Bragg-Brentano geometry

Figure 4 Diffraction patterns (Bragg-Brentano geometry) of three superconducting thin Aims ( 2- im thick) anneaied for different times. The temperatures for 0 resistance and for the onset of superconductivity are noted.

The use of Equation (22) is very general, but it is also possible, with accurate measurements and data treatment, to perform the quantitative phase analysis in semi-crystalline materials without using any internal standard. This procedure is possible only if the chemical compositions of all the phases, including the amorphous one, are known. If the composition of the amorphous phase is unknown, the quantitative analysis without using any internal standard can still be used provided that the chemical composition of the whole sample is available [51]. This approach, until now, has been developed only for the XRD with Bragg-Brentano geometry that is one of the most diffused techniques in powder diffraction laboratories. 

An expression including the diffuse background of a crystalline phase was calculated for a Bragg-Brentano geometry [55] ... 

FTIR spectra were recorded with an Impact 410 (Nicolet) spectrometer. Powder X-ray diffraction data were obtained on a Siemens D 5005 diffractometer in the Bragg-Brentano geometry arrangement using CuKa radiation. Adsorption isotherms of nitrogen at -196 °C... 

Phase identification was performed by X-ray diffraction in Bragg-Brentano geometry with Cu-Ka radiation and a secondary monochromator. A rotating sample holder was used in order to minimize texture effects in the x-y plane and to offset the effects of the rather large grain size. Diffractograms were taken as a function of depth after stepwise removal of layers with an abrasive diamond disk. 

The great majority of the applications of the x-ray diffraction methodology in material characterizations are carried out with the help of diffractometers, which use the Bragg-Brentano geometry. The principal characteristics of the Bragg-Brentano geometry are shown in Figure 1.24. 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

The absorption factor for a sample in the form of a plate located in the sample holder of a Bragg-Brentano geometry powder diffractometer is given by [4]... 

Fig. 5.14. XRD spectra measured in Bragg-Brentano geometry (0 — 2 or lo scans) for ZnO films deposited at the maximum rate (p(O2) = 33 mPa) and different substrate temperatures...

Powder x-ray difftaction data were obtained with a Philips PWl050/25 diffractometer operating in Bragg-Brentano geometry with CrKa radiation, (A, = 2.29 A). Data were collected in the 2Grange 1° - 20" with a step size of 0.05" and dwell time of 6 s per point. 

We show by X-ray diffraction in the Bragg-Brentano geometry that the atomic stmcture of lead sulfide nanoparticles in thin films prepared by wet chemical method is different from the B stmcture, which is the equilibrium phase for bulk single-crystalline PbS. The atomic structure of nanoparticles can be desaibed by the cubic space group Fm-3m with both tetrahedral and octahedral coordinations for sulfur atoms. 

Powder diffraction data were taken on a Philips PW 1710 X-ray powder diffractometer (XRPD) with Bragg Brentano geometry (vertical goniometer) in 0.025 ° step from 5 to 90 29 with 20 s per step. 

Figure 2.48. The illustration of the derivation of Eq. 2.72. The incident beam penetrates into the sample by the distance xj before being scattered by the infinitesimal volume dV. The scattered beam traverses the distance Xs before exiting the sample. In the Bragg-Brentano geometry =...

Another problem with pulverized samples is that their packing density varies as a function of the depth. This is known as the porosity effect, and for the Bragg-Brentano geometry, it may be expressed using two different approaches ... 

A different goniostat with the horizontal orientation of the specimen and Bragg-Brentano geometry is shown in Figure 3.13. The x-ray tube housing is mounted on the movable arm, and both the x-ray source and the detector can be rotated in a synchronized fashion about the common horizontal goniometer axis (also see the schematic in Figure 3.10, middle). 

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.

The second important factor is the absorption of x-rays by the sample. In the Bragg-Brentano geometry the sample should be completely opaque to x-rays. Assuming that the absorption of 99.9 % of the incident beam intensity represents complete opacity, then the beam intensity should be reduced by a factor of 1000 and the following equation can be written (also see Eq. 2.8 in Chapter 2) ... 

No matter how much time has been spent on the sample preparation and how good the resulting specimen is, it always needs to be properly positioned on the goniometer. Consider, for example. Figure 3.26, which shows the effect of sample displacement in the Bragg-Brentano geometry. 

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.

To summarize this section, a high quality specimen for powder diffraction may be difficult to prepare and it is not as simple as it seems. The task requires both experience and creativity. The adverse effects of an improperly prepared sample can be illustrated by the following three figures, shown in Rietveld format, where the experimental powder diffraction data were collected in the Bragg-Brentano geometry from two different specimens prepared from the same material, which was intentionally left in the form of course powder. 

In the transmission geometry the requirements are different. When a flat transmission sample is used, the aperture of the incident beam is defined by the largest Bragg angle of interest, since at 0 = 0 the sample is perpendicular to the incident beam (and not parallel, as in the Bragg Brentano geometry). Equation 3.1 then becomes as follows (where the notation are the same as in Eq. 3.1)... 

In problems 4-8, the data were collected on a powder diffractometer with Bragg-Brentano geometry using Cu Ka radiation. Errors in (/-spacing should not exceed 0.02 A for (/ > 3 A, otherwise they should be less than 0.01 A. 

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