Two-beam approximation

Corrections for primary extinction in the two-beam approximation (the Blackman correction) uses the following relation between the observed and calculated (kinematical) reflection intensity [2] ... 

Diffraction patterns from thin polycrystalline Ge films were measured by the eleetron diffraetometer. After refinement of scale and thermal factors and corrections for the primary extinetion within the two-beam approximation, the parameters k (spherieal deeompression of valence eleetron shell) and multipoles P32- and P40 (anisotropy of electron density) were ealeulated (Table 4). The residual faetor R ealeulated from the experimental and theoretical amplitudes (the latter were ealeulated by the LAPW method, Lu Z.W., et al. Phys.Rev. 1993, B47, 9385) is 2.07% and proofs the high quality of the experimental. 

However, we now make the simplifying assumption that as we approach the condition for Bragg scattering, only two of the amplitudes /Ig are simultaneously large and all the other are negligibly small. The large v4g correspond to g = 0 and g = 1. This is known as the two-beam approximation. For convenience we write these two amplitudes as Aq and Ag. The infinite set of equations now becomes... 

For the two-beam approximation there is only one operating reflection g, and for the deformed crystal the corresponding Fourier coefficient is... 

This effect might be interpreted by the Bethe dynamic potential approximation, which does not take into account the crystal orientation (as in the Blackman correction case) nor crystal thickness parameters. In terms of this approach, the effect of weak beams can be included in two-beam theory by replacing the potential coefficients, Vh, by ... 

The Fourier transform infra-red spectrophotometer incorporates an interferometer (Fig. 3). Its basic components are a beam splitter and two mirrors, perpendicular to each other, one fixed and one which can be moved backwards and forwards at right angles to its plane. Approximately half of the radiation from the source is reflected to the fixed mirror where it is reflected back to the beam splitter which transmits about half (i.e. a quarter of the original) to the detector. The other half of the original radiation passes through the beam splitter to the movable mirror where it is reflected back to the beam splitter and about half (i.e. a quarter of the original) is reflected to the detector. When the two mirrors are equidistant from the beam splitter, the two beams... 

A comparison between the SAED and CBED is given in Figure 3. CBED patterns consist of disks. Each disk can be divided into pixels, each pixel approximately represents one incident-beam direction. For example, the beam P in Figure 3 gives one set of diffraction patterns shown in frill lines. The geometry of the diffraction pattern by the incident-beam P is the same as the selected area diffraction pattern with a parallel illumination. For a second beam P, which comes at a different angle compared to P, the diffraction pattern is displaced from that of P by O /k with a as the angle between the two beams. 

The kinematic approximation breaks down at a certain crystal thickness when the diffracted intensity approaches that of the incident beam. A usefiil criterion for kinematic approximation is r < fg/4, where fg is the extinction distance of the strongest reflection in the diffraction pattern. The extinction distance is orientation dependent. In case only one set of lattice planes is strongly diffracting (the two-beam condition), the extinction distance is given by = h lmeX Vg. 

With optical detection, the overall time resolution is limited by the different velocities of fast electrons and photons in condensed media this results in loss of synchronization as the two beams pass through the sample cell. This desynchronization is approximately 10 ps cm in water [145], so the optical path length has to be reduced proportionally to achieve the improved time resolution provided by subpicosecond pulses. There is thus a compromise between having short time resolution (short optical path) and high absorbance signals (long optical path). 

Kinematical theory is also called two-beam theory, as it includes consideration of only the incident and scattered X-ray beams. This is a very good approximation for thin crystals or for weakly scattering particles (e.g., most X-ray photons and neutrons), as rescattering of the initially scattered beam will be very weak. It is useful for calculating... 

The double-beam system is used extensively for spectroscopic absorption studies. The individual components of the system have the same function as in the single-beam system, with one very important difference. The radiation from the source is split into two beams of approximately equal intensity using a beam splitter, shown in Fig. 2.28. One beam is termed the reference beam, the second beam, which passes through the sample, is called the sample beam. The two beams are then recombined and pass through the monochromator and slit systems to the detector. This is illustrated schematically in Fig. 2.28. In this schematic, there is a cell in the reference beam that would be identical to the cell used to hold the sample. The reference cell may be empty or it may contain the solvent used to dilute the sample, for example. This particular arrangement showing the monochromator after the sample is typical of a dispersive IR double-beam spectrophotometer. There are many commercial variations in the optical layout of double-beam systems. 

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