Procedure to Solve the One-Dimensional Radiative Transfer Equation (1D-RTE) of an Ionized Medium Using Spatial Discretization and an Iterative Process

Abstract

A new solution procedure for the one-dimensional radiation transfer equation, which describes the radiation propagation in semi-transparent media, is proposed. The resolution proposed, in this work for the first time, is based on spatial discretization and an iterative method. The emitter medium is subdivided into several layers and each layer is considered homogeneous and at the local thermodynamic equilibrium. The emission intensity of each layer is deduced from that of the previous layer. Comparisons are made between our method and other methods described in the literature. This solution is then used to model the line profiles of the Balmer H_α line emitted by a plasma created by the breakdown of a pulsed laser in water.

Graphical abstract

This is a visual representation of the abstract.

Introduction

Different physical phenomena such as radiation scattering, absorption, and emission affect the radiation propagation inside an ionized medium, which is the radiative transfer phenomenon.^1,2 Quite often, studies overlook and/or do not take diffusion effects into account. Inside an inhomogeneous plasma medium, this phenomenon of radiative transfer has a major effect on the intensity, width, and profile of spectral lines. Thus, it is necessary to be in the optically thin conditions at the centers of the spectral lines in particular or else it will be necessary to make appropriate corrections during the exploitation and the use of the quantitative analysis data. Large variations in line width and height caused by self-absorption have been observed by several authors.^3,4

Thus, the radiative transfer equation (RTE) is of great interest in many scientific and engineering disciplines and plays a very important part in radiative transfer analysis in many media, e.g., ionized media and plasmas, liquids and semi-transparent solids, gaseous media, and porous and particulate media. Different forms of RTE exist,^5–7 which are suitable for different and various applications. Thus, we find RTEs that are specific for refractive media for different analyses, e.g., RTE under different coordinate systems; several fundamental methods for the resolution of the RTE, among which one can quote the following numerical resolution methods; the finite volumes method, the finite elements method, the spherical harmonics method, and the discrete ordinates method.

The Balmer H_α line is a specific deep-red spectral line with a wavelength of 656.278 nm. Studying this line is one of the simplest means employed by astronomers to trace the content of ionized hydrogen in gas clouds. Filters selecting this line are often used in the observation of the Sun for protuberances, or in astrophotography to highlight the presence of hydrogen in nebulae. They can be used at the focus or the eyepiece.

In this work, we are interested in the one-dimensional RTE (1D-RTE). After the description of our method of resolution, we will apply it to the Balmer H_α spectral line resulting from emission by a plasma generated in water by a focused laser. A description of the experimental setup and different methods of deducing the experimental parameters will also be presented.

One-Dimensional Radiative Transfer Equation (1D-RTE)

During the emission of an ionized medium or a plasma, the quantity that can be measured, almost directly, is the intensity $I(\lambda ,x)$ of the radiation emitted by the medium at a given position, x. By definition, this intensity stands for the power emitted per unit of solid angle, per interval of angular frequency, and per unit area. Essentially, it depends on the propagation direction and not only on the angular frequency. Therefore, it is obligatory to consider the spatial gradient of the intensity through line of sight, to calculate the emitted electromagnetic radiation. The difference between the amount of radiation emitted and the amount of radiation absorbed gives the intensity gradient and this is expressed under local thermodynamic equilibrium (LTE) conditions by the following relationship ⁸ :

$${\displaystyle \frac{\mathrm{d}I(\lambda ,x)}{\mathrm{d}x}={\epsilon }_{\lambda }(x)-K_{\lambda }(x)\times I(\lambda ,x)}$$

(1)

where ${\epsilon }_{\lambda }$ is the emission coefficient, $K_{\lambda }$ is the absorption coefficient, and $\lambda$ is the wavelength. This equation of 1D-RTE in its original form is formulated as a particle density equation^9,10 and gives the variation of the radiation intensity in space, and considers each frequency, the emission, and absorption in any position in the plasma medium. This equation presents real difficulties and complexities of resolution and requires an understanding of the nature of the plasma and also knowledge of its geometry in certain cases. This difficulty is because the radiation intensity is a function of position, frequency, viewing direction, and time. Thus, the resolution of this equation is not feasible in the general case. That which requires one to be interested in the particularities of each plasma. Thus, in many studied plasmas, the time dependence of this equation is insignificant and for isotropic plasmas, the intensity is the same in all directions. These simplifications thus make it possible to find approximate solutions to this equation.

