5th Week: Reaction Rates, Nuclear Networks & NSE

So far we have studied the nuclear reactions and the main thermodynamic properties. Now we move on further to use this knowledge to predict the astrophysical plasma processes such as energy release and composition changes.

Thermonuclear reaction rates

From the definition of nuclear cross section we know that the nuclear cross section between target $i$ and projectile $j$ can be written as

$\sigma ={\frac {r/n_{i}}{n_{j}v}}\ ,$

where $v$ is the relative velocity between targets with the number density $n_{i}$ and projectiles with the number density $n_{j}$ and $r$ is the number of reactions per $cm^{3}$ per $sec$. From here one can find

$r=\sigma vn_{i}n_{j}\ ,$

or more generally when targets and projectiles follow the specific distribution then

$r_{i;j}=\int \sigma \cdot \left|{\vec {v_{i}}}-{\vec {v_{j}}}\right|dn_{i}dn_{j}\ .$

Nuclear Networks

We saw that there could be two types of expressions for the reaction rates

$r_{i,j}={\frac {1}{1+\delta _{ij}}}n_{i}n_{j}<\sigma v>$

Where the delta factor is included to avoid double counting when we have identical particles. And,

$r_{i,j}=\lambda _{i}n_{i}$

Now recall that

$Y_{i}={\frac {n_{i}}{\rho N_{A}}}$