Macro Atom¶

The macro atom is described in detail in [Lucy02]. The basic principal is that when an energy packet is absorbed that the macro atom is on a certain level. Three probabilities govern the next step the P_up, P_down and P_{down emission} being the probability for going to a higher level, a lower level and a lower level and emitting a photon while doing this respectively (see Figure 1 in [Lucy02] ).

The macro atom is the most complex idea to implement as a data structure. The setup is done in ~tardisatomic, but we will nonetheless discuss it here (as ~tardisatomic is even less documented than this one).

For each level, we look at the line list to see what transitions (upwards or downwards are possible). We create a two arrays, the first is a long one-dimensional array containing the probabilities. Each level contains a set of probabilities. The first part of each set contains the upwards probabilities (internal upward), the second part the downwards probabilities (internal downward), and the last part is the downward and emission probability.

each set is stacked after the other one to make one long one dimensional ~numpy.ndarray.

The second array is for book-keeping; it has exactly the length as levels (with an example for the Si II level 15):

Level ID	Probability index	N_up	N_down	N_total
14001015	???	17	5	17 + 5*2 = 27

We now will calculate the transition probabilites, using the radiative rates in Equation 20, 21, and 22 in [Lucy02]. Then we calculate the downward emission probability from Equation 5, the downward and upward internal transition probabilities in [Lucy03].

\[\begin{split}p_\textrm{emission down}&= {\cal R}_{\textrm{i}\rightarrow\textrm{lower}}\,(\epsilon_\textrm{upper} - \epsilon_\textrm{lower}) / {\cal D}_{i}\\ p_\textrm{internal down}&= {\cal R}_{\textrm{i}\rightarrow\textrm{lower}}\,\epsilon_\textrm{lower}/{\cal D}_{i}\\, p_\textrm{internal up}&={\cal R}_{\textrm{i}\rightarrow\textrm{upper}}\,\epsilon_{i}/{\cal D}_{i}\\,\end{split}\]

where \(i\) is the current level, \(\epsilon\) is the energy of the level, and \({\cal R}\) is the radiative: rates.

We ignore the probability to emit a k-packet as TARDIS only works with photon packets. Next we calculate the radiative rates using equation 10 in [Lucy03].

\[\begin{split}{\cal R}_{\textrm{upper}\rightarrow\textrm{lower}} &= A_{\textrm{upper}\rightarrow\textrm{lower}}\beta_\textrm{Sobolev}n_\textrm{upper}\\ {\cal R}_{\textrm{lower}\rightarrow\textrm{upper}} &= (B_{\textrm{lower}\rightarrow\textrm{upper}}n_\textrm{lower}- B_{\textrm{upper}\rightarrow\textrm{lower}}n_\textrm{upper}) \beta_\textrm{Sobolev} J_{\nu}^{b}\\,\end{split}\]

with \(\beta_\textrm{Sobolev} = \frac{1}{\tau_\textrm{Sobolev}}(1-e^{-\tau_\textrm{Sobolev}})\) .

using the Einstein coefficients

\[ \begin{align}\begin{aligned}\begin{split}A_{\textrm{upper}\rightarrow\textrm{lower}} &= \frac{8 \nu^2 \pi^2 e^2}{m_e c^3}~ \frac{g_\textrm{lower}}{g_\textrm{upper}}~f_{\textrm{lower}\rightarrow\textrm{upper}}\\ A_{\textrm{upper}\rightarrow\textrm{lower}} &= \underbrace{\frac{4 \pi^2 e^2}{m_e c}}_{C_\textrm{Einstein}}~ \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{upper}}~f_{\textrm{lower}\rightarrow\textrm{upper}}\\ B_{\textrm{lower}\rightarrow\textrm{upper}} &= \frac{4\pi^2 e^2}{m_e h\nu c}\,f_{\textrm{lower}\rightarrow\textrm{upper}}\\ B_{\textrm{lower}\rightarrow\textrm{upper}} &= \underbrace{\frac{4 \pi^2 e^2}{m_e c}}_{C_\textrm{Einstein}}\frac{1}{h\nu} f_{\textrm{lower}\rightarrow\textrm{upper}}\\\end{split}\\\begin{split}B_{\textrm{upper}\rightarrow\textrm{lower}} &= \frac{4\pi^2 e^2}{m_e h\nu c}\,f_{\textrm{lower}\rightarrow\textrm{upper}}\\ B_{\textrm{upper}\rightarrow\textrm{lower}} &= \underbrace{\frac{4 \pi^2 e^2}{m_e c}}_{C_\textrm{Einstein}}\frac{1}{h\nu}\frac{g_\textrm{lower}}{g_\textrm{upper}}f_{\textrm{lower}\rightarrow\textrm{upper}}\\\end{split}\end{aligned}\end{align} \]