Resolving the Procedure for the 1D-RTE

The plasma studied is considered to have cylindrical symmetry. We subdivide our plasma medium into N layers. Each layer n (for $1\le n\le N$ ) has a thickness, $X_{\mathrm{e}}$ , and it is considered homogeneous. This type of modeling has been described and used in other works.^6,7

By spatial discretization of Eq. 1, the 1D-RTE becomes as follows:

$${\displaystyle \frac{I(n,\lambda )-I(n-1,\lambda )}{X_{\mathrm{e}}}={\epsilon }_{\lambda }(n)-K_{\lambda }(n)I(n,\lambda )}$$

(2)

where $X_{\mathrm{e}}$ is the spatial discretization step that is the length of each layer, n the index of layer number n = 1 for the first layer, n = 2 for layer number 2, etc. Thus,

$$\mathrm{I}(n,\lambda )={\displaystyle \frac{X_{\mathrm{e}}{\epsilon }_{\lambda }(n)+\mathrm{I}(n-1,\lambda )}{1+X_{\mathrm{e}}{K}_{\lambda }(n)}}$$

(3)

The source function $S_{\lambda }(n)$ is given by:

$$S_{\lambda }(n)={\displaystyle \frac{{\epsilon }_{\lambda }(n)}{K_{\lambda }(n)}}$$

(4)

Thus, Eq. 3 becomes:

$$I(n,\lambda )={\displaystyle \frac{X_{\mathrm{e}}S(n,\lambda ){\epsilon }_{\lambda }(n)+S(n,\lambda )I(n-1,\lambda )}{S(n,\lambda )+X_{\mathrm{e}}{\epsilon }_{\lambda }(n)}}$$

(5)

In the case of an LTE plasma, the source function is identical to the Planck function $B_{\lambda }(T(n))$ for the radiation from a black body.

$$S_{\lambda }(n)=B_{\lambda }(T(n))={\displaystyle \frac{2h{c}^2}{{\lambda }^5}{\displaystyle \frac{1}{{\mathrm{e}}^{hc/\lambda KT(n)}-1}}}$$

(6)

Then, Eq. 5 becomes:

$$I(n,\lambda )={\displaystyle \frac{X_{\mathrm{e}}{B}_{\lambda }(T(n)){\epsilon }_{\lambda }(n)+B_{\lambda }(T(n))I(n-1,\lambda )}{B_{\lambda }(T(n))+X_{\mathrm{e}}{\epsilon }_{\lambda }(n)}}$$

(7)

Next, ${\epsilon }_{\lambda }(n)$ is given in several articles^1,2 and it has the following expression:

$${\epsilon }_{\lambda }(n)={\displaystyle \frac{h\times c}{4\times \pi \times \lambda }\times A\times N_{\sup }(n)\times P_{\lambda }(n)}$$

(8)

where ℎ is Planck's constant, A is Einstein's coefficient, N_sup is the density of the upper level,

$P_{\lambda }(n)$ is the normalized line shape, ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{P}_{\lambda }(n)\times \mathrm{d}\lambda =1$ .

In the case of an inhomogeneous medium, the line profile $P_{\lambda }(n)$ and the emission coefficient ${\epsilon }_{\lambda }(n)$ depend on the position. The conditions of temperature and density of electrons of our plasma allow us to consider that the predominant enlargement mechanism is that of the Stark effect. Therefore, the line profile can be considered Lorentzian. This is all truer as the contribution of ions to this Stark effect is weaker. The profile of the spectral lines that we have adopted thus has the following form:

$$P_{\lambda }(n)={\displaystyle \frac{1}{\pi }\times {\displaystyle \frac{\mathrm{\Delta }{\lambda }_{1/2}^L(n)/2}{{(\mathrm{\Delta }{\lambda }_{1/2}^L(n)/2)}^2+{(\lambda -{\lambda }_0)}^2}}}$$

(9)

$\mathrm{\Delta }{\lambda }_{1/2}^L(n)$ : nth layer contribution to the full width half-maximum (FWHM) of the Lorentz line.

