H2 FORMATION ISM - Henry A. Rowland Department of Physics

Ben Waghorn, APRIL 2003
Department of Physics and Astronomy, Johns Hopkins University
[email protected]
The formation of molecular hydrogen in the ISM is not well understood. A widely
accepted method for estimating the formation rate of H2 was through the modeling of the
recombination of hydrogen atoms on dust surfaces. Hollebach et al. (1971) were the first
to use this model by calculating the sticking and mobility of H atoms on the surface of
grains, yielding the steady state rate equation for H2 formation. It was later discovered
that hydrogen recombination takes place on small dust grains, limiting the validity of
these earlier rate equations. Ofer at al. (2001) proposed at master equation for hydrogen
recombination on grain surfaces taking into account both the discrete nature of the H
atom and the fluctuations in the number of atoms on the grain. The equation can then
calculate the recombination rate of H as a function of the grain size and the temperature.
During these developments in theories to understand the formation of H2 other methods
for the formation were identified and examined, in particular the negative ion route (Field
2000). These gas phase reaction were further investigated by Glover (2003) under the
relevant conditions for diffuse molecular clouds in the ISM and were found to be
dominated by H2 formation on dust grain surfaces. In this review article I will be
summarizing the above mentioned theories for molecular hydrogen’s formation rate in
the ISM and will be comparing the results to observational data, primarily obtained using
FUSE obsrervations.
Earlier attempts to answer the question
of the formation rate were made by
considering dust grains that consisted of
perfectly regular grain surfaces, made of
a dirty ice material (Gould and Salpeter
1963). Their method contained an error
by not including the zero-point energy of
the absorbed energy, the results of which
found that the recombination would be
efficient for temperatures below 10˚20˚K. This error was corrected finding
temperatures below 5˚-8˚K but this
method overestimated the zero-point
effect. A more accurate rate equation
wavefunctions and binding energies for
the adsorbed states of a hydrogen atom
on a perfect surface. Hollenbach and
Molecular hydrogen is the most
abundant molecule in the universe and is
the dominant mass component of gas in
regions of star formation. It also forms
important intermediate stages in the
formation of larger molecules and acts as
a coolant in certain conditions,
especially in the early universe.
However the process of H2 formation is
still not completely understood. It is
widely believed that the formation
process for molecular hydrogen in the
ISM is by the recombination of pairs of
absorbed hydrogen atoms on the
surfaces of dust grains.
Ben Waghorn, APRIL 2003
complication to this method namely that
physical-adsorption cannot hold the
hydrogen atoms on the grains for long
enough for them to combine at the
relevant temperatures.
The binding
energies (divided by k) to the edges of
graphite flakes are of the order of
50,000˚K compared to the physical
adsorption energies to the order of 200˚500˚K. However, the activation energy
for formation of molecules is too large to
be formed in the ISM. This problem was
avoided by Hollenbach and Salpeter
(1971) by considering defect sites on the
surfaces of realistic interstellar dust
grains, enhancing the binding energies
giving the reactive atoms greater binding
energies than the molecules. The results,
as are shown later in this paper (§2),
show that H2 recombination is efficient
over a wide range of temperatures.
the grain leads to the calculation of
recombination rates. The master
equation uses experimentally and
parameters to explore the recombination
and other chemical reactions on small
grains, for example reactions involving
oxygen and hydrogen on the grain
Using far-UV absorption spectra
obtained with FUSE H2 formation rates
can be calculated and compared to the
theoretical values (Gry et al. 2002) (§5).
This data is taken for three late B stars
that were shown not to interact with the
interstellar matter responsible for the
absorption and that do not contribute to
the incident radiation field. By taking
previously observed column densities
along the relevant lines of sight
estimates of the H2 formation rates were
This however does not complete the
picture for the formation of molecular
hydrogen in the ISM. These rate
equations describe the diffusion, reaction
and desorption processes on the grain
surface giving a solution for the time
evolution of the average densities of
atoms and molecules on the surface,
neglecting fluctuations. These results are
probably accurate for macroscopic
surfaces but their validity was later
questioned for this study due to the small
grain size and low flux within the ISM.
