A Correcting units from this physics paper?

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The discussion centers on a units discrepancy in a 1973 physics paper by Hora regarding the production of antihydrogen. The poster identifies that the left-hand side of the equation for the number of pairs produced, N_p, is dimensionless, which conflicts with the derived units. They suggest that Equation (25) is missing a factor of E_v, needing correction to include E_v^2 for proper dimensional analysis. Additionally, they note that Equation (28) yields dimensions of length^-2 instead of the expected length^2. The poster seeks assistance in resolving these unit inconsistencies.
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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
 
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