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

Thermonuclear reaction chains generate solar energy. The standard model predicts this energy is produced from the conversion of four protons into

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 4p\Rightarrow \; _{}^{4}\textrm{He}+2e+2\nu_{e} \; \; \;\;\;}

About 98% of the time this occurs through the pp chain, with the CNO cycle contributing the remaining 2%. Reaction cross sections are small due to that center-of-mass energies is way so lower than the Coulomb barriers inhibiting charged particle nuclear reactions. Where a star acts as a slowly but surely reactor producing center-of-mass energies for reacting particles of approx. around 10 keV, in which we have a core temperature somewhere around 1.5x10^7 K.

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{3}}\sqrt{E_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.

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 aA + bB \rightarrow cC + dD \ , }

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 cC + dD \rightarrow aA + bB \ . }

The cross-section of a reverse reaction is

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 \frac{\sigma_i(j,o)_J}{\sigma_m(o,j)_J} = \frac{1+\delta_{ij}}{1+\delta_{om}}\frac{g_og_m}{g_ig_j} \frac{k^2_o}{k^2_j} \ . }

Nuclear Networks

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

1) Reaction rates for two particle reaction:

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}=\frac{1}{1+\delta_{ij}}n_in_j \langle \sigma v\rangle_{i,j} \ ,}

where the delta factor is included to avoid double counting when we have identical particles.

2) Reaction rates for decays, photodisintegrations, electron captures, etc.:

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}=\lambda _i n_i \ . }

If we have the reaction 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+j \rightarrow o+m \,} the change in the densities is given by

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 \left( \frac{\part n_i}{\part t} \right)_{\rho}=\left( \frac{\part n_j}{\part t} \right)_{\rho}=-r_{i,j}}

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 \left( \frac{\part n_o}{\part t} \right)_{\rho}=\left( \frac{\part n_m}{\part t} \right)_{\rho}=+r_{i,j}}

Now recall that the abundances is 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 Y_i=\frac{n_i}{\rho N_A} \ . }

Then taking the derivative with respect to time we have the reaction network given by

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 Y_i=\frac{1}{\rho N_A} \left( \frac{\part n_i}{\part t} \right)_{\rho} \ . }

Reaction Network for two particle reaction

Using the previous equations and the reaction rate definition 1) we can write it 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 \dot Y_i=\frac{r_{i,j}}{\rho N_A}=-\frac{1}{1+\delta_{ij}}\rho N_A \langle \sigma v \rangle_{i,j} Y_iY_j \ .}

This definition is for the case where we destroy a nucleus 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 in the case where we produce a nucleus 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 m}

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 Y_m=\frac{1}{1+\delta_{ij}}\rho N_A \langle\sigma v \rangle_{i,j} Y_iY_j}

Reaction Network for decays, photodisintegrations, electron captures

Now if we use the definition 2) for decays, photodisintegrations, electron captures etc., we have that

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 Y_i=\frac{\dot n_i}{\rho N_A}=\frac{-r_i}{\rho N_A} \ . }

Then

${\dot {Y}}_{i}=-\lambda _{i}Y_{i}\ ,$ for creating a nucleus

${\dot {Y}}_{m}=\lambda _{i}Y_{m}\ ,$ for destroying a nucleus

Since we can have different reactions simultaneously we can write the reaction rate as the sum of the two cases. Then

${\dot {Y}}=\sum _{j}N_{j}^{i}\lambda _{j}Y_{j}+\sum _{j,k}{\frac {N_{j,k}^{i}}{1+\delta _{ij}}}\rho N_{A}\langle \sigma v\rangle _{j,k}Y_{j}Y_{k}\ .$

These concepts of nuclear reaction networks can give us important information about the fundamental properties in the discussion of nucleosynthesis. The most important of these properties are called steady state and equilibrium.

Nuclear Statistical Equilibrium

When temperatures reach about 3-4x109 K nuclei become connected to other nuclei through reaction links. These reaction links can proceed in both a capture reaction and photodisintegration. At high enough temperatures one is able to overcome the coulomb barrier and have capture reactions, creating heavier nuclei. Also at high enough temperatures one has photodisintegration, which is when extremely high-energy gamma rays interact with an atomic nucleus, causing it to enter an excited state, and excited heavy nuclei then decay into two or more lighter daughter nuclei. This situation leads to a complete chemical equilibrium where the particle abundances only depend on atomic mass, density, and temperature. This equilibrium is called Nuclear Statistical Equilibrium.

At chemical equilibrium, proton and neutron capture can be written as

$\;\mu (Z,N)+\mu _{p}=\mu (Z+1,N)$ and $\;\mu (Z,N)+\mu _{n}=\mu (Z,N+1)\ ,$

and so, through capture reaction links we can reach heavier nuclei (Z,N), and the chemical potential will remain in equalibrium

$\;N\mu _{n}+Z\mu _{z}=\mu _{Z,N}\ .$

At high temperatures these nucleons and nuclei will be described by Maxwell-Boltzmann statistics, and the chemical potential for a Boltzmann gas of species $\;i$ is

$\mu _{i}=kTln\left[{\frac {\rho N_{A}Y_{i}}{G_{i}}}\left({\frac {2\pi \hbar ^{2}}{m_{i}kT}}\right)^{\frac {3}{2}}\right]+m_{i}c^{2}\ .$

So by plugging in the chemical potential we can get an equation that looks like this

$N\left(kTln\left[{\frac {\rho N_{A}Y_{n}}{G_{n}}}\left({\frac {2\pi \hbar ^{2}}{m_{n}kT}}\right)^{\frac {3}{2}}\right]+m_{n}c^{2}\right)+Z\left(kTln\left[{\frac {\rho N_{A}Y_{p}}{G_{p}}}\left({\frac {2\pi \hbar ^{2}}{m_{p}kT}}\right)^{\frac {3}{2}}\right]+m_{p}c^{2}\right)=kTln\left[{\frac {\rho N_{A}Y_{Z,N}}{G_{Z,N}}}\left({\frac {2\pi \hbar ^{2}}{m_{Z,N}kT}}\right)^{\frac {3}{2}}\right]+m_{Z,N}c^{2}\ ,$

and by rearranging this we can get

$Nln\left[{\frac {\rho N_{A}Y_{n}}{G_{n}}}\left({\frac {2\pi \hbar ^{2}}{m_{n}kT}}\right)^{\frac {3}{2}}\right]+Zln\left[{\frac {\rho N_{A}Y_{p}}{G_{p}}}\left({\frac {2\pi \hbar ^{2}}{m_{p}kT}}\right)^{\frac {3}{2}}\right]-ln\left[{\frac {\rho N_{A}Y_{Z,N}}{G_{Z,N}}}\left({\frac {2\pi \hbar ^{2}}{m_{Z,N}kT}}\right)^{\frac {3}{2}}\right]={\frac {1}{kT}}\left(m_{Z,N}c^{2}-Nm_{n}c^{2}-Zm_{p}c^{2}\right)\ .$

Where

$\left(m_{Z,N}c^{2}-Nm_{n}c^{2}-Zm_{p}c^{2}\right)=B.E\ .$

Now with the mass relations

$\;A=N+Z\ ,$

$\;m_{n}\approx m_{p}\approx m_{u}\ ,$

and

$\;m_{Z,N}=A\mu _{Z,N}\ .$

We can derive an equation for the abundance of a particular nuclei (Z,N).

The physics and astronomy department at Clemson University offers a NSE calculator that can be found in http://www.webnucleo.org/pages/nse/0.1/