Double Differential Cross-Section for the Ionization of Hydrogenic 2S Metastable State ()

Md. Thowhidul Hoque Chowdhury^{}, Sunil Dhar^{}

Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong, Bangladesh.

**DOI: **10.4236/ojm.2023.131001
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Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong, Bangladesh.

Double differential cross-sections of first Born estimation for ionization of hydrogenic 2S state by electrons are assessed for various kinematics situations in the asymmetric coplanar geometry. A final state wave function of multiple scattering theory is followed in this study. The present outcomes are compared with those of hydrogenic ground state, 2P state and ground state experimental results. Obtained findings show a good qualitative agreement with existing results.

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Chowdhury, M. and Dhar, S. (2023) Double Differential Cross-Section for the Ionization of Hydrogenic 2S Metastable State. *Open Journal of Microphysics*, **13**, 1-13. doi: 10.4236/ojm.2023.131001.

1. Introduction

Ionization process is one of the most significant reactions in atomic collisions. Much information on ionization dynamics has been obtained by assessing the double differential cross sections (DDCS) for emitted electron energy and emitted angles. Ionization by fast particles was first planned quantum mechanically by Bethe [1] . The application of the multi-parameter detection technique, together with the advancement in computational procedures, has made it likely to implement a whole experimentation in which kinematical parameters (like momentum and energies) of all performing particles are determined. This kind of experiment has been successfully used to investigate the fine details of the ionization process both in the ground state [2] - [11] and metastable state [12] - [21] . In atomic ionization collisions, the DDCS covers facts about the angular and energy distribution of secondary electrons [11] . Here, atomic hydrogen is used as a target in order to perceive the ionization mechanism of atomic system by electron impact energy. The emitted electron is detected in coincidence with the scattered electrons and it is a well-known experiment [22] called (e, 2e) experiments. The ionization of atomic hydrogen by electron influence is of major significance and of rare gas atoms, particularly, the cross-sections acquired with ground state ionization, is considered as benchmark facts and before [23] there is a few data existed for atomic hydrogen, though it is the simplest and the most convenient system for theoretical analysis. DDCS of ionization comprises valued info about both the collision dynamics and the internal structure of atomic or molecular systems. Experimental evaluation angle and energy have been obtained [24] [25] [26] [27] [28] and other groups [29] - [39] for DDCS.

In this work, the DDCS for ionization of hydrogenic metastable 2S state by electron impact at 150 eV and 250 eV energies has been calculated. We use a wave function [10] [11] [12] to calculate the DDCS integrated over the scattering angle. It will be interesting here to use the wave function of metastable 2S state hydrogen atoms by electrons. To the best of our knowledge, the DDCS for the ionization of metastable 2S-state hydrogen atoms by electrons at intermediate and high energies were never studied before experimentally. No theoretical calculations of the DDCS of metastable 2S state hydrogen atoms were observed. Therefore, hydrogenic ground state experimental results for ionization of metastable 2S state hydrogen atoms by electrons will be valuable and our outcomes will add a new dimension to the significant study of this field of research.

2. Theory

The direct Transition matrix element for ionization of hydrogen atoms by electron [27] , may be written as,

${T}_{FI}=\langle {\psi}_{F}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)|{V}_{I}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)|{\Phi}_{i}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)\rangle $ (1)

where the perturbation potential ${V}_{I}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)$ is given by

${V}_{I}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)=\frac{1}{{\gamma}_{12}}-\frac{1}{{\gamma}_{2}}$ (2)

For hydrogen atoms nuclear charge is Ze = 1, ${\gamma}_{1}$ is the distance of the atomic electron and ${\gamma}_{2}$ is the distance of projectile electron from the nucleus and ${\gamma}_{12}$ is the distance between the two electrons (Figure 1).

The initial channel unperturbed wave function is given by

${\Phi}_{i}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)=\frac{{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}}{{\left(2\pi \right)}^{\frac{3}{2}}}{\varphi}_{2S}\left({\stackrel{\xaf}{\gamma}}_{1}\right)$ (3)

where ${\varphi}_{2S}\left({\stackrel{\xaf}{\gamma}}_{1}\right)=\frac{1}{4\sqrt{2\pi}}\left(2-{\gamma}_{1}\right){\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}$ .