The numbers of H atoms on the grain
surfaces are expected to be small and
fluctuations become significant. Biham
and Furman (2001) and Biham and
Lipshtat (2002) take the master equation
approach for the formation of H2 (§3).
This equation takes into account both the
discrete nature of the H atoms as well as
the fluctuations. Here the time evolution
of the probability of n atoms occupying
A final formation mechanism that is
addressed in this paper is that of the
relevance of gas-phase H2 formation
(§4). Most of the molecular hydrogen
that forms in the gas phase does so via
the formation of an intermediate H- ion,
as shown later in this paper. Gas-phase
formation is the dominant mechanism
for a number of cases, dominating for a
dust-to-gas ratio less that a critical value.
The relative effects of this process for
the conditions within the ISM are
addressed in this paper.
The rate of molecular hydrogen
formation on dust grains can be written,
RG   v H n g n H  g ,
Ben Waghorn, APRIL 2003
2.2 Surface Mobility
where vH is the average thermal
velocity of hydrogen atoms in the neutral
hydrogen gas,  g is the mean
Now that we know the likelihood of an
atom becoming adsorbed by the grain we
need to look at how the H atom can
move about the surface. By quantum
mechanical methods the diffusion time,
tD, required for an atom to move from
one site on the grain surface to a
neighboring site is finite, even at zero
temperature. For a perfectly regular
wavefunctions are nonlocalized and
form an energy band. If the difference
between the adsorption binding energy,
D, and H binding energy, D′ is much
greater than this energy band the
Interstellar dust grains must therefore
have surface irregularities, due to the
number of dislocations from UV
radiation, soft X-rays and low-energy
cosmic rays. Such irregularities will
increase the binding, showing that for
the enhanced sites the binding energy for
H is larger than the binding energy for
geometric cross-section of a dust
grain, n H and n g are the number
densities of hydrogen atoms and dust
grains, and γ is the recombination
coefficient, defined as that fraction of
atoms, striking a grain, which eventually
forms a molecule on that grain.
2.1 Sticking Coefficient
Firstly, in calculating the recombination
efficiency, γ, for gas atoms on grain
surfaces (Hollenbach and Salpeter 1971),
the number of incident hydrogen atoms
that become adsorbed by the grain,
namely the sticking coefficient S, needs
to be calculated. In general, S is a
function of the grain temperature T, the
gas temperature Tgas, and the adsorption
binding energy D. In practice the gas
temperature is greater than the dust
temperature, both of which are much
smaller than D.
Let N be the number of regular sites, N′
the number of enhanced “irregular” sites,
and N′s the number of scattering sites on
the grain. Most of the atoms that are
adsorbed onto the grain are done so onto
a patch of the regular, smooth, surface. It
takes a transient time ttr for the hydrogen
atom to find and become bound to an
enhanced site, covering a patch of about
(N/N′) sites.
The transient time was found to be,
1/ 2
NN s 
N 1
t tr 
tD   0 ,
where ν0 is the characteristic latticevibrational frequency of the solid.
It is shown that if these circumstances
hold, S is independent of T, with S being
approximated by,
 2  0.8 3
S (Tgas ) 
1  2.4   2  0.8 3
where   c , with Ec being the
characteristic total energy being
transferred to the surface. For given
temperatures that are comparable to that
of regions of atomic hydrogen, Г is of
order unity making the sticking
coefficient of order 0.3. These values are
obtained neglecting quantum effects but
for a first order approximation yield
reliable results.
For a hydrogen atom on the regular
surface (having a limited probability of
Ben Waghorn, APRIL 2003
recombination) the evaporation time is
t ev   0 exp D / kT 
small fraction of time on regular sites
but the probability of evaporation is
much greater during this small fraction
of time. The evaporation time is found to
be sufficiently large so that evaporation
occurs even on regular sites.
With the assumption that the enhanced
sites help in the recombination, the
necessary condition is that the atoms
reach an enhanced site before they
evaporate, namely t tr  t ev .
