# Temperature dependence of binary and ternary recombination of H_ {3}^{+} ions with electrons

**ABSTRACT** We study binary and the recently discovered process of ternary He-assisted recombination of H3+ ions 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 to 330 K. We describe the observed ternary recombination as a two-step mechanism: first, a rotationally excited long-lived neutral molecule H3∗ is formed in electron-H3+ collisions. Second, the H3∗ molecule collides with a helium atom that leads to the formation of a very long-lived Rydberg state with high orbital momentum. We present calculations of the lifetimes of H3∗ and of the ternary recombination rate coefficients for para- and ortho-H3+. The calculations show a large difference between the ternary recombination rate coefficients of ortho- and para-H3+ at temperatures below 300 K. The measured binary and ternary rate coefficients are in reasonable agreement with the calculated values.

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**ABSTRACT:**The chemistry in the diffuse interstellar medium (ISM) initiates the gradual increase of molecular complexity during the life cycle of matter. A key molecule that enables build-up of new molecular bonds and new molecules via proton donation is H. Its evolution is tightly related to molecular hydrogen and thought to be well understood. However, recent observations of ortho and para lines of H2 and H in the diffuse ISM showed a puzzling discrepancy in nuclear spin excitation temperatures and populations between these two key species. H, unlike H2, seems to be out of thermal equilibrium, contrary to the predictions of modern astrochemical models. We conduct the first time-dependent modeling of the para-fractions of H2 and H in the diffuse ISM and compare our results to a set of line-of-sight observations, including new measurements presented in this study. We isolate a set of key reactions for H and find that the destruction of the lowest rotational states of H by dissociative recombination largely controls its ortho/para ratio. A plausible agreement with observations cannot be achieved unless a ratio larger than 1:5 for the destruction of (1, 1)- and (1, 0)-states of H is assumed. Additionally, an increased cosmic-ray ionization rate to 10–15 s–1 further improves the fit whereas variations of other individual physical parameters, such as density and chemical age, have only a minor effect on the predicted ortho/para ratios. Thus, our study calls for new laboratory measurements of the dissociative recombination rate and branching ratio of the key ion H under interstellar conditions.The Astrophysical Journal 05/2014; 787(1):44. · 6.28 Impact Factor - SourceAvailable from: Viatcheslav KokooulineJ. Glosik, R. Plasil, T. Kotrik, P. Dohnal, J. Varju, M. Hejduk, I. Korolov, S. Roucka, V. Kokoouline[Show abstract] [Hide abstract]

**ABSTRACT:**Measurements of recombination rate coefficients of binary and ternary recombination of and ions with electrons in a low temperature plasma are described. The experiments were carried out in the afterglow plasma in helium with a small admixture of Ar and parent gas (H2 or D2). For both ions a linear increase of measured apparent binary recombination rate coefficients (αeff) with increasing helium density was observed: αeff = αBIN + KHe[He]. From the measured dependencies, we have obtained for both ions the binary (αBIN) and the ternary (KHe) rate coefficients and their temperature dependence. For the description of observed ternary recombination a mechanism with two subsequent rate determining steps is proposed. In the first step, in + e (or + e) collision, a rotationally excited long-lived Rydberg molecule (or ) is formed. In the following step (or ) collides with a He atom of the buffer gas and this collision prevents autoionization of (or ). Lifetimes of the formed (or ) and corresponding ternary recombination rate coefficients have been calculated. The theoretical and measured binary and ternary recombination rate coefficients obtained for and ions are in good agreement.Molecular Physics 09/2010; 108(17):2253-2264. · 1.64 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The radiative and ternary association reaction of H+ ions with H2 was studied using a 22-pole RF trap. An overall effective binary rate coefficient (keff) was measured over a broad range of hydrogen densities to determine contributions from the binary and ternary process. The binary rate coefficient of the radiative association was measured for temperature 11-28 K at hydrogen density where ternary process can be neglected. The obtained binary rate coefficient of the radiative association at 11 K is kr(11 K) = (1.7±0.5)×10-16 cm3s-1.Journal of Physics Conference Series 11/2012; 388(1):2041-.

