\(f=\sqrt{\cfrac{c^3}{8\pi}*{density}*{Z}*3.4354}\)
with copper, \(Z=29\) and density \(8.96\,gcm^{-3}\)
\(f=\sqrt{\cfrac{c^3}{8\pi}*8960*29*3.4354}=9.7826e14\,Hz\)
Copper has a work function of \(\Psi=4.65\,eV\), so
\(f=\cfrac{\Psi}{h_{\small{eV}}}=\cfrac{4.65}{4.135667516e-15}=1.1244e15\,Hz\)
and we are off by \(1.15\) times.
For Caesium, \(Z=55\) and density \(1.93\,gcm^{-3}\)
\(f=\sqrt{\cfrac{c^3}{8\pi}*1930*55*3.4354}=6.2526e14\,Hz\)
Caesium has a work function of \(\Psi=2.1\,eV\), so
\(f=\cfrac{\Psi}{h_{\small{eV}}}=\cfrac{2.1}{4.135667516e-15}=5.0778e14\,Hz\)
\(f=\cfrac{\Psi}{h_{\small{eV}}}=\cfrac{2.1}{4.135667516e-15}=5.0778e14\,Hz\)
and we are off by \(0.812\) times.
Not bad...
