Cross dipole selection rules

Measurements with Na+ ions of energies in the 29-590 e V range, corresponding to v/ve from 0.2 to 0.9, were compared to the diabatic SFI spectra of the Na 28f, 28g, and 28h states observed individually by driving resonant microwave transitions from the 28d state.9 These detailed comparisons show clearly that for high velocity, v/ve 0.9, 59% of the 28d— 28 cross section is to the 28f state, the dipole allowed transition. However, at lower values of v/ve, nondipole processes play a more important role. For example, at v/ve = 0.2, only 37% of the cross section is due to the 28d— 28f transition.9 At high velocities the process is predominantly a dipole M = 1 process, but at low velocities the dipole selection rule breaks down. 

The probability of photoionization to an ion state is termed a PE cross section, commonly denoted a. As PE flux is a function of the angle of observation of the PEs, measurements of band intensities are made at a magic angle where the intensity is independent of the angular parameter, / . The cross section is then directly proportional to the band intensity. Direct photoionization is controlled by the dipole selection rule, but as a free electron can assume any... 

Li , Li The reaction lA [yd) He has a very low cross section for gamma radiation of energies up to at least 17-6 MeV [42]. This may be understood in terms of the electric dipole selection rule for isotopic spin in nuclei with T = 0, since the E1 process creates states with T = i and these cannot break up into the even system He + H except as a result of isotopic spin impurity. They can however break up into + He and w + LP as observed. 

The cross-section in Eq. (1 illustrates another distinguishing feature of inelastic neutron scattering for vibrational spectroscopy, i.e., the absence of dipole and polarizability selection rules. In contrast, it is believed that in optical and inelastic electron surface spectroscopies that a vibrating molecule must possess a net component of a static or induced dipole moment perpendicular to a metal surface in order for the vibrational transition to be observed ( 7,8). This is because dipole moment changes of the vibrating molecule parallel to the surface are canceled by an equal image moment induced in the metal. 

For convenience consider the case of diatomic dissociation. Examination of the selection rules shows that when the transition-dipole operators deg and dLje are parallel to the nuclear axis, the two-photon amplitude is nonzero only if Jj—Jj — 2,0. By contrast, in that case the one-photon matrix element (Eu Jh M, dg Ej, Jj, Mj) is nonzero only if Jj —Jf = 1. Since these two conditions are contradictory, Pql2 E) is zero. Hence coherent control over integral cross sections is not possible using the one- vs. two-photon scenario. 

However, numerical estimates of the effect of frequency dependence of the coefficients of depolarization based on the calculation of molecular cross sections Gj(na>, Aw) are rather difficult. But it can be shown that in the cases when different Gj magnitudes are nonzero by symmetry selection rules (neglecting the spin-orbital interaction), they are of the same order of magnitude, since they are determined by the same energy denominators and reduced matrix elements of the operator of the dipole moment. 

Fig. III. 16. In light symmetric top molecules with reasonably large electric dipole moments such as for instance methylfluoride the change of the absorption spectrum due to the translational Zeeman effect occurs at comparatively low perpendicular velocities. The spectrum shown here corresponds to the absorption of a group of molecules moving at 267 m/sec (maximum of the Maxwell-Boltzmann probability distribution) perpendicular to the magnetic field. The dotted line gives the spectrum calculated neglecting the translational Zeeman effect. The Lorentz cross field has caused considerable mixing of Mj substates resulting in considerable changes in the selection rules...

Perturbations affect the rate of absorption and emission of radiation in a fully understood and exactly calculable manner. They also affect the rates of chemical and collisional population/depopulation processes, but in a less easily estimated way. Perturbation effects on steady-state populations can be very large and level-specific. Although collision-induced transitions and chemical reactions are not governed by rigorous selection rules as are electric dipole transitions and perturbation interactions, some useful propensity rules have been suggested theoretically and confirmed experimentally. Gelbart and Freed (1973) suggested that the cross sections for collision-induced transitions between two different electronic states, E and E, are... 

A number of hydrated inorganic salts have also been studied by the inelastic neutron scattering (INS) lechnique. Since the proton scattering cross section is quite large, the INS spectrum reflects mainly the motion of the protons in the crystal. Furthermore, INS spectroscopy has no selection rules involving dipole moments or polarizabilities. Thus it serves as a complementary tool to vibrational spectroscopy in studying the hydrogen vibrations of hydrated salts. 

The selection rules for the Raman effect are obtained by replacing P in Eq. 8.37 by the components of induced dipoles. These components a,y are the nine elements of the polarizability tensor, where i, j = x, y, z. The aij form a basis for the same rep as ij, namely T(ij), so that a particular transition is allowed in the Raman effect if F(m) X F(>/) contains T ij). Many more necessary details regarding the intensities of IR and Raman spectra that are beyond the scope of the present work are given by Wilson, Decius, and Cross (5). 

However it also follows that if there were only dipole forces all the even terms in Zp would be zero for the dipole-allowed transitions. Each term in the Bom series progresses from the initial state to the final by a series of transitions between virtual intermediate states. There are as many transitions as there are orders in the Born term. Each transition changes the parity of the state. Thus an odd number of transitions are required for a dipole-allowed state and therefore only the odd order Born terms contribute. In reality though we only have dipole dominance, not a pure selection rule. Also, first order Born terms are usually not negligible even when forbidden. In any case in the example shown in fig. 5.3, when the sign of Zp is reversed, the cross section 1S-2P remains the same but this is not true of the forbidden transition 1S-2S. 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

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