\(^1\)Department of Physical Electronics, Fac.Sci., Masaryk University, Kotlářská 2, 611 37 Brno, CZ \(^{2}\)Leibniz Institute for Plasma Science and Technology, Felix-Hausdorff-Straße 2, 17489 Greifswald, DE
Zero-dimensional reaction kinetic modeling in Ar is performed in order to evaluate possible methods for determination of the reduced electric field from intensity ratio of Ar excited states. Particular pairs of Ar excited states are identified and selected based on their ability to reflect the value of the reduced electric field in the ratio of their populations.
Balance equations for selected species are combined to a form that is convenient for use in the evaluation of experimental line intensities.
Source terms in the balance equations are analysed and major processes important for the given population are identified.
“Extended reaction kinetics model for non-thermal argon plasmas and its test against experimental data” M. Stankov, M. M. Becker, T. Hoder, D. Loffhagen, accepted in PSST,
23 species (with 408 processes):
\({\tt Ar[1p0]}\)
\({\tt Ar[1s5], Ar[1s4], Ar[1s3], Ar[1s2]}\)
\({\tt Ar[2p10], Ar[2p9], Ar[2p8], Ar[2p7], Ar[2p6], Ar[2p5], Ar[2p4], Ar[2p3], Ar[2p2], Ar[2p1] }\)
\({\tt Ar^*[hl], Ar_2^*[^3S_u^+,v=0], Ar_2^*[^1S_u^+,v=0], Ar_2^*[^3S_u^+,v>>0], Ar_2^*[^1S_u^+,v>>0] }\)
\({\tt Ar_2^+, Ar^+, e}\)
In this work, zero-dimensional modelling is performed, all spatial gradients are considered to be zero:
Equations:
Balance equations: \[ \frac{\partial n_j(t)}{\partial t} = S_j(t). \]
Electron energy density: \[ \frac{\partial w_{\rm e}(t)}{\partial t} = - {\rm e}_0 \Gamma_{{\rm e}}(t)E(t)+P_{{\rm e}}(t), \] \[ \Gamma_{{\rm e}}(t) = - \frac{{\rm e}_0}{m_{\rm e}\nu_{\rm e}} E(t) n_{{\rm e}}(t). \]
Parameters: \(T_{gas}=300\,{\rm K}\), \(N_{gas} = 2.44\times10^{25}\,{\rm m}^{-3}\), \(E(t)\) per case.
Initial conditions: \(n_j=0\), but \(\bigl[{{\tt e}}\bigr]=2\bigl[{{\tt Ar^+}}\bigr]=2\bigl[{{\tt Ar_2^+}}\bigr]=10^{14}\,{\rm m}^{-3}\).
Balance equations together with equation for electron energy density are integrated in time for a given function of \(E(t)\).
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
| 2 | ✗ | ✗ | ✗ | ✗ | 150 | ✗ | 100 | 300 | 100 | |
| 3 | ✗ | ✗ | 150 | 300 | 150 | 300 | 300 | 100 | ||
| 4 | ✗ | 100 | 300 | ✗ | 300 | 200 | 200 | |||
| 5 | ✗ | 150 | ✗ | ✗ | ✗ | 100 | ||||
| 6 | ✗ | ✗ | ✗ | ✗ | ✗ | |||||
| 7 | ✗ | 150 | ✗ | 150 | ||||||
| 8 | ✗ | ✗ | ✗ | |||||||
| 9 | ✗ | ✗ | ||||||||
| 10 | ✗ |
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p1]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{14} \\ &-\bigl[{{\tt Ar[2p1]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{289}+\ldots+k_{292}\right) \\ &-\bigl[{{\tt Ar[2p1]}}\bigr] \left( \nu_{390}+\ldots+\nu_{391}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr]\bigl[{{\tt Ar[1p0]}}\bigr] k_{288} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{406} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p3]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p2]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{13} \\ &-\bigl[{{\tt Ar[2p2]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{293}+\ldots+k_{297}\right) \\ &-\bigl[{{\tt Ar[2p2]}}\bigr] \left( \nu_{386}+\ldots+\nu_{389}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{405} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p3]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p3]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{12} \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{298}+\ldots+k_{307}\right) \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \left( \nu_{383}+\nu_{384}+\nu_{385}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{308} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p2]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{293} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{404} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p3]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p4]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{11} \\ &-\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{308}+\ldots+k_{317}\right) \\ &-\bigl[{{\tt Ar[2p4]}}\bigr] \left( \nu_{379}+\ldots+\nu_{382}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{318} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{298} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{403} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p4]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p5]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{10} \\ &-\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{318}+\ldots+k_{320}\right) \\ &-\bigl[{{\tt Ar[2p5]}}\bigr] \left( \nu_{377}+\nu_{378}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{299} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{309} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{402} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p5]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p6]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{9} \\ &-\bigl[{{\tt Ar[2p6]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{321}+\ldots+k_{323}\right) \\ &-\bigl[{{\tt