Page 1

arXiv:0903.1656v1 [physics.plasm-ph] 9 Mar 2009

Temperature dependence of binary and ternary recombination of H+

3ions with

electrons

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

V. Kokoouline

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

H+

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

molecule H∗

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

I.INTRODUCTION

Electron scattering by the simplest polyatomic ion H+

is of fundamental importance in plasma physics because

H+

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

of H+

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+

3recombination process.

However, the H+

3recombination experiments

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

3

3interaction

3with an electron there

3ion influence

3and D+

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 [18]). However, the experi-

mental plasma results are reproducible and they demand

an understanding and an integration into the full picture

of H+

3recombination with electrons. The present work

discusses and, we hope, clarifies some aspects of this com-

plex problem.

The plasma measurements are usually carried out in

a He/Ar/H2gas mixture (see the review by Plasil et al.

[3]) or in a pure H2 gas [14, 19]. The main question is

how to “reconcile” [16] the rate constant (including its

dependence on experimental conditions) observed in an

H+

3dominated plasma with actual theory and with data

from the storage ring experiments [18]. 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

of H+

3. The fact that the neutral-stabilized recombina-

tion can sometimes play a role was predicted many years

ago by Bates and Khare [20] 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-

nary H+

3recombination rate coefficient [8, 9, 10, 11, 18].

This would also be negligible in comparison with the bi-

Page 2

2

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 [23] 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 [20]. This suggests

that ternary recombination of H+

intrinsic features of the interaction between H+

trons at low collision energies [23]. The recombination

process described by Bates [20] is a ternary process of a

different nature.

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-

cent years.

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 [24], 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

3. The

3and collisions of H∗

3with he-

3recom-

3ions is more effi-

3ions is associated with

3and elec-

3ions. Before pre-

3recombination in

3plasma. A

3-

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

binary (H+

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 [5].

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+

to para-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

3. Molec-

3ions obtained in the process of H+

3forma-

2to H2) producing H+

3occur at nearly

3and H2, which change ortho-H+

3

Page 3

3

(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 [23]. 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 [39]). 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

again later.

Because a recombining 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 [40]).

Third-body perturbation of H+

a plasma was already mentioned in our recent study [23].

In the present study, we show further experimental ev-

idence for the phenomenon and further aspects of the

theoretical interpretation, in conjunction with numerical

calculation.

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

3is sensi-

3then recombination

3distribution towards the ther-

3ions is

3recom-

3than also the condition for thermal

contains neutral

3binary recombination in

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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.

2

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

molecules H∗

3formed in the electron-ion collisions, and es-

timate the rate coefficient for the ternary channel of H+

recombination. Section V summarizes our conclusions.

3plasma, present

3

II.EXPERIMENTS

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 [41].

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

3dominated

Page 4

4

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FIG. 2:

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 [42] is used to

measure the electron density decay downstream from the

hydrogen entry port [24]. 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+

3domi-

2(“benchmark ion” [3, 4])

3

dne

dt

= −αeffnen+−ne

τD

= −αeffn2

e−ne

τD, (1)

where neand n+are electron and ion densities, respec-

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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

experimental conditions.

3ions at

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%).

Page 5

5

III.EXPERIMENTAL RESULTS

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 [47]. 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.

Upper panel:

3

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 [48].

3

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

3ions

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 [48]. 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.

Lower panel:

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

hydrogen density.

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 ×

10−8cm3s−1.

[49].Low

7. 295 K, experiment by Gougousi et al. [15]. 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,

Cryo-FALP, TDT-CRDS).

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

establish ortho/para-H+

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

2

3

Page 6

6

??

?

?

?

?

?

?

?

?

?

?????

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

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. [49] and

Gougousi et al. [15] 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

analysis.

3/para-H+

3state distri-

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+

How-

3ions before recombining

Page 7

7

? ??

