Triangle rule

Applying the symmetry condition Eq. (13) to the third CGC implies that I + l+j = even. Along with the triangle rule this requires that I = I for the isotropic term j = 0, I = l l for j = 1, and I = I 2,0 for j = 2. These deductions determine the nature of the interference implicit in each of the angular parameters The fourth CGC in Eq. (12) further regulates... 

The Clebsch-Gordan coefficients Cj1 U2m2 denote the quantum mechanical probability amplitude that the angular momenta ji with projection mi, and J2 with projection m2 will add to form a total momentum j with projection m. Hence, only those coefficients differ from zero for which we have validity of the triangle rule, namely... 

Wigner s n-j symbols may well look unfriendly and even intimidating on first acquaintance. To overcome this impression, it is help fill to remember that they are only coefficients in the linear expression of a wave function of a quantum system and simply take numerical values in any given application. Furthermore, what is often important in practice is the ability to recognise when they are zero from constraints such as the triangle rule because this leads to useftd selection rules, as we shall see later in this chapter and elsewhere in this book. 

The rank k can take values 0, 1 and 2 by the triangle rule. Of these, the scalar term with k = 0 has no A dependence and hence does not affect the relative positions of the ro-vibrational energy levels. It just makes a small contribution to the electronic energy of the state r], A). The first-rank term produces a second-order contribution to the spin orbit interaction because it is directly proportional to the quantum number A from the 3-j symbol in the first line of (7.119). The contribution to the spin-orbit parameter A(R) which arises in this way is given (in cm-1) by... 

If point X lies in the area of the quadrangle A-A4B-o -e2 (point Xi on Figure 3.32), according to the triangle rule its crystallization path must end in the mechanical mixture of the solid A, A4B, and C (apexes of the triangle A-A4B-C) at the point Pt. Since we are in the crystallization area of component A, at the cooling of the melt crystals, A will appear... 

If the mixture has the composition placed in the triangle AjB-Pt-a (point Xj in Figure 3.32) or A4B-Pb-Pt (point X"), their crystallization paths, according to the triangle rule, must end in the mechanical mixture of solid A4B, B, and C (apexes of the triangle A4B-B-C) in the ternary eutectic point Ct. The crystallization paths, however, will in both cases proceed in a different way. 

For each 3-point pharmacophore, all the combinations of three interatomic distances are determined, considering only those combinations valid that satisfy the triangle rule, that is, the length of each side of a triangle caimot exceed the sum of the lengths of the other two sides, otherwise this would produce a geometrically impossible object. Moreover, redundant pharmacophores related to symmetry are usually eliminated. The resulting number of descriptors is 10549. 

Calculations were done for J 0 and j = 0 In Initial and final channels. Hence Initially and finally one has 1=0 from the triangle rule. Our x hasls was restricted to j = 0 which Implies that 1=0 also In Intermediate channels. As a result ours Is an s-wave model calculation, which Is however converged In other expansions. The selection rule for this case Is... 

Figure 5.6 Vector subtraction represented in terms of the triangle rule...

In this system of coordinates, if a point P has the coordinates x,y,z), then the directed line segment OP, extending from the origin O to point P, corresponds to the vector r. If we apply the triangle rule twice, we obtain ... 

Use the triangle rule in the form c = a + A to derive the form of the cosine formula involving the angle C. 

This method is called the triangle rule. Note that, just as in the lever rule, we assume that the composition scales are linear. If they are not, actual compositions must be used, not distances. However, this is rarely the case in ternary diagrams, which always use a linear scale for the composition axes. 

At fixed T and P, a three-phase ternary has P = 0, which defines a point. On a triangular diagram, a three-phase situation produces three points, each giving the composition of one of the phases. The three points can be connected to form a triangle, and the relative amounts in the three phases can be found by a tie-triangle rule (see Appendix H). 

Analogous results are obtained for N /N and N /N. The result (B.6.17) is a form of the tie-triangle rule for three components in three-phase equilibrium (see H.2). 

When ternary mixtures exhibit three-phase equilibria, a tie-triangle rule can be used to obtain the relative amounts in the three phases. This is analogous to the lever rule for binaries in two-phase equilibria. Consider a ternary mixture of components 1, 2, and 3 in three-phase equilibrium at the overall composition represented by a point P, as in Figure H.2. In the figure, the tie-triangle ABC bounds the three-phase region, the... 

This tie-triangle rule is a consequence of material balances on the system (see (B.6.17) in Appendix B) and therefore it applies to any three-phase equilibrium situation involving ternary mixtures. 

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