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Measurements on the electron-energy distribution function in a low-pressure cesium discharge
Physica
57 (1972) 624-627
o North-Holland
LETTER
Publishing
TO THE
Comment MEASUREMENTS
ON THE
FUNCTION
Co.
EDITOR
on
ELECTRON-ENERGY
IN A LOW-PRESSURE
CESIUM
DISTRIBUTION DISCHARGE
[Physica 49 ( 1970) 77-9 1] A. J. POSTMA Association
Euratom-FOM,
FOM-Instituut
VOOYPlasma-Fysica,
Hijnhuizen,
Jutphaas,
Nederland
and I<. WIESEMANN Fachbereich
Physik
der Universitiit
Marburg
(Lahn),
Gernzanv
Received 20 August 1971
In an article entitled “Measurements on the electron-energy distribution function in a low-pressure cesium discharge” by Postmai), experimental distribution functions obtained from Langmuir-probe measurements are compared with theoretical ones. To explain the difference between the experimental results and maxwellian distribution functions, expected at a large degree of ionization, Postma assumes that reflection of electrons at the probe surface plays an important r81e. For the electron current i, at a repelling probe, the following expression [eq. (4)] is given i, =
c smfc~)~~[ul - R(5)] d5 dx, u
(14
0
where x is the energy of the electrons in the plasma, E the perpendicular energy of the electrons at the probe surface, f(x) the distribution function of the electrons in the plasma, and R(5) the energy-dependent reflection coefficient. The first derivative of i, with respect to the probe voltage U derived in ref. 1 is wrong and, consequently, also the second derivative [eqs. (5) and (6)]. The right expressions are di, O” __ = -C f(x)[l - R(x - U)] dx, dU s u 624
(lb)
(BZ)
'XP
[(n - x)tl - Ilk2-
w/j3 = d? 00
sz9
B3XVHXIcl
Ifh’lISB3
V NI SNOI.LflEII?LLSICI
A321ZINB
NO LNZWW!O3
626
A. J. POSTMA
Fig. B, C: second-derivative only; of the
B:
reflection
angle
of
1. A: theoretical
curves,
corrected
dependent
incidence;
strength
AND
2 x
K. WIESEMANN
distribution for reflection,
on the angle (Cs discharge,
lo-19 Vm2;
function; considering
of incidence; current
30.8 mA;
degree of ionization
reduced
9 x
T(E) and R(E) is given by Do
probe
independent electric
field
10-G).
03 -
u
the repelling
C: reflection
U)[l
-
R(x -
U)]} dx
is u
f(x) dx,
(3)
(if R does not depend on the angle of incidence), i.e., T(E) depends on the distribution function f(x). The success of Postma’s method of using T(E) as an empirical correction for measured distribution functions suggests that this dependence is not too sensitive, however. In using this method one should note that T(E) might be the consequence of a lot of different effects. Our considerations given here and in refs. 1 and 3 are incomplete as they consider the electron repelling probe only. Especially at the space potential there might be an influence from the electron-attraction part of the probe characteristic. When the second derivative is not symmetric around the point where dsi,/dU2 = 0 the measurement of the distribution of the lowenergy electrons and of the plasma potential may be influenced by a “smearing out” of the transition between the different parts of the characteristic. Further possible effects that might be considered are discharge
COMMENT ON ENERGY
DISTRIBUTIONS
IN A CESIUM DISCHARGE
627
noise, effects of the finite signal amplitude for measuring di,/dU or dsi,/dUs, inhomogeneity of the work function of the probe, collisions between electrons and gas atoms in the probe sheath and, last not least, perturbation of the plasma by the probe. To obtain the zero of the energy scale of T(E), eqs. (9) and (12) in ref. 1 must be changed: integrating eq. (5) and taking into account that at the wall potential the ion current equals the electron current, we obtain m
kT,($$$ = j-T(x) e-x!kTedx, u and expression
(12) becomes
Numerical calculations yield within the accuracy of the method that the zero of the energy scale of T(E) remains the same as that reported in ref. 1. This work was performed as part of the research programme of the association agreement of Euratom and the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) with financial support from the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek” (ZWO) and Euratom.
REFERENCES 1) 2) 3) 4) 5) 6)
Postma, A. J., Physica 49 (1970) 77. Shul’man, A. R. and Ganichev, D. A., Soviet Physics - Solid State 4 (1962) 545. Wiesemann, K., Ann. Physik. 27 (1971) 303. Niedermayer, R. and Hijlzl, J., Phys. Status solidi 11 (1965) 651. Fowler, H. A. and Farnsworth, H. E., Phys. Rev. 111 (1958) 103. Heil, H. and Hollweg, J. V., Phys. Rev. 164 (1967) 881.
