## Abstract

We have developed a Monte Carlo model for studying the local degradation of electrons in the energy range 9–10 000 eV in xenon gas. Analytically fitted form of electron impact cross sections for elastic and various inelastic processes are fed as input data to the model. The two-dimensional numerical yield spectrum (NYS), which gives information on the number of energy loss events occurring in a particular energy interval, is obtained as the output of the model. The NYS is fitted analytically, thus obtaining the analytical yield spectrum (AYS). The AYS can be used to calculate electron fluxes, which can be further employed for the calculation of volume production rates. Using the yield spectrum, mean energy per ion pair and efficiencies of inelastic processes are calculated. The value for mean energy per ion pair for Xe is 22 eV at 10 keV. Ionization dominates for incident energies greater than 50 eV and is found to have an efficiency of approximately 65% at 10 keV. The efficiency for the excitation process is approximately 30% at 10 keV.

## 1. Introduction

When electrons interact with atoms or molecules, energy of the incident electron is lost through inelastic collisions with the target. The target atom can undergo ionization or excitation. Secondary electrons released during the ionization process will trigger further collisions. These electrons lose energy via further collisions in the gas. This kind of electron energy degradation process is of great importance in understanding phenomena, like electron beam propagation in the atmosphere, population inversion process in a large group of gas lasers, optical emissions occurring in the upper atmos- phere, such as aurora and airglow, which happen due to the precipitation of high-energy electrons, etc. [1–4].

Xenon is a widely studied atom because of its inert nature and simple ground state atomic structure. The inert nature of xenon makes it possible to study the pristine environment of the early solar system whose traces have been removed by other reactive elements. Noble gases are valuable probes of extraterrestrial environments. Their abundances as well as isotopic compositions are indicators of various processes such as stellar events prior to solar system formation and radioactive decay. Hence their study is vital to understand the sequence of events that led to the formation and subsequent evolution of the solar system. There are various missions that studied xenon in planetary atmospheres. For example, the Galileo mass spectrometer reported a relative abundance of 2.6±0.5 times solar ratios in the Jovian atmosphere [5]. The Pioneer Venus Sounder Probe Neutral Mass Spectrometer measured the upper limit for xenon in the Venusian atmosphere as 120 ppb [6]. In the Earth’s atmosphere xenon is a trace gas, having a concentration of 87 ppb [7].

Electron collision with Xe has a wide range of application. In X-ray and gamma ray detectors, Xe gas counters have been commonly used. In both these detectors, interaction of radiation with Xe atoms results in the production of electrons. Among the various methods of amplifying the obtained skimp signal, a commonly used approach is electron acceleration in which the accelerated electrons excite the xenon atoms through inelastic collisions leading to the production of secondary scintillation, i.e. the emission of detectable light in the vacuum ultraviolet region [8]. The same technique is used in flash lamps which produce high-intensity white light for a short duration. The excited atoms, created by electron–atom collisions, can de-excite by emitting in different wavelengths whose combined effect will give the appearance of white light emission. The electron bombardment technique with xenon is also used in ion thrusters, which is an evolving technology in the field of rocket propulsion. Here, electrically accelerated Xe ions, created using electron impact, are emitted at high speed as exhaust and this will push the spacecraft forward. The efficiency of this kind of ion thrusters is found to be higher than that of conventional chemical propulsion methods and has been used in the NASA missions Deep Space-1 [9] and DAWN [10]. The utmost use of xenon gas for such purposes will be possible only through a thorough understanding of properties associated with microscopic collision processes.

Monte Carlo simulation is a commonly used method for studying the particle energy degradation process [11–16]. The electron collision process in xenon was studied earlier by Date *et al.* [17] using a Monte Carlo method. Rachinhas *et al.* [18] studied the absorption of electrons with energies ≤200 keV in xenon using a Monte Carlo simulation technique. They calculated mean energy per ion pair (*w*-value) and Fano factor in the energy range 20–200 keV and studied the influence of electric field on the results. Absorption of X-ray photons and the drift of resulting electrons under the influence of an applied electric field in xenon was studied by Dias *et al.* [19], by using a Monte Carlo technique. A three-dimensional Monte Carlo method was used by Santos *et al.* [20] to study the drift of electrons in xenon and to calculate various physical properties such as electroluminescence, drift velocities, etc.