The frequency line shape and the wavelength line shape are related by:

$$P_{\lambda }(n)=\left|{\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}\lambda }}\right|\times P_{\omega }(n)$$

(10)

The procedure to calculate the spectrum intensity of emission of the plasma thus constituted by these N layers is based on Eq. 7 by using the following iterative method steps:

• Calculation of radiation intensity of layer 1 (layer number n = 1):

  $I(1,\lambda )={\epsilon }_{\lambda }(1)X_{\mathrm{e}}-{\epsilon }_{\lambda }(1)X_{\mathrm{ee}}^{-X_{\mathrm{e}}({\epsilon }_{\lambda }(1)/B_{\lambda }(T(1)))}$ , which consists of the intensity due to the emission of the first layer from which the part absorbed by this same layer is removed.

• At the exit of the section made up of the N layers, the intensity emitted by the medium capable of being detected is calculated iteratively by Eq. 7: $I(n,\lambda )=(X_{\mathrm{e}}{B}_{\lambda }(T(n)){\epsilon }_{\lambda }(n)+B_{\lambda }(T(n))I(n-1,\lambda ))/(B_{\lambda }(T(n))+X_{\mathrm{e}}{\epsilon }_{\lambda }(n))$ for n = 2 until n = N. In our case, this step is performed using an iterative program written in C / C++. $N_{\sup }(n)$ and $\mathrm{\Delta }{\lambda }_{1/2}^L(n)$ are calculated for each layer n by using the experiment values of the density $N_{\mathrm{e}}(n)$ and the temperature $T(n)$ of electrons. The experimental device and the details of measures are described in the following section.

One of the advantages of the present method compared to the methods of Amamou et al. ⁶ and Rezaei et al. ⁷ is that its mathematical formalism is very easy, and its application does not require heavy numerical processing and/or long calculation times.

Materials and Methods

Application to Experimental Data of the Balmer H_α Emission Line Emitted by the Interaction of a Pulsed Laser With Water

The components of the experimental device (Figure 1) include a pulsed laser (L), focusing lens (F), cell of interaction (C), target where plasma is created (T), optical coupler (IL), optical fiber (OF), spectrometer (S), detector (D), multichannel analyzer (OMA2000), and personal computer.

Figure 1.

Experimental device.

At the center of interaction cell C, where the sample is placed to avoid any external pollution, is focused the neodymium-doped yttrium aluminum garnet laser pulse 1064 nm with a duration of 10 ns and energy between 10 and 115 mJ by a thin lens F of 50 mm focal length. The spot wide is about 62 μm and the plasma emission is sent to a spectrometer using an achromatic doublet IL focused on the input of an optical fiber OF by observation at by observation perpendicularly to the laser trajectory. The spectrometer disperses this light emission that is then intensified using a charge-coupled device, and then analyzed and recorded by an OMA2000 multichannel analyzer.

For our current study, the sample analyzed is water containing impurity traces of Ca and K. This allows to be determined the temperature of the plasma and the electron density by the Stark widths of the line, Ca(I), 422.67 nm, and of the doublet, Ca(II) 393.37 nm and 396.85 nm.

In this work and for more practical reasons, a Jobin-Yvon spectrometer with a dispersion of 240 Å mm⁻¹ is used for the analysis of the H_α line, which is wider, but a high dispersion Chromex 500-IS spectrometer (16 Å mm⁻¹) is used for the analysis of Ca lines which are narrower.

For a homogeneous and stationary hydrogen plasma, the criterion of LTE is $N_{\mathrm{e}}\ge 6\times {10}^{17}\mathrm{c}{\mathrm{m}}^{-3}$ ¹¹ at electron temperature T ≈ 10 000 K.

This criterion is always verified in our observation conditions of the studied spectral lines. But, given that the laser plasmas are time-dependent then equilibration time T_e, checking is also necessary to reach a quasi-steady state, and close to LTE of plasma is sufficiently smaller compared to the observation time of the H_α line. T_e represents the time required for the energy distribution of the excited states of the atom to be a Boltzmann distribution. T_e < 0.2 ns in experimental conditions. ¹¹ N_e ≥ 6 × 10¹⁷ cm⁻³ and T ≤ 10 000 K for the H_α observation times, thus the plasma is quasi-stationary, and it is at LTE.

The profile of Lorentz used for fitting the hydrogen H_α line is very adequate. ¹² The full width at half area (FWHA) is equal to FWHM in the case of a Lorentzian profile. However, results are more accurate for electron density determination using FWHA because the dynamic effect of the ions influences the maximum intensity. But other authors ¹³ found that the usage of FWHM is preferred to FWHA for N_e density measurements ≥1010¹⁷ cm⁻³.

