Non-thermal Doppler-broadened Balmer lines produced by exothermic
reactions in low-pressure hydrogen discharges
J. Loureiro(1) and J. Amorim(2)
(1)
(2)
Instituto de Plasmas e Fusão Nuclear-Laboratório Associado, Instituto Superior Técnico, Universidade
Técnica de Lisboa, 1049-001 Lisboa, Portugal
Laboratório Nacional de Ciência e Tecnologia do Bietanol – CTBE, Caixa Postal 6170, CEP 13083-970
Campinas – SP, Brazil
Abstract: The non-thermal component of the 3-D and 1-D velocity distributions of
hydrogen atoms created by the exothermic reaction H2+ + H2  H3+ + H + ΔE, in which
the energy ΔE ~ 2 eV is transferred to the product species, is calculated using the energy
conservation. Significant deviations relatively to thermal distributions are shown to exist
even in the case of internal energy defects < 2 eV. The profiles of 1-D distributions are
markedly flatter and squarer than Gaussian shapes leading to a broadened component in
the hydrogen Balmer lines as observed in many experiments. The validity of the normal
procedure used for determining the temperature of H atoms from the full width at half
maximum of the spectrum lines is discussed with basis on the non-thermal character of
the distributions. The effects of other reactions producing H atoms are also compared and
discussed.
Keywords: Anomalous Doppler broadening, Balmer lines, hydrogen discharges.
1. Introduction
There is considerable interest in the explanation of
the excessive Doppler broadening of Balmer lines
observed in many low-pressure electrical discharges
[1,2,3]. In those experiments, the development of
extended far wings of the emission lines has been
observed with central Gaussian shapes whose full
width at half maximum (FWHM) allows to predict
kinetic energies for the H atoms as high as tens or
hundreds of eV. The mechanisms responsible for
this excessive broadening have been discussed in
terms of both gas-phase and surface ion-impact
phenomena [4,5]. In principle, the enlargement can
be originated by gas-phase ion-atom and ionmolecule collisions, as a result of simultaneous
neutralization and reflection of ions at the electrode
surface, and/or by sputtering of adsorbed hydrogen.
Several explanations were then offered. In [1], for
example, it has been concluded that the mechanism
for formation of hot H(n=3) atoms in the gas phase
responsible for a Doppler shift in Hα Balmer line
results from ion-neutral charge exchange processes
in the sheaths with both H+ and H3+ ions. On the
contrary in drift tubes, at much lower pressures and
higher E/N values, it has been observed that direct
excitation of H(n=3) by these ions is small compared
to the excitation by neutral fast H atoms and H2
molecules [3,4]. In spite of all variations of the
models used to explain the anomalous Doppler
broadening of hydrogen Balmer lines, the
mechanisms responsible for the excessive
enlargement are not yet totally understood for all
experiments.
With these facts in mind the present work is
nevertheless focused in a different direction. We
pretend to evaluate the validity of the normal
procedure used in the literature to conclude about the
existence of hot H atoms from the FWHM of Balmer
lines, in the positive column of a DC discharge,
while it is clear that the velocity distributions of
atoms are strongly departed from a MaxwellBoltzmann (MB) and the lines do not present a
Gaussian form. The broadest component of the
emitting line is highly non-Gaussian so that the
attribution of a temperature for the atoms
responsible by this part of the spectrum should be
highly questionable.
2. Model for exothermic reactions
The anomalous Doppler broadening may occur due
to collisions in which internal energy from one or
both colliding partners is converted to translational
energy. Although not being the only channel, a
reaction capable to transfer an appreciable amount of
energy to H atoms is: H2+ + H2  H3+ + H + ΔE,
with ΔE = 1.56 eV [6]. The velocity distributions of
both partners may suffer then strong deviations
relatively to MB and the profiles of the emitted H
lines are non-Gaussian. The spectrum of Balmer
lines produced in sequence of a set of reactions
initiated with the above reaction, before thermal
collisions may occur, present a profile dictated by
the conjoint action of these collisions and thermal
collisions. The present approach is based on the
energy conservation and both colliding partners are
initially in equilibrium at the gas bath temperature.
On the other hand, the producing partners have
isotropic distributions, so that the model is invalid as
anisotropic processes dominate the excitation of the
Balmer lines.
two 3-D MB distributions. The velocities and
masses are then referenced to the center-of-mass
frame, by defining the center-of-mass velocity and
the relative velocity. The addition of the energy
defect ΔE is considered in the center-of-mass frame,
i.e. to the kinetic energy of the reduced mass particle
µ. Then the combined probability of the relative
velocity distribution of product species and of the
unchanged center-of-mass velocity are converted
back to the laboratory frame and expressed in terms
of the velocities v3 and v4. This calculation
determines the 3-D velocity distributions of species
X3 and X4, from which the 1-D distributions can also
be obtained by numerical integration using
cylindrical coordinates.
3. Results and discussion
Fig.1 shows the 3-D velocity distribution of H atoms
calculated for T=500 K and the internal energy
defect in the range 0-2 eV. This figure shows that
these distributions are markedly different from MB,
since their maxima (at v4=0 in the case of a MB
distribution) are rapidly pushed away towards higher
energies as ΔE increases.
