Nuclear Statistical Equilibrium


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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 abundance evolutions. Above $\simeq 5 \times10^9$ K it is energetic enough for forward and reverse reactions to be balanced, and abundances are in a state of nuclear statistical equilibrium (NSE). For Maxwell-Boltzmann statistics, the mass fractions $X_i$ of any isotope $i$ in NSE is \begin{equation} 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} \end{equation} 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 \begin{equation} \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} \end{equation} where $\mu_p$ is the chemical potential of the protons, $\mu_n$ is the chemical 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 \begin{equation} \sum_i X_i= 1 \hskip 1.0in Y_e = \sum_i {Z_j \over A_i} X_i \ . \label{eq3} \tag{3} \end{equation} 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 constraints and two unknowns.

The tool in public_nse.tbz puts a 47 isotope netrork into its NSE state. More serious NSE calculations could modify this tool to use more accurate nuclear data (e.g., ground state spins and temperature dependent partition functions), to add more elaborate couplings (e.g., Coulomb corrections), and to increase the number of isotopes. Still, the figures and movies below, which accompany this article, suggest this tool gives reasonable results for the assumptions made.

Abundances vs temperature for varying Ye:
ρ = 103 g cm-3    d1p0e3_yevary_3302_a_pdf.mp4   
ρ = 104 g cm-3    d1p0e4_yevary_3302_a_pdf.mp4   
ρ = 105 g cm-3    d1p0e5_yevary_3302_a_pdf.mp4   
ρ = 106 g cm-3    d1p0e6_yevary_3302_a_pdf.mp4   
ρ = 107 g cm-3    d1p0e7_yevary_3302_a_pdf.mp4   
ρ = 108 g cm-3    d1p0e8_yevary_3302_a_pdf.mp4   
ρ = 109 g cm-3    d1p0e9_yevary_3302_a_pdf.mp4   
Abundances vs Ye for varying temperature :
ρ = 103 g cm-3    d1p0e3_tvary_3302_a_pdf.mp4   
ρ = 104 g cm-3    d1p0e4_tvary_3302_a_pdf.mp4   
ρ = 105 g cm-3    d1p0e5_tvary_3302_a_pdf.mp4   
ρ = 106 g cm-3    d1p0e6_tvary_3302_a_pdf.mp4   
ρ = 107 g cm-3    d1p0e7_tvary_3302_a_pdf.mp4   
ρ = 108 g cm-3    d1p0e8_tvary_3302_a_pdf.mp4   
ρ = 109 g cm-3    d1p0e9_tvary_3302_a_pdf.mp4   

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.