Beam conditioners

We shall discuss beam conditioners in some detail, since no general review applicable to high resolution diffractometry contairrs all the recent developments, though earlier reviews by Beaumont and Hart, Kohra etal. and Wilkirrs and Stevenson are useful. Beam conditioners may in principle contain the following elements  

The intensity at any angrrlar setting restrlting from the two reflections corresponds approximately to the area trader the overlap of the two perfect crystal duMond diagrams, Figrtre 2.12. More precisely, the intensity of each 

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. 

Note that we use the characteristic lines only because they have high intensity. We can work with the Bremsstrahlung but the intensity is low. However, it is a way of exploiting wavelengths that do not correspond to characteristic lines. 

Another way of looking at the traditional double-crystal method is that it measures the difference in reflectivity between the specimen and a reference perfect crystal. This is the original high resolution method and is still the best choice when good reference crystals are available and the specimens to be 

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Better conventional collimation will not do, except for the largest synchrotron radiation installations to obtain snb-arc-second collimation in the laboratory would require a collimator some 100m long with a sealed-tube source, and at this distance the intensity would be impracticably low. The problem is solved by the use of a beam conditioner, which is a further diffracting system before the specimen The measnred rocking cnrve is then the correlation of the plane wave rocking cnrves of the beam conditioner and the specimen crystals, from which most of the diffracting characteristics of the specimen crystal may be deduced. 

We first define the geometry and instrumental parameters common to high resolntion diffractometry. As a reference, we then develop the dnMond diagram for visualisation of X-ray optics and use it to discuss practical beam conditioners. Next we treat the principal aberrations of high resolution diffractometry tilt, curvature and dispersion. We discuss the requirements on X-ray detectors, and finally show how to set up a high resolution measurement in practice. 

This rises to 10 if both the K 1 and K 2 ines are diffracted by the specimen. The effect this has upon the rocking curve depends on the dispersion of the whole system of beam conditioner and specimen, and ranges from zero to very large. This will be discussed below, in section 2.6. 

We therefore need to limit the divergence and wavelength spread of the beam incident upon the specimen. Beam conditioners are used to collimate and to... 

The K line is a doublet with separation abont 10 It is often important to remove the K 2 line, which has approximately half the intensity of the K 1, by the use of a beam conditioner as shown later. 

Since the electrons in storage rings are travelhng at relativistic speeds, the emission of electromagnetie radiation is foreshortened into a cone whose axis is tlie instantaneous direetion of motion of the eleetron. The radiation is therefore intrinsically collimated and is a good mateh to the subsequent beam conditioner. This contrasts favourably with a laboratory somce, in which very little of the more-or-less isotropie emission reaehes the speeimen. The principal characteristics of synchrotron radiation are ... 

The sources are invaluable for the tunability of the radiation, that is where spectroscopic as well as scattering properties are important, and for experiments requiring the polarisation and time structure. However, with recent advances in X-ray tubes, beam conditioners and detectors, many scattering experiments are just as well performed in the convenience of the laboratory. Although it is difficult to attain the same intensities in the laboratoiy, it is in fact easier to achieve good signal-to-noise ratios. If CuK i is suitable for the experiment, it is likely that better productivity will be obtained with a laboratory source. 

Figure 2.4 shows the elements of a high resolution diffractometer. The beam conditioner controls the divergence and wavelength spread of the beam by a combination of diffracting elements and angular-limiting apertures. The latter may also control the spatial width of the beam. This falls upon a specimen,... 

The instmment shown in Figure 2.4 is a double-axis instrument. The first axis is the adjustment of the beam conditioner, the second is the scan of the specimen through the Bragg angle. It is irrelevant to this definition that a practical diffractometer may contain a dozen or more controlled axes , for example, to tune and to align the beam conditioner, to locate the specimen in the beam, to align and to scan the specimen and to control shts. It is the differential movement of the two main axes that make the measnrement and determine the precision and accuracy of the instrument. This is the basic high resolntion diffractometer, which is now widely nsed for measniements of crystal perfection, epilayer composition and thickness. 