57 (1972) 624-627
o North-Holland
LETTER
Publishing
TO THE
Comment MEASUREMENTS
ON THE
FUNCTION
Co.
EDITOR
on
ELECTRON-ENERGY
IN A LOW-PRESSURE
CESIUM
DISTRIBUTION DISCHARGE
[Physica 49 ( 1970) 77-9 1] A. J. POSTMA Association
Euratom-FOM,
FOM-Instituut
VOOYPlasma-Fysica,
Hijnhuizen,
Jutphaas,
Nederland
and I<. WIESEMANN Fachbereich
Physik
der Universitiit
Marburg
(Lahn),
Gernzanv
Received 20 August 1971
In an article entitled “Measurements on the electron-energy distribution function in a low-pressure cesium discharge” by Postmai), experimental distribution functions obtained from Langmuir-probe measurements are compared with theoretical ones. To explain the difference between the experimental results and maxwellian distribution functions, expected at a large degree of ionization, Postma assumes that reflection of electrons at the probe surface plays an important r81e. For the electron current i, at a repelling probe, the following expression [eq. (4)] is given i, =
c smfc~)~~[ul - R(5)] d5 dx, u
(14
0
where x is the energy of the electrons in the plasma, E the perpendicular energy of the electrons at the probe surface, f(x) the distribution function of the electrons in the plasma, and R(5) the energy-dependent reflection coefficient. The first derivative of i, with respect to the probe voltage U derived in ref. 1 is wrong and, consequently, also the second derivative [eqs. (5) and (6)]. The right expressions are di, O” __ = -C f(x)[l - R(x - U)] dx, dU s u 624
(lb)
(BZ)
'XP
[(n - x)tl - Ilk2-
w/j3 = d? 00
sz9
B3XVHXIcl
Ifh’lISB3
V NI SNOI.LflEII?LLSICI
A321ZINB
NO LNZWW!O3
626
A. J. POSTMA
Fig. B, C: second-derivative only; of the
B:
reflection
angle
of
1. A: theoretical
curves,
corrected
dependent
incidence;
strength
AND
2 x
K. WIESEMANN
distribution for reflection,
on the angle (Cs discharge,
lo-19 Vm2;
function; considering
of incidence; current
30.8 mA;
degree of ionization
reduced
9 x
T(E) and R(E) is given by Do
probe
independent electric
field
10-G).
03 -
u
the repelling
C: reflection
U)[l
-
R(x -
U)]} dx
is u
f(x) dx,
(3)
(if R does not depend on the angle of incidence), i.e., T(E) depends on the distribution function f(x). The success of Postma’s method of using T(E) as an empirical correction for measured distribution functions suggests that this dependence is not too sensitive, however. In using this method one should note that T(E) might be the consequence of a lot of different effects. Our considerations given here and in refs. 1 and 3 are incomplete as they consider the electron repelling probe only. Especially at the space potential there might be an influence from the electron-attraction part of the probe characteristic. When the second derivative is not symmetric around the point where dsi,/dU2 = 0 the measurement of the distribution of the lowenergy electrons and of the plasma potential may be influenced by a “smearing out” of the transition between the different parts of the characteristic. Further possible effects that might be considered are discharge
COMMENT ON ENERGY
DISTRIBUTIONS
IN A CESIUM DISCHARGE
627
noise, effects of the finite signal amplitude for measuring di,/dU or dsi,/dUs, inhomogeneity of the work function of the probe, collisions between electrons and gas atoms in the probe sheath and, last not least, perturbation of the plasma by the probe. To obtain the zero of the energy scale of T(E), eqs. (9) and (12) in ref. 1 must be changed: integrating eq. (5) and taking into account that at the wall potential the ion current equals the electron current, we obtain m
kT,($$$ = j-T(x) e-x!kTedx, u and expression
(12) becomes
Numerical calculations yield within the accuracy of the method that the zero of the energy scale of T(E) remains the same as that reported in ref. 1. This work was performed as part of the research programme of the association agreement of Euratom and the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) with financial support from the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek” (ZWO) and Euratom.
REFERENCES 1) 2) 3) 4) 5) 6)
Postma, A. J., Physica 49 (1970) 77. Shul’man, A. R. and Ganichev, D. A., Soviet Physics - Solid State 4 (1962) 545. Wiesemann, K., Ann. Physik. 27 (1971) 303. Niedermayer, R. and Hijlzl, J., Phys. Status solidi 11 (1965) 651. Fowler, H. A. and Farnsworth, H. E., Phys. Rev. 111 (1958) 103. Heil, H. and Hollweg, J. V., Phys. Rev. 164 (1967) 881.