Flipping

Recently commercially available X-ray systems for laminography have a spatial resolution limited to hundred microns, which is not enough for modem multilayer electronic devices and assembles. Modem PCBs, flip-chips, BGA-connections etc. can contain contacts and soldering points of 10 to 20 microns. The classical approach for industrial laminography in electronic applications is shown in Fig.2. 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

B) SINGLET-TRIPLET SPIN-FLIP CROSS SECTION... 

In a coupled spin system, the number of observed lines in a spectrum does not match the number of independent z magnetizations and, fiirthennore, the spectra depend on the flip angle of the pulse used to observe them. Because of the complicated spectroscopy of homonuclear coupled spins, it is only recently that selective inversions in simple coupled spin systems [23] have been studied. This means that slow chemical exchange can be studied using proton spectra without the requirement of single characteristic peaks, such as methyl groups. 

It is known that multivalued adiabatic electronic manifolds create topological effects [23,25,45]. Since the newly introduced D matrix contains the information relevant for this manifold (the number of functions that flip sign and their identification) we shall define it as the Topological Matrix. Accordingly, K will be defined as the Topological Number. Since D is dependent on the contour F the same applies to K thus K = f(F),... 

The fact that there is a one-to-one relation between the (—1) terms in the diagonal of the topological matrix and the fact that the eigenfunctions flip sign along closed contours (see discussion at the end of Section IV.A) hints at the possibility that these sign flips are related to a kind of a spin quantum number and in particular to its magnetic components. 

In case of three conical intersections, we have as many as eight different sets of eigenfunctions, and so on. Thus we have to refer to an additional chai acterization of a given sub-sub-Hilbert space. This characterization is related to the number Nj of conical intersections and the associated possible number of sign flips due to different contours in the relevant region of configuration space, traced by the electronic manifold. 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

In Section IX, we intend to present a geometrical analysis that permits some insight with respect to the phenomenon of sign flips in an M-state system (M > 2). This can be done without the support of a parallel mathematical study [9]. In this section, we intend to supply the mathematical foundation (and justification) for this analysis [10,12], Thus employing the line integral approach, we intend to prove the following statement ... 

If a contour in a given plane surrounds two conical intersections belonging to two different (adjacent) pairs of states, only two eigenfunctions flip sign—the one that belongs to the lowest state and the one that belongs to the highest one. 

In other words, surrounding the two conical intersections indeed leads to the flip of sign of the first and the third eigenfunctions, as was claimed. 

In Sections V and VII, we discussed the possible K values of the D matrix and made the connection with the number of signs flip based on the analysis given in Section IV.A. Here, we intend to present a geomehical approach in order to gain more insight into the phenomenon of signs flip in the Af-state system (M > 2). 

This algebra implies that in case of Eq. (111) the only two functions (out of n) that flip sign are and because all in-between functions get their sign flipped twice. In the same way, Eq. (112) implies that all four electronic functions mentioned in the expression, namely, the jth and the (j + 1 )th, the th and the (/c -h 1 )th, all flip sign. In what follows, we give a more detailed explanation based on the mathematical analysis of the Section Vin. 

In Sections VII and Vm, it was mentioned that K yields the number of eigenfunctions that flip sign when the electronic manifold traces certain closed paths. In what follows, we shall show how this number is formed for various Nj values. 

We briefly summarize what we found in this Nj = 5 case We revealed six different contours that led to the sign flip of six (different) pahs of functions and one contour that leads to a sign flip of all four functions. The analysis of Eq. (87) shows that indeed we should have seven different cases of sign flip and one case without sign flip (not surrounding any conical intersection). 

This conclusion contradicts the findings discussed in Sections V.A.2 and V.A.3. In Section V.A.2, we treated a three-state model and found that functions can n ver flip signs. In Section V.A.3, we treated a four-state case and found that either all four functions flip their sign or none of them flip their sign. The situation where two functions flip signs is not allowed under any conditions. 

Since for any assumed contour the most that can happen, due to C23, is that Yi2(s) flips its sign, the corresponding 2x2 diabatic mahix potential, W(s), will not be affected by that as can be seen from the following expressions ... 

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