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Master Equation for Hydrogen Recombination on Grain Surfaces Master Equation for Hydrogen Recombination on Grain Surfaces
Gianfranco Vidali
Department of Physics, Syracuse University, Syracuse, NY
Ofer Biham
Syracuse University
Itay Furman
The Hebrew University of Jerusalem
Valerio Pirronello
Universita di Catania
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Recommended Citation Recommended Citation
Vidali, Gianfranco; Biham, Ofer; Furman, Itay; and Pirronello, Valerio, "Master Equation for Hydrogen
Recombination on Grain Surfaces" (2000).
Physics
. 509.
https://surface.syr.edu/phy/509
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arXiv:astro-ph/0012267v1 12 Dec 2000
Master Equation for Hydrogen Recombination on Grain Surfaces
Ofer Biham and Itay Furman
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Valerio Pirronello
Dipartimento di Metodologie Fisiche e Chimiche per l’Ingegneria,
Universita’ di Catania, 95125 Catania, Sicily, Italy
and
Gianfranco Vidali
Department of Physics, Syracuse University, Syracuse, NY 13244
ABSTRACT
Recent experimental results on the formation of molecular hyd rogen on astrophysi-
cally relevant surfaces under conditions similar to those encountered in the interstellar
medium provided useful quantitative information about these processes. Rate equation
analysis of experiments on olivine and amorphous carbon surfaces provided the acti-
vation energy barriers for the diffusion and desorption pr ocesses relevant to hydrogen
recombination on these surfaces. However, the suitability of rate equations for the sim-
ulation of hydrogen recombination on interstellar grains, where there might be very few
atoms on a grain at any given time, has been questioned. To resolve this problem, we
introduce a master equation that takes into account both the discrete nature of the H
atoms and the fluctuations in the number of atoms on a grain. The hydrogen recombi-
nation rate on microscopic grains, as a function of grain size and temperature, is then
calculated using the master equation. The results are compared to those obtained from
the rate equations and the conditions under which the m aster equation is required are
identified.
Subject headings: d ust— ISM; abundances — ISM; molecules — molecular processes
1. Introduction
The formation of molecular hydrogen in the interstellar medium (ISM) is a process of funda-
mental importance (Duley & Williams 1984; Williams 1998). It was recognized long ago (Gould
& Salpeter 1963) that H
2
cannot form in the gas phase efficiently enough to account for its abun-
dance. It was thus proposed that dus t grains act as catalysts, where an H atom approaching the
surface of a grain has a probability ξ to become adsorbed. The adsorbed H atom (adatom) spends
an average time t
H
(residence time) before leaving the surface. If during the residence time the
H adatom encounters another H adatom, an H
2
molecule will form with a certain probability.
Various aspects of this process were addressed in extensive theoretical studies (Gould & Salpeter
– 2 –
1963; Williams 1968; Hollenbach & Salpeter 1970, 1971; Hollenb ach et al. 1971; Smoluchowski 1981,
1983; Aronowitz & Chang 1985; Duley & Williams 1986; Pirronello & Averna 1988; Sandford &
Allamandolla 1993; Takahashi et al. 1999; Farebrother et al. 1999). In particular, Hollenbach et al.
calculated the sticking and m ob ility of H atoms on grain surfaces. They concluded that tun neling
between adsorption sites, even at temperature as low as T = 10K, provides the required mobility.
The steady state production rate of molecular hydrogen per unit volume was expressed according
to (Hollenbach et al. 1971)
R
H
2
=
1
2
ρ
H
v
H
σγρ
g
, (1)
where ρ
H
and v
H
are the number density and the speed of H atoms in the gas phase, respectively,
σ is the average cross-sectional area of a grain and ρ
g
is the number density of dus t grains. The
parameter γ is the fraction of H atoms striking the grain that eventually form a molecule, namely
γ = ξη, where η is the probability th at an H adatom on the surface will recombine with another H
atom to form H
2
.
