Apparently there are two issues involved; a general solution of \(a_{\psi}\) from \(a_{\psi\,c}\) to \(a_{\psi\,\pi}\),
Each solution of \(a_{\psi}\) has an associated \(\lambda\) given by \(2\pi a_{\psi}=\lambda\). But according to Bohr Model,
\(2\pi a_{\psi}=n\lambda\) where \(n\) is an integer\(^+\). So,
Across from \(a_{\psi\,c}\) to \(a_{\psi\,\pi}\) of all types of particles charge, gravity, temperature, etc; their energy density type determines their nature. A charge particle will be made of electrical energy density. A gravity particle will be made of gravitation energy density.
Downward, is each type of particle with an associated \(\lambda\), growing bigger, \(1.\lambda\), \(2.\lambda\), \(3.\lambda\)...according to,
\(2\pi a_{\psi}=n\lambda\) where \(n\in\Bbb{Z}^+\)
which gives the spectra lines, absorption and emission, as a single particle grow big and small.
What is \(a_{\psi}=19.34\,nm\)? The frequency \(f_c=3697802.5\,GHz\) was based on \(a_{\psi\,ne}=19.34\,nm\). That the spectral line associated with \(19.34\,nm\) is from the neutral particle making energy transitions in the discharge tube. All charged particles in the tube will have been driven off by the high potential gradients (in this case temperature in the tube and outside).
Just a recap...