We have developed a Monte Carlo model for the local degradation of electrons with energy in the range 9–10 000 eV in neutral xenon gas to understand how the energy of the incident electron will be distributed among various loss channels like ionization and excitation while it is making collisions with neutral xenon atoms. Using our Monte Carlo model, we have calculated numerical yield spectra which are a basic distribution function that contain information regarding the degradation process and can be used to calculate the yield of any excited or ionized states. The numerical yield spectra are then fitted analytically, thus obtaining analytical yield spectra. For practical purposes, the use of the analytical yield spectrum (AYS) simplify the application by substantially reducing the computational time. We have also obtained secondary electron energy distribution, the efficiency of excitation and ionization processes, as well as the mean energy expended for ion pair production.

## 2. Monte Carlo model

In the Monte Carlo simulation, the energy loss process of the electron is treated in a discrete manner. In carrying out the degradation by means of discrete steps, the electron is followed as it undergoes successive collisions. To accomplish the energy degradation in a convenient way, the energy range from the incident energy to cut-off energy is divided into a number of equally spaced bins. Whenever an electron makes an inelastic collision, the collision event is recorded in the corresponding energy bin. This process is continued and the particle, its secondaries, tertiaries, etc., are followed until their energy falls below an assigned cut-off value.

The energy bin size is taken as 1 eV throughout the energy range. To make an inelastic collision with a xenon atom, an electron should have a minimum energy of 8.315 eV as it is the lowest threshold of all the inelastic processes. We have set the cut-off as 9 eV because we have 1 eV energy bins in the model.

Figure 1 shows the flow diagram for our Monte Carlo simulation for electron degradation. The simulation starts by fixing the energy of the incident electron. The direction of movement of electron (*θ*, *ϕ*) is assumed to be isotropic. The distance that the electron has to travel before the collision is calculated as
*R*_{1} is a random number, *N* is the number density of the gas (equal to 10^{10} cm^{−3}) and *σ*_{T} is the total scattering cross section (elastic + inelastic). Next, we decide on the type of collision. The probabilities of the elastic and inelastic events, *P*_{el} (*σ*_{el}/*σ*_{T}) and *P*_{in} (*σ*_{in}/*σ*_{T}), where *σ*_{el} and *σ*_{in} are elastic and inelastic cross sections, respectively, are calculated and compared with a new random number *R*_{2}. Elastic collision occurs if *P*_{el}≥*R*_{2}. The energy loss in elastic collisions Δ*E* due to target recoil is calculated as
*δ* is the scattering angle in the laboratory frame, *v* and *m* are, respectively, the velocity and mass of the incident electron, and *M* is the mass of the target particle. The scattering angle *δ* is determined by using differential elastic cross sections which are fed numerically into the model. The energy lost in the collision is then subtracted from the incident electron energy. After the collision, the deflection angle relative to the direction (*θ*,*ϕ*) is obtained by
*θ*′ and *ϕ*′ are the scattering angles.

If an inelastic collision occurs, the collision event is recorded in the appropriate energy bin corresponding to the energy of the particle. It is further decided whether it is an excitation or ionization event. In the case of an ionization event, the energy of the secondary electron has to be calculated as it can also initiate further inelastic collisions, provided it has sufficient energy. The secondary electron energy is calculated as [21]
*E*_{v} is the incident electron energy; *Γ*_{S}, *Γ*_{A}, *T*_{A}, *T*_{B}, and *T*_{S} are the fitting parameters, and *I* is the ionization threshold. We have used the fitting parameters of Green & Sawada [21]. If this energy is greater than that of the cut-off energy (9 eV) then the secondary electron has to be followed. In order to follow the secondary electron, the parameters of the primary electron, i.e. the energy remaining in the primary, its position and direction of movement are first saved in suitable variables. The secondary electron is then followed in the same method as the primary electron. Once the energy of the secondary is completely degraded, the saved parameters of the primary electron are retrieved and its degradation is continued. Similarly, tertiary, quaternary, etc., electrons are followed in the simulation.

For all inelastic collisions, the collision event is recorded in the corresponding energy bin so that the information on the total number of collisions that occur in each energy bin can be obtained, once the simulation is complete. This is used for calculating the yield spectrum and is described in detail in §4. The number of secondary, tertiary, quaternary, etc., electrons produced during ionization events are also stored in the corresponding energy bins which are used to determine their energy distribution (see §4c). The angle and direction of movement of the electron after each ionization and excitation event are calculated using differential elastic cross sections as described in equation (2.3). After each inelastic collision, appropriate energy is subtracted from the particle energy. If the remaining energy is higher than the cut-off energy, it is again followed in the simulation. The simulation is made for a monoenergetic beam of 10^{6} electrons; each and every electron is followed in a collision-by-collision manner until its energy falls below 9 eV.

