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Title: On modified finite difference method to obtain the electron energy distribution functions in Langmuir probes

Abstract

A modified central difference method (MCDM) is proposed to obtain the electron energy distribution functions (EEDFs) in single Langmuir probes. Numerical calculation of the EEDF with MCDM is simple and has less noise. This method provides the second derivatives at a given point as the weighted average of second order central difference derivatives calculated at different voltage intervals, weighting each by the square of the interval. In this paper, the EEDFs obtained from MCDM are compared to those calculated via the averaged central difference method. It is found that MCDM effectively suppresses the noises in the EEDF, while the same number of points are used to calculate of the second derivative.

Authors:
;  [1]; ; ; ;  [2]
  1. Department of Electrical Engineering, Hanyang University, 222, Wangsimni-ro, Seongdong-gu, Seoul 133-791 (Korea, Republic of)
  2. Seoul Science High School, 63, Hyehwa-ro, Jongno-gu, Seoul 110-530 (Korea, Republic of)
Publication Date:
OSTI Identifier:
22600149
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Plasmas; Journal Volume: 23; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; COMPARATIVE EVALUATIONS; DISTRIBUTION FUNCTIONS; ELECTRIC POTENTIAL; ELECTRONS; ENERGY SPECTRA; FINITE DIFFERENCE METHOD; LANGMUIR PROBE; NOISE; WEIGHT

