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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?

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  • $\begingroup$ looks like SI to me... Are you sure you're getting $\beta^{(2)}$ right? $\endgroup$ – kevinkayaks Oct 3 at 10:19
  • $\begingroup$ 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. $\endgroup$ – arc_lupus Oct 3 at 10:51
  • $\begingroup$ The paper is behind a paywall. $\endgroup$ – G. Smith Oct 3 at 16:07
  • $\begingroup$ Is $S_2$ really dimensionless? $\endgroup$ – G. Smith Oct 3 at 16:13
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    $\begingroup$ 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 $0.618\text{s}^5/(\text{g}^3\text{cm}^5)$. $\endgroup$ – G. Smith Oct 3 at 19:48
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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.

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