Plane wave rocking curve

Figure 2.14 The plane wave rocking curve of Si 220 with CuK i, showing the tails on a logarithmic scale...

We first consider the beam divergence. As discussed in Chapter 2 we must perform a mathematical correlation (often miscalled convolution) between the plane wave rocking curves of the beam conditioner and of the sample crystals. If R (, ) and R 2 ( > j are the reflectivities (in intensity) of the first and... 

The double-crystal rocking curve is symmetric, though the plane wave reflectivity curve is not. This is a consequence of the autocorrelation, since the autocorrelation of any function is an even fimction. 

In this chapter we introduce high resolution diffraction studies of materials, beginning from the response of a perfect crystal to a plane wave, namely the Bragg law and rocking curves. We compare X-rays with electrons and neutrons for materials characterisation, and we compare X-rays with other surface analytic techniques. We discuss the definition and purpose of high resolution X-ray diffraction and topographic methods. We also give the basic theory required for initial use of the techniques. 

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...

As we have seen in Chapter 1, we need something near a plane wave in order to see the finest details of the specimen stracture. A single-axis diffractometer utilises a beam that is very far from a plane wave. Thus, single-crystal rocking curves are broadened due to the beam divergence, and the spectral width of the characteristic X-ray lines. 

The denominator (normalising constant) is the integrated reflectivity of the first crystal. Figure 2.13 shows the plane wave and the double-crystal rocking curve, again for Si 220 with CuK 1. We note the following ... 

If the second crystal is the specimen rather than a beam conditioner element, we shall have got close to the aim of measiuing the plane wave reflectivity of a material. The narrow rocking curve peaks permit us to separate closely matched layer and substrate reflections and complex interference details, as already seen in Figure 1.6. The sensitivity limit depends on the thickness of the layer but for a 1 micrometre layer it is about 50 ppm in the 004 symmetric geometry with GaAs and CuK radiation. This method has been used extensively to study narrow crystal reflections since the invention of the technique. 

Figure 2.13 The plane wave and double-crystal rocking curves for Si 220 with CuK j...

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

If we imagine the diffraction of a plane wave from epilayers we see that there will in general be differences of diffraction angle between die layer and the substrate, whether these are caused by tilt or mismatch f Double or multiple peaks will therefore arise in the rocking curve. Peaks may be broadened... 

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. 

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