Below this temperature Hollenbach and
temperatures and binding energies must
be achieved so that atoms will have
much longer evaporation times that the
time ts between successive striking
events. Also the atom must be either able
to reach equilibrium with an enhanced
site or form a molecule with another
atom in a time less that the evaporation
time from the regular surface.
2.3 Specific Grain Examples
Examining this situation for the
unrealistic case of a perfect surface,
results were found as follows. For a
region with neutral hydrogen densities,
n  1  10 3 cm-3 the mean time for a new
atom to stick is t s  10 4  10 sec. The
for N  10 6
given t tr  10 sec. This shows that
recombination will take place almost
immediately if a second atoms sticks to
the grains whilst the first atom is
adsorbed. However, on a perfect lattice
with the grain temperature T greater that
a critical temperature Tc Tc  13K  the
adsorbed atom will evaporate before the
second atom can stick to form a
molecule. For T between 6.5˚ and 11˚K
this effect will occur.
Following all of the above discussions
for molecular hydrogen formation the
recombination is almost 100% efficient
for H atoms, once they are thermalized
on the grain surface. This assumes that
grain temperatures are below about 25˚K.
The rate equation described in §2 forms
a good background for this topic but has
a number of flaws. It had been noted
(Charnley, Tielens & Rodgers 1997) that
hydrogen recombination takes place on
small grains so the rate equation only
has limited validity. This is due to the
fact that these equations only take into
account average concentrations of H
atoms on the surface, ignoring
fluctuations and the discrete nature of
the H atoms. For the case of very small
grains and low flux, as in diffuse clouds
in the ISM, this property becomes
Now taking the more realistic case
containing enhanced site, there are a
number of possible scenarios. Two such
cases that can be considered are firstly
the high temperature case where the
occupation of N regular surface sites can
be neglected, and secondly the low
irregularities are required for high
recombination efficiencies.
For the high temperature case
Tgr  11K  the atom spends only a
Ben Waghorn, APRIL 2003
significant. The numbers of H atoms
striking the surface of the grain could be
as few as 0, 1 or 2. Clearly
recombination cannot occur without at
least two H atoms simultaneously
occupying the surface so any previous
averages no longer suffice. This is where
the master equation for hydrogen
recombination on grain surfaces (Biham
et al. 2001) becomes important. This
covers the two main problems of the rate
equation, namely the discrete nature of
the H atoms and also fluctuations. The
master equation uses probabilities that
there are a particular number of H atoms,
N H  0,1,2,..... occupying the grain at a
time t, PH(NH). The master equation
provides the time evolution of PH(NH)
from which the recombination rate can
be calculated. This can be used
alongside surface parameters that have
been experimentally determined to
investigate the recombination process on
microscopic grains, taking into account
grain size, flux and surface temperatures
relevant to the ISM.
where n H 2 (t ) is the coverage of H2
molecules at time t. The flux term
represents the flux of incoming atoms
LangmuirHinshelwood rejection term, where H
atoms deposited on top of H atoms or H2
molecules are rejected. This term is
equivalent to the above sticking
A second rate equation shows the change
of the number of H2 molecules on the
surface as a function of μ, the fraction of
H2 molecules remaining on the surface
upon formation, and WH 2 , the H2
desorption coefficient,
dn H 2
 a H n H2  WH 2 n H 2
The H2 production rate, rH 2 , is thus given
rH 2  (1   )aH nH2  WH 2 nH 2
3.2 Master Equation for H2 formation
on small grains
3.1 Rate Equations for macroscopic
I will now show the master equation
(Biham et al 2001) which is required for
conditions in the ISM that are failed by
the rate equations. Due to the small flux
of H atoms it is now more convenient to
use the total number of H atoms and H2
molecules on the grain, NH and N H 2 ,
rather that the amount per unit area
previously. The expectation value for the
number of H atoms on the grain, N H is
In order to follow the theory behind the
master equation it is first necessary to
restate the rate equation in terms of
effective flux of atoms, f H , the
desorption coefficient, WH , and
depletion of H atoms due to
recombinations into H2 molecules.