Here ${\lambda}_{1}=\frac{1}{2}$ and ${\varphi}_{2S}\left({\stackrel{\xaf}{\gamma}}_{1}\right)$ is the hydrogen 2S state wave function and ${\psi}_{F}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)$ is the final three particle scattering state wave function [7] and co-ordinate of the two electrons are ${\stackrel{\xaf}{\gamma}}_{1}$ and ${\stackrel{\xaf}{\gamma}}_{2}$ respectively.

Figure 1. Collision effect amongst two electrons and the nucleus.

Here the approximate wave function ${\psi}_{F}^{(-)}$ the is given by

$\begin{array}{c}{\psi}_{F}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)=N\left({\stackrel{\xaf}{p}}_{1},{\stackrel{\xaf}{p}}_{2}\right)[{\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{i{\stackrel{\xaf}{p}}_{2}\cdot {\stackrel{\xaf}{\gamma}}_{2}}+{\varphi}_{{\stackrel{\xaf}{p}}_{2}}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{2}\right){\text{e}}^{i{\stackrel{\xaf}{p}}_{1}\cdot {\stackrel{\xaf}{\gamma}}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}+{\varphi}_{\stackrel{\xaf}{p}}^{(-)}\left(\stackrel{\xaf}{\gamma}\right){\text{e}}^{i\stackrel{\xaf}{P}\cdot \stackrel{\xaf}{R}}-2{\text{e}}^{i{\stackrel{\xaf}{p}}_{1}\cdot {\stackrel{\xaf}{\gamma}}_{1}+i{\stackrel{\xaf}{p}}_{2}\cdot {\stackrel{\xaf}{\gamma}}_{2}}]/{\left(2\pi \right)}^{3}\end{array}$ (4)

Here $N\left({\stackrel{\xaf}{p}}_{1},{\stackrel{\xaf}{p}}_{2}\right)$ is normalization constant,

$\stackrel{\xaf}{\gamma}=\frac{{\stackrel{\xaf}{\gamma}}_{1}-{\stackrel{\xaf}{\gamma}}_{2}}{2},\text{\hspace{0.17em}}\stackrel{\xaf}{R}=\frac{{\stackrel{\xaf}{\gamma}}_{1}+{\stackrel{\xaf}{\gamma}}_{2}}{2},\text{\hspace{0.17em}}\stackrel{\xaf}{p}={\stackrel{\xaf}{p}}_{2}-{\stackrel{\xaf}{p}}_{1},\text{\hspace{0.17em}}\stackrel{\xaf}{P}={\stackrel{\xaf}{p}}_{2}+{\stackrel{\xaf}{p}}_{1}$

and ${\varphi}_{q}^{(-)}\left(\stackrel{\xaf}{\gamma}\right)$ is Coulomb wave function.

Applying Equations (3) and (4) in Equation (2), we get

${T}_{FI}={T}_{b}+{{T}^{\prime}}_{b}+{T}_{I}-2{T}_{PB}$ (5)

For first Born estimation equation may be written as

$\begin{array}{c}{T}_{b}=\frac{1}{16{\pi}^{2}}{\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\left|\frac{1}{{\gamma}_{12}}-\frac{1}{{\gamma}_{2}}\right|{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\left(2-{\gamma}_{1}\right){\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}\\ =\frac{1}{16{\pi}^{2}}{\displaystyle \int \left[{\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\left(\frac{1}{{\gamma}_{12}}-\frac{1}{{\gamma}_{2}}\right){\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\left(2-{\gamma}_{1}\right){\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}\right]{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}}\\ =\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{12}}2{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}-\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{2}}2{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}\end{array}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.05em}}-\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{12}}{\gamma}_{1}{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.05em}}+\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{2}}{\gamma}_{1}{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}\\ =T{b}_{1}+T{b}_{2}+T{b}_{3}+T{b}_{4}\end{array}$ (6)

Here

$T{b}_{1}=\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{12}}2{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}$

$T{b}_{2}=-\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{2}}2{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}$

$T{b}_{3}=-\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{12}}{\gamma}_{1}{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}$

