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

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

For the nuclei in astrophysical plasma that obey the Maxwell-Boltzmann distribution we can write

$dn_{j}=n_{j}\phi \left({\vec {v_{j}}}\right)d^{3}v_{j}\ ,$

where

$\phi \left({\vec {v_{j}}}\right)=\left({\frac {m_{j}}{2\pi kT}}\right)^{3/2}\exp \left[-{\frac {m_{j}v_{j}^{2}}{2kT}}\right]\ .$

Using this relation we find the following expression for the reaction rate

$r_{i;j}=n_{i}n_{j}\langle \sigma v\rangle _{i;j}=n_{i}n_{j}\int \sigma \left(\left|{\vec {v_{i}}}-{\vec {v_{j}}}\right|\right)\left(\left|{\vec {v_{i}}}-{\vec {v_{j}}}\right|\right)\phi \left({\vec {v_{i}}}\right)\phi \left({\vec {v_{j}}}\right)d^{3}v_{i}d^{3}v_{j}\ .$

Since it is always easier to calculate the cross section in the center of mass and relative coordinates reference frame we can further write

$\langle \sigma v\rangle =\int \sigma (v)v\left({\frac {\mu }{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {\mu v^{2}}{2kT}}\right)d^{3}v\ ,$

where $v$ is the relative velocity and $\mu$ is reduced mass defined as

${\vec {v}}={\vec {v_{i}}}-{\vec {v_{i}}}\ ,$

$\mu ={\frac {\mu _{i}\mu _{j}}{\mu _{i}+\mu _{j}}}\ .$

Now after some manipulation we can see that in Maxwell-Boltzmann distribution $\langle \sigma v\rangle$ depends only on temperature

$\langle \sigma v\rangle \left(T\right){\frac {1}{\sqrt {\mu \pi }}}\left({\frac {2}{kT}}\right)^{3/2}\int _{0}^{\infty }E\sigma (E)\exp \left(-{\frac {E}{kT}}\right)dE\ .$

The experimental application: Determination of reaction rates

The total thermonuclear reaction rate is the sum of all resonant terms and the non-resonant neutron or charged-particle rates. Below we discuss each of them.

Resonant rates

At low energies the number of the excited states in compound nucleus is small and cross sections can be dominated by single resonances. In most case the resonances, n, are very narrow and they act as a delta function at $E_{n}$. The cross section is then defined by Breit-Wigner formula defined in previous lecture.

Non-resonant reactions for neutrons

In this case $s$-waves dominate ($l=0$) and at high energies

$\langle \sigma v\rangle _{nonres}=S(0)\left(1+{\frac {{\dot {S}}(0)}{S(0)}}{\frac {2}{\sqrt {\pi }}}\left(kt\right)^{1/2}+{\frac {{\ddot {S}}(0)}{S(0)}}{\frac {3}{4}}kT\right)\ .$

The constants $S(0)\ ,$ ${\dot {S}}(0)/S(0)$ and ${\ddot {S}}(0)/S(0)$ are determined from experiments.

Non-resonant reactions for charged particles

We again assume that the $s$-wave dominates and using the expression for the cross section of the charged particles defined in previous lecture after some manipulations we arrive at

$\langle \sigma v\rangle _{nonres}=\left({\frac {2}{\mu }}\right)^{1/2}S_{eff}(0){\frac {\Delta }{\left(kT\right)^{3/2}}}\exp \left(-\tau \right)\ ,$

where

$S_{eff}(0)=S(0)\left[1+{\frac {5}{12\tau }}+{\frac {S^{\prime }(0)}{S(0)}}\left(E_{0}+{\frac {35}{36}}kT\right)+{\frac {1}{2}}{\frac {S^{\prime \prime }(0)}{S(0)}}\left(E_{0}^{2}+{\frac {89}{36}}E_{0}kT\right)\right]\ ,$

$E_{0}=\left({\frac {\mu }{2}}\right)^{1/3}\left({\frac {Z_{i}Z_{j}e^{2}\pi kT}{\hbar }}\right)^{2/3}\ ,$

and

$\tau ={\frac {3E_{0}}{kT}}\ ,$

$\Delta ={\frac {4}{\sqrt {3E_{0}kT}}}\ .$

Reverse reactions

A reverse reaction is the reaction that will obtain the original reactants from the products of a nuclear reaction. In other words, the inverse of a nuclear reaction. Not all reactions are reversible. Non-reversible reactions are those in which the reactants are completely consumed in the reaction; the products are formed and the reactants are gone for good. An example of this is the burning of a piece of paper. Once the oxygen and wood combine to create water and CO2, the paper is gone for good. But in a reversible reaction, this is not the case. The reactants can combine to form the products, but then, the products can also revert back to the initial reactants.

aA + bB --> cC + dD, and

cC + dD --> aA + bB

The cross-section of a reverse reaction is :

<math> \frac{\sigma_i(j,o)_J}{\sigma_m(o,j)_J} = \frac{1+\delta_{ij}}{1+\delta_{om}}

Nuclear Statistical Equilibrium

Here is a really cool NSE calculator offered by the physics and astronomy department at Clemson University. http://www.webnucleo.org/pages/nse/0.1/