The employment of the Abel inversion allows us to determine the local values of electron density. An x,y,z micro-positioning system allows high spatial resolutions, i.e., vertical (y-direction perpendicular to the laser beam) and longitudinal (z-direction along the laser beam). Considering a cylindrical symmetry, the observed spectra as a function of y are used to numerically solve the Abel inversion. ⁸ This makes it possible to obtain the local emission coefficients S_λ as a function of the distance r from the plasma center.

The variations of the intensities of the Ca and K lines as a function of the position on the vertical axis (by adding Ca and K traces to pure water) are used for the measurement of the diameter of the plasma. The profiles of temperature and electron density (Figure 2) obtained by Escarguel et al. ¹⁴ are used in this work. The plasma radius (plasma dimension) created and observed is L = 0.4 mm (Figure 3) and considered as a cylindrical symmetry and subdivided into 16 layers of thickness of X_e = 0.025 mm for each layer (Figure 3).

Figure 2.

(o) Electron temperature experimental values versus position. (*) Electron density experimental values versus position. The duration of 500 ns is the time of integration and the duration of 200 ns is the delay from the beginning of plasma formation.

Figure 3.

The 16 layers of plasma section and the created and observed plasma.

Because the center is hotter than the periphery, the electron temperature and the electron density, decrease from the plasma center to its periphery.

Study of the H_α Line of the Balmer Series H_α Emitted by the Interaction of a Pulsed Laser With Water

Among the spectral lines of hydrogen located in the visible part of the electromagnetic spectrum, we find the H_α. It is a specific dark red spectral line of the Balmer series, which is at a wavelength of 656.278 nm and is the first line of this series that corresponds to the transition from energy level with principal quantum number n = 3 to energy level with principal quantum number n = 2. In the literature, there are detailed studies^15,16 of the H_α line broadening.

For the spectral lines of the hydrogenoid particles, the Stark broadening profile can be assumed to be Lorentzian. For each layer n the FWHM is ¹⁷

$$\mathrm{\Delta }{\lambda }_{1/2}^{\mathrm{Stark}}(n)={10}^{-9}\times 0.549\times {\left({\displaystyle \frac{N_{\mathrm{e}}(n)}{{10}^{23}}}\right)}^{0.67965}$$

(11)

where N_e is the electron density (m⁻³), the water density is 1000 kg m⁻³ and the water molecule mass are 3 × 10⁻²⁶ kg. Thus, the neutral hydrogen density is N_z + N_z+1 = N_H = 2/3 × 10²⁹ m⁻³; N₂ is the neutral hydrogen density (m⁻³) and N_z+1 is the density of the first hydrogen ionized state (m⁻³).

The Saha equation gives the singly ionized hydrogen density:

$$N_{z+1}=N_z\times {\displaystyle \frac{1}{N_{\mathrm{e}}}{\displaystyle \frac{Q_{z+1}\times Q_{\mathrm{e}}}{Q_z}\times {\left({\displaystyle \frac{2\times \pi \times {\mathrm{m}}_{\mathrm{e}}\times \mathrm{K}\times \mathrm{T}}{h^2}}\right)}^{3/2}\times \exp \left(-{\displaystyle \frac{E_z-\mathrm{\Delta }{E}_z}{K\times T}}\right)}}$$

(12)

where Q_e = 2 is the electron partition function, Q_z is the neutral hydrogen partition function, Q_z+1 is the singly ionized hydrogen partition function, m_e is the electron mass in kg, E_z is the hydrogen ionization energy in eV, and ΔE_z is the hydrogen ionization energy reduction in eV.

The measurement uncertainties obtained by the analysis of several spectra recorded under the same experimental conditions are 15% for the density of electrons and for the electron temperature at a value of 10% for FWHM.

Considering the fact that ${\sum }_n{N}_n=N_z$ and according to Boltzmann law (for the densities of the energy levels), the normalization factor $Q_z$ is given by the formula^2,8

$$Q_z=\sum _n{g}_n\exp \left(-{\displaystyle \frac{E_n}{kT}}\right)$$

(13)

This normalization factor is called the partition function.