$T{b}_{4}=\frac{1}{16{\pi}^{2}}{\displaystyle \int {\varphi}_{{\stackrel{\xaf}{p}}_{1}}^{(-)*}}\left({\stackrel{\xaf}{\gamma}}_{1}\right){\text{e}}^{-i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}\frac{1}{{\gamma}_{2}}{\gamma}_{1}{\text{e}}^{i{\stackrel{\xaf}{p}}_{i}\cdot {\stackrel{\xaf}{\gamma}}_{2}}{\text{e}}^{-{\lambda}_{1}{\gamma}_{1}}{\text{d}}^{3}{\gamma}_{1}{\text{d}}^{3}{\gamma}_{2}=\frac{1}{2}\frac{\partial}{\partial {\lambda}_{1}}\left(t{b}_{2}\right)$

${T}_{b}$,is the first Born term for TDCS and other terms ${{T}^{\prime}}_{b},{T}_{I},{T}_{PB}$ are calculated in our work of [12] . Afterward, investigative calculation by using the Lewis integral [40] , the above terminologies of Equation (6) have been calculated mathematically and the triple differential cross-sections for T-Matrix element is specified by

$\frac{{\text{d}}^{3}\sigma}{\text{d}{\mu}_{1}\text{d}{\mu}_{2}\text{d}{E}_{1}}=\frac{{p}_{1}{p}_{2}}{{p}_{i}}{\left|{T}_{FI}\right|}^{2}$ . (7)

Later integration of TDCS result [12] of Equation (7), we can acquire the DDCS result using following equation:

$\frac{{\text{d}}^{2}\sigma}{\text{d}{E}_{1}\text{d}{\mu}_{1}}={{\displaystyle \int}}^{\text{}}\frac{{\text{d}}^{3}\sigma}{\text{d}{E}_{1}\text{d}{\mu}_{1}\text{d}{\mu}_{2}}\text{d}{\mu}_{2}$ (8)

Therefore, double differential cross sections (DDCS) have been determined using computer programming language MATLAB, given by Equation (8).

3. Results and Discussion

DDCS are determined here for the ionization of the metastable 2S state hydrogen atoms by electrons at high incident energy *E _{i}* = 250 eV (Figure 2) for emitted electron energies

Again, at intermediate incident energy *E _{i}* = 150 eV (Figure 3), for emitted electron energies

Ionization of hydrogen atoms by electrons from the ground state experimental results [24] and computational results [11] [19] are presented here for assessment. The final state scattering wave function ${\psi}_{F}^{(-)}\left({\stackrel{\xaf}{\gamma}}_{1},{\stackrel{\xaf}{\gamma}}_{2}\right)$ is the continuum state of the atomic hydrogen.

For incident energy *E _{i}* = 250 eV in Figure 2(a), ejection energy

In Figure 2(b), *E*_{1} = 10 eV our result concurs several times at nearly *θ*_{1} = 5˚ , *θ*_{1} = 45˚ , *θ*_{1} = 138˚ and *θ*_{1} = 155˚ with those of [19] , meets almost *θ*_{1} = 20˚ , *θ*_{1} = 60˚ , *θ*_{1} = 100˚ and *θ*_{1} = 177˚ with those of ground state result [11] . Moreover the curve intersects several times with those of [24] and peak flattens as the energy increases and ultimately disappears which indicates good assessment.

(a)(b)(c)(d)(e)

Figure 2. DDCS for the Ionization of atomic hydrogen by 250 eV electron effect as a function of the emitted electron angle *θ*_{1} comparative to the incident electron path. The emitted electron energies are 4 eV, 10 eV, 20 eV, 50 eV, 80 eV respectively. Concept: ( ) curves denote hydrogenic ground-state experimental result [24] , ( ) denote hydrogenic ground-state results [11] , ( ) curves illustrate the results of metastable 2P state [19] and ( ) denote the present outcomes of metastable 2S state.

In Figure 2(c), *E*_{1} = 20 eV the present outcome displays similar shapes with 2P state [19] as well as at *θ*_{1} = 65˚ and *θ*_{1} = 175˚ like hydrogenic ground state experimental results [24] which shows good qualitative assessment.

In Figure 2(d), *E*_{1} = 50 eV the current outcome concurs several times with those of ground state result [11] at *θ*_{1} = 18˚ , *θ*_{1} = 35˚ , *θ*_{1} = 55˚ , *θ*_{1} = 80˚ and *θ*_{1} = 176˚ , crosses four times with the ground state experiment value [24] . Also, it makes a big peak at higher angle which indicates good assessment.

