Graduate Correcting units from this physics paper?

[halcyon_zomboid]

TL;DR

I cannot make units work in this 1973 paper by Hora and need a trustworthy answer.

Hi all,

I've struggled to resolve a units issue in this 1973 paper by Hora:
https://www.academia.edu/23774741/E...tihydrogen_by_lasers_of_very_high_intensities
From the paper:

"
The number N_p of pairs produced in a plasma volume V during a time \tau and a density n_e of electrons is
N_p=\frac{e^8n_e^2}{\pi\hbar^2m_0^2c^5}V\tau\ln^3\frac{\epsilon_{kin}}{m_0c^2}.
"

However, the units do not seem to work out as the LHS is dimensionless.

For what it's worth, I found that Equation (25) is missing one factor of E_v:
\gamma=\frac{e^2\hbar}{\omega m_0^3c^3}E_v\quad(\mathrm{Incorrect})\quad\Rightarrow\quad\gamma=\frac{e^2\hbar}{\omega m_0^3c^3}E_v^2\quad(\mathrm{Correct})

However, I can't tell what factors are missing in this expression. Even in units where k=1/(4\pi\epsilon_0)=1, I end up with dimensions of \mathrm{length}^{-3} where I'm expecting dimensionless units.

FWIW, going back to Equation (28):

\sigma=\frac{e^8}{\pi\hbar^2m_0^2c^6}\ln^3\frac{\epsilon_\mathrm{kin}}{m_0c^2}

I get dimensions of \mathrm{length}^{-2}, not \mathrm{length}^2.

It looks like this is very close to working... please, can someone help me "debug" the units here?

Thanks in advance,
HZ

 

[anuttarasammyak]

To fllow you what physical dimension e has in your estimation ?