# Understanding a Gsponer formula about mi

I'm reading over one of Andre Gsponer's papers on nuclear weapons. I'm confused by section 3.2, particularly formulas 3 and 4. (3) states:

$$\eta_\mathrm{fission} = \frac{\rho R - \omega_c}{\rho R}$$

I believe $$\rho R$$ term roughly equates to "number of atoms between the center and outside", but the $$\omega_c$$ term is an areal density. I am confused about what this means.

The following text states:

$$\omega_c$$ is called the “critical fast-neutron-opacity” ($$\omega_c \approx 100$$ g/cm² for $${}^{239}\mathrm{Pu}$$, and $$160$$ g/cm² for $${}^{235}\mathrm U$$)".

I assume from this terminology that it is a measurement of the point where the fuel becomes opaque to neutrons and thus you have an ~100% chance of any particular neutron causing another reaction. Uncompressed $${}^{235}\mathrm U$$ is about $$19$$ g/cm³, so this seems to be suggesting that a chain reaction is only possible in compressed fuel, which is obviously not true. I think I am missing the meaning of $$\rho R$$.

(4) talks about fusion. I believe the term B, "$$B \approx 6$$ g/cm² for DT , and $$17$$ g/cm² for $${}^6Li_2DT$$", is referring to the density at which the alphas from the DT reaction will be fully captured within the fuel mass and thus self-heat. But again, I am not entirely sure I understand the meaning of $$\rho R$$ here either.

• Areal densities are used (either grams/cm2 or atoms/cm2) when considering scattering interactions from a flux of particles (measured in particles/cm2). This normalizes out any mention of actual material densities - the material could be a gas or a solid, you just want to know how many atoms an incident particle passes by before it might scatter. An actual mean free path, on the other hand, requires knowing the density. Apr 15 at 14:25