The helicity basis

Here the lower component of is discontinuous at the south pole p = (0,0, —p), and the lower component of g is discontinuous at the north pole p = (0,0,p). We note that g (p) are eigenvectors of cr -p. We have 

The operator cr p/p is called the helicity operator. It determines to what extent the momentum of a particle points in the direction of the spin or against it. Next we form the four-dimensional spinors (eigenvectors of P) 

Apply the inverse matrix u(p) to these vectors to obtain 

For each p the four vectors Wpc s,neg(p) form an orthonormal basis in 

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One then derives for the Stokes parameters of the photon beam with individual wavelets (j) in the helicity basis (superscript h) ... 

To obtain the spin-Hall current, it is necessary to analyze the hole distribution function. Within the linear response regime, we find that the linear electric field part of the helicity-basis distribution function, j(p), can be written as a sum of two terms,... 

It is clear that x formally coincide with the three states of spin 1 of a photon. Therefore, one can choose to interpret % ] as the unit vectors of circular polarization with either positive or negative helicity, while Xo gives the linear polarization in the z direction [27]. We note here that to within the sign at % ] the helicity basis (16) coincides with the so-called polarization basis frequently used in optics [57]. 

Derivatized amylose is the basis for the Chiralpak AD CSP. This CSP has been utilized for the resolution of ibuprofen and flurbiprofen, as well as other members of the family of nonsteroidal inflammatory drugs (NSAIDs) [39, 61]. Ibuprofen was not resolved on the Chiralpak AD CSP in LC. Pressure-related effects on stereoselectivity were observed by Bargmann-Leyder et al. on a Chiralpak AD CSP [58]. No corresponding effect of pressure on selectivity was observed with a Chiralcel OD CSP. The authors speculated that the helical conformation of the amylose-based CSP is more flexible than that of the cellulose-based CSP. 

Chemical shift anisotropy evidenced the helical structure of the complexes. NOESY experiment also suggested the presence of a stable secondary structure. On the basis of the NOESY experiments, the... 

The main factor in determining the handedness of the cholesterics induced by bridged 1,1 -binaphtliyls is the helicity (P or M) of the solute, and this observation is the basis of many configurational studies of chiral binaphthyls. All the homochiral (aP)-binaphthyls 15-19 have an M helicity of the core, and all induce, in biphenyl nematics, M cholesterics.65,75 By systematic structural variations of the covalent bridge, it is possible to obtain I J -binaphthalenes with dihedral angles ranging from 60° to 96° (see series 20-24) the handedness of the cholesteric phase always matches the helicity... 

The connectivity is not known for the seven-helix bundle of purple membrane protein (Henderson and Unwin, 1975), but on the basis of its resemblance to other antiparallel a proteins the most likely topologies would be either up-and-down or Greek key (see below). An analysis based on the sequence and the relative electron-densities of the helices (Engelman et ah, 1980) considers a left-handed up-and-down topology as the most probable model. 

X-ray crystallographic analysis also revealed that the crystal of thioamide la was chiral, the space group P2i2i2i. The absolute configuration of (-)-rotatory crystals of la, where the optical rotation was assigned on the basis of CD spectra in a KBr pellet, was determined by the X-ray anomalous scattering method as (-)-(M)-la for the helicity. When (-)-rotatory crystals were irradiated at 0 °C until the reaction conversion reached 100% yield, the asymmetric induction in... 

The chiral-bilayer-effect hypothesis has been evoked for the rationalization of the helical fibers formed from enantiomeric or diastereomeric surfactants (Fig. 54) [373], Different packing of the chiral surfactants in the crystals (head-to-tail) and in bilayer or micellar aggregates (tail-to-tail) is the basis for this postulate. Crystallization from aggregates requires an energetically costly, 180°... 

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