we get

\[ \begin{align}\begin{aligned}\begin{split}{\cal R}_{\textrm{upper}\rightarrow\textrm{lower}} &= C_\textrm{Einstein} \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{upper}}~f_{\textrm{lower}\rightarrow\textrm{upper}} \beta_\textrm{Sobolev}n_\textrm{upper}\\\end{split}\\\begin{split}{\cal R}_{\textrm{lower}\rightarrow\textrm{upper}} &= C_\textrm{Einstein}\frac{1}{h\nu} f_{\textrm{lower}\rightarrow\textrm{upper}} (n_\textrm{lower}-\frac{g_\textrm{lower}}{g_\textrm{upper}}n_\textrm{upper}) \beta_\textrm{Sobolev} J_{\nu}^{b}\\\end{split}\end{aligned}\end{align} \]

This results in the transition probabilities:

\[\begin{split}p_\textrm{emission down}&= C_\textrm{Einstein} \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} \beta_\textrm{Sobolev}n_\textrm{i}\,(\epsilon_\textrm{i} - \epsilon_\textrm{lower}) / {\cal D}_{i}\\ p_\textrm{internal down}&= C_\textrm{Einstein} \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} \beta_\textrm{Sobolev}n_\textrm{i}\,\epsilon_\textrm{lower}/{\cal D}_{i}\\ p_\textrm{internal up}&=C_\textrm{Einstein}\frac{1}{h\nu} f_{\textrm{i}\rightarrow\textrm{upper}} (n_\textrm{i}-\frac{g_\textrm{i}}{g_\textrm{upper}}n_\textrm{upper}) \beta_\textrm{Sobolev} J_{\nu}^{b}\,\epsilon_{i}/{\cal D}_{i}\\,\end{split}\]

and as we will normalise the transition probabilities numerically later, we can get rid of \(C_\textrm{Einstein}\),: \(\frac{1}{{\cal D}_i}\) and number densities.

\[\begin{split}p_\textrm{emission down}&= \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} \beta_\textrm{Sobolev}\,(\epsilon_\textrm{i} - \epsilon_\textrm{lower})\\ p_\textrm{internal down}&= \frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} \beta_\textrm{Sobolev}\,\epsilon_\textrm{lower}\\ p_\textrm{internal up}&=\frac{1}{h\nu} f_{\textrm{i}\rightarrow\textrm{upper}} \underbrace{(1-\frac{g_\textrm{i}}{g_\textrm{upper}}\frac{n_\textrm{upper}}{n_i})} _\textrm{stimulated emission} \beta_\textrm{Sobolev} J_{\nu}^{b}\,\epsilon_{i}\\,\end{split}\]

There are two parts for each of the probabilities, one that is pre-computed by ~tardisatomic and is in the HDF5 File, and one that is computed during the plasma calculations:

\[\begin{split}p_\textrm{emission down}&= \underbrace{\frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} (\epsilon_\textrm{i} - \epsilon_\textrm{lower})}_\textrm{pre-computed} \,\beta_\textrm{Sobolev}\\ p_\textrm{internal down} &= \underbrace{\frac{2\nu^2}{c^2} \frac{g_\textrm{lower}}{g_\textrm{i}}~f_{\textrm{lower}\rightarrow\textrm{i}} \epsilon_\textrm{lower}}_\textrm{pre-computed}\,\beta_\textrm{Sobolev}\\ p_\textrm{internal up} &= \underbrace{\frac{1}{h\nu} f_{\textrm{i}\rightarrow\textrm{upper}}}_\textrm{pre-computed} \beta_\textrm{Sobolev} J_{\nu}^{b}\, (1-\frac{g_\textrm{i}}{g_\textrm{upper}}\frac{n_\textrm{upper}}{n_i}) \,\epsilon_{i}.\end{split}\]