Elliptic conical intersection

Figure 14 Illustration of the general procedure used to locate the initial relaxation direction (IRD) toward the possible decay products, (a) General photochemical relaxation path leading (via conical intersection decay) to three different final structures, (b) Potential energy surface for a model elliptic conical intersection plotted in the branching plane, (c) Corresponding energy profile (as a function of the angle a) along a circular cross section centered on the conical intersection point and with radius d.

Equation [8] is the equation of an elliptic double cone (i.e., with different axes) with vertex at the origin (it will be a circular cone only for the case k = /). Thus, such crossing points are called conical intersections. Indeed, if we plot the energies of the two intersecting states against the two internal coordinates xx and x2 [whose values at the origin satisfy the two conditions and H1 j = H22 and H12 (= H21) = 0], we obtain a typical double-cone shape (see Figure 5). 

Generally the detectors are set up perpendicular to the primary beam, with the intersection of the primary beam at the detector centre. This setting has some advantages the entire Bragg cones are detected and the deviation of the cone projection from an ideal circle is usually small. Sometimes a detector can be placed off-centre and non-orthogonally to the primary beam. This can enlarge the detectable q-space in a very cost effective manner. The downsides are the strongly elliptical conical projections and the loss of the entire azimuthal information of a diffraction cone. 

In Figure 2.20b these transition structures locate the energy ridges that separate the IRD valleys located by Mi and Mg. Thus, although there is no analogue of the transition vector for a conical intersection, the simple case of an elliptic cone shows that the IRDs are still uniquely defined in terms of Ml and Mg. 

The elliptic cone model of the potential energy surface at a conical intersection point discussed above is not general enough to give a correct description of the relaxation in realistic molecules. First, more than two possible IRDs may originate from the tip of the cone. Second, the first-order approximation (i.e., elliptic cone) may break down at larger distances, and some IRDs may lie out of the branching plane because the real... 

Smectic A phases in which the layers are not uniformly parallel to the glass slides confining the sample (i.e. not in a planar orientation) are characterized by fan-like textures (Fig. 5.10b), made up of focal conics (Fig. 5.12). A focal conic is an intersection in the plane of a geometric object called a Dupin cyclide (Fig. 5.13), which results from lamellae forming a concentric roll (like a Swiss roll) being bent into an object based on an elliptical torus of non-uniform cross-section. The straight line that would define the rotation axis of the torus is distorted into a hyperbola in the Dupin cyclide. 

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