From the post "No Nucleus Needed" dated 03 Oct 2016,
\(\Delta E_{2\rightarrow 1}=E_{n_1}-E_{n_2}=h\cfrac{c}{2\pi a_{\psi\,n_1}}-h\cfrac{c}{2\pi a_{\psi\,n_2}}\)
where positive values for \(\Delta E_{n}\) is energy absorbed. And from the post "Speculating About Spectra Series" dated 29 Sep 2016,
\(a_{\psi\,n}=\sqrt[3]{n}.a_{\psi\,c}\) --- (*)
where \(n=1,\,2,\,3,..77\),
we have,
\(\Delta E_{2\rightarrow 1}=h\cfrac{c}{2\pi }\left(\cfrac{1}{a_{\psi\,n_1}}-\cfrac{1}{a_{\psi\,n_2}}\right)=\hbar.c\left(\cfrac{a_{\psi\,n_2}-a_{\psi\,n_1}}{a_{\psi\,n_1}.a_{\psi\,n_2}}\right)\)
Substitute (*) in,
\(\Delta E_{2\rightarrow 1}=\hbar\cfrac{c}{a_{\psi\,c}}\left(\cfrac{\sqrt[3]{n_2}-\sqrt[3]{n_1}}{\sqrt[3]{n_1}\sqrt[3]{n_2}}\right)\)
since,
\(2\pi a_{\psi\,c}=\lambda_{\psi\,c}\) with \(m=1\)
assuming that there is \(m=1\) wavelength along the circular path,
\(\Delta E_{2\rightarrow 1}=h.f_{\psi\,c}\left(\cfrac{\sqrt[3]{n_2}-\sqrt[3]{n_1}}{\sqrt[3]{n_1}\sqrt[3]{n_2}}\right)\)
where \(f_{\psi\,c}=\cfrac{c}{\lambda_{\psi\,c}}\).
Each possible value of \(\Delta E_{2\rightarrow 1}\) due to a pair \(\left(n_2,\,\,n_1\right)\) of particles before and after a collision, corresponds to one spectra line. As more possible tuples of \(\left(n_2,\,\,n_1\right)\) are made available given the set of experimental conditions, more spectra lines appear.
Now for experimental data from the web...
