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Help page for H$_2$ model

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

Description of the H2 model

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. For the H2 model, Yacora considers all the excited levels up to p=10, where p is the principal quantum number related to the electronic states. Further, all the electronic states with p $\leq$ 3 are resolved according to the total angular momentum of the two electrons. In the figure below, the according energy-level diagram of H2 is reported [1]. Since H2 is a homonuclear diatomic molecule with two electrons, the excited states are divided into singlet states and triplet states: no spontaneous transitions are possible between singlet (triplet) and triplet (singlet) states. Furthermore, the diatomic system can vibrate and rotate, thus the electronic levels are split in several vibrational and rotational levels. However, these levels are not considered in Yacora on the web (there are several models based on Yacora solver and some of them include also the vibrational and rotational levels for some electronical states, but for what concerns Yacora on the web up to now only the simplest model is considered).

The state $b^3$ in the triplet system is a non-bounding state, i.e. the potential energy curve (the electronic eigenvalues of the total wavefunction vs the internuclear distance) shows no minimum, thus the internuclear distance of a hydrogen molecule in $b^3$ will increase until dissociation into two atoms takes place. Yacora considers transitions to the state $b^3$ which are called continuum transitions.


Cross sections and rate coefficients

Yacora considers a comprehensive set of reactions in order to determine the population densities of the excited states with $p \leq 10$ by solving the following set of differential equations:

$$\frac{\mathrm{d}n_p}{\mathrm{d}t}= \sum_{q\ne p} \Bigl(X_{q\rightarrow p}n_q - X_{p \rightarrow q}n_p\Bigr)$$

where $X_{q\rightarrow p}$ is the total rate coefficient which includes all the processes from the state $q$ to the state $p$ and $X_{p \rightarrow q}$ is the total rate coefficient which includes all the processes from the state $p$ to the state $q$. The reactions considered by Yacora for the H2 model are:

  • Electron collision from the ground state X1 and the inverse reactions, cross section taken from [2,3,4].
  • Electron collision excitations between electrical resolved states and the inverse reactions [4,5].
  • Spontaneous emission [6,7]: all the transitions are electric dipole. For the transitions from the metastable state c3 to the ground state X1, the electric quadrupole and magnetic dipole transitions are considered, based on the transition probability from [2].
  • Ionization [8].

To solve the previous system of differential equations, the transition probabilities for all the above reactions are required. For what concerns the electron collisions from the ground state X1 (and inverse process), the user has the possibility to select between two databases that contain such transition probabilities: Janev [2] and Miles [3]. The first represents a summary of recent measurements and calculations, the second was created by semiempiric methods based on experimental information and phenomenological extensions of the Born approximation into low-energy region. The results obtained using different databases usually differ of a factor 2-3 and it is not straightforward to understand which database must be use. Anyway, the user can find an extensive discussion about the two databases in [1].

Quenching, dissociative attachment and charge exchange

The reactions described in the previous section are always included in all the calculations. There are instead other reactions (quenching, dissociative attachment and charge exchange) that can be included or not, according to the choice of the user. Quenching is the de-excitation from the state a3 and the metastable state c3 to the ground state by heavy particle collisions. It can be dominant for high molecular densities. For c3 and for a3, the values of 1.88x10-15 m3/s and of 1.15x10-15 m3/s are used respectively. These values are taken from [9]. The effect of quenching is well described in [1]. Dissociative attachment is the collision between an electron and a hydrogen molecule which leads to H2- that is not a stable ion, thus it dissociates in H and H- :

$$\text{H}_2 + e^- \longrightarrow \text{H}_2^- \longrightarrow \text{H} + \text{H}^- $$

This process is of high relevance in volume production based on sources for negative hydrogen ions. For p=2 the rate constant is 10-15 m3/s [10] and for p=3 it is 6x10-11 m3/s [11].

The last reaction is the charge exchange of the excited states of H2 with a positive ion of the hydrogen atom [12]:

$$\text{H}_2 + \text{H}^+ \longrightarrow \text{H}_2^+ + \text{H} $$

This process is of high relevance for the molecular assured recombination (MAR) process. Since this reaction involved the H+ species, when the charge exchange is selected by the user the density of H+ is required as input parameter. The user can choose to specify a separate value of H+ density or keep the same value of the electron density multiplied by a factor that is given by the user.

