NSE Intro

Nuclear statistical equilibrium (NSE) describes a state in which all strong nuclear reactions are in chemical equilibrium, with each forward reaction balanced by its reverse. NSE is achieved when nuclear reaction timescales are much shorter than hydrodynamic timescales, typically at temperatures \(T \gtrsim 6 \times 10^9\,\mathrm{K}\).

Mathematical Description of NSE

Imagine that every isotope is connected to the free nucleons, neutrons and protons, through a series of strong nuclear reactions. Now since in NSE, every strong reaction is in chemical equilibrium, this means that the chemical potential of any isotope can be expressed as the sum of the chemical potentials of the its constituent neutrons and protons. This can be viewed as a net reaction of the form:

\[(A, Z) \rightleftharpoons Z \ \mathrm{p} + N \ \mathrm{n}\]

And chemical equilibrium of such reaction is:

\[\mu_i = Z_i \mu_p + N_i \mu_n\]

Now recall that for gas that follows Maxwell-Boltzmann statistics, as derived in Semi-relativistic ideal gas model, the chemical potential of any isotope is:

\[\mu_i = \mu_i^{\mathrm{kin}} + m_i c^2 + \mu_i^c\]

Combining with the NSE chemical equilibrium condition, we obtain:

\[\mu_i^{\mathrm{kin}} = Z_i (\mu_p^{\mathrm{kin}} + \mu_p^c) + N_i \mu_n^{\mathrm{kin}} + \underbrace{(Z_i m_p + N_i m_n - m_i) c^2}_{B_i} - \mu^c_i\]

Where \(B_i\) is the binding energy of the isotope. The above expression can then be used in the mass fraction equation derived from Maxwell-Boltzmann statistics shown in Semi-relativistic ideal gas model, giving the mass abundance equation for system in NSE.

\[X_{i, \mathrm{NSE}} = \dfrac{(m_i)^{5/2}}{\rho} g_i \left( \dfrac{k_B T}{2\pi \hbar^2} \right)^{3/2} \exp{\left( \dfrac{Z_i (\mu_p^{\mathrm{kin}} + \mu_p^c) + N_i \mu_n^{\mathrm{kin}} + B_i - \mu^c_i}{k_B T} \right)}\]

NSE Constraint Equations

Given the input, \((\rho, T, Y_e)\), the NSE abundance equation has two unknowns, \(\mu_p^{\mathrm{kin}}\) and \(\mu_n^{\mathrm{kin}}\), which can be solved subject two constraint equations:

1. Conservation of mass, or a constraint on the input density, \(\rho\).

\[\sum_i X_i^{\mathrm{NSE}}(\mu_p^{\mathrm{kin}}, \mu_n^{\mathrm{kin}}) - 1 = 0\]

2. Conservation of charge for strong reactions, or a constraint on the input electron fraction, \(Y_e\)

\[\sum_i \frac{Z_i}{A_i} X_i^{\mathrm{NSE}} (\mu_p^{\mathrm{kin}}, \mu_n^{\mathrm{kin}}) - Y_e = 0\]

The corresponding constraint Jacobian is then:

\[\begin{split}\begin{bmatrix} \frac{\partial}{\partial \mu_p^{\mathrm{kin}}} \left(\sum_i X_i^{\mathrm{NSE}} - 1\right) = \sum_i \frac{Z_i}{k_B T} X_i^{\mathrm{NSE}} & \frac{\partial}{\partial \mu_n^{\mathrm{kin}}} \left(\sum_i X_i^{\mathrm{NSE}} - 1\right) = \sum_i \frac{N_i}{k_B T} X_i^{\mathrm{NSE}} \\ \frac{\partial}{\partial \mu_p^{\mathrm{kin}}} \left(\sum_i \frac{Z_i}{A_i} X_i^{\mathrm{NSE}} - Y_e\right) = \sum_i \frac{Z_i^2}{A_i k_B T} X_i^{\mathrm{NSE}} & \frac{\partial}{\partial \mu_n^{\mathrm{kin}}} \left(\sum_i \frac{Z_i}{A_i} X_i^{\mathrm{NSE}} - Y_e\right) = \sum_i \frac{Z_i N_i}{A_i k_B T} X_i^{\mathrm{NSE}} \end{bmatrix}\end{split}\]

The NSE abundance system defined by the constraint equations and Jacobian can be solved using NSENetwork. The primary interface for this calculation is get_comp_nse.