Therefore the rate of change of H atom
coverage, nH (t ) , on the surface is given
given as Sn H , where S is the number of
adsorption sites on the grain, and
similarly for H2. The incoming flux is
now given as FH  Sf H and a H is
replaced by AH  aH / S , which is
approximately the inverse of the time
 f H (1  nH  nH 2 )  WH nH  2a H nH2
Ben Waghorn, APRIL 2003
required for an atom to visit nearly all
the adsorption sites on the surface. The
by RH 2  SrH 2 . Thus the new rate
equations can be written as,
d NH
d N H2
 FH  WH N H  2 AH N H
PH(NH). The final term describes the
recombination effect on the number of
adsorbed H atoms.
The rate of formation of H2 on the
surface (in units of molecules s-1) is thus
given by,
 FH 2  AH N H  WH 2 N H 2
where the term, FH 2 , takes into account
the flux of H2 formed from the gas phase
as shown later in this paper (§4). These
equations hold, as in the previous case,
when the grain size is large but when the
number of atoms on the grain is reduced
another method needs to be taken.
 N N
N H 2
 1PH N H 
The first two equations are ignored as
neither zero nor one H atoms on the
surface of the grain are sufficient to
recombine to form hydrogen.
In order to complete the analysis and
gain an expression for the recombination
rate we now need to find the probability
that there are N H 2 molecules on the
This new method, the master equation,
uses the probability, PH(NH), that there
are NH hydrogen atoms on the grain.
Clearly, summing the probability over
all possible values of occupation yields
unity. The time derivatives of these
probabilities, PH N H  , are calculate by
Biham et al with the generic result,
grain, PH 2 N H 2  . The time evolution of
these probabilities is given by,
 
  
[N  1P N  1
P N ]
P N  1  P N 
PH 2 N H 2  FH 2 PH 2 N H 2  1  PH 2 N H 2
 WH 2
 N H2
+ RH 2
PH N H   FH PH N H  1  PH N H 
 WH N H  1PH N H  1  N H PH N H 
 AH [N H  2N H  1PH N H  2
 N H N H  1PH N H ]
where FH 2 is the flux of H2 molecules
Each of the above equations for
N H  0,1,2,... includes three terms. The
first term describes how the probabilities
vary with incoming flux.
probability of NH occupants increases
when an H atom is adsorbed onto a grain
already containing NH-1 adsorbed atoms,
and decreases when is already contains
NH atoms. The next term shows that an
atom desorbed from a grain with NH
adsorbed atoms decreases the probability
The final step in calculating the
recombination rate is to write the
expectation values for NH and N H 2 in
terms of the respective probabilities, and
then to obtain the new rate equations.
that stick on the grain and WH 2 is the
desorption rate. Again μ is the fraction
of molecules that remain on the surface
after formation.
Ben Waghorn, APRIL 2003
Clearly the expectation value for the
number of H atoms on the grain is given
NH 
N H 0
PH N H 
Thus far we have only considered the
idea of dust grain catalysis for the
formation of molecular hydrogen in the
ISM. This isn’t however the only
possible scenario so this section will be
taken to examine another possible route
for production, namely via the gas-phase
(Field 2000). Firstly I will introduce this
idea and will then look at the conditions
for which this process is relevant.
 FH  W H N H  2 AH N H N H  1
d N H2
 FH 2  AH N H N H  1
FH / 2
for the case of the master equation.
and likewise for the expectation value of
number of molecules. We can know
rewrite the rate equations using the time
evolution probabilities for H atoms and
H2 molecules giving,
d NH
RH 2
 WH 2 N H 2
The net rate at which H2 molecules
desorb into the gas-phase is given by the
recombination rate, R H 2 (molecules s-1),
H2 can be formed in the gas phase,
primarily through the reactions,
RH 2  1   AH N H N H  1  WH w N H 2  FH 2
H e  H 
H  H  H2  e
It is seen from the rate equations listed
above for the master equation that the
previous N H
has been replaced
with some reaction occurring via;
H  H   H 2  
by N H2  N H . This is the key factor
H 2  H  H 2  H 
distinguishing the two approaches. For
macroscopically large grains the two
factors are comparable so either method
would yield accurate results. However,
for the more likely case of a small grain
where N H will be small the two
factors will be considerably different and
the improved master equation is required
to obtain accurate recombination results.
although this second formation process
is slower. Most of the dust formed in the
gas phase does so by the first set of
reactions, with the first of the two
reactions (1) occurring more slowly, thus
achieving a small equilibrium abundance
of H-. Due to this equilibrium there are
two factors that are important to the
formation rate of H2. Firstly we have the
rate at which the H- forms and secondly
the fraction of H- ions that survive to
form H2.
