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Help page for H model

In this page, some useful information about the input parameters for the H model and about Yacora are reported. At the end of the page, you can find some references for further and more detailed explanations.

Yacora

The main target of a collisional radiative model is to determine the population densities (or the population coefficients) of the excited states of specific atoms or molecules in the considered plasma. To determine these population densities of excited states of the hydrogen atom, Yacora solves a system of equations that describes how the density of each excited state evolves in time as a consequence of processes that that populate or depopulate such excited levels:

$$\frac{\mathrm{d}n_p}{\mathrm{d}t}=\sum_{q>p}A_{qp}n_q-\sum_{q<p}A_{pq}n_p+n_e\biggl(\sum_{q\ne p}X_{qp}n_q-\sum_{q\ne p}X_{pq}n_p+(\alpha+\beta n_e)n_+-S_pn_p\biggr)$$

where $p$ indicates the excited state with principal quantum number equal to p. The physical meaning of all the terms in the second member is:

1. Spontaneous emission from the states with $q>p$ to the state $p$.
2. Spontaneous emission from the state $p$ to the state with $q<p$.
3. Electron collision excitation from the states with $q<p$ to the state $p$ and electron collision de-excitation from the states with $q>p$ to the state $p$.
4. Electron collision excitation from the state $p$ to the states with $q>p$ and electron collision de-excitation from the state $p$ to the states with $q<p$.
5. Two-body recombination process which result is a hydrogen atom in the state $p$.
6. Three-body recombination process which result is a hydrogen atom in the state $p$. This is the inverse process of the ionization.
7. Ionization of the atom in the state $p$.
An overview of the reactions considered in Yacora for the hydrogen model is given below.
In some collisional radiative models, the steady-state condition is considered, i.e. $\frac{\mathrm{d}n_p}{\mathrm{d}t}=0$. In that case, the previous system of differential equations becomes a system of algebraic equations and the solution is obtained by inverting a matrix. However, Yacora solves directly the system of differential equations, but because of the stiffness of these equations, ordinary integration techniques, as for example the Runge-Kutta method, are too slow, thus a different solver, CVODE [1] is used.
The self absorption due to optical thickness is not considered [2].

Excitation channel

The Yacora model for the hydrogen atom includes calculations considering different excitation channels, i. e. the coupling of the excited states of H to different particle species [3], but it performs calculation for only one of them at time. Thus, if you want to consider more than one excitation channel, you should repeat the calculation for all of them. In many cases, according to the temperatures and densities of the particles present in the plasma, it is not necessary to consider all the excitation channels, because some of them may have a negligible contribution. In the following figure are reported the excitation channels included to the Yacora CR (collisional radiative) model for atomic hydrogen: (a) all channels; (b) channels relevant in ionizing plasmas; (c) channels relevant in recombining plasmas [3].

Saha states

The Saha ionization equation describes the density of the ionized component of a species according to its temperature and pressure. The Saha states of a Rydberg atom are high-level states, which densities can be approximately estimated by the Saha equation. In particular, Yacora considers as Saha states all the excited states with principal quantum number between 34 and 40. Since their densities depend on the H+ density, if you select "on", the H+ density is required as input parameter and you can choose between insert a separate value for this density or use the electron density multiply for a factor at your choice. The Saha states are always included for recombination and mutual neutralization channels.

Cross section and rate coefficient

Yacora uses a lot of input parameters: some of them are required to the user and others aren't editable. For example, the cross sections, the rate coefficients and the transitions probabilities for each reaction (more than 2300 reactions are considered in Yacora!) aren't available to the user. Anyway, since these are important information, some references are reported in order to allow you to see which reaction probabilities are used by Yacora. If you can provide more recent and accurate values, please send an e-mail to yacora-webmaster@ipp.mpg.de so we can update the database on our website.