Recently, a series of experiments were conducted to measure hydrogen recombination in an
ultra high-vacuum (UHV) chamber by irradiating the sample with two beams of H and D atoms
and monitoring the HD production rate (Pirron ello et al. 1997a,b, 1999). Two different substrates
have been used: a natural olivine (a polycrystalline s ilicate containing Mg
2
SiO
4
and Fe
2
SiO
4
) slab
and an amorphous carbon sample. The substrate temperatures during hydrogen irradiation were
in the range between 5 K and 15 K. The HD formation rate was measured using a quadrupole mass
spectrometer both during irradiation and in a subsequent temperature programmed desorption
(TPD) experiment in which the sample temperature was quickly ramped to over 30 K to desorb
all weakly adsorbed species. It was found that H and D atoms adsorbed on the su rface at the
lowest irradiation temperature of 5 K form m olecules during TPD only above 9 K in the case
of olivine an d above 14 K in the case of amorphous carbon. This indicates that tunn eling alone
does not provide enough mobility to H adatoms to enable recombination, and thermal activation is
required. T he experimental results were analyzed using a rate equation model (Katz et al. 1999).
In this analysis the parameters of the rate equations were fitted to the experimental TPD curves.
These parameters are the activation energy barriers for atomic hydrogen diffusion and desorption,
the barrier for molecular hydrogen desorption and the probability of spontaneous desorp tion of
a hydrogen molecule upon recombination. Using the values of the parameters th at fit best the
experimental results, the efficiency of hydrogen recombination on the olivine and amorphous carbon
surfaces was calculated for interstellar conditions using the same rate equation m odel. It was
found that for both samples the recombination efficiency is strongly dependent on temperature and
exhibits a narrow window of high recombin ation efficiency along the temperature axis.
It was recently pointed out that since hydrogen recombination in the interstellar space takes
place on small grains, rate equations have a limited range of validity (Tielens 1995; Charnley et al.
1997; Caselli et al. 1998; Shalabiea et al. 1998). This is d ue to the fact that these equations take
into account only average concentrations and ignore fluctuations as well as the discrete nature of
the H atoms. These properties become significant in the limit of very small grains and low incoming
flux of H atoms, exactly the conditions en countered in diffuse interstellar clouds where hydrogen
recombination on silicate and carbon surfaces is expected to be relevant. As the number of H atoms
on a grain fluctuates in the range of 0, 1 or 2, the recombination rate cannot be obtained from the
average number alone. This can be easily understood, since the recombination process requires at
– 3 –
least two H atoms simultaneously on the surf ace. Comparisons with Monte Carlo simulations have
shown that the rate equations tend to over-estimate the recomb ination rate. A modified set of rate
equations which exhibits better agreement w ith Monte Carlo simulations was introduced by Caselli
et al. (1998) and applied by Shalabiea et al. (1998) to a variety of chemical reactions. In these
equations the rate coefficients are modified in a semi-empirical way to take into account the effect
of the finite grain size on the recombination process.
In this paper we introduce a master equation that is particularly suitable for the simulation of
chemical reactions on microscopic grains. It takes into account both the discrete nature of the H
atoms as well as the flu ctuations. I ts dynamical variables are the probabilities P
H
(N
H
) that th ere
are N
H
atoms on the grain at time t. The time derivatives
˙
P
H
(N
H
), N
H
= 0, 1, 2, . . . are expressed
in terms of the adsorption, reaction and desorption terms. The master equation provides the time
evolution of P
H
(N
H
), N
H
= 0, 1, 2, . . ., from which the recombination rate can be calculated. We
use it in conjunction w ith the surface parameters obtained experimentally, to explore the hydrogen
recombination pro cess on microscopic grains for grain s izes, fl ux and surface temperatures pertinent
to the conditions in the interstellar medium .
The paper is organized as follows. The rate equation model is described in Sec. 2. The master
equation is introduced in Sec. 3. Computer simulations and results for hydrogen recombination on
microscopic grains under interstellar conditions are presented in Sec. 4. The case of more complex
reactions involving multiple species is considered in Sec. 5 and a summary in Sec. 6.
2. Rate Equations for H
2
Formation on Macroscopic Surfaces
Consider an experiment in which a flux of H atoms is irradiated on the surface. If the tem-
perature is not too low H atoms that stick to the surface perform hops as random walkers and
recombine when they encounter one another. Let n
H
(t) (in monolayers [ML]) be the coverage of H
atoms on the surface and n
H
2
(t) (also in ML) the coverage of H
2
molecules at time t. We obtain
the followin g set of rate equations:
dn
H
dt
= f
H
· (1 − n
H
− n
H
2
) − W
H
n
H
− 2a
H
n
2
H
(2a)
dn
H
2
dt
= µa
H
n
2
H
− W
H
2
n
H
2
. (2b)
The first term on the right hand side of Eq. (2a) represents the flux of H atoms multiplied by the
Langmuir-Hinshelwood rejection term. In this scheme H atoms deposited on top of H atoms or H
2
molecules already on the surface are rejected. The parameter f
H
represents the effective flux of
atoms (in units of MLs
−1
), namely, the (temperature dependent) sticking coefficient ξ(T ) of the
bare surf ace is absorbed into f
H
. The second term in Eq. (2a) represents the desorption of H atoms
from the surface. The desorption coefficient is
W
H
= ν · exp(−E
1
/k
B
T ) (3)
where ν is the attempt rate (standardly taken to be 10
12
s
−1
), E
1
is the activation energy barrier
for desorption of an H atom and T is the temperature. The third term in Eq. (2a) accounts for the
depletion of the H population on the surface due to recombination into H
2
molecules, where
a
H
= ν · exp(−E
0
/k
B
T ) (4)
– 4 –
is the hopping rate of H atoms on the surface and E
0
is the activation energy barrier for H d if-
fusion. Here we assume that diffusion occurs only by thermal hopping, in agreement with recent
experimental results (Katz et al. 1999). We also assume that there is no barrier for recombination.
The first term on the right hand side of Eq. (2b) represents the creation of H
2
molecules. The
factor 2 in the thir d term of Eq. (2a) does not appear here since it takes two H atoms to form one
molecule. The parameter µ represents th e fraction of H
2
molecules that remains on the surf ace
upon formation, while a fr action of (1 − µ) is spontaneously desorbed due to the excess energy
released in the recombination process. The second term in Eq. (2b) describes the desorption of H
2
molecules. T he desorption coefficient is
W
H
2
= ν · exp(−E
2
/k
B
T ), (5)
where E
2
is the activation energy barr ier for H
2
desorption. The H
2
production rate r
H
2
(ML s
−1
)
is given by:
r
H
2
= (1 − µ) · a
H
n
2
H
+ W
H
2
n
H
2
. (6)
This model can be considered as a generalization of the Polanyi-Wigner equation [see e.g. Chan et
al. (1978)]. It provides a description of both first order and second order desorption kinetics f or
different regimes of temperature an d flux.
The model described by Eqs. (2) was us ed by Katz et al. (1999) to analyze the results of the
TPD experiments (Pirronello et al. 1997a,b, 1999). The values of the parameters E
0
, E
1
, E
2
, and
µ, that best fit the experimental results were obtained. For the olivine sample it was found that
E
0
= 24.7 meV, E
1
= 32.1 meV, E
2
= 27.1 meV and µ = 0.33, while for the amorphous carbon
sample E
0
= 44.0 meV, E
1
= 56.7 meV, E
2
= 46.7 meV and µ = 0.413.
The model [Eqs. (2)] was then used in order to describe the steady state conditions that
are reached when both the flux and the temperature are fixed. The steady state solution is then
easily obtained by setting dn
H
/dt = 0 and dn
H
2
/dt = 0 and solving the quadratic equation for n
H
(Biham et al. 1998; Katz et al. 1999). In case th at the Langmuir-Hinshelwood rejection term can
be neglected, the steady-state coverages are
n
H
=
−W
H
+
q
W
2
H
+ 8a
H
f
H
4a
H
(7a)
n
H
2
=
µ
8a
H
W
H
2
W
2
H
+ 4a
H
f
H
− W
H
q
W
2
H
+ 8a
H
f
H
. (7b)
More complicated expressions are obtained when the rejection term is taken into account (Katz et
al. 1999). The recombination efficiency η is defined as the fraction of the adsorbed H atoms that
desorb in the form of H
2
molecules, n amely
η =
r
H
2
f
H
/2
. (8)
Note that under steady state conditions η is limited to the range 0 ≤ η ≤ 1.
By varying the temperatur e and flux over the astrophysically relevant range the domain in
which there is non-negligible recombination efficiency was identified. It was found that the recom-
bination efficiency is highly temperature dependent. For each of the two samples there is a n arrow
window of high efficiency along the temperature axis, which shifts to higher temperatures as the