The recombination efficiency, that is the
fraction of adsorbed H atoms that desorb
in the form of H2 molecules, η, is given
Ben Waghorn, APRIL 2003
The rate at which H- forms H2 depends
predominantly on the rate at which H- is
destroyed by mutual neutralization with
H+ ions,
H   H   2 H (3)
or by photodetachment by the incident
radiation field,
H     H  e (4)
relative to the H2 formation rate. Below
is an example calculation (for more
details and more examples see Glover
2003) of the H2 formation rate via the Hstate, RH , H  . If the formation of H2 via
mentioned H- route and also the rate of
formation due to the H 2 reaction,
RH 2 , gas  RH , H   RH , H 
and also for the grain-catalyzed rate,
written in a different form from §2 but
containing the same information,
 D 
RH 2 ,dust  k dust ntot n H 
 DMW 
Here k dust is the temperature dependant
formation rate, ntot the total particle
number density, D the dust-to-gas ratio,
and DMW the D value in the Milky Way.
reaction (2) occurs much faster that an
destruction of H-, the formation rate is
given by,
RH , H   k1 ne n H
This equation yields a critical dust-togas ratio, Dcr, for which the two
formation rate are equal. Dcr is the ratio
required for grain catalyzed formation to
overtake the gas-phase formation,
RH 2 , gas
Dcr 
k dust ntot n H
where k1 is the reaction rate of reaction
(1). This shows that the H2 formation
rate is approximately equal to the Hformation rate, as expected.
If on the other hand H- destruction via
mutual neutralization (3) dominates over
H2 formation or photodetachment (4),
then the reaction rate equation becomes,
RH , H   1 2 nH2
Further analysis of this function (Glover
2003) shows a strong temperature
dependence to Dcr. At low temperatures
the formation of H2 by dust grain
catalysis is relatively efficient and D
needs only be small (little dust) before
catalysis dominates. This situation
changes above a few hundred ˚K as the
efficiency of grain catalysis drops
significantly and gas-phase formation of
H2 continues to grow. The ISM contains
temperature around 80˚K and low
fractional ionizations x  n H  / n H of
order 10-7. In these conditions grain
catalyzed formation dominates by many
orders of magnitude.
assuming ne  n H , where k i is the
rate, cm 3 s 1 , of reaction i , expressed in
terms of temperature T. These limiting
cases are specific examples of the
general H2 negative ion formation rate
give by,
k 2 nH
RH , H   k1 ne n H
k 2 nH  k3 nH   k 4
For the sake of comparison of gas-phase
formation against dust grain catalysis it
is useful to write the rate equations in the
following form, firstly for the gas-phase
This summarized analysis of the gasphase formation method yields the
conclusion that for the case of the ISM
the conditions are such to greatly favor
Ben Waghorn, APRIL 2003
as nH  , nH 2  and n  nH   2nH 2  .
Without shielding the photo-dissociation
rate in the Solar Neighborhood is  0 , G
the radiation field value and S the
shielding factor including H2 selfshielding and dust extinction.
H2 formation by the method of dust
grain catalysis. The master equation in
§3 takes into account the gas-phase
formation of H2 via the term FH 2 .
Now nR can be calculated by integrating
the above equation over the line of sight,
for constant n ,
1 f
nR 
0 S .
2 1 f
Here f is the molecular hydrogen
fraction: f  2 N H 2  / N total where N total
is the total column density and can be
derived from the extinction and S the
mean shielding factor.
Now that I have looked at a number of
theoretical predictions for the formation
rates of H2 in the ISM it is useful to use
formation rates can be calculated, as a
comparison. Gry et al. (2002) addressed
this issue by using far-UV absorption
spectra obtained with FUSE looking at
three late type B stars. H2 formation and
excitation had previously been studied
by observing mid-infrared transitions
between rotational levels of the
vibrational ground state of H2 but only
for warm photo-dissociation regions.
