Angle Bragg

X rays and so that the angle between the difiracting plane and the incident X rays is equal to the Bragg angle For a single crystal or epitaxial thin film, there is only one specimen orientation for each (hkl) plane where these difiraction conditions are satisfied. 

Fig. 4-9. This diagram shows the intensity variation with angle for a rock salt crystal in the region near the Bragg angle, 0q, for an incident monochromatic beam. The area under the mosaic crystal curve could be thirty times greater than the ideal. (After Renninger, Z. Krist. 89, 344.)...

A series of powder photographs was taken in a Norelco camera 360 mm. in circumference which holds the film in the Straumanis arrangement and permits a maximum Bragg angle of 87-6°. The copper K radiation was filtered through 0-001 in. nickel foil. Eastman No-Screen X-ray film was used throughout. All photographs were taken at room temperature, 26 + 2° C. 

For the alloys containing less than 70 atomic percent thallium the powder photographs showed the reflection 620. For the other alloys in the A1 phase the photographs contained reflections only as far as 600 and 442. The presence of the reflection 620 at Bragg angles varying from 80-7 to 85-5° significantly improves the... 

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

The spectrometer is set to the appropriate Bragg angle 0 of the requisite characteristic wavelength, and only these X-rays will reach the detector and be counted. The detector employed is the gas proportional counter, which can operate at much faster count rates than the EDS crystal detector. 

Figure 14 Angles y and iji defining the position of the normal to a given hkl crystal plane in the sample reference system. Ix, incident X-ray beam lhkl, diffracted intensity 6B, Bragg angle. Adapted from Lafrance et al. [82]. Reprinted with permission of John Wiley 8t Sons, Inc.

The vector s ( s s) can be expressed in Cartesian (s, S2,s-s) or polar (s,polar angle61 and the azimuthal angle62 j/. The customary symbol 9 should not be used for the polar angle in a treatise on scattering because of a likelihood of confusion with the Bragg angle 0. 

The line width of the X-ray source is on the order of 1 eV for A1 or Mg Ka sources but can be reduced to better than about 0.3 eV with the use of a monochromator. A monochromator contains a quartz crystal which is positioned at the correct Bragg angle for A1 Ka radiation. The monochromator narrows this line significantly and focuses it onto the sample. It also cuts out all unwanted X-ray satellites and background radiation. An important advantage of using a monochromator is that heat and secondary electrons generated by the X-ray source cannot reach the sample. 

Unlike the case of diffraction of light by a ruled grating, the diffraction of x-rays by a crystalline solid leads to the observation that constructive interference (i.e., reflection) occurs only at the critical Bragg angles. When reflection does occur, it is stated that the plane in question is reflecting in the nth order, or that one observes nth order diffraction for that particular crystal plane. Therefore, one will observe an x-ray scattering response for every plane defined by a unique Miller index of (h k l). 

Total reflection x-ray fluorescence (TXRF) has become very popular for the conduct of microanalysis and trace elemental analysis [77-79]. TXRF relies on scatter properties near and below the Bragg angle to reduce background interference, and to improve limits of detection that can amount to an order of magnitude or moreover more traditional XRF measurements. As illustrated in Fig. 7.18, if x-rays are directed at a smooth surface at a very small angle, virtually all of the radiation will be reflected at an equally small angle. However, a few x-rays will excite atoms immediately at the surface, and those atoms will emit their characteristic radiation in all directions. One obtains very clean... 

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

The wavelength of the wave associated with accelerated electrons is much shorter than the neutron or X-ray wavelength (about a hundred times). As a result, the Bragg angles are very small (a few tenths of degrees) and the diffraction pattern is concentrated around the transmitted beam. 

---