View source for 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 

:<math>4p\Rightarrow \; _{}^{4}\textrm{He}+2e+2\nu_{e} \; \; \;\;\;</math> 
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 <math>i</math> and projectile <math>j</math> can be written as

:<math> \sigma = \frac{r/n_i}{n_jv} \ , </math>

where <math>v</math> is the relative velocity between targets with the number density <math>n_i</math> and projectiles with the  number density <math>n_j</math> and <math>r</math> is the number of reactions per <math>cm^3</math> per <math>sec</math>. From here one can find 

:<math>r = \sigma v n_i n_j \ , </math>

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

:<math> r_{i; j} = \int \sigma \cdot \left|\vec{v_i}-\vec{v_j}\right| dn_i dn_j \ . </math>

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

:<math> dn_j = n_j \phi \left(\vec{v_j}\right) d^3 v_j \ , </math>

where 

:<math>\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 ] \ .</math>

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

:<math>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  \ .</math>

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

:<math>\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 \ , </math>

where <math> v </math> is the relative velocity and <math>\mu</math> is reduced mass defined as

:<math>\vec{v} = \vec{v_i}- \vec{v_i} \ , </math>

:<math>\mu = \frac{\mu_i \mu_j}{\mu_i+\mu_j} \ . </math>

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

:<math>\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 \ . </math>

=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 <math>E_n</math>. The cross section is then defined by [[4th_Week:_Decays,_Tunneling_and_Cross_Sections#Resonance_reaction_in_a_compound_nucleus|Breit-Wigner formula]] defined in previous lecture.

==Non-resonant reactions for neutrons==

In this case <math>s</math>-waves dominate (<math>l=0</math>) and at high energies

:<math>\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) \ . </math>

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

==Non-resonant reactions for charged particles==

We again assume that the <math>s</math>-wave dominates and using the expression for the [[4th_Week:_Decays,_Tunneling_and_Cross_Sections#Nuclear cross section|cross section of the charged particles]] defined in previous lecture after some manipulations we arrive at

:<math>\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) \ , </math>

where
 
:<math> 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]\ ,  </math>

:<math>E_0 = \left(\frac{\mu}{2}\right)^{1/3} \left(\frac{Z_iZ_j e^2 \pi kT}{\hbar}\right)^{2/3} \ , </math>

and

:<math> \tau = \frac{3E_0}{kT} \ ,</math>

:<math>\Delta = \frac{4}{\sqrt{3}}\sqrt{E_0kT} \ .</math>

=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.

:<math>aA + bB \rightarrow cC + dD \ , </math> 

and

:<math>cC + dD \rightarrow aA + bB \ . </math>

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}}\frac{g_og_m}{g_ig_j} \frac{k^2_o}{k^2_j} \ . </math>

=Nuclear Networks=

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

'''1)''' Reaction rates for two particle reaction:  

:::<math> r_{i,j}=\frac{1}{1+\delta_{ij}}n_in_j \langle \sigma v\rangle_{i,j} \ ,</math> 

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

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

:::<math>r_{i,j}=\lambda _i n_i \ . </math>

If we have the reaction <math>i+j \rightarrow o+m \,</math> the change in the densities is given by

:<math> \left( \frac{\part n_i}{\part t} \right)_{\rho}=\left( \frac{\part n_j}{\part t} \right)_{\rho}=-r_{i,j}</math>
:<math> \left( \frac{\part n_o}{\part t} \right)_{\rho}=\left( \frac{\part n_m}{\part t} \right)_{\rho}=+r_{i,j}</math>

Now recall that the abundances is defined as

:<math> Y_i=\frac{n_i}{\rho N_A} \ . </math>

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

:<math>\dot Y_i=\frac{1}{\rho N_A} \left( \frac{\part n_i}{\part t} \right)_{\rho} \ . </math>

==Reaction Network for two particle reaction==

Using the previous equations and the reaction rate definition  '''1)''' we can write it as

:<math>\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 \ .</math>

This definition is for the case where we destroy a nucleus <math>i</math> and in the case where we produce a nucleus <math>m</math> 

:<math>\dot Y_m=\frac{1}{1+\delta_{ij}}\rho N_A \langle\sigma v \rangle_{i,j} Y_iY_j</math>

==Reaction Network for decays, photodisintegrations, electron captures==

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

:<math> \dot Y_i=\frac{\dot n_i}{\rho N_A}=\frac{-r_i}{\rho N_A} \ . </math>

Then

:<math> \dot Y_i=-\lambda _i Y_i \ ,</math> for creating a nucleus

:<math>\dot Y_m=\lambda _i Y_m \ ,</math> 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

:<math> \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_jY_k \ . </math>

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

:<math>\;\mu(Z,N)+\mu_p=\mu(Z+1,N)</math> and <math>\;\mu(Z,N)+\mu_n=\mu(Z,N+1) \ , </math>

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

:<math>\;N\mu_n+Z\mu_z = \mu_{Z,N} \ .</math>

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

:<math>\mu_i= kTln\left[\frac{\rho N_A Y_i}{G_i}\left(\frac{2\pi\hbar^2}{m_ikT}\right)^\frac{3}{2}\right] + m_ic^2 \ .</math>

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

:<math>N\left(kTln\left[\frac{\rho N_A Y_n}{G_n}\left(\frac{2\pi\hbar^2}{m_nkT}\right)^\frac{3}{2}\right] + m_nc^2\right) + Z\left(kTln\left[\frac{\rho N_A Y_p}{G_p}\left(\frac{2\pi\hbar^2}{m_pkT}\right)^\frac{3}{2}\right] + m_pc^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 \ ,</math>

and by rearranging this we can get

:<math>Nln\left[\frac{\rho N_A Y_n}{G_n}\left(\frac{2\pi\hbar^2}{m_nkT}\right)^\frac{3}{2}\right] + Zln\left[\frac{\rho N_A Y_p}{G_p}\left(\frac{2\pi\hbar^2}{m_pkT}\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_nc^2 - Zm_pc^2\right) \ . </math>

Where 

:<math>\left(m_{Z,N}c^2 - Nm_nc^2 - Zm_pc^2\right) = B.E \ . </math>

Now with the mass relations

:<math>\;A=N+Z \ , </math>

:<math>\;m_n\approx m_p\approx m_u \ ,</math>

and 

:<math>\;m_{Z,N}=A\mu_{Z,N} \ . </math>

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/