Atomic radii electrons, relationship between

The basic relationship between the intensity profile I(s) and the particle given by a number of atoms N and number of electrons per atom n and a characteristic radius r is given by the following ... 

In Bohr s model of the hydrogen atom, the circular orbits were determined by the quantum number more accurately, by the square of the quantum number n. No other orbits were allowed. By changing the orbits from circles to ellipses, Sommerfeld introduced a second radius, which gave him another variable to play with. So it was that Sommerfeld generalized Bohr s quantum condition for electron orbits in terms of the two quantum numbers n and k. His analysis led him to establish a relationship between the two quantum numbers namely, the quantum number n set the upper limit on the quantum number k, but k could have smaller values as follows ... 

This justifies Bohr s assumption of quantized angular momentum. Therefore, each quantum state of the Bohr model corresponds to a different number of wavelengths of the electron at the orbital radius. Bohr s model is extraordinarily tempting in its simplicity and its power to predict the atomic spectra. The hidden strength of the model is that it accurately portrays the relationship between the electron energy and its characteristic v and r values. It is all the more important, therefore, that we be aware of its shortcomings. 

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