Modelling the electron energy degradation primarily requires a set of electron impact excitation and ionization cross sections for the atom. These cross sections are essential for electron energy deposition schemes and are presented below in detail.

## 3. Cross sections

### (a) Total elastic cross sections

Total elastic scattering cross sections for Xe have been measured or calculated by many authors, like Mayol & Salvat [22], Gibson *et al.* [23], Adibzadeh & Theodosiou [24], Vinodkumar *et al.* [25] and McEachran & Stauffer [26]. In this model, we have used the analytically fitted theoretical cross sections of McEachran & Stauffer [26], which are calculated using a relativistic optical potential method. These cross sections are in good agreement with the measured values of Mayol & Salvat [22] and also with the theoretical values of Adibzadeh & Theodosiou [24] and Vinodkumar *et al.* [25] in the energy range 50–1000 eV. However, at energies between 15 eV and 50 eV, cross sections of Vinodkumar *et al.* [25] are lower with a maximum deviation of 50% at 30 eV. Calculations by McEachran & Stauffer [26] are higher than the cross sections of Gibson *et al.* [23] and Adibzadeh & Theodosiou [24] at 1–10 eV. The maximum deviation (25%) is found at 6 eV. We have extended the analytical fit of McEachran & Stauffer [26] to 10 keV to calculate the cross section at higher energies. This extension is valid as it agrees with the cross sections calculated by Garcia *et al.* [27] using a scattering potential method.

### (b) Differential elastic cross sections

The direction in which an electron is scattered after each collision is calculated using differential elastic scattering cross sections (DCSs). For this work, the DCS of Adibzadeh & Theodosiou [24] is used, in which values for the energy range 1–1000 eV are given for a finer energy grid (1 eV). These values are in good agreement with the DCS values of McEachran & Stauffer [28] and Sienkiewicz & Baylis [29]. For energies greater than 1000 eV, linearly extrapolated values of differential cross sections are used, as measurements are not available. DCS values at various energies are shown in table 1.

### (c) Ionization cross sections

Both single and multiple ionization cross sections of xenon have been measured by Schram [30], Nagy *et al.* [31], Stephan & Märk, [32], Wetzel *et al.* [33], Lebius *et al.* [34], Krishnakumar & Srivastava [35], Almeida [36] and Rejoub *et al.* [37]. Up to 1000 eV, we have used the recent measurements of Rejoub *et al.* [37], which are in good agreement with the work of Rapp & Englander-Golden [38], Schram [30], Nagy *et al.* [31] and Stephan & Märk [32]. At energies greater than 1000 eV, the measurements of Schram [30] have been used as they are the only available measurements at higher energies. These cross sections are fitted using the empirical formula of Krishnakumar & Srivastava [35]
*A* and *B*_{i} are fitting coefficients, *I* is the ionization threshold, *E* is the electron energy and *i* is the number of terms *N* required to fit the data. Fitting parameters had to be adjusted as the cross sections of Krishnakumar & Srivastava [35] are higher than that measured by Rejoub *et al.* [37] and Schram [30] by about 20% at the maximum. Fitting parameters used in this study are given in table 2.

In our model, we have considered only up to the fifth ionization state of xenon. Higher states have very low ionization cross sections and the total yield will remain more or less the same even if they are taken into account. Ionization cross sections used in the model are shown in figure 2. These partial ionization cross sections are then used to calculate gross ionization cross section as

As xenon is a gas that is capable of multiple ionization, the total ionization cross section will be the charge weighted sum of partial ionization cross sections [38]. It is this gross ionization cross section that is used in the model.

### (d) Excitation cross sections

Xenon (*Z*=54) has a ground state configuration of 5p^{6}. Electron impact excitation can result in configurations like 5p^{5}*n*s, 5p^{5}*n*p, 5p^{5}*n*d, etc. Each of these excited configurations will be composed of different levels which occur due to the coupling between the core angular momentum *J*_{c} and the angular momentum of the excited electron. For example, the 5p^{5}6s configuration is composed of four levels which are represented as 1s_{2}, 1s_{3}, 1s_{4} and 1s_{5} (in decreasing order of energy) in Paschen notation with *J* values 1, 0, 1 and 2, respectively. Excitation cross sections for the various excitation levels of Xe are available in the literature. However, individual cross sections of various levels in each configuration have not been calculated.

