The Path Also Taken

If we use,

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

and

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

$\cfrac{N+1}{N}=\left(\cfrac{ave.\,E\alpha}{E\beta}\right)^3$

$\cfrac{N+1}{N}=(\cfrac{1.54184}{1.39222})^3=1.36$

$N=9$

since,

$\cfrac{9+1}{9}=1.37$

So,

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

$a_{\psi\,c}=\cfrac{1.39222}{\sqrt[3]{9}.2\pi }$

$a_{\psi\,c}=0.10493e-10=0.010493\,\,nm$

This is the first time we have actually obtained $a_{\psi\,\,c}$ from experimental data.  It is however, way smaller than expected, because,

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

when $N=77$

$a_{\psi\,\small{N}}=4.4639e-11\,\,m$

which smaller than the values for $a_{\psi}$ obtained from hydrogen spectra lines experiments, in the post "Sizing Them Up" dated 3 Dec 2014.

Nonetheless,

$f_{res}=0.061\cfrac { c }{ a_{\psi} }$

$f_{res}=0.061\cfrac { 299792458 }{ \sqrt[3]{9}*0.10493e-10 }$

$f_{res}=0.83786*10^{18}\,\,Hz$

which is nine times less than the previously calculated value of $f_{res}=7.473*10^{18} Hz$.  And so the current that would cause resonance is,

$I_{res}=\cfrac{1.1974}{9}=0.13304\,\,A$

This current reduced by an integer divisor will also cause resonance but slower.  Currents higher by an integer multiplier than $I_{res}$ will not caused resonance.  But it is,

$\sqrt[3]{N}.a_{\psi\,c}=\sqrt[3]{9}*0.10493e-10=2.1826e-11$

that is resonating.  What is this entity than is about nine times bigger than the previous value,

$a_{\psi}=0.2447e-11$

Does,

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

make sense in the first place?  Or,

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

What is happening here?  What are we setting into resonance?  $a_{\psi\,c}$, $\sqrt[3]{N}a_{\psi\,c}$ or $Na_{\psi}$?

The lower values for $a_{\psi\,c}$ suggests that,

$\cfrac{a_{\psi\,1}}{a_{\psi\,1}}=\left(\cfrac{N_1}{N_2}\right)^3$

maybe wrong;  that when particles merge, the total volume of $\psi$ may not be conserved.

The expression, $N.a_{\psi}$ suggests that,

the particles, $a_{\psi}$ are stacked up. Which would imply that X-ray emission is along an axis where $N$ particles stack up; perpendicularly outwards, across $N$.

If $a_{\psi}$ being stack up is true, then it would be impossible to obtain X ray from atomized $Cu$ vapor nor non-homogeneous alloys of $Cu$ where $a_{\psi}$ cannot stack up.