???

?

?

?

?

??? ???

??????

?

?

?

??????

??

???????

?

?

?

?

?

?

?

??

?

?

?

???

??

???????

???

?

?

??? ???

?

?

?

?

?

?

?

?

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 [16], Laube et al. [49], Leu et al. [51]

(see description in the text). The horizontal line indicates the

theoretical value for the binary dissociative recombination of

H+

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

panel:

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 [53].

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

[48].

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].

3

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. [48]).

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–

230 K.

9. 250–260 K, AISA & FALP. Compilation of data

from several experiments (for details see Refs. [23,

54]).

10. 305 K, Cryo-FALP. The data were obtained by

directly changing the helium pressure in the flow

tube.

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. [51]. 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. [51].

13. 300 K, Laube et al. [49], a FALP (FALP-MS) ex-

periment.The measured value is αeff = 7.8 ×

Page 8

8

10−8cm3s−1(see Fig. 4). He pressure was ∼ 70 Pa.

Used hydrogen density corresponds to the satu-

rated region.

14. 300 K, Smith & Spanel [16] 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 [16]). 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. [15]. 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. [49],

Smith & Spanel [16], and Gougousi et al. [15], 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 [51]. 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 [14]. 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

H+

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 [8]. The data for 250 K

was partially obtained [3] also before the calculations.

The pressure dependence of the 250 K data set was mea-

sured after [23] 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 [14] 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+

2ions

Page 9

9

?

?

?

?

?

?

?

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.

3with elec-

[3, 4].

dependent because O+

(electron–ion) process. We have also studied the pressure

dependence of recombination rate coefficients for HCO+

and DCO+ions [54] using the probe. For both cases

(O+

pendence. Our HCO+and DCO+rate coefficients are

in good agreement with results of Leu et al. [51] and

Amano [14] measured by different techniques. Leu et al.

[51] used a microwave technique to determine electron

densities but obtained results are consistent with our ob-

servation.

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

2

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-

ror bars.

5. 250–260 K, FALP & AISA. These data were ob-

tained in several experiments. A detailed descrip-

tion was given in our previous publication [23] but

there is one key difference: In that study, [23] 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. [51]. (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.

Page 10

10

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

coefficient.

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,

H+

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,

the H∗

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.

[23]:

3

3play an important

3+ e−recombination [7], because almost always, such

3

3ion and

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 [60]): 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

3;

3leading to the eventual dissociation of H∗

3.

3+ e−collision in

3+ e−

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 [60]:

3+ e−system.

3+ e−ioniza-

kl∆tji

2E/mσji(E),(4)

∆tji= R

?

−i?1

Sji

dSji

dE

?

, (5)

where Sjiis an element of the scattering matrix for the

i → j process. Notice a difference in conventions in the

present paper and Ref.[60]: Here, the first index in

each matrix denotes the final channel. The second index

denotes the incident channel. In Ref. [60], 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

σji(E) =

?2π

2mE|Sji|2. (6)

Using equations (4, 5, 6) and taking the sum over the

final ionization channels j, we obtain the three-body rate

coefficient K3d

i

if the initial state of the H+

is i:

3+ e−system

K3d

i

= kl

no

?

j=1

R

?

−i?1

Sji

dSji

dE

??

no

?

2E/m?2π

2mE|Sji|2

= kl(2E)−1/2m−3/2?2π

j=1

R

?

−i?1

Sji

dSji

dE

?

|Sji|2

The sum in the second line can be simplified as [60]

no

?

j=1

R

?

−i?1

Sji

dSji

dE

?

=

no

?

j=1

R

?

S+dS

−i?S†

ji

dSji

dE

?

= −i?

?

dE

?

ii

= Qii, (7)

where Qii is a diagonal element of the lifetime matrix

introduced by Smith [60]

Q = −i?S†dS

dE

(8)

Page 11

11

???

???

?

?

?

???

???

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.

3collisions.

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. [60]. Therefore, for K3d

have

dEin the above equation is the regu-

i

we

K3d

i

= klm−3/2?2

π

√2EQii, (9)

or in atomic units

K3d

i

= kl

π

√2EQii. (10)

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

?

iK3d

iwiexp

?

−Ei

−Ei

kBT

?

?

iwiexp

?

kBT

?

,(11)

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

i

does

2

?π(kBT)3

?∞

0

K3d

i(E)exp

?

−

E

kBT

?√EdE . (12)

Combining the above equations, we obtain

?K3d

or using the lifetime matrix element Qii,

i

?=

2?

i

?∞

?π(kBT)3?

0K3d

iwiexp

?

−E+Ei

kBT

?

?√EdE

−Ei

iwiexp

kBT

?

,(13)

?K3d

i

?=

?

2π

(kBT)3

kl

?

i

?∞

?

0Qiiwiexp

?

−E+Ei

−Ei

kBT

?

?

dE

iwiexp

?

kBT

.

(14)

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

contributions Q(N)

ii

from different N should be accounted

in Eq. (14), namely as

3+ e−system, which is the sum

Qii=

1

2N++ 1

?

N

(2N + 1)Q(N)

ii

. (15)

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 [61].

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 [61] 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

of H+

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 [8] 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

ortho-H+

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+

3+ e−

For

3are not

3with electrons is much slower for ortho-H+

3

3the ratio ortho/para-H+

3could

Page 12

12

?

?

??

?

?

?

?

?

?

FIG. 9: Calculated thermally-averaged three-body rate co-

efficient

˙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

the H+

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-

Thus,

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 [20].

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 [60]). 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+

molecule (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.

3+ e−

3ion. The delay time at

3. During that collision time, the H∗

3+ e−complex) can collide with a helium

3

Acknowledgments

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

(202/07/0495,205/09/1183,202/09/0642,202/08/H057)

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-

puting resources.

[1] T. Oka, Philos. Trans. R. Soc. London, Ser. A 364, 2847

(2006).

[2] T. Geballe and T. Oka, Science 312, 1610 (2006).

Page 13

13

[3] R. Plaˇ sil, J. Glos´ ık, V. Poterya, P. Kudrna, J. Rusz,

M. Tich´ y, and A. Pysanenko, Int. J. Mass Spectrom. 218,

105 (2002).

[4] R. Johnsen, J. Phys.: Conf. Ser. 4, 83 (2005).

[5] M. Larsson and A. Orel, Dissociative Recombination of

Molecular Ions (Cambridge University Press, Cambridge,

2008).

[6] V. Kokoouline, C. Greene, and B. Esry, Nature 412, 891

(2001).

[7] V. Kokoouline and C. H. Greene, Phys. Rev. A 68,

012703 (2003).

[8] S. dos Santos, V. Kokoouline, and C. Greene, J. Chem.

Phys. 127, 124309 (2007).

[9] C. Jungen and S. T. Pratt, Phys. Rev. Lett. 102, 023201

(pages 4) (2009).

[10] H. Kreckel, M. Motsch, J. Mikosch, J. Glos´ ık, R. Plaˇ sil,

S. Altevogt, V. Andrianarijaona, H. Buhr, J. Hoffmann,

L. Lammich, et al., Phys. Rev. Lett. 95, 263201 (2005).

[11] B. J. McCall, A. J. Huneycutt,

T. R. Geballe, N. Djuric, G. H. Dunn, J. Semaniak,

O. Novotny, A. Al-Khalili, A. Ehlerding, et al., Nature

422, 500 (2003).

[12] J. Glosık, R. Plaˇ sil, V. Poterya, P. Kudrna, and M. Tich` y,

Chem. Phys. Lett 331 (2000).

[13] J. Glos´ ık, R. Plasil, V. Poterya, P. Kudrna, M. Tich` y,

and A. Pysanenko, J. Phys. B: At. Mol. Opt. Phys 34,

L485 (2001).

[14] T. Amano, J. Chem. Phys 92, 6492 (1990).

[15] T. Gougousi, R. Johnsen, and M. F. Golde, Int. J. Mass

Spectrom. 149-150, 131 (1995).

[16] D. Smith and P.ˇSpanˇ el, Int. J. Mass Spectrom. 129, 163

(1993).

[17] V. Poterya, J. Glos´ ık, R. Plaˇ sil, M. Tich´ y, P. Kudrna,

and A. Pysanenko, Phys. Rev. Lett. 88, 044802 (2002).

[18] M. Larsson, B. McCall, and A. Orel, Chem. Phys. Lett.

462, 145 (2008).

[19] M. Feh´ er, A. Rohrbacher, and J. P. Maier, Chem. Phys.

185, 357 (1994).

[20] D. Bates and S. Khare, Proc. Phys. Soc. 85, 231 (1965).

[21] Y. Cao and R. Johnsen, J. Chem. Phys. 94, 5443 (1991).

[22] G. Gousset, B. Sayer, and J. Berlande, Phys. Rev. A 16,

1070 (1977).

[23] J. Glosik, I. Korolov, R. Plasil, O. Novotny, T. Kotrik,

P. Hlavenka, J. Varju, I. A. Mikhailov, V. Kokoouline,

and C. H. Greene, J. Phys. B: At. Mol. Phys. 41, 191001

(2008).

[24] J. Glos´ ık, O. Novotn` y, A. Pysanenko, P. Zakouril,

R. Plaˇ sil, P. Kudrna, and V. Poterya, Plasma Sources

Sci. Technol. 12, S117 (2003).

[25] P. Macko, G. B´ an´ o, P. Hlavenka, R. Plaˇ sil, V. Poterya,

A. Pysanenko, O. Votava, R. Johnsen, and J. Glos´ ık, Int.

J. Mass Spectrom. 233, 299 (2004).

[26] P. Macko, G. B´ an´ o, P. Hlavenka, R. Plaˇ sil, V. Poterya,

A. Pysanenko, K. Dryahina, O. Votava, and J. Glosık,

Acta Phys. Slovaca 54, 263 (2004).

[27] P. Kudrna, R. Plasil, J. Glosik, M. Tichy, V. Poteriya,

and J. Rusz, Czech. J. Phys. 50, 329 (2000).

[28] R. Johnsen and J. Mitchell, Advance in Gas-Phase Ion

Chemistry, vol. 3 (Elsevier, 1998).

[29] N. Adams, D. Smith, and E. Alge, J. Chem. Phys 81,

1778 (1984).

[30] P. Macko, R. Plasil, P. Kudrna, P. Hlavenka, V. Poterya,

A. Pysanenko, G. Bano, and J. Glosik, Czech. J. Phys.

52, 695 (2002).

R. J. Saykally,

[31] D. Strasser,

M. Lange, A. Naaman, D. Schwalm, A. Wolf, and D. Za-

jfman, Phys. Rev. A 66, 032719 (2002).

[32] N. Adams, V. Poterya, and L. Babcock, Mass Spectrom.

Rev. 25, 798 (2006).

[33] J. Glosik, P. Hlavenka, R. Plasil, F. Windisch, D. Gerlich,

A. Wolf, and H. Kreckel, Philos. Trans. R. Soc. A-Math.

Phys. Eng. Sci. 364, 2931 (2006).

[34] I. Korolov, R. Plasil, T. Kotrik, P. Dohnal, O. Novotny,

and J. Glosik, Contrib. Plasma Phys. 48, 461 (2008).

[35] D. Trunec, P.ˇSpanˇ el, and D. Smith, Chem. Phys. Lett.

372, 728 (2003).

[36] M. Cordonnier, D. Uy, R. Dickson, K. Kerr, Y. Zhang,

and T. Oka, J. Chem. Phys. 113, 3181 (2000).

[37] D. Gerlich, F. Windisch, P. Hlavenka, R. Plaˇ sil, and

J. Glosik, Philos. Trans. R. Soc. London, Ser. A 364,

3007 (2006).

[38] L. Pagani, C. Vastel, E. Hugo, V. Kokoouline, C. Greene,

A. Bacmann, E. Bayet, C. Ceccarelli, R. Peng, and

S. Schlemmer, Astron. Astrophys. 494, 623 (2009).

[39] P. Atkins, Physical Chemistry (Oxford University Press,

1988).

[40] D. Gerlich and S. Horning, Chem. Rev. 92, 1509 (1992).

[41] A. Florescu-Mitchell and J. Mitchell, Phys. Rep. 430,

277 (2006).

[42] J. Swift and M. Schwar, Electrical Probes for Plasma Di-

agnostics (Iliffe, London, 1970).

[43] R. Plasil, I. Korolov, T. Kotrik, P. Dohnal, G. Bano,

Z. Donko, and J. Glosik, Eur. Phys. J. D (2009), in print.

[44] I. Korolov, T. Kotrik, R. Plasil, J. Varju, M. Hejduk, and

J. Glosik, Contrib. Plasma Phys. 48, 521 (2008).

[45] E. McDaniel, J. Mitchell, and M. Rudd, Atomic colli-

sions, heavy Particles Projectiles (John Wiley & Sons,

New York, 1993).

[46] M. Skrzypkowski, R. Johnsen, R. Rosati, and M. Golde,

Chem. Phys. 296, 23 (2004).

[47] O. Pysanenko, O. Novotny, P. Zakouril, R. Plasil,

V. Poterya, and J. Glosik, Czech. J. Phys. 52, 681 (2002).

[48] J. Glosık,R. Plasil, A. Pysanenko,

P. Hlavenka, P. Macko, and G. Bano, J. Phys.: Conf.

Ser. 4, 104 (2005).

[49] S. Laub´ e, A. Le Padellec, O. Sidko, C. Rebrion-Rowe,

J. Mitchell, and B. Rowe, J. Phys. B: At. Mol. Opt. Phys.

31, 2111 (1998).

[50] O. Novotny, R. Plasil, A. Pysanenko, I. Korolov, and

J. Glosik, J. Phys. B-At. Mol. Opt. Phys. 39, 2561 (2006).

[51] M. Leu, M. Biondi, and R. Johnsen, Phys. Rev. A 8, 413

(1973).

[52] K. Hiraoka and P. Kebarle, J. Chem. Phys. 62, 2267

(1975).

[53] R. Plasil, P. Hlavenka, P. Macko, G. B´ an´ o, A. Pysanenko,

and J. Glos´ ık, J. Phys.: Conf. Ser. 4, 118 (2005).

[54] I. Korolov, R. Plasil, T. Kotrik, P. Dohnal, and J. Glosik,

Int. J. Mass Spectrom. 280, 144 (2009).

[55] B. McCall, A. Huneycutt, R. Saykally, N. Djuric,

G. Dunn, J. Semaniak, O. Novotny, A. Al-Khalili,

A. Ehlerding, F. Hellberg, et al., Phys. Rev. A 70, 052716

(2004).

[56] N. Adams and D. Smith, Reactions of Small Transient

Species (Academic Press, New York, 1983).

[57] J. Glosik, P. Zakouril, V. Hanzal, and V. Skalsky, Int. J.

Mass Spectrom. Ion Process. 150, 187 (1995).

[58] J. Glosik, G. Bano, E. Ferguson, and W. Lindinger, Int.

J. Mass Spectrom. 176, 177 (1998).

L. Lammich,H. Kreckel,S. Krohn,

O. Novotny,

Page 14

14

[59] I. Mistr´ ık, R. Reichle, U. M¨ uller, H. Helm, M. Jungen,

and J. A. Stephens, Phys. Rev. A 61, 033410 (2000).

[60] F. T. Smith, Phys. Rev. 118, 349 (1960).

[61] A. P. Hickman, Phys. Rev. A 18, 1339 (1978).

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