Convenient analytical solution for vibrational distribution function of molecules colliding with a wall
Abstract
We study formation of the Vibrational Distribution Function (VDF) in a molecular gas at low pressure, when vibrational levels are excited by electron impact and deactivated in collisions with walls and show that this problem has a convenient analytical solution that can be used to obtain VDF and its dependence on external parameters. The VDF is determined by excitation of vibrational levels by an external source and deactivation in collisions with the wall. Deactivation in wall collisions is little known process. However, we found that the VDF is weakly dependent on the functional form of the actual form of probability $${\gamma }_{\upsilon ^{\prime} \to \upsilon }$$ for a vibrational number $$\upsilon ^{\prime} $$ to transfer into a lower level $$\upsilon $$ at the wall. Because for a given excitation source of vibrational states, the problem is linear the solution for VDF involves solving linear matrix equation. The matrix equation can be easily solved if we approximate probability, in the form: $${\gamma }_{\upsilon ^{\prime} \to \upsilon }=(1/\upsilon ^{\prime} )\theta (\upsilon ^{\prime} \upsilon ).$$ In this case, the steadystate solution for VDF($$\upsilon $$) simply involves a sum of source rates for levels above $$\upsilon ,$$ with a factor of $$1/(\upsilon +1).$$ As an example of application, we study the vibrational kinetics in a hydrogen gas and verify the analytical solution by comparing with a full model.
 Authors:

 Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Donghua Univ., Shanghai (China); Dalian Univ. of Technology (China)
 Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
 Dalian Univ. of Technology (China)
 Publication Date:
 Research Org.:
 Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
 Sponsoring Org.:
 USDOE
 Contributing Org.:
 China Scholarship Council (CSC), National Magnetic Confinement Fusion Science Program, China (Grant No. 2015GB114000), National Key R&D Program of China (Grant No. 2017YFE0300106) and Initial Research Funds for Young Teachers of Donghua University.
 OSTI Identifier:
 1663369
 Grant/Contract Number:
 AC0209CH11466; 2015GB114000; 2017YFE0300106
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Plasma Sources Science and Technology
 Additional Journal Information:
 Journal Volume: 28; Journal Issue: 10; Journal ID: ISSN 13616595
 Publisher:
 IOP Publishing
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY
Citation Formats
Yang, Wei, Khrabrov, Alexander V., Kaganovich, Igor D., and Wang, YouNian. Convenient analytical solution for vibrational distribution function of molecules colliding with a wall. United States: N. p., 2019.
Web. doi:10.1088/13616595/ab4310.
Yang, Wei, Khrabrov, Alexander V., Kaganovich, Igor D., & Wang, YouNian. Convenient analytical solution for vibrational distribution function of molecules colliding with a wall. United States. https://doi.org/10.1088/13616595/ab4310
Yang, Wei, Khrabrov, Alexander V., Kaganovich, Igor D., and Wang, YouNian. Tue .
"Convenient analytical solution for vibrational distribution function of molecules colliding with a wall". United States. https://doi.org/10.1088/13616595/ab4310. https://www.osti.gov/servlets/purl/1663369.
@article{osti_1663369,
title = {Convenient analytical solution for vibrational distribution function of molecules colliding with a wall},
author = {Yang, Wei and Khrabrov, Alexander V. and Kaganovich, Igor D. and Wang, YouNian},
abstractNote = {We study formation of the Vibrational Distribution Function (VDF) in a molecular gas at low pressure, when vibrational levels are excited by electron impact and deactivated in collisions with walls and show that this problem has a convenient analytical solution that can be used to obtain VDF and its dependence on external parameters. The VDF is determined by excitation of vibrational levels by an external source and deactivation in collisions with the wall. Deactivation in wall collisions is little known process. However, we found that the VDF is weakly dependent on the functional form of the actual form of probability ${\gamma }_{\upsilon ^{\prime} \to \upsilon }$ for a vibrational number $\upsilon ^{\prime} $ to transfer into a lower level $\upsilon $ at the wall. Because for a given excitation source of vibrational states, the problem is linear the solution for VDF involves solving linear matrix equation. The matrix equation can be easily solved if we approximate probability, in the form: ${\gamma }_{\upsilon ^{\prime} \to \upsilon }=(1/\upsilon ^{\prime} )\theta (\upsilon ^{\prime} \upsilon ).$ In this case, the steadystate solution for VDF($\upsilon $) simply involves a sum of source rates for levels above $\upsilon ,$ with a factor of $1/(\upsilon +1).$ As an example of application, we study the vibrational kinetics in a hydrogen gas and verify the analytical solution by comparing with a full model.},
doi = {10.1088/13616595/ab4310},
journal = {Plasma Sources Science and Technology},
number = 10,
volume = 28,
place = {United States},
year = {2019},
month = {10}
}
Works referenced in this record:
A theoretical calculation of dissociation rates of molecular hydrogen in electrical discharges
journal, March 1977
 Capitelli, M.; Dilonardo, M.; Molinari, E.