Ar[2p6]}}\bigr] \left( \nu_{374}+\nu_{375}+\nu_{376}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p8]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{331} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p7]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{324} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{319} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{310} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{300} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{401} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p6]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p7]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{8} \\ &-\bigl[{{\tt Ar[2p7]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{324}+\ldots+k_{330}\right) \\ &-\bigl[{{\tt Ar[2p7]}}\bigr] \left( \nu_{370}+\ldots+\nu_{373}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p8]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{332} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p6]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{321} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{311} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{301} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{400} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p7]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p8]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{7} \\ &-\bigl[{{\tt Ar[2p8]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{331}+\ldots+k_{338}\right) \\ &-\bigl[{{\tt Ar[2p8]}}\bigr] \left( \nu_{367}+\nu_{368}+\nu_{369}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p9]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{339} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p7]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{325} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p6]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{322} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{320} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{312} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{302} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{399} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p8]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p9]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{6} \\ &-\bigl[{{\tt Ar[2p9]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{339}+\ldots+k_{344}\right) \\ &-\bigl[{{\tt Ar[2p9]}}\bigr] \left( \nu_{366} \right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p8]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{333} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p7]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{326} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p6]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{323} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{313} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{303} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{398} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p9]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p10]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{5} \\ &-\bigl[{{\tt Ar[2p10]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{345}+\ldots+k_{348}\right) \\ &-\bigl[{{\tt Ar[2p10]}}\bigr] \left( \nu_{362}+\ldots+\nu_{365} \right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p8]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{334} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p9]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{340} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{397} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p10]})} \end{align*}\]
Electric field \(E(t)\in\{10,\ldots400\}\,\)Td is considered, it’s value is constant in time. Simulation time is chosen so that the initial transient state passes and a steady development of the system is established.
The first figure show the development of density of selected species in time for reduced electric field of 100 Td. Steady development of densities can be observed.
The second figure shows the ratio population densities of selected species for reduced electric field of 100 Td.
As the specie densities tend to increase exponentially, ratios of their densities converge.
Energy: electron energy converges at a given value of the reduced electric field on a time scale of \(10^{-11}s\). Converged mean electron energy at corresponding values of \(E/N\) are shown
RKM: mean electron energy obtained from current reaction kinetic model
Boltzmann equation: BOLSIG+ solver ver. 03/2016, www.lxcat.net,
Siglo: SIGLO database, www.lxcat.net, retrieved on July 15, 2022
IST: IST-Lisbon database, www.lxcat.net, retrieved on July 15, 2022
BSR: BSR database, www.lxcat.net, retrieved on July 15, 2022
Biagi: Biagi-v7.1 database, www.lxcat.net, retrieved on July 12, 2022
Ratio: Selected population ratios at given value of \(E/N\)
\({\tt Ar[2p3]/Ar[2p2]}\) and \({\tt Ar[2p4]/Ar[2p3]}\) exhibit desired variation with \(E/N\)
Boundary values for electron energy at given \(E/N\) from a collection of curves on Energy figure are used to determine uncertainty band of ratio curves.
