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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and projectile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{r/n_i}{n_jv} \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is the relative velocity between targets with the number density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} and projectiles with the number density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_j} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the number of reactions per Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cm^3} per Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sec} . From here one can find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sigma v n_i n_j \ , }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dn_j = n_j \phi \left(\vec{v_j}\right) d^3 v_j \ , }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \left(\vec{v_j}\right) = \left( \frac{m_j}{2 \pi k T} \right)^{3/2} \exp\left[-\frac{m_jv_j^2}{2kT}\right ] \ .}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{i;j} = n_i n_j \langle \sigma v\rangle_{i;j} = n_in_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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \sigma v \rangle = \int \sigma(v) v \left(\frac{\mu}{2 \pi k T}\right)^{3/2}\exp\left(-\frac{\mu v^2}{2kT}\right)d^3 v \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } is the relative velocity and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is reduced mass defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \vec{v_i}- \vec{v_i} \ , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \frac{\mu_i \mu_j}{\mu_i+\mu_j} \ . }

Now after some manipulation we can see that in Maxwell-Boltzmann distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \sigma v\rangle} depends only on temperature

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n} . The cross section is then defined by Breit-Wigner formula defined in previous lecture.

Non-resonant reactions for neutrons

In this case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} -waves dominate (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l=0} ) and at high energies

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(0) \ , } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{S}(0)/S(0)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{S}(0)/S(0)} are determined from experiments.

Non-resonant reactions for charged particles

We again assume that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]\ , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0 = \left(\frac{\mu}{2}\right)^{1/3} \left(\frac{Z_iZ_j e^2 \pi kT}{\hbar}\right)^{2/3} \ , }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = \frac{3E_0}{kT} \ ,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta = \frac{4}{\sqrt{3E_0kT}} \ .}

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/