Phase information

For the special case of a two-state systems, a Flermitean phase operator was proposed, [143], which was said to provide a quantitative measure for phase information. )... 

Phase information can also be obtained by Multiwavelength Anomalous Diffraction experiments... 

X-ray data are recorded either on image plates or by electronic detectors The rules for diffraction are given by Bragg s law Phase determination is the major crystallographic problem Phase information can also be obtained by Multiwavelength Anomalous Diffraction experiments... 

The single most severe drawback to reflectivity techniques in general is that the concentration profile in a specimen is not measured directly. Reflectivity is the optical transform of the concentration profile in the specimen. Since the reflectivity measured is an intensity of reflected neutrons, phase information is lost and one encounters the e-old inverse problem. However, the use of reflectivity with other techniques that place constraints on the concentration profiles circumvents this problem. 

Independent arrays of telescopes have been discussed for decades but have generally not been successful, except for radio telescopes, where interferometry is a key virtue, aided by the fact that the individual telescope signals can be amplihed and combined while preserving phase information. This is not practical in the optical, thus there are significant inefficiencies in sensitivity by coherently combining the light from an array of optical telescopes. Instrumentation for an array of telescopes has also been a cause of difficulty. Perhaps the best known successful array has been the VLT with four 8-m telescopes, each with its own suite of science instruments, and the capacity to combine all telescopes together for Interferometric measurements. 

Fig. 1—Profile measurement technique of Champper 2000+. A surface measurement is made with a linearly polarized laser beam that passes to translation stage which contains a penta-prism. The beam then passes through a Nomarski prism which shears the beam into two orthogonally polarized beam components. They recombine at the Nomarski prism. The polarization state of the recombined beam includes the phase information from the two reflected beams. The beam then passes to the nonpolarizing beam splitter which directs the beam to a polarizing beam splitter. This polarizing beam splitter splits the two reflected components to detectors A and B, respectively. The surface height difference at the two focal spots is directly related to the phase difference between the two reflected beams, and is proportional to the voltage difference between the two detectors. Each measurement point yields the local surface slope [7].

Suppose the first pulse resulted in the creation of a phase coherence across the Ai transition between the aa and a/3 states (Fig. 1.44). It is possible to transfer this phase information from the a)3 state to the )3/3 state by applying a selective it pulse across the Xi transition. The two successive pulses would therefore transfer the phase of the aa state to the )8)3 state, with the two states now becoming phase coherent with one another. 

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. 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

The present chapter has no ambition to cover all these topics. We focus solely on the information content of the two-pathway coherent control approach, where the energy-domain, single quantum states approach to the control problem simplifies the phase information and allows analysis at the most fundamental level. We regret having to limit the scope of this chapter and thus exclude much of the relevant literature. We hope, however, that this contribution will entice the reader to explore related literature of relevance. 

The general computational mechanism of Bayesian crystal structure determination in presence of various sources of partial phase information was first outlined by... 

Pecora noticed that the phase information of C is lost in the original constraints [i.e. P2 = P Tr P = N], but found it not at all clear . Here, we showed in which way one might take into account the loss of the phase information in C when calculating the number of conditions to uniquely determine C one has to impose, over and above the constraints arising from fixing the projector, the conditions to determine a particular unitary transformation in the TV-dimensional subspace, apart from the phases of the basis functions which are physically meaningless in the context of Quantum Mechanics. 

The intrinsic dissolution rate is the rate of mass transfer from the solid phase to the liquid phase. Information on the intrinsic dissolution rate is important in early drug product development. It has been suggested that drugs with intrinsic dissolution rates of less than 0.1 mg/(min cm2) will have dissolution rate-limited absorption, while drugs with intrinsic dissolution rates greater than 0.1 mg/ (min cm2) are unlikely to have dissolution rate problems. 

The reason for this is that our experiments are phase-sensitive. What do we mean by this You will remember that in the DEPT and APT spectra the CH/CHj and CH2 peaks are in one case positive (up) and in the other negative (down), which we also refer to as in opposite phase. Here in COSY and NOESY our experiments include such phase information, which is read off from the way the signals look in the plot. 

More recent work has focused on understanding the mechanism or mechanisms of selectivity. Some of these studies have been performed on well-characterized catalysts about which particle size information is available. Still others have been performed on single crystals. So conclusions may be reached about the effects on chemoselectivity of planes, edges, and corners that are related to particle size (structure sensitivity). A number of these studies, mostly on Pt, are summarized in Table 2.6. Since these studies have usually been performed in the vapor phase, information about solvent effects and their possible influence on chemoselectivity is unavailable. 

For Comparison Notions of Normal Scattering. As the electron density is assumed to be a real quantity, it directly follows the central symmetry of scattering patterns known by the name Friedel s law [244], Friedel pairs are Bragg reflections hkl and hkl that are related by central symmetry. Concerning their scattering amplitudes, Friedel pairs have equal amplitude Aha = A-g and opposite phase (phki = -scattering intensity the phase information on the structure factor is lost. 

The three space groups Cmc2, C2cm and Cmcm have the same systematic absences and cannot be distinguished from diffraction data. However, their projection symmetries are different (see Table 1). Since HRTEM images maintain the phase information, it is... 

For wave functions like = exp[if x,t)], the squared operator would mask the phase information, since = <3> 2, and to avoid this, a linear Schrodinger operator would be preferred. This has the immediate advantage of a wave equation which is linear in both space and time derivatives. The most general equation with the required form is... 

The so-called direct methods rely on the principles that phase information is included in the intensities, that electron density is always positive, and that the crystal contains atoms that are or may be considered equal. Phase relations based on probability theory have been formulated and applied to clusters of reflections. Direct methods are still under development for proteins, although they are standard techniques for determining phase angles in smaller molecules. 

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

Solubility and kinetics methods for distinguishing adsorption from surface precipitation suffer from the fundamental weakness of being macroscopic approaches that do not involve a direct examination of the solid phase. Information about the composition of an aqueous solution phase is not sufficient to permit a clear inference of a sorption mechanism because the aqueous solution phase does not determine uniquely the nature of its contiguous solid phases, even at equilibrium (49). Perhaps more important is the fact that adsorption and surface precipitation are essentially molecular concepts on which strictly macroscopic approaches can provide no unambiguous data (12, 21). Molecular concepts can be studied only by molecular methods. 

Note that we need F, a complex quantity, to find the electron density. However, from the intensity we can only derive the amplitude of F. The lack of phase information requires specific methods for solution, beyond the scope of this book. 

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