To Assume Not Much

In the post "X Ray, Inner Electron Cloud And Just As Shocking TWO" dated 15 Oct 2017, the expression,

$N.2\pi a_{\psi}=\lambda_\beta$

was formulated before $a_{\psi\,c}$ was discovered. We might have,

$\sqrt[3]{N}.2\pi a_{\psi\,c}=\lambda_\beta$  

from,

$\sqrt[3]{\cfrac{1}{N}}=\cfrac{a_{\psi\,c}}{a_{\psi\,\small{N}}}$

and so,

$\cfrac{ave.\,E\alpha}{E\beta}=\sqrt[3]{\cfrac{N+1}{N}}$

From which we may find $a_{\psi\,c}$, but for the resonance of the electron cloud that we seek, we need the electron cloud $a_{\psi\,cloud}$,

not  $\sqrt[3]{N}.a_{\psi\,c}$   nor   $a_{\psi\,\,c}$

The rationale for making the estimate $N$ and $N+1$ is that given,

$N.2\pi a_{\psi}=\lambda_\beta$

and

$(N+1)2\pi a_{\psi}=\lambda_{ave\,\,\alpha}$

no matter what $a_{\psi}$ is, the addition of one more particle pushes $a_{\psi}$ outwards and the allowable energy states splits into two narrowly spaced energy states.  When the resonance frequency is applied using an appropriate current, the particles injected resonate.  X-ray radiations at the $E\alpha$ values will dominate and is adjusted as the current is reduced by an integer divisor.

This way, there is no need to assume that $a_{\psi}=a_{\psi\,c}$.  The only assumption is that the $E\alpha$'s and $E\beta$ are consecutive, due to the injection and emission of one particle $a_{\psi}$.

Furthermore,

$N.2\pi a_{\psi}=\lambda_\beta$

$(N+1)2\pi a_{\psi}=\lambda_{ave\,\,\alpha}$

$(\cfrac{\lambda_\beta}{2\pi a_{\psi}}+1)2\pi a_{\psi}=\lambda_{ave\,\,\alpha}$

$\lambda_\beta+2\pi a_{\psi}=\lambda_{ave\,\,\alpha}$

If $f \propto \cfrac{1}{m}$ (the approximation part) such that,

$m.f=\,\,constant$

$m_\beta c.f_{\beta}.\lambda_\beta+m_\psi c.f_{\psi}.2\pi a_{\psi}=(m_\beta+m_\psi) c.f_{ave\,\,\alpha}.\lambda_{ave\,\,\alpha}$

that the total work done in going around $\lambda_\beta$ and around the additional particle $a_{\psi}$, is the work done going around $\lambda_{ave\,\,\alpha}$.

Goodnight.