Overview of the reactions included in the CR model for the hydrogen atom [4]

Process Reaction Reference
Excitation by e- collision H(q) + e- → H(p>q) + e- [3,5]
De-excitation by e- collision H(q) + e- → H(p<q) + e- [3,5], Detailed balance
Spontaneous emission H(q) → H(p<q) + hν [6,7]
Ionization H(q) + e- → H+ + e- [5]
Recombination of H+

H+ + e- → H(p) + hν

H+ + 2e- → H(p) + e-

[8]

[8]

Dissociation of H2 H2 + e- → H(p) + H(1) + e- [8]
Dissociation of H2+ H2+ + e- → H(p) + H+ + e- [8]
Dissociative recombination of H2+ H2+ + e- → H(p) + H(1) [5]
Dissociative recombination of H3+ H3+ + e- → H(p) + H2 [5,9]
Mutual neutralization

H+ + H- → H(p) + H

H2+ + H- → H(p) + H2

[10]

[5,11]

Input parameters

For almost all the input parameters, there are three possible choices:

1. Fixed: The related parameter is kept fixed during the calculation.
2. Range: The related parameter varies in the specific range during the calculation. The admissible maximum number of points is 100.
3. Values: The related parameter assumes the given values and Yacora performs a calculation for each of them. The values must be separated by a semicolon.

NB: The total maximum number of calculations must be less than 10000 (100*100).

The following tables show which parameters must be given by the user for the different excitation channels and what are the allowed values.

Input parameters for each excitation channel

H

H+

H-+H+

H-+H2+

H2

H2+

H3+

Te

Required

Required Required Required Required Required Required

ne

Required Required Required Required Required Required Required

T(H)

Required Required Required Required Required Required Required

n(H)

Required

T(H+)

Required Required

n(H+)

Required

Optional (*)

Optional (*)

T(H-)

Required Required

n(H-)

Required Required

T(H2)

Not required (**)

n(H2)

Required

T(H2+)

Required   Not required (**)

n(H2+)

Optional (*)   Required

T(H3+)

Required

n(H3+)

Required
(*) In these cases, you can choose if you want to insert a separate value for the density or use the electron density multiply for a factor at your choice. (**) Yacora uses directly the rate coefficients, thus the temperature is not needed.

Admissible values for each input parameter

Te

[1,50] eV max. 100 points Used in all the excitation channels to calculate the rate coefficients for electron collision excitation and de-excitation.

ne

[1e14,1e22] m-3 max. 100 points Used in all the excitation channels to calculate the reaction rate for electron collision excitation and de-excitation.

T(H)

[300,57971] K max. 100 points Used in all the excitation channels to calculate the rate coefficients for electron collision excitation and de-excitation. Since the hydrogen mass is much greater than the electron mass, this parameter doesn't change the results so much. If you don't known the value, you can insert the gas temperature.

n(H)

[1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients.

T(H+)

[300,57971] K max. 100 points

n(H+)

[1e14,1e22] m-3 max. 100 points This value is also required if you want to consider the Saha states. In the case of H+ excitation channel, you can also set the value to 1 if you are interested only in the population coefficients.

T(H-)

[300,57971] K max. 100 points

n(H-)

[1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients.

T(H2)

[300,57971] K max. 100 points

n(H2)

[1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients.

T(H2+)

[300,57971] K max. 100 points

n(H2+)

[1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients.

T(H3+)

[300,57971] K max. 100 points

n(H3+)

[1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients.

Output quantities

There are three possible output quantities: population density, population coefficient and density balance (that is in principle not a quantity, as explained below). You can choose more than one quantity in the same submission. The maximum excited state of H that is included here is the state with principal quantum number equal to 20. For every chosen quantity, a file is generated. At the end of the calculation, all the files are automatically uploaded in your folder and an email is sent to you, as notification.

Population density

The population density of the considered species is obtained integrating the first system of differential equations, which takes into account the different processes that populate or depopulate the considered excited level. As seen before, if you want to consider more than one excitation channel, you have to repeat the calculation as many times as the chosen excitation channels.

Population coefficient

In order to understand the meaning of the population coefficient, you are invited to see [4,12]. The population coefficients are useful quantities that were introduced with the purpose to find the solution of the first system of differential equations when the steady-state solution (no time dependence) is considered. In particular, they are defined as

$$R_{0p}= \frac{n_p}{n_e n_0}$$

where $n_0$ denotes the density of the ground state or another species with quasi-constant density (that you have specified when you have chosen the excitation channel).

Mutual neutralization of negative hydrogen ions splits up in two channels: Mutual neutralization of H- with H+ and mutual neutralization of H- with H$_2^+$, i.e. in both cases two species with quasi-constant density are involved. In these cases, the term $n_0$ in the population coefficient refers to $n$(H-) while the values for $n$(H+) or $n$(H$_2^+$) have to be defined by the user when generating the input for the calculation.

