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# Help page for He model

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

### Description of the He model

Helium is a two electron system, thus according to the possible orientations of the electron spins, the energy levels split up into a singlet system (antiparallel configuration), which is called parahelium, and a triplet system (parallel configuration), which is called orthohelium. In the figure below, an energy-level diagram for He [1] is shown. As reported in this diagram, the Yacora model for helium includes all the states with principal quantum number $p\leq 4$ and the singly ionized positive ion.

Since for helium model only dipole transitions are considered, there are two metastable states: $2\ ^1S$ in the singlet system and $2\ ^3S$ in the triplet system. In the "Diffusion of the metastable states" section, it will be explained how these states are managed by Yacora and which are the possible choices for the user.

### Cross section and rate coefficients

Yacora considers a comprehensive set of reactions in order to determine the population densities of the excited states with $p\leq 4$ 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 He model are the electron excitation processes [2,3], with the inverse processes (calculated using the detailed balance principle [4]), the spontaneous emissions [5] and the ionization process [6]. The self-absorption due to the optical thickness [7] is neglected.

### Diffusion of the metastable states

As already mentioned before, the He atom has two metastable states. For high electron densities, the dominant depopulating process for these states is the excitation and the de-excitation by electron collisions, but in plasma with low electron density, the transport of particles in the metastable states can take over. This is the reason why in Yacora two possible mechanisms are provided: fix the density of such states, i.e. treat them in the same way as the ground state, or include the diffusion. You can choose between the two possibilities just selecting "off" or "on" in the field called "Diffusion". According to your choice, the densities of $2\ ^1S$ and $2\ ^3S$ or the normal diffusion length and the molecular diffusion length [8] are required (the turbulent diffusion is not included in the model). For high neutral density also the quenching (excitation due to collision with heavy particles) can play an important role, but it is not considered in the He model.

Before explaining what the mean diffusion lengths are, it is necessary to give a simple introduction to the diffusion theory of the neutral particles, in particular for He. Considering a gas of He in a vessel, The "wall" confinement time which is approximately the time that a particle takes to reach the wall is

$$\tau_w = \frac{\int n\;\mathrm{d}V}{\oint \vec{j_w}\cdot\mathrm{d}\vec{A}}$$

where $\vec{j_w}$ denotes the net flux to the wall element $\mathrm{d}\vec{A}$. Now, the mean free path of the He atoms is given by

$$\lambda_n=\frac{1}{n\sigma_n}$$

where $n$ denotes the helium density. The collisional cross section $\sigma_n$=1.3x10-19 m2 considered in Yacora is taken for collision of helium atoms in a helium background from [9]. For $\lambda_n$ small compared to the vessel dimensions (fluid regime), the transport to the walls is governed by the Fick's law

$$\vec{j_w}= -D\nabla n$$

where $D$ is the diffusion coefficient [8] given by

$$D= \frac{3\sqrt{\pi}}{8} \lambda_n \sqrt{\frac{k_B T_g}{m}}$$

where $m$ is the He mass and $T_g$ its temperature. For simple geometry, the confinement time can be determined analytically and the solution can be written as

$$\tau_d = \frac{\Lambda^2}{D}$$

where $\Lambda$ is called mean diffusion length and it is the one of the two parameters required by Yacora. Just to make an example, for a cylindrical vessel with radius $\rho$ and length $2l$ and assuming perfectly sticking walls, $\Lambda$ is given by

$$\Lambda = \Bigl(\frac{8}{\rho^2}+\frac{3}{l^2}\Bigr)^{-1}$$

In [8], some other examples are reported and the user can find (or calculate) the $\Lambda$ value that proper suits with his or her requirement.

If the mean free path is large compared to the vessel dimensions (free fall situation), the mean confinement time is given by

$$\tau_f=\frac{\bar{\Lambda}}{v_{th}}$$

where $v_{th}$ is the thermal velocity

$$v_{th}=\sqrt{\frac{8 k_B T_g}{\pi m}}$$

and $\bar{\Lambda}$ denotes an average connection length from the locus of production to the wall and it is the second parameter required by Yacora. Assuming perfectly sticking wall, you can set this parameter as

$$\bar{\Lambda}=2 d$$

where

$$d=\frac{V}{A}$$

is the characteristic linear dimension of the vessel. Again, the user is invited to see [8] in order to determine the $\bar{\Lambda}$ that better suits with his or her requirement.

Since, according to the plasma regime, the diffusion can be molecular or laminar, to implement a smooth transition between the two conditions, Yacora sums the two confinement times:

$$\tau_w=\tau_d+\tau_f$$

The user must pay attention that the previous expression is valid only if the pumping is not considered, as in Yacora on the Web.

### 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

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(He) [300,57971] K max. 100 points Used in order to calculate the rate coefficient for reactions for which the cross section is available.
n(He) [1e14,1e22] m-3 max. 100 points You can also set this value to 1, if you are interested only in the population coefficients. You should not fix this value to 1 if you consider the diffusion, because it depends on the helium density.

### 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. 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 each state.

#### Population coefficient

To better understand the meaning of the population coefficients, we invite the users to see [10,11]. 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 He, $n_e$ is the electron density and $n_p$ is the density of the excited state $p$.

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

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

### Final notes

#### Time trace

In the output box you can choose if you want 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 you select 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.

#### 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] 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] F. J. De Heer, Critically Assessed Electron-Impact Excitation Cross Sections for He (11S), IAEA Nuclear Data Section Report, INDC(NDS)-385, IAEA: Vienna, Austria, 1998, www-nds.iaea.org/publications/indc/indc-nds-0385/.

[3] Y. V. Ralchenko, R. K. Janev, T. Kato D. V. Fursa, I. Bray and F. J. de Heer, Cross Section Database for Collision Processes of Helium Atom with Charged Particles, Research Reports NIFS DATA, NIFS-DATA-59, NIFS: Toki, Japan, 2000.

[4] R. H. Fowler, Statistical equilibrium with special reference to the mechanism of ionization by electronic impacts, Philos. Mag. 1926, 47, 257-277.

[5] W. F. Drake Gordon, Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Science+Business Media, Inc.: New York, NY, USA, 2006; pp. 199-216.

[6] T. Fujimoto, A collisional-radiative model for helium and its application to a discharge plasma, Quant. Spectrosc. Radiat. Transfer 21, 439 (1979), doi:10.1016/0022-4073(79)90004-9.

[7] 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.

[8] W. Möller, Plasma and Surface Modelling of the Deposition of Hydrogenated Carbon Films from Low-Pressure Methane Plasmas, Appl. Phys. A 56, 527-546 (1993), doi:10.1007/BF00331402.

[9] B. M. Smirnov, Reference Data on Atomic Physics and Atomic Processes, Springer Series on Atomic, Optical and Plasma Physics 51, 2008; p.102, doi:10.1007/978-3-540-79363-2.

[10] 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.

[11] 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.