Line conic

E. M. Pugh, R.J. Eichelberger, and N. Rostoker, Theory of Jet Formation by Charges with Lined Conical Cavities, J. Appl. Phys. 23, 532-536 (1952). 

No 5, p 658(1951) (On the diminution of brisance of hollow charges caused by rotation) 33) E.M. Pugh et al, JApplPhys 22, 487-93(1951) (Application of Kerr-Cell to photographing of metal jets squirted from lined conical HE s charges) 34) D.C. 

The plane dual of a point conic is called a line conic - Figure 3.17. We state without proof that the tangents to a point conic is a line conic (Veblen Young, 1910). 

D.W. Rowbotham, J. H. Laird, and G.S.G. Beveridge, Design and operation of glass lined conical dryer blenders with reference to the safety aspects. Seminar on Safety in Drying, IChemE, Scottish Branch, Dairy, Scotland, April 10 (1979). 

It is a requirement of the food laws that areas of the pressure vessel in contact with the product during the extraction of natural substances for human consumption must be made of stainless steel. This necessitates the use of austenitic steel. As a special feature, the pressure vessel shown in Figure 8.5 is equipped with a lined conical bottom constructed of a special steel which retains its strength at low temperatures. This precaution was considered necessary since it is possible to produce very low temperatures when depressurising the vessel at the end of the extraction (see section 8.4.3). 

Other examples of cylindrical lined vessels include blast furnaces, blast furnace stoves, basic oxygen furnaces, torpedo ladles, shaft kilns, multiple hearth furnaces, and rotary kilns. Some refractory lined vessels have refractory lined conical bottoms. When heated, the conical lining will want to displace upward. Therefore, the top mortar joint of the conical lining is critical. The cylindrical lining will be critical in restraining the upward expansion of the conical lining. At the ends of rotary kilns, restraint must be provided to contain and keep the kiln lining within the kiln shell. 

Lin et al. [70, 71] have modeled the effect of surface roughness on the dependence of contact angles on drop size. Using two geometric models, concentric rings of cones and concentric conical crevices, they find that the effects of roughness may obscure the influence of line tension on the drop size variation of contact angle. Conversely, the presence of line tension may account for some of the drop size dependence of measured hysteresis. 

Comparison between the first and last lines of the table shows that the sign of the ground-state wave function has been reversed, which implies the existence of a conical intersection somewhere inside the loop described by the table. 

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

The emphasis in our previous studies was on isolated two-state conical intersections. Here, we would like to refer to cases where at a given point three (or more) states become degenerate. This can happen, for example, when two (line) seams cross each other at a point so that at this point we have three surfaces crossing each other. The question is How do we incorporate this situation into our theoretical framework ... 

Figure 8. The representation of an open contour T in terms of an open contour Fi2 in the vicinity of the conical intersection at C12 and a closed contour F23 at the vicinity of a conical intersection at C23 F = F12 + F23, It is assumed that the intensity of Xu is strong along F12 (full line) and weak along F23 (dashed line), ia) The situation when C23 is outside the closed contour F23, (6) The situation when C23 is inside the closed contour F23.

Recently, Xu et al. [11] studied in detail the H3 molecule as well as its two isotopic analogues, namely, H2D and D2H, mainly with the aim of testing the ability of the line integral approach to distinguish between the situations when the contour surrounds or does not surround the conical intersection point. Some time later Mebel and co-workers [12,72-74,116] employed ab initio non-adiabatic coupling teiins and the line-integral approach to study some features related to the C2H molecule. 

In this series of results, we encounter a somewhat unexpected result, namely, when the circle surrounds two conical intersections the value of the line integral is zero. This does not contradict any statements made regarding the general theory (which asserts that in such a case the value of the line integral is either a multiple of 2tu or zero) but it is still somewhat unexpected, because it implies that the two conical intersections behave like vectors and that they arrange themselves in such a way as to reduce the effect of the non-adiabatic coupling terms. This result has important consequences regarding the cases where a pair of electronic states are coupled by more than one conical intersection. 

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

Figure 12, Results for the C2H molecule as calculated along a circle surrounding Che 2 A -3 A conical intersection, The conical intersection is located on the C2v line at a distance of 1,813 A from the CC axis, where ri (=CC distance) 1.2515 A. The center of the circle is located at the point of the conical intersection and defined in terms of a radius < . Shown are the non-adiabatic coupling matrix elements tcp((p ) and the adiabatic-to-diabatic transformation angles y((p i ) as calculated for (ii) and (b) where q = 0.2 A (c) and (d) where q = 0.3 A (e) and (/) where q = 0.4 A. Also shown are the positions of the two close-by (3,4) conical intersections (designated as X34).

The numerical part is based on two circles, C3 and C4, related to two different centers (see Fig. 13). Circle C3, with a radius of 0.4 A, has its center at the position of the (2,3) conical intersection (like before). Circle C4, with a radius 0.25 A, has its center (also) on the C v line, but at a distance of 0.2 A from the (2,3) conical intersection and closer to the two (3,4) conical intersections. The computational effort concentrates on calculating the exponential in Eq, (38) for the given set of ab initio 3 x 3 x matrices computed along the above mentioned two circles. Thus, following Eq, (28) we are interested in calculating the following expression ... 

Going back to our case and recalling that x(

conjugate functions, namely, iTn((p) where nr((p) = V T 2 + Tjj + T 3- In Figure 13a and b we present tn(conical intersections and they occur at points where the circles cross their axis line. 

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