The terms of the sum in Eq. 13 are the energies of all levels of this ionization state z. But the values of some energy levels are unknown, and this sum is often divergent. Thus, the disturbance is considered to only reach energy levels close to the ionization level, which means that only k terms are considered in this sum so that $E_k\le E_z-\mathrm{\Delta }{E}_z$ . The main cause of this decrease in the energy of ionization is the strong disturbance of the energy levels close to the state of ionization. The experimental determination of this reduction is difficult because of its low value and because its value is poorly predicted by theory. However, there are some theoretical formulas for calculating this reduction in the ionization energy. Two methods exist; one is described by Unsöld ¹⁸ and the other is given by Griem ⁸ and Ecker and Kroll. ¹⁹ Two different approaches for its calculation are used by these two methods.

Unsöld’s method is based on the fact that the emitter particle is disturbed by the ion's nearest neighbor. His reasoning gives the following relation:

$$\mathrm{\Delta }{E}_z=3\times z^{2/3}\times {\displaystyle \frac{e^2}{r_{\mathrm{e}}}}$$

(14)

In this formula, the average interelectronic distance $r_{\mathrm{e}}$ is given by:

$$r_{\mathrm{e}}={\left({\displaystyle \frac{4}{3}\times \pi \times N_{\mathrm{e}}}\right)}^{-(1/3)}$$

(15)

Griem's reasoning uses thermodynamic arguments and arrives at the relation:

$$\mathrm{\Delta }{E}_z=(z+1)\times {\displaystyle \frac{e^2}{{\rho }_{\mathrm{D}}}}$$

(16)

In this formula, the Debye length ${\rho }_{\mathrm{D}}$ is given by:

$${\rho }_{\mathrm{D}}={\displaystyle \frac{1}{e}\times \sqrt{{\displaystyle \frac{K\times T}{4\times \pi \times \left[N_{\mathrm{e}}+{\sum }_z{\sum }_i{(z+1)}^2\times N_{z+1,i}\right]}}}}$$

(17)

where e is the elementary charge of the electron, z is the state of ionization (neutral state: z = 1, first ionization state: z = 2,…), and N_z,i is the density of the particle i in the ionization state z.

By comparison, the values obtained using Griem's formula are slightly lower than those obtained using Unsöld’s formula.

For the hydrogen lines, the parameters used are ²⁰ $Q_z=2$ , $Q_{z+1}=1$ , $E_z=13.5984\mathrm{eV}$ , and $\mathrm{\Delta }{E}_z=7.7464\times {10}^{-13}\times$ ${10}^{-13}\times (N_{\mathrm{e}}{)}^{1/3}(\mathrm{eV})$ .

In this work, we considered the temperature and the density of electrons for a time of integration of 500 ns and a delay of 200 ns from the beginning of plasma. We also considered the contribution of the continuous radiation, as about half the intensity of the H_α line, which agrees with the results of Parigger and Oks. ²¹

Results and Discussion

Figure 4 shows the normalized intensities evolution of the hydrogen H_α line emitted by plasma from the first layer to the last layer. Only the emission of the last layer can be detected. We notice that the width of the line increases as we move away from the center of the plasma. This is because the plasma is less dense and colder as one moves away from its center and therefore the atoms absorb more. ¹⁴

Figure 4.

Evolution of normalized intensity of the hydrogen H_α line emitted by plasma from the first layer to the last layer.

Figure 5 illustrates the normalized intensity of the hydrogen H_α line emitted by plasma after the first layer. Figure 6 illustrates the normalized intensity of the hydrogen H_α line emitted by plasma after the last layer. The comparison of the two figures clearly shows that the width of the line has considerably increased after crossing the outer layers of the plasma. The physical phenomenon which is responsible for this enlargement is that of self-absorption.

Figure 5.

Normalized intensity of the hydrogen H_α line emitted by the first layer of plasma.

Figure 6.

Normalized intensity of the hydrogen H_α line emitted by the last layer of plasma.

Figure 7 shows the evolution of the normalized intensity maximum of the hydrogen H_α line emitted by plasma from the first layer to the last layer. This intensity increases maximum as we move away from the center, but we notice that when we reach the eighth layer (or 0.2 mm) it begins to decrease. The increase in intensity from the center to the periphery is normal because we add the emissions from the layers of the first layer to the layer in question (obviously considering their absorptions). We will give later explanations for the reasons for this decrease.

Figure 7.

Evolution of normalized intensity maximum of the hydrogen H_α line emitted by plasma from the first layer to the last layer.

