The first spectra line is not the line with double intensity, but
\(\Delta E_{1\rightarrow n=large}=h.f_{\psi\,c}\left(\cfrac{\sqrt[3]{n_{\small{large}}}-\sqrt[3]{1}}{\sqrt[3]{1}\sqrt[3]{n_{\small{large}}}}\right)=h.f_{\psi\,c}\left(\cfrac{\sqrt[3]{n_{\small{large}}}-1}{\sqrt[3]{n_{\small{large}}}}\right)\) --- (*)
To see what is \(\left(\cfrac{\sqrt[3]{n_{large}}-{1}}{\sqrt[3]{n_{large}}}\right)\), we plot
As \(n_{\small{large}}\rightarrow 77\), the increment in\(\left(\cfrac{\sqrt[3]{n_{large}}-{1}}{\sqrt[3]{n_{large}}}\right)\) decreases but the plot is not asymptotic towards a steady value around \(n=70\approx 80\). (When \(n\rightarrow \infty\), we have an asymptote towards \(y=1\).)
Given that \(n\) takes on integer values (\(n+1\) being the number of constituent basic particles the big particle has before its breakage into a basic particle (\(n=1\)) and another big particle, \(n\)), and that expression (*) at high consecutive values of \(n\) has very close values, spectra lines involving high values of \(n\) will seem to split into numerous close lines.
This could be the explanation for split spectra lines. The above is an emission spectra line plot, NOT an absorption spectra line plot.
To be sure that the first spectra line is given by (*), we look at other coalescence/breakage possibilities. When big particles of \(n+1\), \(n+2\) and \(n+3\) constituent basic particles break into pairs of \((n,\,1)\), \((n,\,2)\) and \((n,\,3)\) particles respectively,
Since the particle pairs \((n=3,\,n+3)\) are at closer energy levels than \((n=1,\,n+1)\), the transition from \(n+3\) to \(n=3\) requires less energy than the transition fro \(n+1\) to \(n=1\). As such breakage into a small particle of higher \(n\) involve lower absorption energy. Such lines will be higher up in the absorption spectrum plot when this plot is reflected about \(y=0\).
After considering other possible particle pairs, \((n=i,\,n+i)\), the first spectra line is still given by (*).
But how large is \(n_{\small{large}}\)? \(n_{\small{large}}=77\)? \(n_{\small{large}}=74\)?...
