Transition probabilities electric quadrupole radiation

We are also interested in the total power radiated since this will enable us to derive an expression for the transition probability for electric quadrupole radiation in section 7.2. From equations ( 2.96) and (2.97) we see that we require integrals over products of the cartesian compo-... 

The transition probabilities for magnetic dipole and electric quadrupole radiation are important since they can be combined with measurements of the absolute and relative intensities of forbidden lines emitted by nebulae, the aurora, or the solar corona to yield estimates of the number density, composition, and temperature existing in these various sources. We therefore proceed to obtain explicit expressions for these transition probabilities, making use of the expressions for the power radiated from the corresponding classical current and charge distributions which we obtained in sections 2.10 and 2.11. 

The second term in the expansion of the classical vector potential for an oscillating distribution of current and charge, equation (2.81), contains contributions from both magnetic dipole and electric quadrupole distributions, as shown in sections 2.10 and 2.11, We therefore expect these different distributions to radiate at similar rates. Thus, whenever the electric dipole transition probabilities from a given level are identically zero we must consider the possibility of decay by electric quadrupole radiation in addition to the magnetic dipole radiation discussed in... 

Unfortunately no generally applicable method of measuring the transition probabilities of forbidden lines has been developed. This is mainly because the lifetimes of the metastable levels which can decay only by magnetic dipole or electric quadrupole radiation in the visible region of the spectrum are of the order of 10 s or longer. 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

Fig. 1. On the left is a simplified energy-level diagram for l Hg+. The 281.5 nm quadrupole "clock" transition can be observed by monitoring the 194 nm fluorescence. If the ion has made a transition from the Si to the 5/2 level the 194 nm flourescence disappears. For the figure on the right, on the horizontal axis is plotted the relative detuning from line center in frequency units at 281.5 nm. On the vertical axis is plotted the probability that the fluorescence from the 6s Si - 6p pi first resonance transition, excited by laser radiation at 194 nm, is on immediately after the 281.5 nm pulse. The electric-quadrupole-allowed S-D transition and the first-resonance S-P transition are probed sequentially in order to avoid light shifts and broadening of the narrow S-D transition. The recoilless absorption resonance or carrier (central feature) can provide a reference for an optical frequency standard. (From ref. 11)...

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