Symmetrical reflection geometry

Guinier [6], p. 181), with the cross-section, F0, of the incident X-ray beam, the angle of incidence on the sample surface, a, and the angle of exit with respect to the sample surface being ae. For symmetrical-reflection geometry (a = Ofe = 6) the irradiated volume becomes Fq/ (2id), and 1 /2fi is the penetration depth into the sample. We thus have... 

When reflection geometries are set up in modern scattering applications to study the structure of thin layers, the simplifying assumption of infinite sample thickness is not allowed, and the absorption correction becomes more difficult. Moreover, symmetrical-reflection geometry is utilized less frequently than asymmetrical-reflection geometry with fixed incident angle. Thus both cases are of practical interest. 

Figure 7.4. Sketch for the deduction of the intensity, It, transmitted into the detector for symmetrical-reflection geometry. The photon is scattered in a depth of x. Integration direction is indicated by a straight dashed arrow... 

Figure 7.4 presents a sketch for the deduction of the intensity in symmetrical-reflection geometry. F is the footprint area of an incident microbeam on the sample surface, and dlt is the related contribution to intensity. Again utilizing the absorption law Eq. (7.1) we have... 

For very thin sample thickness t and a scattering angle 29 that is well above the critical angle of total reflection, the exponential factor is approximately unity and a simple background subtraction without consideration of absorption is allowed. Symmetrical-reflection geometry is only a special case of asymmetrical-reflection geometry. 

On the other hand, if the primary beam is illuminating the complete sample surface (case 2), the absorption factor for symmetrical-reflection geometry becomes... 

For symmetrical-reflection geometry the modulus of the true scattering vector is... 

Figure 8.2. WAXS curves from semicrystalline and amorphous poly(ethylene terephthalate) (PET). Separation of the observed intensity into crystalline, amorphous, and machine background (laboratory goniometer Philips PW 1078, symmetrical-reflection geometry)... 

Let us consider a nanostructured thin film built from lamellar particles [84], If the principal axis of layer stacks is oriented normal to the film surface, the scattered intensity measured in symmetrical-reflection geometry (SRSAXS) is... 

According to [18,19] and our experience, it is usual that the smaller FWHM for symmetrical reflections (measuring mostly tilt mosaicity) corresponds to a larger FWHM for asymmetrical reflections (measuring together twist and tilt mosaicity). For example, for the MOCVD GaN layer which possessed a 00.2 RC of only 40 arc sec, the 10.2 reflection exhibited an FWHM of 740 arc sec [18], Similarly, for a 0.76 pm MBE layer, Amano et al [19] reported 48 arc sec for the 00.2 reflection and 5226 arc sec for the 10.0 reflection (grazing incidence geometry). The other sample (1.7 pm) possessed 365 arc sec and 1581 arc sec, respectively, for those two reflections. 

Figure 2.2. The classical setup for the static measurement of X-ray scattering in symmetrical-reflection geometry. 26 is the scattering angle. If the sample is turned upright and thus transmitted instead, the corresponding geometry becomes the symmetrical-lransniission geometry...

Suppose a beam of cross-sectional area A falls on a sheet sample of thickness t in the symmetrical reflection geometry, as in Figure 2.24a. We assume, for the sake of simplicity, that the rays in the beam are all parallel to each other. Now consider a layer of thickness dx inside the sample, at depth x below the flat surface, where the irradiated volume is equal to dx A/ sin 0. Before reaching this depth the beam has traveled distance Z within the sample, where l is equal to x/sin 0, and has suffered attenuation by a factor exp(—/x/), p being the linear absorption coefficient. The scattered beam must travel the same distance within the sample again on its way out. If i 20) is the intensity of scattering per unit volume of the sample, then the contribution dl(20) to the total scattering intensity by the layer dx at depth x is... 

Figure 2.24 Geometry in the calculation of the absorption factor in the (a) symmetrical reflection and (b) symmetrical transmission mode with a sheet sample.

Figure 3.19 Geometry of sample orientation in the diffractometer in (a) symmetric transmission technique and (b) symmetric reflection technique.

Figure 7.5. Relationship between symmetrical (

reflection geometry. Bold bars symbolize the sample in symmetrical (dashed) and asymmetrical (solid) geometry. Incident and scattered beam are shown by dashed-dotted arrows, the incident angle is a = 0 + scattering vector s. For the tilted sample the sample-fixed scattering vector S3 is indicated (after [84])... 

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