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I am trying to follow the equations in the paper "Multiphoton absorption coefficients in solids: a universal curve". Here the authors state that they use the following equation for calculating the value for $\beta^{(2)}$: $$\beta^{(2)}=\frac{2^{11/2}\pi e^4}{3c^2}S_2f_2(\zeta)$$ with $$f_2(\zeta)=\left(\frac{2\zeta-1}{\zeta^5}\right)^{1.5}$$ The equation for $S_2$ (a scaling factor) is $$\left(\frac{p_{vc}^2}{m^2}\frac{\left(m^\star\right)^{5/2}}{m_1^2n^2}\frac{1}{E_g^{7/2}}\right) $$ with $E_g$ the bandgap in eV, $\frac{p_{vc}^2}{m^2}\approx\frac{3E_g}{4m^\star}$, $n$ the refractive index, $m^\star$ the reduced effective mass and $m_1$ the effective mass of the conduction band.
Thus, the dimension of $S_2$ is $\left[\frac{1}{J^{5/2}kg^{1/2}}\right]$, assumed I did not make a mistake in the calculation.
The authors also give the value for $S_2=0.618$ for ZnSe and $\zeta=0.69$. The result for $\beta^{(2)}$ should be $0.049\cdot10^{-8}$.

Now, when putting in the values, I get $$\frac{2^{11/2}\pi e^4}{3c^2}=209.36\cdot10^{-12}$$ $$\frac{e^4}{c^2}=4.4178\cdot10^{-12}$$ Assuming $c$ is equivalent to the speed of light, I get $$e^4=397053$$ or $$e=25.1$$ which is neither equivalent to the electric charge or the number $e$. Thus, I was wondering if another unit system was used, but which? Or is there another mistake I did not see?

Thomas Fritsch
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arc_lupus
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  • looks like SI to me... Are you sure you're getting $\beta^{(2)}$ right? – kevinkayaks Oct 03 '19 at 10:19
  • In the table given in the paper (table 2) they state that $\beta^{(2)}=0.049$ cm/MW. Regardless if I use $\beta^{(2)}=0.049$ or $\beta^{(2)}=0.049\cdot10^{-8}$, I do not get useful results. – arc_lupus Oct 03 '19 at 10:51
  • The paper is behind a paywall. – G. Smith Oct 03 '19 at 16:07
  • Is $S_2$ really dimensionless? – G. Smith Oct 03 '19 at 16:13
  • @G.Smith Not sure how I can make it open for everyone. Will check if $S_2$ is dimensionless, though – arc_lupus Oct 03 '19 at 18:29
  • You don’t need to make it open. You just need to provide sufficient information. Without $S_2$ having dimensions, I don’t think the formula can be dimensionally correct. – G. Smith Oct 03 '19 at 18:34
  • @G.Smith: I added the full equation and the units – arc_lupus Oct 03 '19 at 18:47
  • From your equation for $S_2$, I find that its SI units would be $\text{s}^5/(\text{kg}^3\text{m}^5)$. This is different from what you found. – G. Smith Oct 03 '19 at 19:27
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    The equation is dimensionally consistent in CGS units, not in SI units. However, I don't get anywhere near the stated value for $\beta^{(2)}$ if $S_2$ is $0.618\text{s}^5/(\text{g}^3\text{cm}^5)$. – G. Smith Oct 03 '19 at 19:48
  • Email the author ? – kevinkayaks Oct 03 '19 at 19:56
  • I get $6.55\times10^{-44}$ cm/MW. – G. Smith Oct 03 '19 at 20:01
  • In CGS units, the elementary charge $e$ is $4.8\times 10^{-10}$ statcoulombs, and $1\text{ statcoulomb}=1\text{ g}^{1/2}\text{ cm}^{3/2}\text{s}^{-1}$. – G. Smith Oct 03 '19 at 20:04
  • Sorry, your units for $S_2$ are the same as mine. – G. Smith Oct 03 '19 at 22:14
  • @kevinkayaks: Not sure if he even is still alive, the paper was published in 1983, and the author is no longer listed on the web pages of the university – arc_lupus Oct 06 '19 at 11:54
  • https://www.if.ufrj.br/en/humberto-siqueira-brandi/ ? He was publishing in 2017. I'm sure you could find one of these recent papers and get his email. But yes, maybe not. In my experience authors usually appreciate brief questions when preceded by a statement of your appreciation of the work. – kevinkayaks Oct 06 '19 at 20:57

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Given your equations, they must be in some version of CGS units, such as electrostatic units to be dimensionally consistent.

For example, in these units, the elementary charge $e$ is $4.8\times 10^{-10}\text{ statC}$, which is $4.8\times 10^{-10}\text{ g}^{1/2}\text{cm}^{3/2}\text{s}^{-1}$. If you use these units, the result for $\beta^{(2)}$ is dimensionally consistent with being length/power.

The clue that the equation for $\beta^{(2)}$ can't be in SI units is that there is no vacuum permittivity $\epsilon_0$. Since $\beta^{(2)}$ and $S_2$ and $c$ are non-electrical quantities, in SI units the elementary charge $e$ would have to appear in the combination $e^2/\epsilon_0$ in order to "de-electricize" things.

G. Smith
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