Cococubed.com Nuclear Statistical Equilibrium

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.

Below $\simeq 10^6$ K it is not energetic enough for nuclear reactions. Up to $\simeq 5 \times10^9$ K one uses a nuclear reaction network to follow the abundances. Above $\simeq 5 \times10^9$ K it is energetic enough for forward and reverse reactions to be balanced. In this case abundances are in a state of nuclear statistical equilibrium (NSE). For a Maxwell=Boltzmann statistics, the mass fractions $X_i$ of any isotope $i$ is given by $$X_i(A_i,Z_i,T,\rho) = {A \over N_A \rho} \omega(T) \left ( 2\pi kT M(A_i,Z_i) \over h^2 \right )^{3/2} \exp \left [ { \mu(A_i,Z_i) + B(A_i,Z_i) \over kT } \right ] \ , \label{eq1} \tag{1}$$ where $A_i$ is the atomic number (number of neutrons + protons on the nulceus), $Z_i$ is the charge (number of protons), $T$ is the temperature, $\rho$ is the mass density, $N_A$ is the Avogardo number, $\omega(T)$ is the temperature dependent partition function, $M(A_i,Z_i)$ is the mass of the nucleus, $B(A_i,Z_i)$ is the binding energy of the nucleus, and $\mu(A_i,Z_i)$, in the simplest case, is the chemical potential of the isotope $$\mu(A_i,Z_i) = Z_i\mu_p + N_i\mu_n = Z_i\mu_p + (A_i-Z_i) \mu_n \ , \label{eq2} \tag{2}$$ where $\mu_p$ is the chemical potential of the protons, $\mu_n$ is the chemical potential of the neutrons, $\mu_C$ is the chemcial potential of the neutrons.

The mass fractions of equation $\ref{eq1}$ are subject to two constraints, conservation of mass (baryon number) and charge, which are expressed as $$\sum_i X_i= 1 \hskip 1.0in Y_e = \sum_i {Z_j \over A_i} X_i \ . \label{eq3} \tag{3}$$ Given the triplet of input values $(T, \rho, Y_e)$, an NSE solution boils down to a two-dimensional root find for the chemical potentials of the protons $\mu_p$ and neutrons $\mu_n$. Two equations and two unknowns.

The instrument in the bzip2 tarball public_nse.tbz illustrates how to put any reaction network into its NSE state. If you are going to get serious about NSE calculations, then you'll want to modify this public code to use more accurate nuclear data (e.g., ground state spins and temperature dependent partition functions), add more physics (Coulomb corrections to equation $\ref{eq2}$), and increase the number of isotopes. Still, the two figures below suggest the public code gives reasonable results for the assumptions made.

 Most abundant isotopes as a function of temperature Most abundant isotopes as a function of Ye

The movies below accompany "Proton-Rich Nuclear Statistical Equilibrium". Each movie shows the isotope abundance on the vertical axis and either the temperature or electron fraction Ye on the horizontal axis. Limits for the abundance vertical axes are 0.01 (i.e., major abundances only; lower limits of 10-4 are avaliable upon request). The movies cover the temperature range 3 ≤ T9 ≤, density range 103 to 109 g cm-3, and Ye range 0.4 to 0.6.

 ρ = 103 g cm-3 Abundance vs Ye ρ = 103 g cm-3 Abundance vs Temp ρ = 104 g cm-3 Abundance vs Ye ρ = 104 g cm-3 Abundance vs Temp ρ = 105 g cm-3 Abundance vs Ye ρ = 105 g cm-3 Abundance vs Temp ρ = 106 g cm-3 Abundance vs Ye ρ = 106 g cm-3 Abundance vs Temp ρ = 107 g cm-3 Abundance vs Ye ρ = 107 g cm-3 Abundance vs Temp ρ = 108 g cm-3 Abundance vs Ye ρ = 108 g cm-3 Abundance vs Temp ρ = 109 g cm-3 Abundance vs Ye ρ = 109 g cm-3 Abundance vs Temp

Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.