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 table shows the allowed values for each input parameter.

Admissible values for each input parameter

  Admissible values Comment
Te [1,50] eV max. 100 points Used to determine the rate coefficients for electron collision excitation and de-excitation.
ne [1e14,1e22] m-3 max. 100 points Used to determine the reaction rate for electron collision excitation and de-excitation.
T(H2) [300,57971] K max. 100 points Used in order to calculate the rate coefficient for reactions for which the cross section is available.
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.
n(H+) [1e14,1e22] m-3 max. 100 points This parameter is required only if you want to consider the charge exchange in the calculation.

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). The user can choose more than one quantity in the same submission. The maximum excited state of H2 that is included here is the non resolved state with $p$=10. All the states with $p\leq$3 are resolved according to the angular momentum of the two electrons. For every chosen quantity, a file is generated. At the end of the calculation, all the files are automatically uploaded in the user folder and an email is sent to the user, as notification.

Population density

The population density of the considered species is obtained integrating the system of differential equations reported above which takes into account the processes that populate or depopulate such states. The processes included in Yacora were illustrated before.

Population coefficient

To better understand the meaning of the population coefficients, we invite the users to see [13,14]. The population coefficients are useful quantities that were introduced with the purpose to find the solution of the above system of differential equations when the steady-state solution (no time dependence) is considered. In particular, they are defined as

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

where $n_0$ denotes the ground state density of H2, $n_e$ is the electron density and $n_p$ is the density of the excited state $p$, where in this case with $p$ is indicated both the resolved and the unresolved excited states of H2 .

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 from the state $p$ to the state $q$ and $X^\text{eff}_{pq}$ is the effective emission rate coefficient:

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

The Einstein coefficient $A_{pq}$ can be found in [6] or [7], but the interested user has to remember that the model here included is non-vibrationally and non-rotationally resolved, thus only the intensity line emissions between electronic states can be determined and the Einstein coefficients that must be used in the previous equation are given by averaging the Einstein coefficients between the vibrational and rotational levels of the considered electronic states.

Final notes

Time trace

In the output box the user can choose if he or she wants that the time trace is calculated 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 the user selects a range of values for the electron temperature, then the time trace is related to the maximum value of the temperature in that range. Generally, the user doesn't need this information and the default choice is to not upload this file.


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.


[1] 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.

[2] R. K. Janev, D. Reiter, U. Samm, Report Jül-4105, Forschungzentrum Jülich: Jülich, Germany, 2003.

[3] W. T. Miles, R. Thompson, A.E.S. Green, Electron-impact cross sections and energy deposition in molecular hydrogen, J. Appl. Phys. 1972, 43, 678-686, doi:

[4] R. Celiberto (Bari, Italy). Private communication, 2004.

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

[6] U. Fantz and D. Wünderlich, Franck-Condon factors, transition probabilities, and radiative lifetime for hydrogen molecules and their isotopomeres, Atomic Data and Nuclear Data Tables 92 (2006) 853–973, doi:

[7] U. Fantz and D. Wünderlich, Franck-Condon factors, transition probabilities, and radiative lifetime for hydrogen molecules and their isotopomeres,

[8] D. Wünderlich, Vibrationally resolved ionization cross sections for the ground state and electronically excited states of the hydrogen molecule, Chem. Phys. 2011, 390, 75-82, doi:

[9] A. B. Wedding and A. V. Phelps, Quenching and excitation transfer for $c^3\Pi_u^-$ and $a^3\Sigma_g^+$ states of H2 in collisions with H2 , J. Chem. Phys. 1988, 89, 2965-2974, doi:

[10] J. R. Hiskes, Molecular Rydberg states in hydrogen negative ion discharges, Appl. Phys. Lett. 69, 755 (1996), doi:

[11] P. G. Datskos, L. A. Pinnaduwage and J. F. Kielkopf, Photophysical and electron attachment properties of ArF-excimer-laser irradiated H2 , Phys. Rev. A 55 (1997) 4131, doi:

[12] R. K. Janev. Private communication, 2003.

[13] 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.

[14] 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: