arXiv:0903.1656v1 [physics.plasm-ph] 9 Mar 2009
Temperature dependence of binary and ternary recombination of H+
J. Glos´ ık, R. Plaˇ sil, I. Korolov, T. Kotr´ ık, O. Novotn´ y, P. Hlavenka, P. Dohnal, and J. Varju
Charles University, Mathematics and Physics Faculty, Prague 8, Czech Republic
Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
Chris H. Greene
Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440, USA
(Dated: March 9, 2009)
We study binary and the recently discovered process of ternary He-assisted recombination of
3ions with electrons in a low temperature afterglow plasma. The experiments are carried out
over a broad range of pressures and temperatures of an afterglow plasma in a helium buffer gas.
Binary and He-assisted ternary recombination are observed and the corresponding recombination
rate coefficients are extracted for temperatures from 77 K to 330 K. We describe the observed
ternary recombination as a two-step mechanism: First, a rotationally-excited long-lived neutral
3collisions. Second, the H∗
atom that leads to the formation of a very long-lived Rydberg state with high orbital momentum.
We present calculations of the lifetimes of H∗
para and ortho-H+
coefficients of ortho and para-H+
rate coefficients are in reasonable agreement with the calculated values.
3is formed in electron-H+
3molecule collides with a helium
3and of the ternary recombination rate coefficients for
3. The calculations show a large difference between the ternary recombination rate
3at temperatures below 300 K. The measured binary and ternary
Electron scattering by the simplest polyatomic ion H+
is of fundamental importance in plasma physics because
3ions are dominant in many hydrogen-containing plas-
mas, including astrophysically-relevant plasmas in par-
ticular. For quantum theory, the electron-H+
is important because the process involves the simplest
polyatomic ion that can be studied from first principles
without adjustable parameters. In the long history of
research on recombination of H+
have been numerous exciting, contradictory and some-
times unaccepted results. We mention here only a few
recent reviews devoted to H+
3recombination at low tem-
peratures (see e.g. [1, 2, 3, 4, 5]). A successful theory
3recombination with electrons at scattering ener-
gies below 1 eV was developed relatively recently when
the Jahn-Teller non-Born-Oppenheimer coupling was in-
cluded into theory [6, 7, 8, 9]. The measurements of the
recombination rate constant in different storage ring ex-
periments have also converged to the same value recently,
after it became understood that the internal rotational
and vibrational degrees of freedom of the H+
the recombination rate [8, 10, 11]. The theoretical devel-
opments and improvements in storage ring experiments
have resulted in a reconciliation of theory and experiment
for the binary electron-H+
However, the H+
carried out in afterglow plasmas [3, 4, 5, 12, 13, 14, 15,
16, 17] have repeatedly given rate coefficients very differ-
ent from the ones obtained in the aforementioned storage
ring experiments and the theoretical calculations. Until
now, the plasma studies have not been fully understood
3with an electron there
and, as a result, they have been frequently rejected be-
cause they do not mesh with the present understanding
of the binary DR process (see, e.g., the very recent re-
view discussing this subject ). However, the experi-
mental plasma results are reproducible and they demand
an understanding and an integration into the full picture
3recombination with electrons. The present work
discusses and, we hope, clarifies some aspects of this com-
The plasma measurements are usually carried out in
a He/Ar/H2gas mixture (see the review by Plasil et al.
) or in a pure H2 gas [14, 19]. The main question is
how to “reconcile”  the rate constant (including its
dependence on experimental conditions) observed in an
3dominated plasma with actual theory and with data
from the storage ring experiments . The plasma ex-
periments are typically carried out with helium buffer gas
at pressures in the range 50–2000 Pa. It has been gen-
erally accepted that such pressures are too small to pro-
duce appreciable ternary helium-assisted recombination
3. The fact that the neutral-stabilized recombina-
tion can sometimes play a role was predicted many years
ago by Bates and Khare  and confirmed for some ions
(but not for H+
3) by experiments [21, 22]. The typical
value for the three-body recombination rate coefficient
KHe with helium as an ambient gas obtained in exper-
iment and estimated theoretically [21, 22] is of order of
10−27cm6s−1at 300 K. Thus, at pressures of 1300Pa, the
corresponding apparent binary recombination rate coeffi-
cient is smaller than 10−9cm3s−1, which would therefore
be negligible in comparison with the now-accepted bi-
3recombination rate coefficient [8, 9, 10, 11, 18].
This would also be negligible in comparison with the bi-
nary rate coefficients (about 10−7cm3s−1at 300 K) for
the majority of molecular ions. This is the reason why
the ternary recombination of H+
glected in FALP (flowing afterglow) and SA (stationary
afterglow) experiments carried out at 50–2000 Pa.
In our recent study  we have shown that at 260 K
a significant fraction of the H+
to formation of long-lived (up to tens of picoseconds)
rotationally-excited neutral Rydberg molecules H∗
formation of long-lived H∗
lium atoms can influence the overall process of recombi-
nation of the H+
3dominated afterglow plasma. By mea-
suring the helium pressure dependence of the H+
bination rate coefficient we have found that at 260 K the
ternary recombination rate is comparable with the binary
rate, already at pressures of several hundred Pa. The
observed ternary recombination of H+
cient by a factor of 100 than the ternary recombination
rate predicted by Bates and Khare . This suggests
that ternary recombination of H+
intrinsic features of the interaction between H+
trons at low collision energies . The recombination
process described by Bates  is a ternary process of a
This study presents a further extension of our measure-
ments to a broader range of temperatures and it extracts
the temperature dependence of the binary and ternary
recombination rate coefficients of H+
senting these new results and their interpretation, some
of us (the Prague group) wish to provide a few comments
concerning the plasma experiments made in Prague in re-
Previously, we have studied the H+
decaying plasmas formed from a He/Ar/H2 gas mix-
ture using several afterglow experiments based on several
modifications of flowing and stationary afterglow appara-
tuses (FALP , AISA–Advanced Integrated Stationary
Afterglow [3, 13, 17], TDT-CRDS–Test Discharge Tube
with Cavity Ring-Down Spectroscopy [25, 26]). We have
systematically investigated the dependence of the recom-
bination process on hydrogen partial pressure. Initially
(in 2000), our intention was to explain the influence of the
internal excitation of H+
3ions formed in a sequence of ion-
molecule reactions on the measured recombination rate
coefficient. Experimental conditions were such that the
plasma was decaying for a long time (up to 60 ms), so the
internal excitation was quenched in collisions with He and
H2. As a rule, we skip the first 10 ms of the decay process
considering it as a formation time. At low H2densities,
[H2] < 1012cm−3, and helium pressures of 100–300 Pa,
we have observed a decrease of the recombination rate co-
efficient from ∼ 10−7cm3s−1to 10−8cm3s−1when the H2
density was decreased from ∼ 1012cm−3to ∼ 1011cm−3.
At high hydrogen densities, [H2] > 1012cm−3, the mea-
sured recombination rate coefficient was independent of
[H2].We will consider this again below when discuss
equilibrium conditions for recombining H+
similar dependence on D2density was observed in a D+
3had previously been ne-
3+ e−collisions leads
3and collisions of H∗
3ions is more effi-
3ions is associated with
3ions. Before pre-
dominated afterglow plasma (see e.g. [3, 17]).
In our early experimental publications [12, 27], we de-
rived the recombination rate coefficient from experimen-
tal measurements assuming that the process is strictly
sumption was based on the then-existing level of knowl-
edge about H+
3recombination (see the recent and older
reviews in Refs. [4, 5, 14, 18, 28, 29]). However, we
soon realized that the observed dependence on hydrogen
density is coupled to the multistep character of the re-
combination process in a plasma. The theory of binary
dissociative recombination of H+
applicable to the storage ring experiments. Therefore we
denoted the plasma recombination rate constant as an
effective deionization rate constant αeff[3, 13]. Later ex-
periments using absorption spectroscopy for ion density
measurements and for the identification of the recombin-
ing ions (TDT-CRDS) [25, 26, 30] did not clarify the
mechanism of recombination in an afterglow plasma.
The interpretation of data from storage ring [5, 10, 11,
31] experiments and from afterglow [3, 5, 14, 15, 16, 17,
24, 25, 32] experiments was at this point not reconciled
[4, 5, 18, 33]. We stress here one principal difference be-
tween the two types of experiments: In a storage ring
experiment, the recombining ions and electrons have a
small adjustable relative velocity; the measured cross-
section corresponds to the binary electron-ion recombi-
nation process. In a plasma experiment an ambient gas
must be used. Typically, the pressure of the ambient gas
in the afterglow experiments is 50–2000 Pa. The ions and
electrons in afterglow plasma undergo multiple collisions
with neutral particles (He and H2 in the experiments
discussed here) prior to their mutual collisions; these are
collisions at low energy (around ∼ 0.025 eV). In a stor-
age ring, collisions of H+
3with neutral particles are also
possible, but when an ion collides with a background gas
molecule or atom, it is removed from the beam and does
not contribute any further to the observable recombina-
tion events .
Plasma experiments measure the thermally averaged
rate constant. It means that the afterglow plasma should
ideally be in thermal equilibrium with respect to all de-
grees of freedom, in particular with respect to the rota-
tional, vibrational and nuclear spin states of H+
ular hydrogen present in the recombining plasma equi-
librates the ortho/para-H+
3ratio and should make the
“kinetic” temperature of ions and electrons to coincide
with the He temperature [34, 35]. An internal excita-
tion of H+
tion [3, 4] is also quenched in collisions with He atoms,
but the population of the lowest energy levels (lowest or-
tho and para-states) is governed by collisions with H2.
One has to have in mind that proton transfer reactions
(from ArH+or H+
the rate of elastic collisions, but the rate coefficients for
reactions between H+
3and vice versa (“state changing collisions”)
are a factor of 5–10 smaller, ksc ∼ 3 × 10−10cm3s−1
3+ e−) at the low-pressure limit.The as-
3is more immediately
3ions obtained in the process of H+
2to H2) producing H+
3occur at nearly
3and H2, which change ortho-H+
(see Refs. [36, 37, 38]). If recombination of H+
tive to the nuclear spin state of H+
drives the H+
3ions out of ortho/para thermal equilib-
rium. In contrast, the “state-changing collisions” with
H2shift the ortho/paraH+
mal equilibrium. In this case, thermal equilibrium means
a population of rotational states corresponding to ther-
modynamic equilibrium at the given temperature. Note
that we have already demonstrated that in a plasma at
260 K the recombination of ortho and para-H+
very different . The recombination frequency (num-
ber of events per unit time) is νrec ∼ αeffne and the
“state changing frequency” is νsc∼ ksc[H2]. The condi-
tion νsc> νrecrequired for thermal equilibrium leads to
ksc[H2] > αeffne. If values typical for FALP experiments
(ne∼ 2 × 109cm−3and αeff ∼ 1.5 × 10−7cm3s−1) are
used in this inequality, we obtain the condition on the hy-
drogen density: [H2] > 1012cm−3. This is an important
density value for the interpretation of data in the plasma
experiments (see also the definition of the rate coefficient
in plasma as it is discussed in ). It is presumably
not accidental that the density 1012cm−3is the same as
the density at which the measured effective H+
bination rate coefficient changes its character. We have
in mind that at temperatures below 300 K ksc and αeff
are dependent on rotational excitation (ortho and para
states) of H2and H+
equilibrium is more complex. We will come to this point
Because arecombining plasma
molecules and atoms, collisions between ions and elec-
trons are perturbed by the neutral particles, which can
influence the observed effective plasma recombination
rate.The perturbation could be significant if in a
electron-ion collision a long-lived highly excited interme-
diate neutral molecule is formed (see the discussion in
Refs. [23, 28]). If the third particle is an atom from the
buffer gas, the probability of such a collision is propor-
tional to its pressure and, as a consequence, the apparent
recombination rate coefficient will be pressure dependent.
Similar phenomena are well known from studies of ion-
molecule association reactions, which can have a binary
(e.g. radiative association) and/or three-body character
depending on the lifetime of the complex. The rate of
these processes can depend on pressure and temperature
(e.g. associative reactions, see the discussion in ).
Third-body perturbation of H+
a plasma was already mentioned in our recent study .