Cross sections for the excitation into 5p^{5}7p levels from the ground level as well as from the 5p^{5}6s levels of xenon were measured by Jung *et al.* [39]. Sharma *et al.* [40] theoretically calculated the cross sections for the excitation into the 5p^{5}7p levels. Excitation cross section from the ground state to the 5p^{5}6s level was measured by Fons & Lin [41]. Puech & Mizzi [42] reported cross sections for the 13 excited levels of xenon where the excitation cross sections for the forbidden and allowed transitions were calculated separately. They made use of Born–Bethe approximation to calculate the cross sections at high-incident electron energies and a low-energy modifier to extend the calculations down to threshold energies. These semi-empirical expressions which are valid from threshold to relativistic energies are used in the current model. The excitation cross section for an allowed level is calculated as
*a*_{0} is the Bohr radius, *R* the Rydberg constant, *m* the rest mass of electron and *β* is the velocity of incident electron in units of light velocity *c*. *W*_{j} is the excitation threshold of the *j*th level and *F*_{oj} is the oscillator strength. For forbidden states, cross sections are calculated as
*F*_{j} is a constant. To calculate cross sections at energies near the threshold region, equations (3.3) and (3.4) have to be multiplied by a low-energy modifier
*a*_{j}, *b*_{j} and *c*_{j} are fitting parameters. Values of these parameters for the different excitation levels are shown in table 3. Cross sections for various excitations are added together to obtain the total excitation cross section.

### (e) Total cross sections

The total inelastic cross section is calculated by adding the total excitation cross section and the gross ionization cross section. These total inelastic cross sections and elastic cross sections are added up to obtain total scattering cross sections. Our calculated total scattering cross sections are in good agreement with the values of Kurokawa *et al.* [43], Zecca *et al.* [44] and Vinodkumar *et al.* [25]. Figure 3 shows various cross sections that are used in our model.

## 4. Results

### (a) Yield spectrum

Yield spectrum, *U*(*E*,*E*_{0}), for an incident electron energy *E*_{0} and spectral energy *E*, is defined as the number of discrete energy loss events that happened in an energy interval *E* and *E*+Δ*E*.
*N*(*E*) is the number of inelastic collisions and △*E* is the energy bin width which is 1 eV in our model. This yield spectrum can be used for calculating the population (*J*) of any state *j*, which is the number of inelastic events of type *j* caused by an electron while degrading its energy from *E*_{0} to cut-off as
*W*_{th} is the threshold for the *j*th process; *P*_{j}(*E*) is the probability of the *j*th process at energy *E*, which can be calculated as *P*_{j}(*E*)=*σ*_{j}(*E*)/*σ*_{in}(*E*); *σ*_{in}(*E*) is the total inelastic collision cross section at energy *E*.

The numerical yield spectrum (NYS), obtained as the output of the model, can be represented in an analytical form as [13]
*H* is the Heavyside function, *E*_{m} is the minimum threshold of the processes considered and *δ*(*E*_{0}−*E*) the Dirac delta function which accounts for the collision at source energy *E*_{0}. Green *et al.* [15] have given a simple analytical representation for *U*_{a}(*E*,*E*_{0}) as
*ξ*=*E*_{0}/1000 and *ϵ*=*E*/*I* (*I* is the lowest ionization threshold, and *A*_{1}, *A*_{2}, *t*, *r* and *s* are fitting parameters). The fitting parameters for xenon gas are *A*_{1}=0.035, *A*_{2}=1.75, *t*=0.0, *r*=−0.065 and *s*=−0.085.

Figure 4 shows NYS as well as AYS for five different incident electron energies. Rapid oscillations seen in the yield spectrum at energies close to the incident electron energy are not taken into account in our analytical fit. These oscillations occur due to the fact that energy loss processes are discrete in nature. For an electron having incident energy *E*_{0}, an inelastic collision with threshold energy *E*_{m} will bring the energy down to a value of *E*_{0}−*E*_{m}. No energy value in the region between *E*_{0} and *E*_{0}−*E*_{m} can be acquired by the electron. This is known as the Lewis effect [45]. The heavyside function in equation (4.3) accounts for the Lewis effect.