Balance equations for selected species:
the first three terms listed on RHS are convenient because the unknowns can be cancelled out in population ratios.
indented terms are not-convenient, because they contain unknowns that cannot be be cancelled out but are (see further) shown to be non-negligible.
other, neglected terms are not listed.
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p2]}}\bigr]}{{\mathrm d}t} = &\bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{13} \\ &-\bigl[{{\tt Ar[1p0]}}\bigr] \bigl[{{\tt Ar[2p2]}}\bigr]\left(k_{293}+\ldots+k_{297}\right) \\ &-\bigl[{{\tt Ar[2p2]}}\bigr]\left(\nu_{386}+\ldots+\nu_{389}\right)\\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^*[hl]}}\bigr]\nu_{405}\\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p2]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p3]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{12} \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{298}+\ldots+k_{307}\right) \\ &-\bigl[{{\tt Ar[2p3]}}\bigr] \left( \nu_{283}+\nu_{284}+\nu_{285}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{308} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p2]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{293} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{404} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p3]})} \end{align*}\]
\[\begin{align*} \frac{{\mathrm d}\bigl[{{\tt Ar[2p4]}}\bigr]}{{\mathrm d}t} = & \bigl[{{\tt e}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{11} \\ &-\bigl[{{\tt Ar[2p4]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] \left(k_{308}+\ldots+k_{317}\right) \\ &-\bigl[{{\tt Ar[2p4]}}\bigr] \left( \nu_{379}+\ldots+\nu_{382}\right) \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p5]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{318} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar[2p3]}}\bigr] \bigl[{{\tt Ar[1p0]}}\bigr] k_{298} \\ &\quad\quad\quad\quad+\bigl[{{\tt Ar^{*}[hl]}}\bigr] \nu_{403} \\ &\quad\quad\quad\quad\quad\quad\quad\quad+\hbox{ (neglected 31 source terms for {\tt Ar[2p4]})} \end{align*}\]
Marked terms are to be analyzed for their contribution to the value of the corresponding population ratio.
\[\begin{align*} \frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p2]}}\bigr]}=\left(\frac{k_{12}}{k_{13}}\right) \frac{\displaystyle \nu_{293...297} +\nu_{386...389} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p2]}}\bigr]}}{{\mathrm d}t} -\overbrace{\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p2]}}\bigr]}\nu_{405}}^{\displaystyle (d)}+\ldots} {\displaystyle \nu_{298..307} +\nu_{283...285} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} -\underbrace{\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{308}}_{\displaystyle (a)} -\underbrace{\frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{293}}_{\displaystyle (b)} -\underbrace{\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{404}}_{\displaystyle(c)}+\ldots } \end{align*}\]
\[\begin{align*} \frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}=\left(\frac{k_{11}}{k_{12}}\right) \frac{\displaystyle \nu_{298...307} +\nu_{283...285} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} -\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{308} -\frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{293}+\ldots} {\displaystyle \nu_{308..317} +\nu_{379..382} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p4]}}\bigr]}}{{\mathrm d}t} -\underbrace{\frac{\bigl[{{\tt Ar[2p5]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}\nu_{318}}_{\displaystyle (a^{\prime})} -\underbrace{\frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}\nu_{298}}_{\displaystyle (b^{\prime})} -\underbrace{\frac{\bigl[{{\tt Ar^{*}[hl]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}\nu_{403}}_{\displaystyle(c^{\prime})}+\ldots } \end{align*}\]
Source terms for \(E/N=100\,\)Td. Threshold for importance of individual processes is set to \(10^{-3}\) for relative value of the source term. Processes selected as important are considered in equations above.
![Sources for Ar[2p2] at 100 Td](../figs/souces-100-Gain-Ar%5B2p2%5D.png) |
 |
|---|---|
![Gain for Ar[2p3] at 100 Td](../figs/souces-100-Gain-Ar%5B2p3%5D.png) |
![Loss for Ar[2p3] at 100 Td](../figs/souces-100-Loss-Ar%5B2p3%5D.png) |
![Gain for Ar[2p4] at 100 Td](../figs/souces-100-Gain-Ar%5B2p4%5D.png) |
![Loss for Ar[2p4] at 100 Td](../figs/souces-100-Loss-Ar%5B2p4%5D.png) |
![Ratio Ar[2p3]/Ar[2p2]](../figs/rat-plot-cal-2p3-2p2.png)
Circles show population ratio given by complete RKM.
