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...
