# Which unit system is used in the paper: “Multiphoton absorption coefficients in solids”?

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?

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

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.