Emission lines and the Rydberg equation

Rydberg (1897) modified the above equation for hydrogen to fit these emission lines for sodium by introducing two further parameters a and fi  

If we substitute n = 3 into this equation (i.e., assume we are dealing with a 3s-3p transition), we obtain  

the energy difference between the 3s and the 3p orbital in the neutral sodium atom corresponds to light with a wavelength of 589 nm. If, therefore, the atom receives quanta of energy greater than or equal to  

However, in the sodium atom, An = 0 is also allowed. Thus the 3s — 3p transition is allowed, although the 3s — 4s is forbidden, since in this case A/ = 0 and is forbidden. Taken together, the Bohr model of quantized electron orbitals, the selection rules, and the relationship between wavelength and energy derived from particle-wave duality are sufficient to explain the major features of the emission spectra of all elements. For the heavier elements in the periodic table, the absorption and emission spectra can be extremely complicated - manganese and iron, for example, have about 4600 lines in the visible and UV region of the spectrum. 

As a final note, closer inspection of the emission lines from Na shows that most emission lines are not, in fact, single lines, but are closely spaced doublets or triplets - for example, the strong yellow line discussed above at 589.3 nm is composed of two separate lines at 589.0 and 589.6 nm. This is termed fine structure, and is not predictable from the Bohr model of the atom. It is addressed in the Bohr-Sommerfield model, and is the result of a quantum mechanical interaction, known as spin-orbit coupling, further discussion of which is not necessary for this volume. 

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