Many types of beam conditioners have been designed, since the original double-ciystal experiments of Ehrenberg, Ewald and Mark in the early 1920s. These... 

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. 

Accordingly, a number of beam conditioners have been developed which both collimate and monochromate the beam, for general-purpose high resolution diffractometers. 

Figure 2.17 The duMond-Hart-Bartels four crystal (+n, -n, -n, +n) beam conditioner design...

The advantage of an increased input divergence is apparent when we plot the CuK 1 line at high magnification (Figure 2.19). Indicated on the plot are the angular widths of the symmetric and asymmetric Si 220 reflections as used for the beam conditioner shown in Figure 2.20. The symmetric reflection clearly... 

The complexity of the geometry usually necessitates numerical analysis by ray tracing in order to design such beam conditioners and to predict their aberrations. The design principles are straightforward ... 

Figure 2.19 A detailed plot of the spectrum from a Cu X-ray tube, in the vicinity of the K lines, showing the area selected by the high resolution and high intensity settings of the beam conditioner shown in Figure 2,20...

Figure 2.20 The Loxley-Tanner-Bowen combined high resolution and high intensity beam conditioner based on the duMond principle, illustrated for Si 202 with CuK j radiation, (a) Geometric arrangement for high resolution, (b) geometric arrangement for high intensity...

Wavelength dispersion arising from the bandpass of the beam conditioner... 

The theory of this aberration was worked out in the 1920s by Schwarzchild. For simphcity we shall discuss the case of a beam conditioner comprising a single crystal and an aperture as in the classic double-crystal arrangement. If the Bragg planes are tilted about an axis contained in the incidence plane and the... 

Figure 2.22 A three-dimensional plot of the diffracted intensity as a function of specimen rotations both parallel ( ) and perpendicular ( j to the dispersion plane. The larger peak is that from the substrate GaAs and the smaller from the GaAIAs layer. CuK i, slit-limited from a single GaAs beam conditioner...

The dispersive (+ n, - m ) mode has already been seen clearly with the duMond diagrams, Figure 2.10. Here, the curves are no longer identical and the crystals must be displaced from the parallel position in order to get simultaneous diffraction. As the crystals are displaced, so the band of intersection moves up and down the curve. When the curves become very different, the K 1 and K 2 intensities are traced out separately. Then the peaks are resolved in the rocking curve, and if no better beam conditioner is available it is important in such cases to remove the K 2 component with a slit placed after the beam conditioner. A slit placed in front of the detector, with the detector driven at twice the angular speed of the specimen, also works very well. This is in effect a low resolution triple-axis measurement. 

In the general multiple-crystal beam conditioner case, there is no universal formula for broadening. Rather, the duMond diagram is constructed for the beam conditioner and the shape of the passed band in and is determined. The specimen crystal is then represented on the duMond diagram and scanned... 

Determine the take-off angle from the X-ray tube. This controls the beam size, and a resultant beam size of approximately 0.5x0.5 mm is about right. Use the maximum take-off angle consistent with this size. Ahgn the instrument so that the beam conditioner points at the source at this take-off angle. 

The beam conditioner may have some angular, tilt and translation adjustments. These should be set so that the beam delivered... 

For symmetric reflections the peak search may now begin. For asymmetric reflections, the specimen must be rotated about its normal until the desired diffraction vector lies in the incidence plane of the beam conditioner. This is normally the diffractometer surface. An accurate knowledge of the orientation of the specimen in two axes is required to set asymmetric reflections this is usually taken from the position of the orientation flat or groove. 

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. 

For laboratory-based systems the instrument function given by the effect of the beam conditioner must now be introduced. In Chapter 2 we discussed beam conditioners in detail and showed that they may be characterised in terms of an intensity which is a function of both divergence and wavelength. 

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... 

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