Correcting units from this physics paper?

[halcyon_zomboid]

TL;DR Summary

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 ?