Figure 8 illustrates the evolution of the FWHM of the hydrogen H_α line emitted by plasma from the first layer to the last layer. FWHM increases linearly as we move away from the center up to layer number 8. But this linear growth begins to flatten out thereafter. We will give later explanations for the reasons for this flattens.

Figure 8.

Evolution of the FWHM of the hydrogen H_α line emitted by plasma from the first layer to the last layer.

The FWHM values of the hydrogen H_α line are in good agreement with those found by the authors.^6,14,15,22

To understand this change in the variation of intensity maximum and of line width from the eighth layer (0.2 nm), we have also presented, the spectral line intensities at the exit of each of the 16 layers of the plasma. From these presentations, we have constated that from layer number eight, the spectral line begins to flatten, and from the 12th layer it begins to self-reverse. The inversion of the spectral lines has as one of its causes the inhomogeneity of the plasma. This inversion of the spectral lines is manifested by a hollow observed in the center of these lines, as has already been proven in other works.^6,23

Our model is a very interesting model for many plasma types if there is no technical limitation for the measurements of the density of electrons and the temperature for layers with a very small thickness. But it is quite sufficient for plasmas which are not very inhomogeneous; that is the case of thermal plasmas.

Figure S1a (Supplemental Material) shows the theoretical H_α line, ⁶ Figure S1b (Supplemental Material) shows the theoretical H_α line (present work), and Figure S1c (Supplemental Material) shows the experimental line. ²¹

We found that the results obtained by these two works which use different methods are very similar or even identical.

The results from which this experimental line is derived are shown in Table I.

Table I.

Experimental values of FWHM and N_e obtained by Parigger and Oks. ²¹

Hα width (Å)	Ne (1017 cm−3)
254 ± 35	70–100
54.5 ± 5	7–12
14.9 ± 1.5	1.3
8.67 ± 0.8	0.56

The conditions of these works are similar. By comparison of these lines, we can see that are very similar.

The H_α line that we obtain in our work is for a density of electrons of 18.4 × 10¹⁷ cm⁻³ (as can be deduced from Figure 2) and the associated FWHM value is 145 Å (as can be deduced from Figure 8). Table I presents the FWHM of the H_α line measured for different electron densities by Parigger and Oks ²⁰ (for different experimental conditions of integration time and the delay from the beginning of plasma formation). From this table, we can deduce for a density of 18.4 × 10¹⁷ cm⁻³ the FWHM is between 54.4 and 254 Å, which allows us to affirm that our value is coherent. By comparing these results, we find that there is a very obvious consistency between the two results. Thus, we can say that our method leads to similar results to the experimental and theoretical results of other authors.

It can also be deduced that the inhomogeneities of the ionic field and the static quadratic Stark effect (to a lesser extent) are the main causes of the asymmetries of the H-line profiles observed in certain experimental spectra.

Conclusion

For the diagnostics of physical plasmas, self-absorption phenomenon and homogeneity effects or/and their importance are largely neglected. These two effects can be overcome by solving the equation of radiative transfer. However, the analytical resolution of such an equation in the general case remains very thorny. In this present work, we have proposed a new solving procedure for the one-dimensional radiation transfer equation. The resolution proposed, in this work for the first time, is based on spatial discretization and an iterative method. The emitter medium is subdivided into several layers and each layer is considered homogeneous. The emission intensity of each layer is deduced from the previous layer. This method does not require heavy digital processing compared to other resolution methods. This solution is then used to model the Balmer H_α spectral line resulting from emission by a plasma generated in water. The line profiles obtained are in very good agreement with those of other authors. The FWHM of the Balmer H_α spectral line is also in very good agreement with that observed experimentally. From the results presented in this work, we can affirm that our procedure resolution of the equation of radiative transfer is a very good method that gives results very much in adequacy with other theories and with the experimental results.

Supplemental Material

sj-docx-1-app-10.1177_27551857241228616 - Supplemental material for Procedure to Solve the One-Dimensional Radiative Transfer Equation (1D-RTE) of an Ionized Medium Using Spatial Discretization and an Iterative Process

Supplemental material, sj-docx-1-app-10.1177_27551857241228616 for Procedure to Solve the One-Dimensional Radiative Transfer Equation (1D-RTE) of an Ionized Medium Using Spatial Discretization and an Iterative Process by Hssaïne Amamou, André Bois and Belkacem Ferhat in Applied Spectroscopy Practica