section 2.3.1.1.1. Geometrical instrument parameters

The powder diffractometer is basically a single-axis goniometer with a large-diameter precision gear and worm drive. The detector and receiving-slit assembly are mounted on an arm attached to the gear in a radial position. The specimen is mounted in a holder carried by a shaft precisely positioned at the centre of the gear. 2/1 reduction gears drive the specimen post at one-half the speed of the detector. Some diffractometers have two large gears, making it possible to drive only the detector with the specimen fixed or vice versa, or to use 2/1 scanning. Synchronous motors have been used for continuous scanning for rate-meter recording and stepping motors for step-scanning with computer control.

The geometry of the method requires that the axis of rotation of the diffractometer be parallel to the X-ray tube focal line to obtain maximum intensity and resolution. The target is normal to the long axis of the tube; vertically mounted tubes require a diffractometer that scans in the vertical plane while a horizontal tube requires a horizontal diffractometer. The X-ray optics are the same for both. The incident angle θ and the reflection angle 2θ are defined with respect to the central ray that passes through the diffractometer axis of rotation O.

The axis of rotation of the specimen is the central axis of the main gear of the diffractometer, as shown in Fig. 2.3.1.3. The centre of the specimen is equidistant from the source F and receiving slit RS. The instrument radius R_DC = F − O = O − RS. The radius of commercial instruments is in the range 150 to 250 mm, with 185 mm most common. Changing the radius affects the instrument parameters and a number of the aberrations. Larger radii have been used to obtain higher resolution and better profile shapes. For example, the asymmetric broadening caused by axial divergence is decreased because the chord of the diffraction cone intercepted by the receiving slit has less curvature. However, if the same entrance slit is used, moving the specimen further from the source proportionately increases the length of specimen irradiated and decreases the intensity.

The imaginary specimen focusing circle SFC passes through F, O and the middle of RS and its radius varies with θ: The specimen holder is set parallel to the central ray at 0° and the gears drive the RS-detector arm at twice the speed of the specimen to maintain the θ–2θ relation at all angles. The source F is the line focus of the X-ray tube viewed at a take-off angle ψ. The actual width, , is foreshortened to In a typical case, = 0.4 mm and, at ψ = 5°, = 0.03 mm and the projected angular width is 0.025° for R = 185 mm. The angular aperture α_ES of the incident beam in the equatorial (focusing) plane is determined by the entrance slit width ES_w (also called the `divergence slit' since it limits the divergence of the beam) and its distance D₁ from F: Because the beam is divergent, the length of specimen irradiated S_l in the direction of the incident beam normal to O varies with θ: where α is in radians and is the distance from F to the crossover point before ES and is given by . The approximate relation is close enough for practical purposes (Parrish, Mack & Taylor, 1966). The intensity is nearly proportional to ES_α but the maximum aperture that can be used is determined by S_l and the smallest angle to be scanned 2θ_min, as shown in Fig. 2.3.1.4 . The entrance-slit width may be increased to obtain higher intensity at the upper angular range; for example, ES = 1° for the forward-reflection region and 4° for back-reflection.

Figure 2.3.1.4| top | pdf | Length of specimen irradiated, Sl, as a function of 2θ for various angular apertures. Sl = αR/sin θ, R = 185 mm.

Some slit designs are shown in Fig. 2.3.1.5 . The base is machined with a pair of rectangular shoulders whose separation A is the sum of the diameters of the two rods (a) or bar widths (b) and the central spacers on both ends that determine the slit opening. The distance P between the centre of the slit opening and the edge of the slit frame is kept constant for all slits to avoid angular errors when changing slits. The rods may be molyb­denum or other highly absorbing metal and are cemented in place. It may also be made in one piece (c) using machinable tungsten (Parrish & Vajda, 1966).

Figure 2.3.1.5| top | pdf | Slit designs made with (a) rods, (b) bars, and (c) machined from single piece. (d) Parallel (Soller) slits made with spacers or slots cut into the two side pieces (not shown) to position the foils.

A variable ES whose width increases with 2θ so that the irradiated length is about the same at all angles has been described by Jenkins & Paolini (1974). It is called a θ-compensating slit in which a pair of semicircular cylinders with a fixed opening is rotated around the axis of the opening by a linkage attached to the specimen shaft of the diffractometer to vary the aperture continuously with θ. The observed intensities must be corrected to obtain the relative intensities and the angular dependence of the aberrations is different from the fixed aperture slit.

Another way to irradiate constantly the entire specimen length is to use a self-centring slit which acts as an entrance and antiscatter slit (de Wolff, 1957). A 1 mm thick brass plate with rounded edge is mounted above the centre of the specimen and is moved in a plane normal to the specimen surface so that the aperture is proportional to sin θ. It can only be used for forward reflections.

Owing to the beam divergence, the geometric centre of the irradiated specimen length shifts a small amount during the scan (see also §2.3.5.1.5). It is generally advisable to centre the beam at the smallest 2θ to be scanned. Below about 20°, the irradiated length increases rapidly and it is essential to use small apertures and to align the entrance and antiscatter slits carefully. Failure to do this correctly could cause (a) errors in the relative intensities owing to the primary beam exceeding the specimen area, (b) cutoff by the walls of the specimen holder for low-absorbing thick specimens, and (c) increased background from scattering by the specimen holder or the primary beam entering the detector. The transmission specimen method (Subsection 2.3.1.2) has advantages in measuring large d's.

The beam converges after reflection on the receiving slit RS, whose width defines the reflection and profile width. Only those rays that are within the θ–2θ setting are in sharp convergence, i.e. `in focus'. The reflections become broader with increasing distance from the RS, and, therefore, this method is not suited for position-sensitive detectors. The RS aperture is the dominant factor in determining the intensity and resolution. For RS_w = 0.1 mm and R = 185 mm, α_RS = 0.031°.

Antiscatter slits AS are slightly wider than the beam and are essential in this and other geometries to make certain the detector can receive X-rays only from the specimen area. They must be carefully aligned to avoid touching the beam.

The use of the long X-ray source makes it necessary to reduce the axial divergence, which would cause very large asymmetry. This is done with two sets of thin (25 to 50 μm) parallel metallic foils PS (`Soller slits'; Soller, 1924) placed before and after the specimen. If a monochromator is used, the set on the side of the monochromator is not essential because the crystal reduces the divergence. The angular aperture of a set of slits is The overall width of the set and δ determine the width of the specimen irradiated in the axial direction, which remains constant at all 2θ's. The construction is illustrated in Fig. 2.3.1.5(d). The aperture δ is usually 2 to 5°. Each set of parallel slits reduces the intensity; for example, with 12.5 mm long foils with 1 mm spacings, the intensity is about one-half of that without the parallel slits. The aperture can be selected with any combination of spacings and lengths but the greater the length, the fewer foils are needed, and the less is the intensity loss due to thickness of the metal foils (usually 0.025 mm). These slits can be made as interchangeable units of different apertures.