In the present study, we show further experimental ev-
idence for the phenomenon and further aspects of the
theoretical interpretation, in conjunction with numerical
The rest of the article is organized in as follows. First,
we briefly describe the experiments in section II. Then
we present new experimental results in section III, where
we interpret the observed dependencies of the apparent
binary recombination rate coefficients (αeff) on hydrogen
and helium pressure, and from that analysis, derive the
3distribution towards the ther-
3than also the condition for thermal
3binary recombination in
FIG. 1: Cryo-FALP. Plasma created in a microwave discharge
is carried along the flow tube by helium carrier/buffer gas
(from left to right in the drawing). After addition of Ar (via
the indicated gas inlet) the plasma is converted from He+
dominated to Ar+dominated. Further downstream H2 is in-
jected to an already-relaxed plasma (with thermalized elec-
trons) and an H+
density decay downstream from the hydrogen entry port is
monitored by an axially movable Langmuir probe.
3dominated plasma is formed. The electron
binary and ternary recombination rate coefficients. Fi-
nally, in section IV we introduce our theoretical descrip-
tion of ternary recombination of an H+
our calculation of the lifetimes of the long-lived neutral
3formed in the electron-ion collisions, and es-
timate the rate coefficient for the ternary channel of H+
recombination. Section V summarizes our conclusions.
For measurements of pressure and temperature depen-
dences of the H+
3recombination rate we have built the
Cryo-FALP apparatus, designed to operate in the range
77–300 K and at helium pressures adjustable from ∼ 400
to ∼ 2000 Pa. UHV technology and high buffer gas pu-
rity (the level of impurities is < 0.1 ppm) are used in the
Cryo-FALP (Fig. 1).
The Cryo-FALP apparatus is a low temperature high
pressure variant of the standard FALP apparatus .
Here we will just describe the essential features of the
new construction.In Cryo-FALP, plasma is created
in a microwave discharge (10–30 W) in the upstream
glass section of the flow tube (at 300 K). Because of
the high pressure, He+ions formed by electron impact
then react in a three-body association reaction with
two atoms of helium, and a He+
formed. Downstream from the discharge region, Ar is
added to He+
2dominated afterglow plasma to remove he-
lium metastable atoms (Hemin Fig. 2) formed in the
discharge and to form Ar+dominated plasma (see de-
tails in [3, 4, 12, 13]). Via the second entry port situated
approximately 35 ms downstream from the Ar entry port,
hydrogen (diluted in He) is introduced into the plasma,
which at this point is already cold. Note that the po-
sition in the flow tube is linked to the decay time. In
a sequence of ion-molecule reactions, an H+
plasma is formed shortly after the second entry port.
We have used a numerical model to simulate the pro-
cess of formation of H+
3dominated plasma. Examples
2dominated plasma is
mation and plasma decay in FALP after addition of Ar
(7.7×1012cm−3) and H2 (5×1012cm−3). Argon was added
in the flow tube 33 ms before H2. The time scale origin is
set at the position of the hydrogen injection port. The right
panel focuses on a narrow time interval corresponding to a
transition from Ar+dominated to H+
ter adding hydrogen.
Left panel:Calculated dependences of ion for-
3dominated plasma af-
of ion density evolutions calculated within the model
for conditions typical for the present Cryo-FALP exper-
iments are shown in Fig. 2. Similar calculations were
carried out for all of our experiments presented in this
work. Argon also plays a role in plasma relaxation and
in formation of H+
nated plasma has already been formed, Ar does not play
any role (in contrast with He) because its density is at
least four orders of magnitude lower than the helium den-
sity. Downstream from the Ar entry port the flow tube
is thermally insulated and cooled to the desired temper-
ature by liquid nitrogen. Because theory suggests a very
strong temperature dependence of the process of inter-
est, we monitored carefully the temperature of the flow
tube. An axially movable Langmuir probe  is used to
measure the electron density decay downstream from the
hydrogen entry port . Electron energy distribution
functions (EEDF) were checked along the flow tube to
characterize plasma relaxation during the afterglow time
[34, 43]. Recombination of O+
was used for calibration of the Langmuir probe over a
broad range of pressures.
An advanced analysis was used to fit the decay curves
[3, 17, 44] with the purpose to obtain recombination rate
coefficients from the measured decay of electron densi-
ties. In the analysis we have assumed that once an H+
dominated plasma is formed, the plasma decay can be
described by a single value of the recombination rate co-
efficient, which we call the effective apparent binary re-
combination rate coefficient, αeff. The plasma decay can
then be described by the balance equation with a recom-
bination and a diffusion loss terms:
3ions. However, when an H+
2(“benchmark ion” [3, 4])
where neand n+are electron and ion densities, respec-
FIG. 3: Examples of electron density decay curves measured
in the H+
Upper panel: Cryo-FALP experiment. Lower panel: FALP
experiment. The effective recombination rate coefficients ob-
tained at different hydrogen densities are indicated. For com-
parison, both panels also show the decay curves (rectangles)
obtained in the Ar+dominated plasma with [H2] = 0.
3dominated plasma at several hydrogen densities.
tively. We assume that plasma is quasineutral (ne= n+).
The constant τD characterizes the ambipolar diffusion
during the afterglow. The recombination of H+
temperatures below 300 K depends strongly on the ro-
tational excitation of ions. The assumption about the
constant value of αeff must be discussed for particular
In the present experiments we use helium densities in
the range [He] ∼ 6 × 1016– 6 × 1017cm−3and hydro-
gen densities in the range [H2] ∼ 1011– 1014cm−3. In
the Cryo-FALP and other experiments discussed we use
normal hydrogen (n-H2) i.e. the mixture of ortho and
para-H2corresponding to 300 K (approximately 25% of
para H2). The variation of the ortho/para-H2ratio with
temperature in the interval 77 K–300 K is not significant
for the results of the present experiments. (In thermal
equilibrium at 100 K the fraction of para H2 is ∼ 38%,
and at 77 K the fraction is ∼ 50%).
We monitored decay of the afterglow plasma in a
He/Ar/H2 mixture at different temperatures and over
a broad range of helium and hydrogen densities. Ex-
amples of decay curves measured at 77 K and at 250 K
at several hydrogen densities are plotted in Fig. 3. The
dependence of the decay rate on hydrogen density is ev-
ident. The obtained apparent recombination rate coeffi-
cients (αeff) depend on all three parameters αeff(T, [H2],
[He]). It clearly indicates that the observed “deionization
process” is not pure binary dissociative recombination.
In Fig. 3 we have also plotted the decay curves measured
in a He/Ar afterglow plasma dominated by Ar+ions at
77 K and at 250 K in otherwise very similar conditions.
At 250 K (see lower panel) the decay curve is exponential
(straight line in the semilogarithmic plot) because recom-
bination of these atomic ions is very slow and the decay
is governed by ambipolar diffusion. At 77 K (see upper
panel of Fig. 3) we observe a faster decay during the early
afterglow (at higher electron densities). We assume that
this faster decay is primarily due to collisional radiative
recombination (CRR) [45, 46] and partly also due to the
formation of Ar+
2ions (in three-body association at low
temperatures) followed by immediate recombination of
these ions . The rate of the decay is comparable with
the rate calculated for the CRR at ∼ 85 K.
The apparent binary recombination rate coefficients
(αeff) obtained from measured decay curves in the H+
dominated plasma are plotted as functions of hydrogen
density in Fig. 4. Examples of the data obtained in other
experiments (FALP, AISA and TDT-CRDS) are also in-
cluded in Fig. 4. Below we summarize the data plotted
in Fig. 4.
1. 77 K, Cryo-FALP. At a fixed flow tube temperature
(T = 77 K) and at fixed [He] = 1.9 × 1017cm−3,
the dependence of αeffon H2density was measured
from [H2] ∼ 2 × 1011cm−3to ∼ 1013cm−3. Then,
for several other helium densities the recombina-
tion rate coefficient was only measured in a lim-
ited range of hydrogen densities close to [H2] ∼
1013cm−3. We plot only a few examples.
2. 100 K, TDT-CRDS. The H+
measured by using laser absorption spectroscopy
(CRDS technique). During the discharge pulse and
during the recombination dominated afterglow, the
ion temperature was determined from the Doppler
broadening of an absorption line. The details of
such experiments can be found in .
ion density was
3. 330 K, TDT-CRDS. The absorption spectroscopy
technique (CRDS) was used to measure the tem-
perature and density of the recombining H+
[25, 26, 48]. Relatively high hydrogen density was
used in these experiments. Therefore, an extrap-
olation had to be carried out in order to obtain
αeff for lower hydrogen densities (see the discus-
sion below). For details of this extrapolation see
the discussion in [24, 47]. In some experiments, the
He temperature was determined by measuring the
Doppler broadening of the H2O line . We have
also assumed that the temperatures of H2O and
He are identical. The H2O density is very low but
sensitivity of CRDS is very high for this molecule.
4. 130 and 230 K, AISA. Examples of data measured
with AISA. The data were measured at a fixed tem-
perature and at a fixed pressure as a function of
5. 170, 195, 250 K, FALP. Three different versions of
FALP designed to work at high He pressures were
used in these experiments.
6. 300 K, Experiment by Laube et al.
pressure FALP experiment: [H2] = 2×1014cm−3,
[He] = 1.6 × 1016cm−3, obtained αeff is 7.8 ×
7. 295 K, experiment by Gougousi et al. . In this
FALP experiment the dependence of the recombi-
nation rate coefficient on [H2] was observed. The
helium pressure was about 130 Pa.
8. 300 K, theory [8, 38]. The value calculated for bi-
nary dissociative recombination (for [H2] = 0).
Notice that in certain experiments the rate coeffi-
cients were measured over a limited range of hydro-
gen densities. In our experiments, we have covered the
1010cm−3< [H2] < 1016cm−3range (AISA, FALP,
In the measured dependences shown in Fig. 4 we distin-
guish three clearly different regions of hydrogen densities
that exhibit specific behavior of αeffas a function of [H2].
We indicate these regions as N < 1, N > 1, and N ≫ 1.
The “vertical shifts” for some dependences in Fig. 4 (such
as the data presented with open and full triangles in the
lower panel) are discussed below. We characterize the
three regions as follows.
1. [H2] < 1012cm−3, N < 1. At these hydrogen den-
sities αeffdecreases with decreasing hydrogen den-
sity. At such conditions the H+
an exothermic proton transfer (from ArH+or H+
to H2) do not have enough collisions with H2 to
3thermal equilibrium prior
to their recombination (see e.g. the discussion in
[36, 37, 38]).The number of these H2 + H+
collisions that a H+
3ion will undergo prior to its
recombination with an electron (at typical condi-
tions of the discussed afterglow experiments) is in-
dicated in the Fig. 4 by number N. If N < 1,
the decay of the plasma caused by the recombina-
tion (at a given electron density) is faster than the
3ions formed by
FIG. 4: Measured dependencies of the effective recombina-
tion rate coefficient of H+
data obtained by Cryo-FALP and TDT-CRDS are plotted in
the upper panel. The data obtained in other experiments are
plotted in the lower panel. We have also plotted examples
of data obtained in our previous studies using: AISA, TDT-
CRDS and FALP. The FALP data by Laube et al.  and
Gougousi et al.  are plotted for comparison. Additionally,
we show the value of the theoretical recombination rate coef-
ficient calculated for binary dissociative recombination (DR),
αBin(300 K) [8, 38]. Note that for this theoretical result, only
the binary DR process is considered, and as such it cannot
depend on the hydrogen density.
3ions on hydrogen density.The
rate of rethermalization. Therefore, in this non-
equilibrium regime, the individual state composi-
tion of H+
3afterglow plasma is different along the
flow tube because the effective recombination rate
depends on the absolute value of electron density,
which varies along the length of the tube. By “state
composition” we mean not only the kinetic energy
distribution, which is established in collisions with
He atoms with a nearly collisional rate, but also
the rotational and ortho-H+
bution. A quantitative description of this particu-
lar regime would require a much deeper theoretical
2. 1012cm−3< [H2] < 5 × 1013cm−3, N > 1.
In this regime, the measured rate coefficients are
nearly independent of [H2]. On the basis of the
same arguments mentioned above, it is clear that
at [H2] > 1012cm−3, the H+
ter hydrogen is injected) has several collisions with
H2prior to its recombination with an electron. Be-
cause only some collisions lead to a change in ro-
tational excitation of H+
3or in ortho↔para transi-
tions [36, 37]), we assume that if [H2] > 1012cm−3
the recombining ions in the flow tube will be in
thermal equilibrium corresponding to the hydrogen
temperature, which is assumed to be equal to the
helium temperature. Because of the independence
of αeff on [H2] we will call this region the “sat-
urated region”. The boundaries of the saturated
region depend on actual helium pressure, temper-
ature and electron density.
enough to find conditions where the value of αeffis
nearly constant (the plateau part of the αeff([H2])
dependence). Depending on experimental condi-
tions we have covered either the whole saturated
region or in some cases just a part of it.
3ion (formed shortly af-
The region is broad
3. [H2] > 5 × 1013cm−3, N ≫ 1. Here the mea-
sured recombination rate coefficient increases with
hydrogen pressure. This behavior is caused by a
formation of H+
5ions and their fast recombination.
The process is temperature and pressure depen-
dent (see details in [24, 50, 51, 52]). Because of
the strong temperature and pressure dependence
of ternary association, the onset of this region de-
pends on these parameters.
At first sight the “vertical shifts” of the dependences
plotted in Fig. 4 are very chaotic. We demonstrate below
that they arise because the apparent binary recombina-
tion rate coefficient (αeff) depends not only on hydrogen
density and temperature but also on the helium density.
In addition, we also show that the temperature depen-
dence of αeffis not monotonic.
We will not discuss the N < 1 region here.
ever, in connection with the data obtained at 77 K we
want to point out one difference from the data obtained
at higher temperatures. In all high temperature experi-
ments made at [H2] < 1012cm−3, we have observed a fast
decrease of αeff with decreasing [H2] (see e.g. the FALP
data measured at 250 K in the lower panel of Fig. 4).
In measurements at 77 K using Cryo-FALP (see the up-
per panel of Fig. 4) the decrease is substantially smaller.
The difference can be partly associated with the effect
of collisional radiative recombination (CRR) at 77 K. At
[H2] ∼ 1013cm−3when the overall recombination rate
coefficient αeff(77 K) > 1.0×10−7cm3s−1we can neglect
the influence of these processes (see the upper panels of
Fig. 3 and Fig. 4).
Figure 4 has a great deal of information, because it ac-
tually shows αeffas a function of three variables T, [H2],
and [He]. By analyzing the data we found the form of
the function αeff(T,[H2],[He]) in the “saturated region”,
i.e. for the plasma where the H+
3ions before recombining
FIG. 5: The effective binary recombination rate coefficients
(αeff) measured at the stated temperatures are shown as func-
tions of the helium density. Upper panel: Low temperature
data – Cryo-FALP (77 K), TDT-CRDS & Cryo-FALP (100 K)
and AISA & FALP (130 K). The horizontal line indicates the
theoretical value for binary dissociative recombination at 77 K
[8, 38]. For details, see the summary in the text. Lower panel:
Higher temperature data – Cryo-FALP (305 K), FALP (250–
260 K, 170 and 195 K), TDT-CRDS (330 K), AISA (250–260,
230 and 150 K). Individual points measured in other laborato-
ries: Smith and Spanel , Laube et al. , Leu et al. 
(see description in the text). The horizontal line indicates the
theoretical value for the binary dissociative recombination of
3at 300 K [8, 38].
undergo a sufficient number of “state changing collisions”
with H2to reach thermal equilibrium. For a better anal-
ysis of the experimental data we plotted αeff measured
at a fixed temperature as a function of helium density.
The rate coefficients measured in the three principally
different afterglow experiments at several different tem-
peratures at hydrogen densities corresponding to the sat-
urated region are plotted as functions of helium density
in Fig. 5.
We briefly summarize the data plotted in Fig. 5. Upper
1. 77 K, Cryo-FALP. The data from measurements at
[H2] ∼ 1013cm−3at different [He] (see upper panel
of Fig. 4). The experiment was intended as a mea-
surement of the pressure dependence at a fixed tem-
perature. The straight line is the best fit through
the measured data.
2. 100 K, TDT-CRDS. The values plotted were ob-
tained from the measured dependences of αeff on
hydrogen density as limits approaching the satu-
rated region (see upper panel of Fig. 4). More de-
tails are given in .
3. 100 K, Cryo-FALP. The present measurements.
4. 130 K, AISA. The data were obtained from the
measured dependence of αeff on hydrogen density
(see lower panel of Fig. 4). More details is given in
5. 130 K, Cryo-FALP. The present measurements.
6. 77 K Theory. The horizontal line indicates the the-
oretical value of the recombination rate coefficient
for binary dissociative recombination (DR) of H+
at 77 K [8, 38].
Lower panel of Fig. 5:
7. 150, 230 and 250 K, AISA. The values shown were
obtained from measured dependencies of αeffon hy-
drogen density as limits approaching the saturated
region (see upper panel of Fig. 4 and Ref. ).
The accuracy of the values is high because the val-
ues were obtained from a number of measurements.
8. 170 and 195 K, FALP & AISA. The FALP data
were obtained similarly to the AISA data, i.e.
as limits approaching the saturated region. The
straight lines connect the measured FALP points
with the AISA points, which are obtained by an ex-
trapolation of the data measured by AISA at 130–
9. 250–260 K, AISA & FALP. Compilation of data
from several experiments (for details see Refs. [23,
10. 305 K, Cryo-FALP. The data were obtained by
directly changing the helium pressure in the flow
11. 330 K, TDT-CRDS. The values were obtained as
limits approaching the saturated region (see upper
panel of Fig. 4).
12. 210 and 300 K, Leu et al. . In the experiment a
microwave technique was used to measure the elec-
tron density. The values 2.3 × 10−7cm3s−1and
3.3 × 10−7cm3s−1for 300 K and 210 K respec-
tively (pressure ∼ 2.6 kPa) were taken from Figs.
2 and 4 of Ref. .
13. 300 K, Laube et al. , a FALP (FALP-MS) ex-
periment.The measured value is αeff = 7.8 ×
10−8cm3s−1(see Fig. 4). He pressure was ∼ 70 Pa.
Used hydrogen density corresponds to the satu-
14. 300 K, Smith & Spanel  a FALP experiment.
We show the value from their plot of the recombina-
tion rate coefficient as a function of position along
the flow tube (in Fig. 4). In the conditions that
arise shortly after injection of hydrogen, Smith &
Spanel measured the value αeff∼ 6×10−8cm3s−1
for a relatively long time (see Fig. 4 in ). Fur-
ther downstream, they obtained a lower value of the
recombination rate coefficient. The helium pressure
was ∼ 260 Pa.
15. 300 K, Gougousi et al. . We did not include
their value in the graph because their measure-
ments were at hydrogen densities too high and
therefore out of the saturated region. We only men-
tion their experiment in order to show that we in-
cluded this study in our considerations.
With regard to data obtained by Laube et al. ,
Smith & Spanel , and Gougousi et al. , we have
not analyzed their experiments in full detail. However,
it is clear from their studies that, in agreement with our
experimental data, there is a general trend: The effective
rate constant αeff increases with the increase of helium
density (in the temperature interval covered). In Refs.
[15, 16, 49] the authors used relatively low helium pres-
sures. Therefore, the increase of αeffwith [He] was not as
large as we observe in our measurements, such as Cryo-
FALP. Very high pressure was used in . As a result,
they obtained large αeff in agreement with the trend.
The experimental data plotted in Fig. 5 show that the
apparent effective binary recombination rate coefficient
αeff measured at a fixed temperature depends linearly
on helium density. We will discuss a possible recombina-
tion mechanism below, but at this point we can assume
that the process has a binary character at very low [He],
whereas with increasing [He] the helium assisted ternary
process begins to contribute substantially to the overall
recombination deionization process. Therefore, we can
write for the observed linear dependence
αeff= αBin(T) + KHe(T)[He](2)
in terms of the rate coefficient αBin(T) for binary recom-
bination and the rate coefficient KHe(T) for ternary He-
assisted recombination. The two coefficients depend on
temperature. Note that for some temperatures we have
the FALP data (e.g. 77 K), for others we used AISA &
FALP data, and also the TDT-CRDS data. The AISA
data were obtained at low helium densities. The FALP
data were taken at high helium densities and in most
cases we could vary the He pressure in the FALP exper-
iments. We extrapolate the data from AISA to temper-
atures 170 K and 195 K. Then we plot a straight line
through these new points (obtained by the extrapola-
tion) and the corresponding points measured by FALP
FIG. 6: The H+
sured in the present study (circles). The theoretical values
shown (the dashed line) are calculated for relative populations
of para and ortho H+
3corresponding to the thermal distribu-
tion at the stated T [8, 38]. The dotted line indicates values
calculated from cross sections obtained in storage ring exper-
iments [10, 11, 55] using cold ion sources. The present values
are obtained as limits approaching the saturated region at low
helium density from plots in Fig. 5. We also show the data
(squares) obtained by Amano in pure hydrogen . Further
details are given in the text.
3binary recombination rate coefficient mea-
(see lower panel in Fig. 5). Using the straight line fits we
obtain experimental values for αBin and KHe at 170 K
and 195 K.
The obtained values of the binary recombination rate
coefficients αBin(T) are plotted in Fig. 6 as a function of
temperature. We also show on Fig. 6 thermal rate coeffi-
cients calculated for binary dissociative recombination of
3[8, 38]. The agreement is very good. Note that the
data for 170 K and 195 K (at high [He]) were published
[24, 48] before the calculation . The data for 250 K
was partially obtained  also before the calculations.
The pressure dependence of the 250 K data set was mea-
sured after  the calculations were published. Some
data for 100 K and 300 K were also measured earlier,
using CRDS [25, 53].
Figure 6 also presents the data obtained by Amano
using absorption spectroscopy  in experiments made
with pure hydrogen at ∼ 40 Pa ([H2]∼ 1016cm−3) used
as a buffer gas. The large values of the measured re-
combination rate coefficients indicate that the molecular
hydrogen is a more effective three body partner (with an
effective ternary rate constant of order of 10−23cm6s−1)
than helium, which is not surprising because H2has ro-
tational and vibrational degrees of freedom.
We have also checked that the observed linear pres-
sure dependence of αeffis not associated with the Lang-
muir probe operating at different pressures from 40 to
2600 Pa. For this test we have used the same probe to
measure the recombination rate coefficient for O+
FIG. 7: The measured ternary recombination rate coefficient,
KHe(T), for helium assisted recombination of H+
trons. The legend indicates the experiments used to extract
the data. The plotted line is shown merely to guide the eye.
dependent because O+
(electron–ion) process. We have also studied the pressure
dependence of recombination rate coefficients for HCO+
and DCO+ions  using the probe. For both cases
pendence. Our HCO+and DCO+rate coefficients are
in good agreement with results of Leu et al.  and
Amano  measured by different techniques. Leu et al.
 used a microwave technique to determine electron
densities but obtained results are consistent with our ob-
The ternary rate coefficients obtained from the data
plotted in the Fig. 5 are presented in Fig. 7 as a func-
tion of buffer gas temperature. The values obtained for
300 K are: αBin(300 K) = (4.7 ± 1.5) × 10−8cm3s−1,
KHe(300 K) = (2.5 ± 1.2) × 10−25cm6s−1. The figure
shows that the ternary rate coefficient has a maximum at
∼ 170 K. Towards lower temperatures the rate coefficient
is decreasing. This is a surprising result if one takes into
account studies of ternary association processes. For such
processes the ternary rate coefficients decrease monoton-
ically with temperature (see Refs. [56, 57, 58]).
We briefly summarize the data plotted in Figure 7:
Our assumption was that αO+
2recombination is a direct binary
is pressure in-
2and HCO+/DCO+) we observed no pressure de-
1. 77 K, Cryo-FALP. The data obtained from the pres-
sure dependence measured in the present experi-
ments at 77 K. See upper panels of Figs 3, 4, and
5. The value of KHe was obtained from measure-
ments that were repeated many times.
2. 100 K, Cryo-FALP & TDT-CRDS. The data ob-
tained by a combination of the values measured
in two experiments. The TDT-CRDS values are
based on measurements of the ion density evolu-
tion during the afterglow using CRDS absorption
spectroscopy. (see the upper panels of Figs. 4 and
5). In order to calculate the ion density from the
absorption signal, thermodynamic equilibrium was
assumed. The assumption should be valid at the
hydrogen densities used. The Cryo-FALP values
were measured in the present experiments (upper
panel of Fig. 5).
3. 130, 170, and 195 K, AISA & FALP. Data obtained
in two experiments (Fig. 5).
4. 150 and 230 K, AISA. AISA was used to measure
the dependence of αeff on hydrogen density over
a broad range including the “saturated region”,
(lower panel of Fig. 4). From these measurements,
average values of the recombination rate coefficient
in the “saturated region” have been obtained for
several temperatures (see Fig. 5). These data are
very accurate because the measurements have been
done many times at a fixed temperature. As we
did not measure the dependence on helium pres-
sure in the AISA experiments, we calculated KHe
using Eq. (2) with the current theoretical value
for αBin(T). The data have been obtained at low
pressures, 160–330 Pa. Therefore, the ternary rate
coefficients extracted from this data have large er-
5. 250–260 K, FALP & AISA. These data were ob-
tained in several experiments. A detailed descrip-
tion was given in our previous publication  but
there is one key difference: In that study,  the
“260 K” plot also included values obtained at tem-
peratures close to but different from 260 K; the
data shown there for αeff(260 K) were recalculated
from the measured values, assuming a T−0.5depen-
dence, which is a small correction. Because of the
additional experiments with Cryo-FALP the recal-
culation procedure was not necessary in the present
work. Now it is clear that αeff depends on both
temperature and pressure; in the vicinity of 260 K
the temperature dependence is steeper than T−0.5.
6. 305 K, Cryo-FALP. In this experiment the pressure
dependence was measured (see Fig. 5).
7. 330 K, TDT-CRDS. The measurements were sim-
ilar to those at 100 K (upper panel of Fig. 4 and
lower panel of Fig. 5). Two different absorption
lines were used in these studies. We have obtained
the same value of the rate coefficient αeff using ei-
ther absorption line.
8. 210 and 300 K, Data from Leu et al. . (see
Fig. 5).We have utilized the current theoreti-
cal binary recombination rate coefficients for 210
and 300 K and Eq. 2 to obtain the corresponding
ternary rate coefficients.
IV. THEORETICAL MODEL FOR
HELIUM-ASSISTED NEUTRALIZATION OF
THE AFTERGLOW PLASMA
It is possible to estimate theoretically the rate coef-
ficient, Eq. (2), of the He-assisted recombination of H+
with electrons in the following way. To stress that it is an
estimated value, we use symbol K3dfor the theoretical
He-assisted recombination is treated as a two step pro-
cess. In electron-H+
3scattering, rotational autoionization
resonances of the neutral molecule H∗
role (see, for example Ref. [7, 59]). The star next to H3
refers to the unstable character of these autoionizing res-
onances. Such resonances with angular momentum l = 1
could be very quite broad due to the high probability
to capture an electron into the l = 1 partial wave, but
they normally contribute relatively little to the two-body,
resonances decay back into a free electron and an H+
ion. As we demonstrate below, lifetimes of some of these
resonances can be quite long. If, during their lifetime,
3molecule collides with a helium atom, the colli-
sion can lead to a change in the electronic state of the
outer electron or in the rotational state of the H+
therefore, can make the autoionization process impossi-
ble (or, at least, much less probable than the dissociation
of H3). The overall rate coefficient for such He-assisted
recombination is given by the formula derived in Ref.
3play an important
3+ e−recombination , because almost always, such
K3d= kl∆tα∗, (3)
where α∗is the rate coefficient for the formation of H∗
klis the rate coefficient for l-changing collisions between
He and H∗
Finally, ∆t is the delay time in the H+
the sense introduced by Smith ): it is an additional
time that the electron spends near the ion in the modi-
fied Coulomb potential compared to the collisional time
in a pure Coulomb potential. The delay time and the
coefficient α∗ are substantially different from zero only
near resonances. Therefore, the three-body rate coeffi-
cient discussed above is expected to vary resonantly as a
function of collisional energy.
We stress here that the rate constants kl, and α∗de-
scribed above depend on the corresponding scattering en-
ergies (and defined as usual as cross-sections multiplied
by relative velocities). They are not yet thermally av-
eraged over the Maxwell-Boltzmann distribution. Cor-
respondingly, the ternary rate constant K3din Eq. (3)
depends on two energies: the energy E of the H+
collision due to the dependence of ∆t and α∗on E; and
the energy EHeof collision between He and H∗
dependence of klon EHe. In this approach, however, the
coefficient kl(EHe) is considered to be constant over the
energy interval of interest. The rate coefficient α∗in our
estimation is evaluated as vσ(E), where v = (2E/m)1/2
is the relative velocity, σ(E) is the cross-section for the
3leading to the eventual dissociation of H∗
3+ e−collision in
3due to the
process leading to the delay time, m is the reduced mass
of the H+
For the following discussion, we assume that at a given
energy E, there could be several open H+
tion channels (i = 1,2...no). Such a situation is possi-
ble for the conditions of the present experiment. If there
are several open channels, the three-body coefficient K3d
above should be averaged over the incident channels and
summed over the final ones. If the incident ionization
channel (before collision) is denoted by the index i, and
the final one (following the collision) is j, then the corre-
sponding rate coefficient for the three-body recombina-
tion during the i → j inelastic collision is given by
with σji(E) being the cross-section for the inelastic col-
lision. The delay time ∆tjifor the process is an element
of the delay-time matrix ∆t as it is introduced and dis-
cussed by Smith :
where Sjiis an element of the scattering matrix for the
i → j process. Notice a difference in conventions in the
present paper and Ref.: Here, the first index in
each matrix denotes the final channel. The second index
denotes the incident channel. In Ref. , the convention
adopted was the opposite, i.e., the first index ∼ incident
channel, second index ∼ final one.
The cross-section σji(E) is given by
Using equations (4, 5, 6) and taking the sum over the
final ionization channels j, we obtain the three-body rate
if the initial state of the H+
The sum in the second line can be simplified as 
= Qii, (7)
where Qii is a diagonal element of the lifetime matrix
introduced by Smith 
Q = −i?S†dS
FIG. 8: Diagonal elements Qii of matrix Q for the three low-
est (rotational) incident channels for the e−+ H+
The rotational channels are (N+,K+) = (11),(10), and (22).
Each maximum in Qii corresponds to an autoionization reso-
nance. The lifetime of a resonances is given by Qii/4 eval-
uated at the maximum if there is only one channel open,
Qii = Q.
and the product S† dS
lar matrix product. Due to the aforementioned difference
in matrix index conventions, the order of the product is
opposite to the one in Ref. . Therefore, for K3d
dEin the above equation is the regu-
or in atomic units
The next step in the evaluation of the three-body rate
coefficient is the thermal average over incident ionization
channels i and over the Maxwell-Boltzmann velocity dis-
tribution for a given temperature T. The average over
incident channels is given by
where wi= (2N+
the incident channel, N+
tum and the nuclear spin of the H+
is the energy of the incident channel (rotational energy
of channel i). The average over the velocity (energy) dis-
tribution should be performed over the two energy vari-
ables, E and EHe, in K3d
not depend on EHe, the thermal averaging is reduced to
the familiar two-body averaging integral over E only
i+1)(2Ii+1) is the statistical weight of
iand Iiare the angular momen-
3ion respectively, Ei
i(E,EHe). Because K3d
Combining the above equations, we obtain
or using the lifetime matrix element Qii,
The lifetime matrix Q is calculated using Eq. (8) from
the scattering matrices obtained numerically [7, 8]. In
practice, the scattering matrix S(N)(and matrix Q(N))
are calculated for a fixed value of the total angular mo-
mentum N of the H+
of the ionic N+and electronic l angular momenta. The
from different N should be accounted
in Eq. (14), namely as
3+ e−system, which is the sum
(2N + 1)Q(N)
The sum in the above equation runs over all N for which
the incident channel i enters into the scattering matrix.
