Equation of a Conic

In this section a common equation of a cone in an arbitrary coordinate system is used. A common equation of a conic can be obtained from the common equation of the cone in the coordinate system related to the cutting plane by setting z = 0, where z is the coordinate normal to the cutting plane. 

Here X is the position vector of any point X on the cone. After squaring both sides and regrouping the terms  

Opening the parentheses and regrouping the terms accordingly to their order in the components of the vector X = x, y, zj we get  

A conic is the intersection of a cone with a plane. Introducing a system coordinate related to the plane in a such way that the z-axis is perpendicular 

Substituting in Equation (10) gives for the second order terms  

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Another possibility of applying this method is using a general equation of a conic. " The general equation of a conic can be mathematically expressed by a second degree polynomial ... 

Equation of a Conic in the Receiving Slit Plane (Coordinate System CS)... 

Substituting this in Equation (10) and simplifying we obtain the equation of a conic in the receiving slit plane in implicit form ... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

Cone-bottom vertical vessels are sometimes used where solids are anticipated to be a problem. Most cones have either a 90 apex (a = 45 ) or a 60 apex ia = 30 ). These are referred to respectively as a 45 or 60 cone because of the angle each makes with the horizontal. Equation 12-4 is for the thickness of a conical head that contains pressure. Some operators use internal cones within vertical vessels with standard ellipsoidal heads as shown in Figure 12-2. The ellipsoidal heads contain the pressure, and thus the internal cone can be made of very thin steel. 

Capillary condensation can be illustrated by the model of a conical pore with a totally wetting surface (Fig. 2.12). Liquid will immediately condense in the tip of the pore. Condensation continues until the bending radius of the liquid has reached the value given by the Kelvin equation. The situation is analogous to that of a bubble and we can write... 

If there is a finite point in the xy plane which is symmetrically, situated with respect to a conic section, the latter is called a central conic (ellipse, circle, hyperbola). If this point is taken as the origin, the equation of the conic contains no terms of the first degree in x and y. The parabola is a non-central conic, since the centre is at an infinite distance. 

The position vector V is determined by the point A2 (on the sample), and vector U is a vector from point Ai (on the source) to point 2- Equations (12a-12f) together with Equations (13a-13c) can be used in an arbitrary coordinate system to obtained the equation of the conic in form Equation (7). In the next section... 

In our laboratory a microwave discharge was generated in helium by a 100 watt Raytheon microtherm generator and sampled through a pinhole leak at the apex of a conical probe into a quadrupole mass filter. Typical variations in the intensities of He+ and He2+ with pressure are given in Figure 1. If the He2+ is formed by the three-body process (Equation 11) the rate constant would have to be about 10"28 cc.2/mole-cule2/sec. which far exceeds the measured value (84). We conclude then that the diatomic ion is formed principally by the chemi-ionization process (Equation 10). 

Lord Kelvin realized that, instead of completely drying out, moisture is retained within porous materials such as plants and vegetables or biscuits at temperatures far above the dew point of the surrounding atmosphere, because of capillary forces. This process was later termed capillary condensation, which is the condensation of any vapor into capillaries or fine pores of solids, even at pressures below the equilibrium vapor pressure, Pv. Capillary condensation is said to occur when, in porous solids, multilayer adsorption from a vapor proceeds to the point at which pore spaces are filled with liquid separated from the gas phase by menisci. If a vapor or liquid wets a solid completely, that is the contact angle, 0= 0°, then this vapor will immediately condense in the tip of a conical pore, as seen in Figure 4.8 a. The formation of the liquid in the tip of the cone by condensation continues until the cone radius, r, reaches a critical value, rc, where the radius of curvature of the vapor bubble reaches the value given by the Kelvin equation (r = rc). Then, for a spherical vapor bubble, we can write... 

Combined with appropriate asymptotic conditions, these are the nuclear motion equations which must be solved when two electronically adiabatic PESs are involved in the process. As for the one-state approximation, the existence of a conical intersection between the eftq) and Ejd(q) PESs requires that additional restrictions be imposed on x d and Xjd-These will be discussed in Sec. III.B. 

Equation (31) is the standard nuclear Schrodinger equation in the absence of a conical intersection, while Eq. (30) evinces the changes attributable to the geometric phase effect that result from a conical intersection. This changes were termed the Molecular-Aharonov-Bohm (MAB) effect.The 6, term in Eq. (31), the adiabatic correction, modifies the Born-Oppenheimer potential energy surfaces and makes them mass dependent. 

In the absence of a conical intersection, both Eq. (15) and Eq. (16) lead to the standard adiabatic nuclear Schrodinger equation... 

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