Rocking curve calculation

Figure 6 DCD rocking curves—measured (dashed) and calculated (solid)—of the (400)...

Figure 2. Calculated CBED rocking curves for Si[ 110], a primary beam energy of 193.35 keV and a crystal thickness of 369nm. The three curves shown in the figure were calculated using 80 Bloch waves (circle+solid line) 20 Bloch waves (star solid line) and 5 Bloch waves (dotted line) and the curves correspond to the line of Figure 1 along A-D.

Figure 4. Calculated CBED rocking curves within the (000) and the (ill) disks in a Si[l 10] zone axis CBED pattern. All curves shown in the figure were calculated for a crystal thickness of 250 nm, and a primary beam energy of 196.35 keV., and correspond to the line scan A-B of Figure 1.

The goodness-of-fit between the experimental and theoretically calculated CBED rocking curves is described by a merit function, and in the present study we use the chi-square merit function defined as... 

Figure 5. Calculated CBED rocking curves. This figure is essentially the same as Figure 4, except that all calculations were made for a crystal thickness of 500 nm.

Figure 9. Energy-filtered experimental and fitted Si[l 10] CBED rocking curves for (a) a line scan along the [111] direction and (b) a line scan along the [002] direction (see Figure 1). The calculations were made for a primary beam energy of 195.35keV and a crystal thickness of 369 nm.

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

Figure 1.2 Calculated plane wave X-ray rocking curves, (a) Si 004 with CuK i (0.154 nm), FWHM=3.83 arcsec, (b) Si 333 with MoK j (0.071 nm), FWHM=0.73 arcsec, (c) Ge 111 with CuK i, FWF1M=16.69 arcsec, (d) GaAs 004 witih CuK FWHM=8.55 arcsec...

Figure 3.8 The effect of thickness on layer peak intensity, (a) Rocking curves, (b) integrated intensity ratio between substrate and epUayer peaks. Calculated curves for AIq 3Gao 7AS on GaAs. CuK radiation 004 reflection...

Thus we calculate the reflectivity of a whole layered material from the bottom up, using the amplitude ratio of the thick crystal as the input to the first lamella, the output of the first as the input to the second, and so on. At the top of the material the amplitude ratio is converted into intensity ratio. This calculation is repeated for each point on the rocking curve, corresponding to different deviations from the Bragg condition. This results in the plane wave reflectivity, appropriate for synchrotron radiation experiments and others with a highly collimated beam from the beam conditioner. 

Match the main features of the rocking curve first, using plane wave calculations and a single (sigma) polarisation. It shonld be possible to fit all the main peaks accnrately in spacing and approximately in intensity. Then begin refinement to match the intensities and widths. 

Fig. 3.5. S180 vs. 87Sr/86Sr (A) and Nb/Zr (B) diagrams for IAP rocks. Curved full lines are calculated mixing trends between Roman ultrapotassic magma and limestone (i.e. magma contamination trend). Dashed line is a mixing trend between mantle and marly sediments (source contamination). Numbers along lines indicate amounts of sediments involved in the mixing.

The rocking curve full width at half maximum intensity (fwhm) was characterized at the 100/001 diffraction for pzt 52/48 thin films with gixrd geometry. Figure 6.12 shows the calculated diffraction pattern distribution for pzt 52/48 thin films based on the lattice parameters obtained from xrd-rsms shown in Figure 6.6. As shown in Figure 6.12, the 1001001 diffraction includes 100 diffraction of c-domains, the 001 diffraction of a-domains and the 100 diffraction of the pseudocubic phase. Therefore, fwhm obtained by this measurement indicates a twist of all these phases. 

Figure 6.13 Incident angle dependence on the rocking curve FWHM of Pb(Zro.52Tio.48)03 thin films and the penetration depth (calculation).

The only observational constraint on the internal structure of satellites is density. The bulk composition (rock to ice ratio) of icy satellites is obtained by comparing tlieir density vs. radius relationship witli model curves calculated by Lupo and Lewis [6] as shown in Fig. 9.2. From this comparison, we can estimate tlie rock to ice (to pore) ratio of the satellites. 

Fig. 4. Fractionation of PGE pattern (Pd/Ir)n v. AI2O3 for progressive melt removal and the effects of sulphide addition as a result of melt-rock interaction. Calculation of the melting trends follows the method of Lorand et al. (1999). Tick marks denote 5% melting intervals. The two curves shown depict starting compositions with 250 ppm (model 1) and 300 ppm (model 2) total sulphur. The trends terminate when residual sulphide disappears at 25 and 30% melting, respectively. The sulphide addition trends are general trends taken from Rehkamper et al. (1999).

Lorentz and polarization factors. The Lorentz factor, l/sin(2<9), is associated with the fact that the calculations are performed in rectangular coordinates while the measurements are all performed with spectrometers that are controlled in an angular coordinate system. For most experimental configurations, the Lorentz factor has the familiar form L = l/sin(26) when the rocking curves are measured by scanning the motor. However, other functional forms are found for different spectrometers (Feidenhans l 1989). 

Here, Ij and aj are the intensity and uncertainty of the j th data point in the rocking curve (the uncertainty is assumed to be due to counting statistics, whereby the measurement of N photons is uncertain by V7V) Icaic is the calculated intensity for a particular set of parameters (peak position, width and intensity, etc.) the sum is performed over all data... 

To provide a more quantitative description of the surface defect structure and distribution, we use a model for randomly distributed surface defects (Lu and Lagally 1982). With this model, the rocking curve width and its variation with Q can be calculated from a few parameters that describe the probability of encountering a defect (i.e., a step) and the phase change in encountering that defect. (For specular reflectivity, this phase change reflects the height difference across the step.) Within this model, an approximately Lorentzian line shape is reproduced. 

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