In the case of a reaction of the type:
X1 + X2  X3 + X4 + ΔE,
the non-MB velocity distributions of energetic
reaction products may be determined by energy
conservation, assuming that two particles of masses
m1 and m2, with velocities v1 and v2, collide at
temperature T, producing new particles of masses m3
and m4, with m3+m4 = m1+m2, and velocities v3 and
v4. Since the energy difference between the internal
energies of two reactants and product species is ΔE
> 0, this energy defect necessarily appears as an
increase in the kinetic energies of both product
species.
The development of this calculation can be followed
in [7,8]. It starts with the probability of finding the
particles in the double three-dimensional (3-D)
velocity element dv1 and dv2 given by the product of
Fig. 1 – 3-D velocity distribution of H atoms calculated for T =
500 K and the energy defects ΔE = 0 (MB), 0.05, 0.1, 0.2, 0.5, 1
and 2 eV.
The integration of the above 3-D distributions on the
perpendicular velocity using cylindrical coordinates
allows to obtain the distribution along one direction,
F(vz). Although these latter present their maxima at
vz=0 they are considerably flatter and squarer than
Gaussians.
temperature really exist, because the profiles are
strongly departed from a Gaussian shape.
From the 1-D distributions we can derive now the
profiles of a Doppler line. Fig.2 shows the
symmetrical intensity profiles of the Hα Balmer line
centered at 656.28 nm, calculated for ΔE = 0, 0.2
and 2 eV and T = 300, 500 K and 800 K. The
profiles are normalized to the spectrum line obtained
at ΔE = 0 and T = 300 K.
This conclusion can still be reinforced by inspection
of Figs.3 and 4 where the 1-D and 3-D velocity
distributions calculated for ΔE = 2 eV and T = 500
K are compared with the MB distribution at 25 906
K, i.e. with the MB distribution at the temperature
obtained from the half-maximum of the former nonGaussian 1-D distribution.
Fig. 2 – Intensity of Hα Balmer line calculated for ΔE = 0, 0.2
and 2 eV, and T = 300 (full curves), 500 K (dashed curves) and
800 K (broken curves).
Fig. 3 – 1-D velocity distribution of H atoms calculated for ΔE =
2 eV and T = 500 K (full curve) and Maxwell-Boltzmann
distribution (broken curve) calculated at the temperature 25 906
K obtained from the FWHM of the former distribution.
The calculated profiles of the Hα line reveal the
existence of anomalous Doppler broadening with
non-Gaussian profiles, which become more and
more pronounced as ΔE increases. These profiles
have been obtained assuming that the H atoms only
suffer collisions dictated by the above reaction.
However, if other collisions have also been included,
such as the collisions of momentum transfer, a
weighted combination of Gaussian and nonGaussian profiles should be considered and final
profiles with shapes close to the measured lines
would be obtained [8].
The profiles shown in Fig.2 are markedly squarer
and flatter than Gaussian-type. This fact indicates
that the FWHM of the lines cannot be used to deduct
the temperature of H atoms. In the case of ΔE = 2
eV and T = 500 K, for example, the temperature
derived from the usual expression for the halfmaximum of a spectral line is 25 906 K, but we
cannot conclude that atoms with such high
Fig. 4 – 3-D velocity distribution of H atoms and MB
distribution calculated at the same conditions as in Fig.3.
Fig.3 and Fig.4 are completely different from MB
distributions. This is still more visible in the case of
the 3-D distribution, since this distribution is
centered at ~ ½ m4 v42 = ¾ ΔE. These conclusions
show that in the case of a spectral line with
anomalous Doppler broadening, and if the
enlargement is due to an exothermic reaction of the
type we consider here, a simple line width
measurement cannot be used to derive the
temperature of the emitting species since the lines
are much flatter and squarer than Gaussians. These
predicted velocity distributions will relax to MB
distributions due to collisions, but they are
detectable in low-pressure discharges where the
radiative lifetime of the emitting state is significantly
smaller than the mean time between collisions.
Finally, Fig.5 shows for comparison the 1-D velocity
distributions of H atoms produced by the above
mentioned reaction and by the reaction of charge
exchange, H2 + H+  H2+ + H + ΔE, assuming the
same energy defects for the two reactions: ΔE = 0,
0.2 and 2 eV.
the distributions must be seriously questioned. In the
case of a fit of a measured spectrum by a series of
Gaussians, the line width of each individual
Gaussian cannot be used to conclude about the
existence of species with such temperature, because
the global line shape cannot be interpreted as a sum
of different Gaussians. The total velocity distribution
cannot be seen as well as a sum of various MB
distributions. These facts applied to the
interpretation of the excessive Doppler broadening
of Balmer lines observed in many low-pressure
hydrogen discharges should lead to take care in
attributing the enlargement to the presence of hot H
atoms.
This work will be pursued by considering energydependent cross sections in the calculations, since
we believe that even more rectangular profiles can
be obtained for particular shapes of the cross section
conserving the same value of energy defect ΔE.
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Fig. 5 – 1-D velocity distribution of H atoms produced by the
reaction H2+ + H2  H3+ + H + ΔE (full curve) and by H2 + H+
 H2+ + H + ΔE (broken curve), calculated for ΔE = 0, 0.2 and
2 eV and T = 500 K.
As expected only slightly modifications are observed
as the mass of reactants m1+m2 changes from 4 to 3.
4. Conclusions
In this paper we show that the energy transferred to
the product species by an exothermic reaction of
constant probability produces a strong deviation of
the velocity distributions relatively to MB
distributions. Due to this fact the profiles of the
spectrum lines emitted by radiative species before
undergoing thermal collisions are not Gaussian and
the use of the line width to derive the temperature of
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