Slow Neutron Captures


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

The tool minis.tbz evolves an educational version of an s-process reaction network. One hundred isotopes are evolved until a chosen ending time. The initial abundance of the first isotope, notionally 56Fe, is taken equal to one. Guidance on what this tool does seems prudent. Let $R$ be the reaction rate for (n,g) reactions. In general $R$ is temperature, density, and composition dependent - but not here. The ordinary differential equations describing the change in the abundances $y$ of the $m$ isotopes are: \begin{equation} \frac{{\rm d}y_{1}}{{\rm d}t} = -y_{1} - R_{1} \hskip 0.5in \frac{{\rm d}y_{i}}{{\rm d}t} = y_{i-1} \ R_{i-1} - y_{i} \ R_{i} \ , \ i=1,2\ldots,m-1 \hskip 0.5in \frac{{\rm d}y_{m}}{{\rm d}t} = y_{m-1} \ R_{m-1} \label{eq1} \tag{1} \end{equation} For the implicit first-order accurate Euler method, each abundance is updated over a timestep h as $y_{i}^{{\rm new}} = y_{i} + \Delta y_{i}$. The change in the abundances over a time step $\Delta y_{i}$ is obtained from solving the system of linear equations $({\bf I}/h - \tilde{{\bf J}}) \cdot \Delta {\bf y} = \dot{\bf y}$, which is simply the familar $\tilde{{\bf A}} \cdot {\bf x} = {\bf b}$. With only (n,g) reactions, Jacobian matrix $\tilde{{\bf J}}$ has the simple form \begin{equation} \left[\begin{array}{rrrrrr} -R_{1} & & & & & \\ R_{1} & -R_{2} & & & & \\ & R_{2} & -R_{3} & & & \\ & & & \ldots & & \\ & & & & R_{m-1} & 0 \\ \end{array}\right] \label{eq2} \tag{2} \end{equation} This system of linear equations can be easily solved by hand: \begin{equation} \Delta y_1 = \frac{-y_1 R_1}{1/h + R_1} \hskip 0.5in \Delta y_i = \frac { y_{i-1} R_{i-1} - y_i R_i } {1/h + R_i} \ , \ i=1,2\ldots,m-1 \hskip 0.5in \Delta y_m = \frac{-y_{m-1} R_{m-1}}{1/h} \label{eq3} \tag{3} \end{equation} Thus the succint evolution loop implemented in minis.tbz. Here are some results:

image All exposures equal to one
image A middle exposure at 0.1
image A middle exposure at 10.0

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 of the publication if appropriate. At best, you'll love these programs so much that you'll send great wads of cash to me.