Contrast transfer function equation

Since Fp is a real quantity and thus iatV is imaginary, no contrast would be visible without the transfer function of the microscope (Equation 6). The lens aberrations result in a transfer of some imaginary part information at the exit plane into the real part at the image plane, which can be imaged by HRTEM. 

Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be taken. The first involves models based on first principles, called physical or first principles models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated from data. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. 

This equation shows that only limited information is preserved. In particular, depending on the spatial frequency Mo, no information is transferred at all at the zeroes of the phase-contrast function sin(x). The loss of information is even more serious when the phase object approximation holds and for ideal imaging in that case the phase information is completely lost in the Gaussian image of the object and special methods the so-called phase-contrast [94,95] methods should be employed in order to partly recover this information. 

By contrast, for iodide 18 having the triple bond activated by a phenyl group, conversion to the cyclic organozinc species 25 occurred effectively and the latter could be efficiently functionalized, provided that traces of moisture were excluded by pre-treatment of zinc powder with Mel. The substituted benzylidene cyclopentanes 26 and 27 were respectively obtained after iodinolysis and palladium-catalyzed cross-coupling reaction with benzoyl chloride (equation 10). However, it could not be assessed whether the formation of organozinc 25 was attributable to an anionic or a radical cyclization pathway (or both) as, had iodide 26 been produced by a radical iodine atom-transfer, it would have been converted to 25 by reaction with metallic zinc due to the presence of the activating phenyl group21. 

The electronic nature of silylsilver intermediate was interrogated through inter-molecular competition experiments between substituted styrenes and the silylsilver intermediate (77).83 The product ratios from these experiments correlated well with the Hammett equation to provide a p value of —0.62 using op constants (Scheme 7.19). Woerpel and coworkers interpreted this p value to suggest that this silylsilver species is electrophilic. Smaller p values were obtained when the temperature of the intermolecular competition reactions was reduced [p = — 0.71 (8°C) and —0.79 (—8°C)]. From these experiments, the isokinetic temperature was estimated to be 129°C, which meant that the product-determining step of silver-catalyzed silylene transfer was under enthalpic control. In contrast, related intermolecular competition reactions under metal-free thermal conditions indicated the product-determining step of free silylene transfer to be under entropic control. The combination of the observed catalytically active silylsilver intermediate and the Hammett correlation data led Woerpel and colleagues to conclude that the silver functions to both decompose the sacrificial cyclohexene silacyclopropane as well as transfer the di-terf-butylsilylene to the olefin substrate. 

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