Questions And More Questions Popping Charges

The expression, per atom view,

$\cfrac{1}{2}(m_{\rho}*Z)v^{ 2 }_{max}=E_{input/atom}=\cfrac{1}{2}c^{ 2 }*{density}*{Z}*3.4354$

gets into trouble because,

$\cfrac{1}{2}m_{\rho}v^{ 2 }_{max}=E_{input/particle}$

but

$\cfrac{1}{2}c^{ 2 }*{density}*{Z}*3.4354=\cfrac{1}{2}c^{ 2 }*\cfrac{density}{Z}*Z^2*3.4354$

$E_{input/atom}=E_{input/particle}*Z^2$

and so,

$\cfrac{1}{2}(m_{\rho}*Z)v^{ 2 }_{max}=E_{input/particle}*Z\ne E_{input/atom}$

what happened?  Furthermore, either,

$\cfrac{1}{2}(m_{\rho}*Z)v^{ 2 }_{max}=\cfrac{1}{2}c^2*\cfrac{8\pi}{c^3}f^2$

or

$\cfrac{1}{2}(m_{\rho}*Z^2)v^{ 2 }_{max}=\cfrac{1}{2}c^2*\cfrac{8\pi}{c^3}f^2$

suggests,

 $f\propto v_{max}$,

not,

$f\propto KE$  or  $f\propto v^2_{max}$

And between

$\cfrac{1}{2}c^{ 2 }*\cfrac{density}{Z}*3.4354$  and  $\cfrac{1}{2}c^{ 2 }*{density}*{Z}*3.4354$

we have

$\cfrac{1}{2}c^{ 2 }*{density}*3.4354$

what of this expression?

The first issue seems to be resolved by considering the scaling factor $Z$ as applied to $v_{max}$ and not to $m_{\rho}$ that,

$\cfrac{1}{2}m_{\rho}\left(v_{max}*Z\right)^{ 2 }=E_{input/atom}=E_{input/particle}*Z^2$

would this make sense?  Whether of $Z$ particles or just one, mass density $m_{\rho}$ remains constant.  So the formulation,

$\cfrac{1}{2}(m_{\rho}*Z)v^{ 2 }_{max}=E_{input/atom}$

is wrong.  But why would $v_{max}\rightarrow v_{max}*Z$?

Good night...