Density balance

The density balance option shows the rate of all the considered reactions that populate or depopulate the given state. The rate is positive if the reaction populates the state, vice versa it is negative if the reaction depopulates such state. In the output files you can find a comment for each reaction that gives you more information about the relative process.

NB: Creating the density balance needs a lot of calculation time and disk space. Please use it only for a very reduced number of calculations.

Line emission intensity

Starting from the population densities $n_p$, it is possible to determine the absolute intensity line emissions (in units of $m^{-3}s^{-1}$):

$$I_{pq}= n_p A_{pq}=n_e n_0 R_{0p} A_{pq}=n_e n_0 X^\text{eff}_{pq}$$

where $A_{pq}$ is the Einstein coefficient [6,7] from the state $p$ to the state $q$ and $X^\text{eff}_{pq}$ is the effective emission rate coefficient:

$$X^\text{eff}_{pq}\equiv R_{0p} A_{pq}\ .$$

If all the excitation channels are taken into account, the absolute intensity line emission is given by:

$$I_{pq}= n_e (n_\text{H} R_{\text{H},p}+n_{\text{H}_2} R_{\text{H}_2,p}+n_{\text{H}^+} R_{\text{H}^+,p}+n_{\text{H}^+_2} R_{\text{H}^+_2,p} +n_{\text{H}^+_3} R_{\text{H}^+_3,p}+n_{\text{H}^- }R_{\text{H}^-,p})A_{pq}$$

where the $n_0$ has been substituted with the quasi constant density of the involved species.

Final notes

Time trace

In the output box you can choose if you want to calculate the time trace or not. The time trace is reported in a homonym file, which contains the evolution in time of the population density of all the considered species and excited states. The time trace is available only for the very last used set of parameters; i.e. if you select a range of values for the electron temperature, then the time trace is related to the maximum value of the temperature in your range. Generally, the user doesn't need this information and the default choice is to not upload this file.

Comment

Another note concerns the comment box: you should always put some comments about the calculation that you are doing. This is a very good habit (not only in this case), because in few months you will not be able to remember what you have done, especially if in your folder there are a lot of calculations.

References

[1] C. D. Cohen and A. C. Hindmarsh, CVODE a stiff/nonstiff ODE solver in C, Comput. Phys. 1996, 10:138-43, https://computation.llnl.gov/casc/nsde/pubs/u121014.pdf.

[2] K. Behringer and U. Fantz, The influence of opacity on hydrogen excited-state population and applications to low-temperature plasmas, New Journal of Physics 2 (2000), doi:10.1088/1367-2630/2/1/323.

[3] D. Wünderlich and U. Fantz, Evaluation of State-Resolved Reaction Probabilities and Their Application in Population Models for He, H and H2 , Atoms 2016, 4, 26, doi:10.3390/atoms4040026.

[4] D. Wünderlich, S. Dietrich and U. Fantz, Application of a collisional radiative model to atomic hydrogen for diagnostic purposes, J. Quant. Spectrosc. Radiat. Transfer 110 (2009), doi:10.1016/j.jqsrt.2008.09.015.

[5] R. K. Janev, D. Reiter and U. Samm, Report Jül-4105, Forschungszentrum Jülich, 2003.

[6] L. C. Johnson, Approximations for collisional and radiative transition rates in atomic hydrogen, Astrophysics J. 1972, 174:227-36.

[7] NIST: physics.nist.gov/PhysRefData/ASD

[8] K. Sawada and T. Fujimoto, Effective ionization and dissociation rate coefficients of molecular hydrogen in plasma, J. Appl. Phys. 1995, 78:2913-24, doi: 10.1063/1.360037.

[9] S. Datz, G. Sundström et al., Branching processes in the dissociative recombination of H3+, J. Phys. B 1979, 12:L501-4, doi:10.1103/PhysRevLett.74.896.

[10] M. Stenrup, Å Larson et al, Mutual neutralization in low-energy H++H− collisions: A quantum ab initio study, Phys. Rev. A 2009, 79:012713, doi:10.1103/PhysRevA.79.012713.

[11] M. J. J. Eerden, M. C. M. van de Sanden et al., Cross section for mutual neutralization reaction H2+ + H- , calculated in a multiple-crossing Landau-Zener approximation, Phys. Rev. A 1995, 51:3362-5.

[12] T. Fujimoto, Plasma Spectroscopy, Springer Berlin Heidelberg, Series on Atomic, Optical and Plasma Physics 44, 2008, pp 29-49, doi:10.1007/978-3-540-73587-8_3.