Inclusion of two channels the Lu-Fano graph

The theory described in section 3.3 is called single channel QDT. It reduces an unperturbed Rydberg series to a small number of constants, which include /z. Were this the only achievement of QDT, it would not have assumed much importance in atomic physics, because most series in the spectra of many-electron atoms are in fact perturbed, so that a constant value of p is a rarity. 

When a series is unperturbed, /z is constant. Suppose the unsorted levels have been observed in an experiment. In order to assign them to one or other series, the procedure is to calculate quantum defects referring them to both possible limits, and then to choose the best fit to two series, one of constant pi and the other of constant 2, converging to the limits 

Ii and I2 respectively. As a matter of convenience, each of the Rydberg members i would have two numbers p and p associated with it, and we can plot all the points on a plane using these two coordinates. 

The result is the two-dimensional graph pictured in fig. 3.2(a) since the principal quantum number n is not relevant to the determination of the fractional part of p, the graph may be plotted modulo 1 on both axes. For the case we have chosen (constant quantum defects), the points will lie on two intersecting straight lines, and we can use such a plot to read off directly which lines belong to which series. 

For series members close to the avoided crossing, the distance to the two branches in the graph may easily become equal. It is then impossible to decide which series they belong to, and we say that they are completely mixed. 

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