(a)(b)(c)(d)(e)

Figure 3. DDCS for the ionization of atomic hydrogen by 150 eV electron effect as a function of the emitted electron angle *θ*_{1} relative to the incident electron direction. The emitted electron energies are 4 eV, 10 eV, 20 eV, 30 eV, 50 eV. Concept: ( ) curves denote hydrogenic ground-state experimental result [24] , ( ) denote hydrogenic ground-state results [11] , ( ) denote the results of metastable 2P state [19] and ( ) denote the current results of metastable 2S state.

Lastly, in Figure 2(e), *E*_{1} = 80 eV it is seen that the experimental result, the theoretical result, 2P state result and the current result show similar nature in shape at ejection angles up to *θ*_{1} = 80˚ and at higher angle, the present outcome and 2P state results [19] displays a peak with big magnitude which indicate the good assessment. It displays a good assessment with the theoretical data as well as hydrogenic ground state experimental results.

For incident energy *E _{i}*

In Figure 3(b), *E*_{1} = 10 eV, our curve intersects several times with those of 2P state results [19] , relatively closer to the ground state results [11] at lesser angles, lies above to the experimental values. Approximately, *θ*_{1} = 10˚ meets with the ground state results [11] , it creates maximum at *θ*_{1} = 150˚ and at higher ejection angle, *θ*_{1} the curve meets with those of ground state results [11] .

In Figure 3(c), *E*_{1} = 20 eV our present outcome coincides various times at *θ*_{1} = 32˚ , *θ*_{1} = 55˚ , *θ*_{1} = 100˚ and *θ*_{1} = 175˚ with those ground state results [11] , accordingly passes at *θ*_{1} = 62˚ , *θ*_{1} = 100˚ and *θ*_{1} = 165˚ with ground state experiment results [24] . Our result creates a peak at *θ*_{1} = 140˚ and passing nearer to both the experimental and the theoretical results. It displays a good qualitative judgement.

Again considering *E*_{1} = 30 eV in Figure 3(d), the theoretical result forms two peak nearby *θ*_{1} = 40˚ , *θ*_{1} = 145˚ and lower deep at about *θ*_{1} = 75˚ . Comparison of these present outcomes with ground state results [11] , 2P state results [19] , and ground state experiment results [24] show a suitable qualitative exhibition.

At last, we consider ejection energy as *E*_{1} = 50 eV, in Figure 3(e), our curve overlay four times at *θ*_{1} = 14˚ , *θ*_{1} = 55˚ , *θ*_{1} = 97˚ and *θ*_{1} = 177˚ those of [19] and at higher angles between 70˚ to 110˚ a lower deep formed where the experimental curve-runs relatively closer to the current outcome which shows a suitable judgement.

Table 1. DDCS results for emitted angles *θ*_{1} corresponding to various scattering angles *θ*_{2} for four different values of emitted electron energies are *E*_{1} = 4 eV, *E*_{1} = 20 eV, *E*_{1} = 50 eV and *E*_{1} = 80 eV in ionization of hydrogen atoms for 250 eV electron.

It is observed that the present first Born result coincides at *θ*_{1} = 60˚ with result once only and similarly concurs two time each at lower and higher ejection angle *θ*_{1} with experimental result of Shyn as well as those of the Roy, Mandal and Sil. We also see that results one may look carefully to the Table 1 where values of the different ejection angles *θ*_{1} are presented for different values of the scattering angles *θ*_{2} for four values of emitted electron energy *E*_{1}, in the case *E _{i}* = 250 eV.

4. Conclusions

In this study, DDCS for ionization of metastable 2S-state hydrogen atoms by 150 eV and 250 eV electron effect has been estimated.

To understand these configurations of the DDCS, the current calculation applying the multiple scattering theory [11] provides a significant contribution in the field of metastable 2S-state ionization problems. Due to the absence of any experimental data for the DDCS results of the hydrogenic metastable 2S-state ionization process, it is not possible to compare the present computational results with the experimental findings. Therefore, hydrogenic ground state experimental results for ionization of metastable 2S state hydrogen atoms by electrons will be valuable, present computational work gives a significant contribution in the field of ionization problems for further research.

Acknowledgements

The computational works have been performed in the Simulation Lab of the Department of Mathematics, Chittagong University of Engineering and Technology Chittagong-4349, Bangladesh.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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