Citation Formats

Kang, Hyun-Ju, Chung, Chin-Wook, E-mail: joykang@hanyang.ac.kr, Choi, Hyeok, Kim, Jae-Hyun, Lee, Se-Hun, and Yoo, Tae-Ho. On modified finite difference method to obtain the electron energy distribution functions in Langmuir probes. United States: N. p., 2016. Web. doi:10.1063/1.4951020.
Kang, Hyun-Ju, Chung, Chin-Wook, E-mail: joykang@hanyang.ac.kr, Choi, Hyeok, Kim, Jae-Hyun, Lee, Se-Hun, & Yoo, Tae-Ho. On modified finite difference method to obtain the electron energy distribution functions in Langmuir probes. United States. doi:10.1063/1.4951020.
Kang, Hyun-Ju, Chung, Chin-Wook, E-mail: joykang@hanyang.ac.kr, Choi, Hyeok, Kim, Jae-Hyun, Lee, Se-Hun, and Yoo, Tae-Ho. 2016. "On modified finite difference method to obtain the electron energy distribution functions in Langmuir probes". United States. doi:10.1063/1.4951020.
@article{osti_22600149,
title = {On modified finite difference method to obtain the electron energy distribution functions in Langmuir probes},
author = {Kang, Hyun-Ju and Chung, Chin-Wook, E-mail: joykang@hanyang.ac.kr and Choi, Hyeok and Kim, Jae-Hyun and Lee, Se-Hun and Yoo, Tae-Ho},
abstractNote = {A modified central difference method (MCDM) is proposed to obtain the electron energy distribution functions (EEDFs) in single Langmuir probes. Numerical calculation of the EEDF with MCDM is simple and has less noise. This method provides the second derivatives at a given point as the weighted average of second order central difference derivatives calculated at different voltage intervals, weighting each by the square of the interval. In this paper, the EEDFs obtained from MCDM are compared to those calculated via the averaged central difference method. It is found that MCDM effectively suppresses the noises in the EEDF, while the same number of points are used to calculate of the second derivative.},
doi = {10.1063/1.4951020},
journal = {Physics of Plasmas},
number = 6,
volume = 23,
place = {United States},
year = 2016,
month = 6
}
  • Electron energy distribution functions (EEDFs) were determined from probe characteristics using a numerical ac superimposed method with a distortion correction of high derivative terms by varying amplitude of a sinusoidal perturbation voltage superimposed onto the dc sweep voltage, depending on the related electron energy. Low amplitude perturbation applied around the plasma potential represented the low energy peak of the EEDF exactly, and high amplitude perturbation applied around the floating potential was effective to suppress noise or distortion of the probe characteristic, which is fatal to the tail electron distribution. When a small random noise was imposed over the stabilized provemore » characteristic, the numerical differentiation method was not suitable to determine the EEDF, while the numerical ac superimposed method was able to obtain a highly precise EEDF.« less
  • In plasma diagnostics with a single Langmuir probe, the electron temperature T{sub e} is usually obtained from the slope of the logarithm of the electron current or from the electron energy probability functions of current (I)-voltage (V) curve. Recently, Chen [F. F. Chen, Phys. Plasmas 8, 3029 (2001)] suggested a derivative analysis method to obtain T{sub e} by the ratio between the probe current and the derivative of the probe current at a plasma potential where the ion current becomes zero. Based on this method, electron temperatures and electron densities were measured and compared with those from the electron energymore » distribution function (EEDF) measurement in Maxwellian and bi-Maxwellian electron distribution conditions. In a bi-Maxwellian electron distribution, we found the electron temperature T{sub e} obtained from the method is always lower than the effective temperatures T{sub eff} derived from EEDFs. The theoretical analysis for this is presented.« less
  • It is shown that a simple circuit consisting of a semiconductor diode, a resistor, and a dc voltage source can model a narrow-energy group of electrons in a plasma for the purpose of calibration of a Langmuir probe. The calibration is appropriate when the probe is used for measurement of the electron energy distribution function (EEDF). This simple circuit allows real-time determination of sensitivity, energy resolution, and signal-to-noise ratio for probe measurements of the EEDF.
  • By using a rf compensated Langmuir probe and optical emission spectroscopy, the effects of driving frequency (13.56-50 MHz) on the electron energy probability function (EEPF), electron density, electron temperature, and the vibrational and rotational temperatures in capacitively coupled nitrogen discharge were investigated. Measurements were performed in the pressure range 60-200 mTorr, and at a fixed voltage of 140 V (peak-to-peak). With increasing the driving frequency, the dissipated power and electron density markedly increased along with the intensity of the optical emission lines belonging to the 2nd positive (337.1 nm) and 1st negative systems (391.4 nm) of N{sub 2}. The EEPFmore » at low pressure 60 mTorr is two-temperature (bi-Maxwellian) distribution, irrespective of the driving frequency, in contrast with argon and helium discharges in the similar conditions. The mechanism forming such bi-Maxwellian shape was explained by two combined effects: one is the collisionless sheath-heating effect enhancing the tail electron population, and the other is the collision-induced reduction of electrons at the energy 2-4 eV where the collision cross-section for the vibrational excitation has a resonantly large peak. The two-temperature EEPF structure was basically retained at moderate pressure 120 mTorr and high pressure 200 mTorr. The vibrational temperature T{sub vib} and rotational temperature T{sub rot} are measured for the sequence ({Delta}{nu}=-2) of N{sub 2} second positive system (C{sup 3}{Pi}{sub u}{yields}B{sup 3}{Pi}{sub g}) using the method of comparing the measured and calculated spectra with a chi-squared minimization procedure. It was found that, both of T{sub vib} and T{sub rot} are a weakly dependent on driving frequency at low pressure 60 mTorr. At higher pressure (120 and 200 mTorr), T{sub vib} rises monotonically with the driving frequency, whereas the T{sub rot} slightly decreases with frequency below 37 MHz, beyond which it relatively increases or saturated.« less
  • The method proposed to determine the electron energy distribution is based on the numerical simulation of the effect induced by a sinusoidal perturbation superimposed to the direct current voltage applied to the probe. The simulation is generating a multiple harmonic components signal over the rough experimental data. Each harmonic component can be isolated by means of finite impulse response filters. Then, the second derivative is deduced from the second harmonic component using the Taylor expansion. The efficiency of the method is proved first on simple cases and second on typical Langmuir probes characteristics recorded in the expansion of a microwavemore » plasma containing argon or nitrogen-hydrogen gas mixture. Results obtained using this method are compared to those, which are determined using a classical Savitzsky-Golay filter.« less