Using the yield spectrum, the population of various excitation states can be calculated through equation (4.2). This is useful to determine various properties of gas, like mean energy per ion pair and efficiencies of different loss channels which are described in the following sections.

### (b) Mean energy per ion pair

Mean energy per ion pair, also known as the *w*-value, is the average energy lost by the incident electron in forming an electron–ion pair. The *w*-value for an incident electron energy *E*_{0} is calculated as
*J*(*E*_{0}) is the population of the ionization events. Figure 5 shows the mean energy per ion pair value calculated for neutral xenon and for the various ionization channels of Xe. At high-incident electron energies *w* approaches a constant value. As the incident electron energy decreases, the ionization population also decreases since the excitation process starts dominating due to the higher cross section at these energies. Thus, *w* increases as the incident particle energy decreases. This behaviour of *w* agrees well with the previous calculations of Combecher [46], Date *et al.* [17], Dayashankar [47] and Dias *et al.* [19]. Mean energy per ion pair calculated for neutral xenon and for the various ionization states of Xe at two different incident energies, 10 keV and 300 eV, are shown in table 4. Date *et al.* [17] reported a *w*-value of 21.7 eV at 10 keV. Combecher [46] measured the *w*-value for electrons in xenon and obtained a value of 22 eV for high energy electrons. Dias *et al.* [19] obtained a value of 22 eV at 10 keV, while Dayashankar [47] calculated a value of 23.1 eV for energy >200 eV. Our calculated value of mean energy per ion pair is in good agreement with those reported previously.

### (c) Secondary electron distribution

Secondary electrons generated during ionization can also cause inelastic collisions, provided they have sufficient energy. Energies of secondary electrons are calculated using equation (2.4), and the number of secondary, tertiary, quaternary, etc. electrons is recorded in appropriate energy bins. The energy distribution of secondary electrons for different incident electron energies is shown in figure 6. Also shown in the same figure is the distribution of tertiary and quaternary electrons for an incident energy of 10 keV. It is clear from the figure that during degradation, an electron with 10 keV energy will produce at least one secondary or tertiary electron whose energy is less than 34 eV, which is still sufficient to cause an inelastic collision.

### (d) Efficiency

The efficiency of each of the various inelastic processes *j* can be calculated as
*W*_{th} is the threshold for the *j*th process.

Figure 7 shows the efficiencies of the various ionization channels. Efficiencies are calculated using both numerical yield as well as analytical yield and are compared with each other. A good match is observed between the values obtained using the two methods. Throughout the energy range, the Xe^{+} ionization channel is found to have the maximum efficiency due to its high cross section. At 10 keV, Xe^{+} has an efficiency of 40.5%. Xe^{2+}, Xe^{3+}, Xe^{4+} and Xe^{5+} have efficiencies of 11.5%, 7.4%, 3.2% and 1.8%, respectively.

Efficiencies of various levels in the 1s configuration are shown in figure 8. For an incident electron energy of 10 keV, approximately 10% of the energy is spent in the 1s configuration. As seen in the figure, the allowed excitation 1s_{4} has the highest efficiency throughout the energy range with a value of 4.5% at 10 keV and the lowest efficiency is for the forbidden excitation 1s_{3} with 0.4% efficiency. The other two excitations, 1s_{2} and 1s_{5}, have efficiencies of 2.3% and 2.9%, respectively. Figure 9 shows the efficiencies of the 2p configuration. The 2p_{9}+2p_{8} level has an efficiency of 1.6% at 10 keV and is the highest among various levels in the 2p configuration. Out of approximately 4% efficiency of the 2p configuration at 10 keV, 0.9% is channelled into the 2p_{10} level, 0.8% into the 2p_{7}+2p_{6} and 0.8% into the 2p_{4}+2p_{3}+2p_{2}+2p_{1} levels. Efficiencies of the remaining excitation channels are shown in figure 10. The 3d_{5} level has a very low efficiency of 0.1% at 10 keV. A combination of various forbidden levels, 3d_{6}+2p_{5}+3d_{4}+3d_{3}+3d_{4}+3d^{′′}+3d_{1}, has an efficiency of 4.7%. The upper allowed excitation levels (3s−9s) consume around 5.9% of the incident electron energy. The 3d_{2} and 2s_{5}+2s_{4} levels have efficiencies of 4% and 1% at 10 keV, respectively. Efficiencies of various inelastic processes at two different incident energies 300 eV and 10 keV are shown in table 5.

