Quantum numbers, selection rules

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

At this point we need to consider that there is another process operating in this system. When the populations of the spin states have been disturbed from their equilibrium values, as in this case by irradiation of the proton signal, relaxation processes will tend to restore the populations to their equilibrium values. Unlike excitation of a spin from a lower to a higher spin state, relaxation process are not subject to the same quantum mechanical selection rules. Relaxation involving changes of both spins simultaneously (called double-quantum transitions) are allowed in fact they are relatively important in magnitude. The relaxation pathway labeled W2 in Fig. 4.6 tends to restore equilibrium populations by relaxing spins from state N4 to Ni. We shall represent the number of spins that are relaxed by this pathway by the symbol d. The populations of the spin states thus become as follows ... 

The majority of atoms in a flame are in the ground state (Eq), therefore, many electronic transitions originate from this state. Such transitions are limited in number, since by quantum-mechanical selection rules some energy levels are not directly accessible from the ground state. 

Fig. 14.1). Only certain electronic transitions are permitted by quantum-mechanical selection rules, which are described in various text books on atomic physics. The x-ray spectral lines are designated by symbols such as Ni K i, Fe K 02. Sn Laa, and U Mcci. The symbol of an x-ray line represents the chemical element (Ni, Fe, Sn, and U) the notations K, L, or M indicate that the lines originate by the initial removal of an electron from the K, L, or M shell, respectively a particular line in the series is designated by the Greek letter a, j8, etc. (representing the subshell of the outer electron involved in the transition), plus a numerical subscript. This numerical subscript indicates the relative strength of each line in a particular series—for example, K i is more intense than Kota. Because there are a limited number of possible inner-shell transitions, the x-ray spectrum is much simpler than the complex optical spectrum that results from the removal or transition of valence electrons in addition. 

Raman spectroscopy also has selection rules. The gross selection rule for a Raman-active vibration is related to the polarizability of the molecule. Polarizability is a measure of how easily an electric field can induce a dipole moment on an atom or molecule. Vibrations that are Raman-active have a changing polarizability during the course of the vibration. Thus, a changing polarizability is what makes a vibration Raman-active. The quantum-mechanical selection rule, in terms of the change in the vibrational quantum number, is based on a transition moment that is similar to the form of M in equation 14.2. For allowed Raman transitions, the transition moment [a] is written in terms of the polarizability a of the molecule ... 

Light emission requires that electrons and holes recombine in a single step, satisfying requisite conservation laws and with allowed quantum-mechanieal selection rules. The rate at which radiative recombination, Riadiative. takes place between free electrons and holes across the energy gap of a semiconductor depends upon the product of the number of electrons and the number of holes in the material. The rate at which these interact depends also upon their thermal velocity Vu, (the velocity of carriers with a kinetic energy near keT), the cross section for the recombination process Sr, and a factor due to internal reflection in the material. The final relationship is ... 

The solutions can be labelled by their values of F and m.p. We say that F and m.p are good quantum. num.bers. With tiiis labelling, it is easier to keep track of the solutions and we can use the good quantum numbers to express selection rules for molecular interactions and transitions. In field-free space only states having the same values of F and m.p can interact, and an electric dipole transition between states with F = F and F" will take place if and only if... 

One of the consequences of this selection rule concerns forbidden electronic transitions. They caimot occur unless accompanied by a change in vibrational quantum number for some antisynnnetric vibration. Forbidden electronic transitions are not observed in diatomic molecules (unless by magnetic dipole or other interactions) because their only vibration is totally synnnetric they have no antisymmetric vibrations to make the transitions allowed. 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

In the case of atoms, deriving states from configurations, in the Russell-Saunders approximation (Section 7.1.2.3), simply involved juggling with the available quantum numbers. In diatomic molecules we have seen already that some symmetry properties must be included, in addition to the available quantum numbers, in a discussion of selection rules. 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

In non-linear polyatomic molecules the process of deterioration of quantum numbers continues to such an extent that only the total electron spin quantum number S remains. The selection rule... 

Strategy Use the selection rules for the four quantum numbers to find the sets that could not occur. For the valid sets, identify the principal level and sublevel... 

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