Gry et al. readdress the question of H2
formation, now within the diffuse ISM,
by analyzing FUSE observations of three
stars. IRAS maps show that these stars
do not heat the matter responsible for
In practice a model in needed to
distribution of H2 molecules over it rovibrational levels as a function of depth
into the cloud. With the model f can be
calculated, and therefore values of nR
found for each star. Now all that is
needed to get an estimate of the
formation rate R is an estimate of the gas
density n . Seeing as the assumption for
this procedure is that clouds are in
thermal balance, the fact that gas density
and temperature are uniquely related can
be used. Tracing the temperature
observationally therefore also traces the
density. A temperature diagnostic used
in this case is to obtain the column
density ratio of the first two
levels, N J  1 / N J  0 , from which
the gas density can be determined.
The first method in the process to
calculate, observationally, H2 formation
rates is to determine the product nR . R is
the formation rate that is ultimately
being searched for and n is the mean
effective H2 density along each line of
sight. To effectively calculate the rate of
formation of H2 we need to assume an
equilibrium environment where the H2
formation balances its photo-dissociation.
Such a situation can be expressed as,
Gry et al. (2002) evaluate this data with
their results displayed below in table 1.
The values of R for the three lines of
sight are close to each other and with
significantly lower uncertainties than the
nH nR  nH 2  0 GS
The atomic, molecular and total
Ben Waghorn, APRIL 2003
similar previous values found by Jura
(1975). These values can be converted
directly into an H2 formation timescale
of 1 / nR   2 x10 7 yrs . This model tells
us that the photodissociation timescale is
larger that the H2 formation timescale,
especially in the shielded layers of the
absorbing cloud.
Table 1. H2 formation rate R from the product nR and the density n estimated
from N H 2 , J  1 / N H 2 , J  0 . Gry et al. 2002.
 
ncm 
Rcm s 
nR s
HD 102065
2.3 x10 15
HD 108927
0.87 x10 15
HD 96675
2.0 x10 15
4.5 x10 17
3.1x10 17
4.0 x10 17
requirements on the regular surface of a
dust grain and on the surface of
irregularities to produce the observable
high recombination efficiency.
In this paper I have addressed the
question of the formation rate of
molecular hydrogen in the ISM. The two
primary processes, namely dust-grain
catalysis and gas-phase production, were
both examined and compared for the
relevant diffuse cloud conditions. It was
found that the gas-phase formation
process via the negative ion route, and to
a lesser extent via H 2 , was important
only at temperature greater than
expected in the ISM. Results from gasphase formation analysis show that, in
practice, it is slower than dust-grain
catalysis, either because of a shortage of
free electrons and protons, or because
the incident radiation field destroys the
ions before they have a chance to form
This method however fails to account for
the microscopic size of grains and small
incident flux of H atoms striking the
grain. The master equation, summarized
in §3, introduces a new approach for the
simulation of hydrogen recombination
on microscopic dust grains in the ISM.
This approach uses probabilities of N H
hydrogen atoms occupying a grain
surface, thus accounting for fluctuations
in numbers.
The master equation can be extended to
describe more complex situation
involving chemical reactions with
multiple species, for example oxygen.
Calculations for finding the formation
rate of OH using the master equation are
shown in Biham et al (2001). More
details into solving the master equation
were emitted from this paper but have
been reviewed (Biham and Lipshtat 200)
in detail giving expressions for
Therefore, the dominant effect for H2
formation is via dust-grain catalysis.
This paper summarized two such models,
initially showing the important effects in
calculating the recombination rates.
Such earlier methods discuss the
Ben Waghorn, APRIL 2003
probability function and recombination
efficiencies in terms of flux and hopping
uncertainties of conditions within the
ISM, such as dust grain sizes and shapes,
there is still work to be done as
uncertainties are contained in the present
The master equation successfully models
H2 formation in the ISM but due to
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2002, A&A, 391, 675.
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