Figure 11 shows how the incident electron energy is divided among ionization and excitation processes. From 50 eV onwards, ionization is the dominant inelastic process. More than 50% of incident energy is spent into ionization at these energies. Above 1000 eV, ionization efficiency attains a constant value of approximately 64%. Excitation dominates at energies less than 30 eV. In the energy range were only elastic and excitation collision can occur, excitation efficiency is found to be around approximately 90% which is consistent with the results of Dias *et al.* [19] and Santos *et al.* [20]. At incident electron energies of 10 keV around 30% of the energy is spent on excitation events.

## 5. Dependence of model results on cross sections

To test the dependence of the model results on electron impact cross sections which are used as input to the model, we made a test run of the simulation for an incident electron energy of 200 eV. We have run the model by using ionization cross section measurements for the Xe^{+} state by Wetzel *et al.* [33], which are higher than that of Rejoub *et al.* [37] (which we have used in the model) to a maximum of 20%. The *w*-value obtained in this case is 23.1 eV, while ionization efficiency increases from 55% to 57% and excitation efficiency decreases from 34% to 32%. Similarly, when the excitation cross sections for the 1s_{3} state is replaced by National Institute of Fusion Science recommended cross sections [48], which are higher than the cross sections of Peuch & Mizzi [42] (which we have used in the model) by a factor of 2, the change in the *w*-value as well as ionization and excitation efficiencies is less than 1%.

We also tried doubling or halving the major ionization and excitation cross sections to see the impact on model results. The four cases considered are for incident electron energy of 200 eV:

*case 1*: The ionization cross section for Xe^{+}state is doubled.*case 2*: The ionization cross section for Xe^{+}state is halved.*case 3*: The excitation cross section for 1s_{3}is doubled.*case 4*: The excitation cross section for 1s_{3}is halved.

The *w*-values and efficiencies obtained for each case is shown in table 6. Doubling or halving the Xe^{+} cross sections results in a difference of approximately 5% in *w*-value and approximately 3% in ionization efficiency. The variation in excitation efficiency in this case is only approximately 1%. As expected, variation of 1s_{3} cross sections is not having much effect on *w*-value or efficiencies.

We also observed that, when all the ionization cross sections are doubled keeping excitation cross sections unchanged, the *w*-value shows a decrease of 12% (20.7 eV) and ionization efficiency increases to 63%. When all the excitation cross sections are doubled without changing ionization cross sections, the *w*-value increases by 22% (29.2 eV) and ionization efficiency decreases to 45% and excitation efficiency increases to 44%.

## 6. Conclusion

We have developed a Monte Carlo model for degradation of electrons with energy ≤10 keV in neutral xenon gas. Electron impact cross sections for elastic and various inelastic processes were compiled based on the recent experimental and theoretical studies. Analytically fitted forms of these cross sections are used as input data to the model. The NYS calculated using Monte Carlo simulation is analytically represented through equation (4.4), thus generating AYS. A good agreement is observed between NYS and AYS. From these results the mean energy per ion pair and the efficiency of inelastic processes have been calculated. The value of mean energy per ion pair is 22 eV for an incident energy of 10 keV, which is consistent with the values obtained in earlier studies [17,19,46,47]. Secondary electron energy distribution is shown in figure 6. Efficiency calculations showed that the ionization process dominates for incident energies greater than 50 eV and is found to have an efficiency of approximately 65% at 10 keV. Efficiency of excitation is approximately 30% at 10 keV incident energy. Our results are consistent with the previous calculations of Santos *et al.* [20] and Dias *et al.* [19].

Results presented in this paper will be useful to understand electron energy degradation process in xenon. The AYS derived using the Monte Carlo model can be used to calculate steady state electron flux in a medium like planetary atmospheres [49,50], ion thrusters, etc. as well as to calculate excitation rates or emission intensities [51,52]. Efficiencies can be used to calculate volume production rate multiplying by electron production rate and integrating over energy.

## Author' contributions

V.M. developed the model, ran the simulation and drafted the manuscript. A.B. supervised the work, helped in analysing and interpreting the results, and corrected the manuscript.

## Competing interests

We have no competing interests.

## Funding

V.M. was supported by ISRO Research Fellowship during the period of this work.

## Acknowledgements

V.M. gratefully acknowledge ISRO for the research fellowship provided during the period of this work.

- Received October 21, 2015.
- Accepted January 29, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.