Tilted fields

The Freedericksz transition for a tilted magnetic field is similar to the classical Freedericksz transitions. We restrict our attention to the case when Xa 0 and H is applied across the sample at a fixed angle a, as shown in Fig. 3.13. The director and magnetic field now take the forms, respectively, 

A similar calculation to that involved in deriving the equilibrium equation (3.111) for the first classical Preedericksz transition shows that the equilibrium equation for this example is 

As before, this equation can be multiplied throughout by 0 and integrated to find 

Notice that if = a, then the alignment of the director is parallel to the 

As in the pretilt boundary example in the previous Section, iox H 0 there is generally no constant solution to the equilibrium equation (3.183) which satisfies the boundary conditions (3.181) for a 0. However, it is straightforward to use the result in (3.187) and suitably adapt the method used in equations (3.129) to (3.133) to find that 

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Figure 9. Diagram of the antiferromagnetism vector L (left-hand) and the magnetization M (right-hand) turn at sublattice flipping in tilted field (ip < ipc)- The arrows show the turn directions in the I and II phases. The dashed line encloses the angles, corresponding to the regions of phase instability and realization of domain boundaries [1,2].

The various intersubunit binding domains of the oligomer would be expected to cleave under different conditions to yield trimers or dimers. Formation of dimers and trimers in the presence of SDS has been reported, but details of conditions were not described 140). In addition, the porcine enzyme in 33% dioxane retains 50% of its activity for several days and exhibits a molecular weight of 140,000 (5<9). Propellor shaped trimers have been seen in electron micrographs (136,137,139) and micrographs of tilted fields containing these particles reveal that they consist of three subunits, presumably formed by cleavage between layers of trimers (139). 

The above discussion of the Fredericks transition can be extended to include (1) pretilt at the boundaries where the boundary conditions (p 0) = (j) L) = 0 are replaced by (p 0) = (p L) = < o (2) the tilted fields where the magnetic field is applied across the sample at a fixed angle (3) weak anchoring where the director is weakly anchored to both boundary plates. A more detailed discussion can be found in Stewart (2004) and Virga (1994). 

In this case, we can conclude that the small sensor is lightly tilted with an angle of 0,25 degrees. We have concluded, during experimentations, that the measurement of the magnetic field is very sensitive to the angle of inclinaison of the sensor. In this way, we validate the computation of the incident field E (r). We can also expect some difficulties for the validation of the forward problem by experimental data. 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

The conventional hand of a particular isochiral cluster of tubes can be deduced from dark field diffraction contrast tilting experiments [26]. 

Parker [55] studied the IN properties of MEH-PPV sandwiched between various low-and high work-function materials. He proposed a model for such photodiodes, where the charge carriers are transported in a rigid band model. Electrons and holes can tunnel into or leave the polymer when the applied field tilts the polymer bands so that the tunnel barriers can be overcome. It must be noted that a rigid band model is only appropriate for very low intrinsic carrier concentrations in MEH-PPV. Capacitance-voltage measurements for these devices indicated an upper limit for the dark carrier concentration of 1014 cm"3. Further measurements of the built in fields of MEH-PPV sandwiched between metal electrodes are in agreement with the results found by Parker. Electro absorption measurements [56, 57] showed that various metals did not introduce interface states in the single-particle gap of the polymer that pins the Schottky contact. Of course this does not imply that the metal and the polymer do not interact [58, 59] but these interactions do not pin the Schottky barrier. 

Figure 8. Eigenvalues of a 3D mapping system with zero field. Top to bottom piston and tilts measured by the LGSs, piston not measured and neither piston nor tilts measured. Curves are normalized to 1, 10 and 100 from bottom to top.

Figure 10. Eigenvalues of a 4 LGSs 3D mapping system with a /O field. Neither piston nor tilts are measured from the LGSs. Dashed line without natural guide star. Solid line with a NGS to measure tilts, defocus and astigmatisms (Le Louam and Tallon, 2002).

For zero field of view (Fig. 8), as expeeted, there are respeetively 2, 4 and 8 singular modes when the piston and the tilts are measured from the LGSs, when the piston is not but the tilt is (ease of the polychromatie LGS, see 15.3), and when neither the piston nor the tilts are measured (monoehromatie LGS ease). Even and odd modes eorrespond respeetively to the high and low eigenvalues (lowest and highest modes). 

Figure 15. Sky coverage for a LGSs + 1 deformable mirrors AO device. Field is 6 wide. 1 NGS is required to sense the tilt. Same symbols as Fig. 14.

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

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