Lines show population ratio given the the equation above with (a),(b),(c),(d) terms considered or not.
Terms (a) and (b) are shown to be of high importance and need to be considered.
Term (c) shows importance that is only limited. This is fortunate because \({\tt Ar^{*}[hl]/Ar[2p4]}\) is experimentally hardly accessible value.
In conclusion, terms (a) and (b) must be considered:
(a) requires acquisition of \({\tt Ar[2p4]}\)
(b) is already known (just it’s inverse of \({\tt Ar[2p3]/Ar[2p2]}\)
Simplified formula:
\[\begin{align*} \frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p2]}}\bigr]}=\left(\frac{k_{12}}{k_{13}}\right) \frac{\displaystyle \nu_{293...297} +\nu_{386...389} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p2]}}\bigr]}}{{\mathrm d}t} +\ldots} {\displaystyle \nu_{298..307} +\nu_{283...285} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} -\underbrace{\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{308}}_{\displaystyle (a)} -\underbrace{\frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{293}}_{\displaystyle (b)} +\ldots } \end{align*}\]
![Ratio Ar[2p4]/Ar[2p3]](../figs/rat-plot-cal-2p4-2p3.png)
|
Circles show population ratio given by complete RKM.
Lines show population ratio given the equation above with (a’),(b’),(c’) terms considered or not.
Terms (a’) and (b’) are shown to be of high importance and need to be considered.
Inclusion of (c’) cases slight overshoot compared to RKM, this is because \({\tt Ar^{*}[hl]/Ar[2p3]}\) has been omitted in nominator of eq. for \({\tt Ar[2p4]/Ar[2p3]}\).
In conclusion, terms (a’) and (b’) must be considered:
(a’) requires acquisition of \({\tt Ar[2p5]}\)
(b’) is already known (just it’s inverse of \({\tt Ar[2p4]/Ar[2p3]}\)
Simplified formula:
\[\begin{align*} \frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}=\left(\frac{k_{11}}{k_{12}}\right) \frac{\displaystyle \nu_{298...307} +\nu_{283...285} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p3]}}\bigr]}}{{\mathrm d}t} -\frac{\bigl[{{\tt Ar[2p4]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{308} -\frac{\bigl[{{\tt Ar[2p2]}}\bigr]}{\bigl[{{\tt Ar[2p3]}}\bigr]}\nu_{293}+\ldots} {\displaystyle \nu_{308..317} +\nu_{379..382} +\frac{{\mathrm d}\ln{\bigl[{{\tt Ar[2p4]}}\bigr]}}{{\mathrm d}t} -\underbrace{\frac{\bigl[{{\tt Ar[2p5]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}\nu_{318}}_{\displaystyle (a^{\prime})} -\underbrace{\frac{\bigl[{{\tt Ar[2p3]}}\bigr]}{\bigl[{{\tt Ar[2p4]}}\bigr]}\nu_{298}}_{\displaystyle (b^{\prime})} +\ldots } \end{align*}\]
Ratios of \({\tt Ar[2p3]/Ar[2p2]}\) and \({\tt Ar[2p4]/Ar[2p3]}\) are identified as potentially interesting for diagnostics to provide estimation of \(E/N\) in Argon discharge.
Pattern is found:
simplified equation for \({\tt Ar[2p3]/Ar[2p2]}\) requires knowledge of \({\tt Ar[2p4]/Ar[2p3]}\)
simplified equation for \({\tt Ar[2p4]/Ar[2p3]}\) requires knowledge of \({\tt Ar[2p5]/Ar[2p4]}\)
\(\ldots\)
More density ratios needed to be determined form optical emission diagnostics.
This is inconvenience, but also it opens door to determination of \(E/N\) by using multiple density ratios at the same time. The evaluation of this method needs to be performed.
Presented results are for constant values of \(E/N\).