Since the principal contribution to the rotational captur-
ing of the electron comes from the l = 1 partial wave, the
sum has three or less terms.
The collisional l-changing process is known to be rel-
atively effective for excited atomic Rydberg states .
Because the H∗
collisions have a large principal quantum number (n ∼
40–100, see Fig. 8), it is reasonable to use the atomic
rates for l-changing collisions in our estimates.
Na∗+ He l-changing collisions  the experimental rate
is 2.3 × 10−8cm3s−1. In our estimation, we use this
value for kl. Using the procedure described above and
the value above for klwe have calculated thermally av-
eraged rate coefficient for the He-assisted recombination
3. The coefficient is shown in Fig. 9 as a function
of temperature. The overall agreement with the experi-
mental rate coefficient shown in Fig. 7 is reasonable given
the approximative approach that we used here. Below
150 K, the experimental curve drops down, but the the-
oretical (dashed) curve continues to grow. Interestingly,
the curve for pure ortho-H+
3(dot-dashed curve) has a be-
havior similar to the experiment. In fact, it is possible
that in the experiment the ortho and para-H+
in thermal equilibrium at low temperatures. Our previ-
ous calculation  showed that the binary recombination
rate of H+
than for para-H+
3: At low temperatures, due to fast re-
combination of para-H+
be larger than the one at thermal ortho/para equilib-
rium. In such a situation the averaged theoretical rate
coefficient K3din Fig. 9 would be closer to the curve for
3, i.e. it would be lower at small T similar to the
experimental behavior (Fig. 7).
3(l = 1) molecules formed in the H+
3with electrons is much slower for ortho-H+
3the ratio ortho/para-H+
FIG. 9: Calculated thermally-averaged three-body rate co-
˙K3d¸. The rate coefficients calculated separately
for ortho and para-H+
ing plasma is not in thermal equilibrium with respect to or-
tho/para ratio, the averaged rate coefficient (dashed) could
be very different from the one shown.
3are very different. If the recombin-
V.CONCLUSIONS AND DISCUSSION
We have studied recombination of H+
glow plasma, in the presence of a helium buffer gas with
a small admixture of molecular hydrogen. The helium
densities were in the range 0.5 – 6 × 1017cm−3and hy-
drogen densities 1–100 × 1012cm−3. In such conditions
3ions formed in the plasma have several collisions
with H2 before they recombine with electrons.
we assume that in these conditions the ions are in or-
tho/para thermal equilibrium. The apparent binary re-
combination rate coefficient αeff was measured as func-
tion of hydrogen and helium densities for several tem-
peratures in the 77–330 K range. From the experimental
data we have derived the binary and ternary recombina-
tion rate coefficients and their dependences on tempera-
ture. The measured binary recombination rate coefficient
is in good agreement with recent theoretical calculation
over the whole covered temperature range (77–330 K).
Therefore, for the first time, the recombination rate co-
efficients obtained in afterglow plasma experiments agree
with storage ring experiments and with theoretical val-
ues. As we have demonstrated in the present study, re-
sults from previous afterglow plasma experiments were
previously interpreted without taking into account the
role of the buffer gas and, as a result, those experiments
seemed to disagree with the storage ring experiments and
with theoretical calculation. The present work reconcile
observation data from the plasma and storage ring ex-
periments and the theoretical result.
3ions in an after-
The obtained binary rate coefficient at 300 K is αBin=
(4.7±1.5)×10−8cm3s−1. The observed ternary recombi-
nation (KHe(300 K) = (2.5±1.2)×10−25cm6s−1) is fast
and at pressures of hunderds of Pa it is already dom-
inant over the binary process. The dependence of the
measured ternary recombination rate coefficient on tem-
perature has a maximum at ∼ 130–170 K. The observed
ternary process is more effective by factor about 100 than
the process of ternary recombination described by Bates
and Khare .
To explain the process of fast ternary recombination
we have developed a theoretical model for the process. In
particular, we have calculated the delay time in H+
collisions (∆t as introduced by Smith ). We found
that the delay time is sensitive to the rotational and
nuclear-spin states of the H+
collision energies E ∼ 150 cm−1can be of order of
100 ps for para-H+
atom, which enhances the overall plasma neutralization.
The calculated delay time was used to derive the ternary
recombination rate coefficient. The derived ternary coef-
ficient as a function of temperature is smaller than the ex-
perimental value by a factor of order 2-10, which is plau-
sible agreement for such a sophisticated process (from
a theoretical ab initio point of view) within the some-
what heuristic theoretical method employed. Theory can
in principle be further improved. In particular, the co-
efficient for l-changing collisions can be evaluated more
accurately. At temperatures below 130 K there is a qual-
itative difference between measured and calculated val-
ues of the ternary rate coefficient: the experimental rate
constant decreases with temperature, the theoretical one
continues to grow. It is possible that at low temperature
ortho and para-H+
3ion are not in thermal equilibrium in
this afterglow plasma because of the very different binary
rate constants αBin(T) that have been predicted for low
temperatures. Nevertheless, for a first semiquantitative
study of this kind, the agreement between theory and ex-
periment for the ternary rate coefficients is encouraging.
3ion. The delay time at
3. During that collision time, the H∗
3+ e−complex) can collide with a helium
This work is a part of the research plan MSM
0021620834 financed by the Ministry of Education of
the Czech Republic and was partly supported by GACR
by GAUK 53607, GAUK 124707 and GAUK 86908. It
has also benefitted from support from the National Sci-
ence Foundation, Grants Nos. PHY-0427460 and PHY-
0427376, and from an allocation of NERSC supercom-
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