Intensity, and Polarization of Spectral Lines

In the preceding sections we have characterized the levels E. (a) and E (b) of a two-level system by two sets a and b of quantum numbers without further specification of the individual quantum numbers. A closer inspection of the intensity and polarization of emitted or absorbed radiation has to take into account the orientation of the atoms or molecules with respect to a specified direction, called the quantization axis. We therefore describe the two levels E. and Ej by two sets (a, J, M) and (a, J, M ) of quantum numbers, where J stands for the total angular momentum and M for its projection on the quantization axis. All other quantum numbers are represented as a whole by a. 

The first factor, called the 3J-symbol, depends on M, M and m = M - M. It can be expressed by the Clebsch-Gordon coefficients which describe the coupling of angular momenta for a system initially in a state (J,M). The photon transfers an angular momentum of with projection mji (m = 0, 1) and brings the system into a state (J M ). The second factor 

The intensity of the fluorescence emitted into the direction e against the quantization axis is obtained as the product of the emission rate dn/dt of individual photons and the appropriate angular distribution of the electric dipole field. In the far field zone at a distance p x one obtains 

The Clebsch-Gordon coefficients CG (J, M, J, M ) also contain a selection rule on AJ = J - J, For electric or magnetic dipole radiation El or Ml the evaluation of the CG coefficients yields 

Unless the Zeeman sublevels M of the upper state (a J ) can be selectively populated one observes the total fluorescence of a line (a J ) - (a J). The 

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