Reflectance function reconstructed

Fig. 8.3. Reconstruction of the reflectance function of duraluminium from magnitude only V(z) data, measured at 320 MHz. (a) Steps in the reconstruction of P(9)R(9) after (i) 1 (ii) 3 (iii) 10 and (iv) 30 iterations of the phase retrieval algorithm, (b) Reconstructed R 8) (Fright etal. 1989).

As we have discussed already in the previous section on Bleistein inversion, the goal of inversion in many applications is the reconstruction of the reflectivity function, which provides information about the distribution of the reflecting boundaries in the medium under investigation. This problem can also be addressed by using the Kirehhoff approximation (14.66), which we reproduce here for convenience ... 

When we make these same comparisons for an internuclear separation of 20 bohr, we obtain the coefficients shown in Table 2.5 and the weights shown in Table 2.6. Now the orthogonalized AOs give the asymptotic ffinction with one configuration, while it requires three for the raw AOs. The energies are the same, of course. The EGSO weights imply the same situation. A little reflection will show that the three terms in the raw VB function are just those required to reconstruct the proper HI5 orbital. 

One of such methods is based on spectra obtained from normai and obiique incidence reflection and transmission US, which are piotted as a function of six non-dimensionai parameters of the coating determined from two types of spectra. The six non-dimensionai parameters aiiow the reconstruction process to be transformed from one search in a six-dimensionai space to two searches in three-dimensionai spaces (one search for normai incidence and one for obiique incidences). Thickness density and longitudinai and shear eiastic moduii of the coating and attenuations can be determined for thicknesses iess than the uitrasonic waveiength from the non-dimensionai parameters [43]. it shouid be noted that US-based examination of coatings is particuiariy usefui for poiymer-coated systems as the impedance of metais is about tenfoid more than that of common poiymers. 

At metallic surfaces, STS spectra are generally not as structured as at semiconductors. This probably explains why STS has had much less impact upon metals [69]. STS has nevertheless been successfully attempted on Au(lOO), Au(lll) and Pd(lll) [70-72]. On Au(lll), imaging the surface near the surface state gives a better contrast [73]. On Ni(lOO), islands of NiO were detected by STS [2]. Very nice results have recently been obtained on Al(lll) after adsorption of various species [74]. Hasegawa and Avouris [75] have imaged on reconstructed Au(lll) the standing wave pattern formed by the electron density. Such a phenomenon, observed at steps or around adsorbates, stems from interferences between the incident and the reflected wave functions of electrons in 2-D states on this surface. 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

The original function, in our case the electron-density distribution in the crystal, can be reconstructed by performing the inverse Fourier transformation, that is by summing together the corresponding density waves for all reflections (see Figure 30.11). However, in order to make this summation, we need to know not only the amplimde of the density wave, but also its relative position with respect to all other... 

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