 Chemical Physics, Vol. 20, Issue 3
Chemical kinetics of low pressure high density hydrogen plasmas: application to negative ion sources for ITER
journal, October 2014
 Gaboriau, F.; Boeuf, J. P.
 Plasma Sources Science and Technology, Vol. 23, Issue 6
Benchmarking and validation of global model code for negative hydrogen ion sources
journal, November 2018
 Yang, Wei; Averkin, Sergey N.; Khrabrov, Alexander V.
 Physics of Plasmas, Vol. 25, Issue 11
A joint vibroelectronic mechanism in the dissociation of molecular hydrogen in nonequilibrium plasmas
journal, November 1978
 Cacciatore, M.; Capitelli, M.; Dilonardo, M.
 Chemical Physics, Vol. 34, Issue 2
Global model analysis of negative ion generation in lowpressure inductively coupled hydrogen plasmas with biMaxwellian electron energy distributions
journal, March 2015
 Huh, SungRyul; Kim, NamKyun; Jung, BongKi
 Physics of Plasmas, Vol. 22, Issue 3
A simple formula for diffusion calculations involving wall reflection and low density
journal, August 1987
 Chantry, P. J.
 Journal of Applied Physics, Vol. 62, Issue 4
Vibrational and RotationalState Dependence of Dissociative Attachment in $e$  ${\mathrm{H}}_{2}$ Collisions
journal, December 1978
 Wadehra, J. M.; Bardsley, J. N.
 Physical Review Letters, Vol. 41, Issue 26
A Global Enhanced Vibrational Kinetic Model for HighPressure Hydrogen RF Discharges
journal, June 2015
 Averkin, Sergey N.; Gatsonis, Nikolaos A.; Olson, Lynn
 IEEE Transactions on Plasma Science, Vol. 43, Issue 6
Kinetics of electrons and neutral particles in radiofrequency transformer coupled plasma H ^{−} ion source at Seoul National University
journal, October 2016
 Chung, K. J.; Dang, J. J.; Kim, J. Y.
 New Journal of Physics, Vol. 18, Issue 10
Dissociative electron attachment to vibrationally and rotationally excited ${\mathrm{H}}_{2}$ and HF molecules
journal, April 2000
 Xu, Y.; Gallup, G. A.; Fabrikant, I. I.
 Physical Review A, Vol. 61, Issue 5
Effect of Vibrational and Rotational Exictation on Dissociative Attachment in Hydrogen
journal, December 1978
 Allan, M.; Wong, S. F.
 Physical Review Letters, Vol. 41, Issue 26
Analysis of the H _{2} vibrational distribution in a hydrogen discharge
journal, February 1989
 Hiskes, J. R.; Karo, A. M.
 Applied Physics Letters, Vol. 54, Issue 6
NegativeIonSource Modeling: From Expansion to Extraction Region
journal, August 2008
 Taccogna, Francesco; Longo, Savino; Capitelli, Mario
 IEEE Transactions on Plasma Science, Vol. 36, Issue 4
Generation of negative ions in tandem high‐density hydrogen discharges
journal, October 1984
 Hiskes, J. R.; Karo, A. M.
 Journal of Applied Physics, Vol. 56, Issue 7
Lowenergy $e{\mathrm{H}}_{2}$ scattering: Separation of dissociative attachment and dissociation channels
journal, December 2000
 Xu, Y.; Kazansky, A. K.; Fabrikant, I. I.
 Physical Review A, Vol. 63, Issue 1
Production of hydrogen negative ions in an ECR volume source: balance between vibrational excitation and ionization
journal, July 2018
 Aleiferis, S.; Svarnas, P.; Béchu, S.
 Plasma Sources Science and Technology, Vol. 27, Issue 7
Negative hydrogen ion production mechanisms
journal, June 2015
 Bacal, M.; Wada, M.
 Applied Physics Reviews, Vol. 2, Issue 2