Goniometer radius

Overall, powder diffractometers equipped with point detectors offer the best resolution of the resulting powder diffraction data. While the instrumental resolution increases with the increasing goniometer radius, the intensity of the diffracted beam unfortunately decreases because the incident beam produced by an analytical x-ray tube is always divergent. Therefore, typical goniometer radii vary between 150 and -300 mm. 

When the goniometer radius increases, the size of a flat specimen, needed to maintain high intensity in the Bragg-Brentano geometry, becomes unreasonably large, see section 3.5.3. 

Figure 3.24. The length of the projection of the incident beam, L, on the surface of the flat sample in Bragg-Brentano geometry. F - focal point of the x-ray source, DS - divergence slit, R - goniometer radius,

Figure 3.25. Irradiated lengths, /, and I2, of the flat specimen in the Bragg-Brentano geometry as functions of Bragg angle calculated using Eq. 3.1 for different angular divergences of the incident beam assuming goniometer radius, R = 285 mm.

Figure 3.28. The effect of sample displacement, s, on the observed Bragg angles calculated from Eq. 3.4 assuming Bragg-Brentano geometry and goniometer radius R = 285 mm. 0 is the observed Bragg angle, 0 is the Bragg angle in the absence of sample displacement.

Figure 3.34. The set of x-ray powder diffraction patterns collected from the nearly spherical LaNi4 gsSno.is powder (see Figure 3.32, inset) on a Rigaku TTRAX rotating anode powder diffractometer using Mo Ka radiation. Goniometer radius R = 285 mm receiving slit RS = 0.03° flat specimen diameter d = 20 mm. Incident beam apertures were 0.05, 0.17, 0.25, 0.38, 0.5, 0.75, 1, 1.5, 2° and completely opened ( 5°), respectively. An automatic variable scatter slit was used to reduce the background. The data were collected with a fixed step A20 = 0.01°, and the sample was continuously spun during the data collection.

Unlike the goniometer radius and radiation wavelength, both the diffracted beam aperture and the step size (or sampling step), are easily controlled by the experimentalist. Their effects on the resolution of the powder diffraction pattern were examined in section 3.6, which should be consulted for proper selection of beam apertures. 

B) Sealed x-ray tube source, Bragg-Brentano goniometer, radius 185 mm, scintillation detector. 

Both Eqs. 5.48 and 5.49 introduce one additional parameter in the least squares refinement, i.e. sample displacement divided by the goniometer radius, s R, or zero shift, 50o, respectively. Regardless of the fact that the contribution from both parameters is nonlinear, the linear least squares technique can still be applied after the following simplifications. 

X = 1.540593 A. Peak positions are already corrected for the sample shift -0.15 mm assuming goniometer radius of 250 mm. 

A white crystalline powder, prepared by hydrothermal treatment at 200°C of a mixture of molybdic acid, H2M0O4, and methylammonium ma) chloride, CH3NH3CI, taken in a 1 2 molar ratio and acidified with hydrochloric acid, HCl, to pH = 3.5, resulted in a complex powder diffraction pattern shown in Figure 6.29. It was indexed in the monoclinic crystal system as was discussed in section 5.12.2. The space group C2/c (or its acentric subgroup Cc) was established from the analysis of the systematic absences, and the unit cell dimensions were refined using 120 resolved reflections below 20 = 60° a = 23.0648(6) A, b = 5.5134(2) A, c = 19.5609(5) A, p = 122.931(1)°, and the sample displacement 8 = -0.098(3) mm for a 250 mm goniometer radius. The unit cell volume is 2087.8 A. ... 

The sample displacement parameter Ss calculated for each particular case from the displacement (s, in mm) obtained during unit cell refinement as Ss = -36000s/(7tR) or Ss = -144s/7T for the goniometer radius R = 250 mm. 

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