Orbital momentum selection rule

The probabilities of absorption by structures larger than an atom are determined in essentially the same way. For diatomic and linear polyatomic molecules the orbital momentum selection rule becomes AA = 0, 1, where A is the symbol for molecular total orbital angular momentum. For systems with appropriate symmetry, the rules are g m, -I- <-> H-, —. For non-linear polyatomic molecules... 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

As a result of the atomic nature of the core orbitals, the structure and width of the features in an X-ray emission spectrum reflect the density of states in the valence band from which the transition originates. Also as a result of the atomic nature of the core orbitals, the selection rules governing the X-ray emission are those appropriate to atomic spectroscopy, more especially the orbital angular momentum selection rule A1 = + 1. Thus, transitions to the Is band are only allowed from bands corresponding to the p orbitals. 

The subscript A on a perimeter orbital defined in Equation (2.2) can be viewed as a quantum number related to the z component of orbital angular momentum associated with an electron in the orbital The selection rule for one-electron transitions between perimeter orbitals is similar to that familiar from atoms the quantum number k is allowed to increase or decrease by I. Thus promotions from to, and to, are allowed,... 

Now, according to the orbital angular momentum selection rules (which determine what transitions the electron is allowed to make), the only electronic transitions that are allowed are those in which either the orbital... 

These angular momentum selection rules figure prominently in the fine structure of alkali atom spectra. The filled-shell core electrons have zero net orbital and spin angular momentum, so the term symbols 2 Si/2,... 

There does not seem to be any selection rule such as conservation of spin or orbital angular momentum which this reaction does not satisfy. It is also not clear that overall spin conservation, for example, is necessary in efficient reactions (5, 16, 17, 20). Further, recent results (21) seem to show a greatly enhanced (20 times) reaction rate when the N2 is in an excited vibrational state (vibrational temperature 4000 °K. or about 0.3 e.v.). This suggests the presence of an activation energy or barrier. A barrier of 0.3 e.v. is consistent with the low energy variation of the measured cross-section in Figure 1. 

The spin rule is satisfied, but the orbital angular momentum rule is not. The reaction is apparently fast at low ion energies (4) hence, if there is an important selection rule in the combination of reactants, it is seemingly the spin rule. Conservation of spin in combining reactants is probably more likely than conservation of orbital angular momentum, since the latter will be more strongly coupled to collision angular momentum. 

An electronic transition must involve a change in the orbital angular momentum quantum number such that A = 1. Thus a Is to 2p transition is allowed and a Is to 3p transition is allowed, but a Is to 2s or Is to 3d transition is forbidden. This rule is sometimes called the Laporte selection rule. 

We now consider many-electron atoms. We will assume Russell-Saunders coupling, so that an atomic state can be characterized by total electronic orbital and spin angular-momentum quantum numbers L and S, and total electronic angular-momentum quantum numbers J and Mj. (See Section 1.17.) The electric-dipole selection rules for L, J, and Mj can be shown to be (Bethe and Jackiw, p. 224)... 

In Chapter 1, we introduced the concept of parity, the response of the wave function to an operation in which the signs of the spatial coordinates were reversed. As we indicated in our discussion of a decay, parity conservation forms an important selection rule for a decay. Emission of an a particle of orbital angular momentum / carries a parity change (— l/ so that 1+ —0+ or 2 0+ a decays are forbidden. In general, we find that parity is conserved in strong and electromagnetic interactions. 

In order to establish more detailed selection rules, we have to examine the tensorial structure of the respective operators. Let us start with selection rules for orbital momentum L. The operator of Ek-radiation (4.12) contains the tensor C k whereas that of M/c-radiation (4.16) contains... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

Note that m is an abbreviation for mf.) It can be seen that in general there exist two photoionization channels which differ in the orbital angular momentum ( of the photoelectron and can interfere. Equ. (2.9a) is the dipole selection rule for the... 

The last transition is forbidden because the demands from the angular momentum coupling and the parity requirement are mutually exclusive the coupling of the orbital angular momenta requires the vector addition L + = 0 with L = 1 and hence also = 1 on the other hand, the parity selection rule requires = even, and both conditions cannot be fulfilled simultaneously. Therefore, only five transitions are expected for the K-LL Auger spectrum in neon, and these can be identified in Fig. 3.3. 

---