about 0.145 nucleons/fm$^3$. As a check, we can compute the number of nucleons which have entered the hot zone as a function of time. $$N_i^{\rm part}(t) = \int \frac{d^3x dp_{iz}}{2\pi} f(z,p_{1z},t) \, \Theta
(d_i)
= 2\pi R^2 \gamma n_0 v t$$ The step function fixes the limits on the $z$ integration. The number of participating nucleons grows linearly with time, and at time $t_0$ we get $N_i^{\rm part}(t_0) = \pi R^2 L n_0$, which is the total number of nucleons in the nucleus.

The number of elementary nucleon-nucleon collisions can now be expressed as $$\begin{aligned}
N_{1+2 \rightarrow X} &=& \pi R^2 \gamma^2 n_0^2 \int_0^{t_0} dt
\int_{-vt}^{vt} dz \int_0^1 \frac{dx_1}{x_1} \int_0^1 \frac{dx_2}{x_2}
\, v_{12} \, \sigma_{1+2 \rightarrow X}(s_{12}) \nonumber \\
&\,& \sum_{N_1 = 0}^{\infty} \sum_{N_2 = 0}^{\infty} H(x_1,N_1)\,S(N_1,d_1)\,
 H(x_2,N_2)\,S(N_2,d_2) \, .\end{aligned}$$ Since the nucleons’ velocities are antiparallel the velocity factor is $$v_{12} = \frac{x_1 p_0}{\sqrt{x_1^2 p_0^2 + m_N^2}}
+ \frac{x_2 p_0}{\sqrt{x_2^2 p_0^2 + m_N^2}} \, ,$$ where $p_0$ is the beam momentum in the center-of-momentum frame.

The survival function $S(N,d)$ is characterized by the mean free path $\lambda$ of nucleons in the hot and dense hadronic matter. For a dilute gas the inverse of the mean free path is the sum 