Diffractometer intensity equations

Any powdered material consists of a set of randomly oriented crystallites of the material under test. The line intensity of a powder XRD pattern obtained in a Bragg-Brentano geometry diffractometer for a pure sample, comprised of three-dimensional crystallites with a parallelepiped form (see Equation 1.64), is given by [3,4,22,24,28]... 

The relative anatase content in the samples was calculated from the intensity of the X-ray diffraction band with maximum at 2 = 25,4° (the spectrum was registered by DRON-3M diffractometer, copper Ka-line). Here, commercial Ti02 P25 (Degussa Corp.), containing 75% of anatase [11, 12], was used as a reference standard. The average size of Ti02 crystallites was estimated from the diffraction band of (101) lattice plane using Scherer s equation. 

For each Bragg reflection, the raw data normally consist of the Miller indices (h,k,l), the integrated intensity I(hkl), and its standard deviation [ a[I) ]. In Equation 7.2 (earlier), the relationship between the measured intensity / [hkl] and the required structure factor amplitude F[hkl) is shown. This conversion of I hkl) to F hkl) involves the application of corrections for X-ray background intensity, Lorentz and polarization factors, absorption effects, and radiation damage. This process is known as data reduction.The corrections for photographic and diffractometer data are slightly different, but the principles behind the application of these corrections are the same for both. 

The use of a monochromator produces a change in the relative intensities of the beams diffracted by the specimen. Equation (4-19), for example, was derived for the completely unpolarized incident beam obtained from the x-ray tube. Any beam diffracted by a crystal, however, becomes partially polarized by the diffraction process itself, which means that the beam from a crystal monochromator is partially polarized before it reaches the specimen. Under these circumstances, the usual polarization factor (1 - - cos 26)12, which is included in Eqs. (4-19) through (4-21), must be replaced by the factor (1 + cos 2a cos 20)/(l -I- cos 2a), where 2a is the diffraction angle in the monochromator (Fig. 6-16). Since the denominator in this expression is independent of 6, it may be omitted the combined Lorentz-polarization factor for crystal-monochromated radiation is therefore (1 + cos 2a cos 20)/sin 6 cos 6. This factor may be substituted into Eqs. (4-19) and (4-20), although a monochromator is not often used with a Debye-Scherrer camera, or into Eq. (4-21), when a monochromator is used with a diffractometer (Sec. 7-13). But note that Eq. (4-20) does not apply to the focusing cameras of the next section. 

The chief problem in determining particle size from line breadths is to determine B from the measured breadth fiv/ of the diffraction line. Of the many methods proposed, Warren s is the simplest. The unknown is mixed with a standard which has a particle size greater than 1000 A, and which produces a diffraction line near that line from the unknown which is to be used in the determination. A diffraction pattern is then made of the mixture in either a Debye camera or, preferably, a diffractometer. This pattern will contain sharp lines from the standard and broad lines from the unknown, assumed to consist of very fine particles. Let Bg be the measured breadth, at half-maximum intensity, of the line from the standard. Then B is given, not simply by the difference between B and B, but by the equation... 

To find the relation between diffracted intensity and concentration, we must go back to the basic equation for the intensity diffracted by a powder specimen. The form of this equation depends on the kind of apparatus used, namely, camera or diffractometer we shall consider only the diffractometer here. The exact expression for the intensity diffracted by a single-phase powder specimen in a diffractometer is... 

As with angle-dispersive neutron diffractometers, the design of TOE powder diffractometers can be optimised for either high resolution or high intensity or some compromise of both. An understanding of the factors that affect the resolution is therefore important. The relative uncertainty in (i-spacing, hdjd, may be determined from the equation ... 

Powder X-ray diffraction (XRD) patterns were obtained at room temperature from 1.5 to 100° 20 with a Bruker-AXS D8 Advance X-ray Diffractometer setup using Co-Ka radiation. The average Ni3(N03)2(0H)4, NiO and Ni crystallite sizes were calculated according to the Scherrer equation using the most intense lines at 20 = 14.9°, 50.8° and 52.2° 20, respectively. The calcination of sample S/SD was followed with an Enraf Nonius PDS 120 diffractometer using Co-Kui and equipped with an Anton-Paar XRK reaction chamber. The in situ XRD patterns were recorded from room temperature till 300 °C (heating rate = 1 °C min ) with intervals of 20 °C. 

The intensity distributions calculated for the atomic models (using equation 8) are shown in fig. 6, together with the diffractometer traces overlaid for direct comparison. The calculated data are for a distribution of chain lengths centered on M=10, and have been corrected for the Lorentz and polarization effects. The intensity agreement is very good, especially for the peaks at d v sA, and... 

The crystallinty index of the fibres was studied using a wide angle X-ray diffractometer Bruker model D8 Advance, equipped with a scintillation counter and a linear amplifier was used. The diffraction intensities were recorded between 5 and 60 (26 angle range). The crystallinty index was determined using the equation ... 

Physico-chemical characterizations were performed on the finished ceria-doped silicas. Surface area measurements (BET) and mesopore size distribution (BJH) were carried out by means of Sorptomatic 1900 (Carlo Erba) instrument. X-ray diffraction patterns were recorded with a D 5005 X-Ray Diffractometer (SIEMENS) using Cu Ka radiation coupled with a graphite monochromator. The crystallite sizes of ceria phase were calculated from the line broadening of the most intense reflection using the Scherrer equation [13]. 

The X-ray diffraction patterns have been recorded with a Philips diffractometer equipped with a proportional counter by using a Nl-flltered CuKa radiation. The samples have been examined without any previous pretreatment. The crystallinity degree has been determined by a procedure developed in our laboratory (ref. 8). The method is based on a comparison of the integrated intensities of two different spectral ranges affected respectively by the crystalline and amorphous fractions of the solid. In this manner the use of standards with a known crystallinity can be avoided. The crystallite size has been determined from the half height width using the Scherrer equation (ref. 8) after the corrections for the Kal,a2 doublet and the instrumental broadening. 

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