A general method for the inclusion of radiation chemistry in astrochemical models

Christopher N. Shingledecker *a and Eric Herbst ab
aDepartment of Chemistry, University of Virginia, P.O. Box 400319, Charlottesville, VA 22904, USA. E-mail: shingledecker@virginia.edu; Tel: 434 831 6240
bDepartment of Astronomy, University of Virginia, Charlottesville, VA 22904, USA

Received 29th August 2017 , Accepted 11th October 2017

First published on 11th October 2017

In this paper, we propose a general formalism that allows for the estimation of radiolysis decomposition pathways and rate coefficients suitable for use in astrochemical models, with a focus on solid phase chemistry. Such a theory can help increase the connection between laboratory astrophysics experiments and astrochemical models by providing a means for modelers to incorporate radiation chemistry into chemical networks. The general method proposed here is targeted particularly at the majority of species now included in chemical networks for which little radiochemical data exist; however, the method can also be used as a starting point for considering better studied species. We here apply our theory to the irradiation of H2O ice and compare the results with previous experimental data.

1 Introduction

Interstellar space is permeated by highly energetic particles known as cosmic rays. These particles, which are over 90% protons and typically have energies of MeV–TeV, are formed in supernovæ1 or active galactic nuclei (see review by Blasi2 and references therein). Unlike external UV photons, cosmic rays are not quickly attenuated in regions with high visual extinctions (>10) and can undergo a large number of collisions before being fully stopped.3,4 The interaction between these ionizing particles and the gas and dust of the interstellar medium can have significant effects on the physical and chemical properties of a source5–10 and are thus of great astrochemical interest.

In addition to the production of gas-phase ions, which leads to ion–molecule chemistry, cosmic rays have two main roles in astrochemical simulations. The more important is in generating internal UV photons, following the work of Prasad and Tarafdar,11 which drive photoionization and photodissociation. In many astrochemical models, cosmic rays can also non-thermally desorb grain-surface species,12 following the work by Hasegawa and Herbst.9 In addition to these functions, however, cosmic rays can also have a substantial impact on the chemistry of interstellar ices, particularly in cold cores such as TMC-1 where there is very little thermal energy to drive the normal diffusive chemistry of grain species.

In this work, we will refer to the chemical changes induced by ionizing particles by the historical and still widely used, though somewhat confusing, name of “radiation chemistry”,13 which was originally coined by Burton14 and meant to distinguish the field from photochemistry. One key characteristic of radiation chemistry is the production of “secondary electrons” which are produced in ionizing collisions in an irradiated material and typically have energies below ∼50 eV.15

A large body of work in laboratory astrophysics exists on the action of ionizing radiation on astrophysically relevant ices,16,17 and some experiments have shown that complex organic molecules, even amino acids, can be produced in irradiated ice mixtures.18,19 However, despite the abundance of such studies, astrochemical models largely omit any solid-phase radiation chemistry. This deficiency in models has been due, in part, to the complexity of the microscopic processes that contribute to the ultimately measured values in experiments, and the lack of radiochemical data for the majority of the many species now included in astrochemical networks. For instance, the chemical network used in recent work by Ruaud et al.12 contains 717 different species.

Recently, in Abplanalp et al.,19 we made an initial attempt to examine the effects of radiation chemistry in astrochemical simulations. This was, to the best of our knowledge, the first time such processes were incorporated into an astrochemical network, and our results showed that the additions improved the agreement between observed and calculated abundances. A major obstacle to a more thorough implementation of radiation chemistry in astrochemical networks has been the lack of a general theory that can be used to reduce the detail needed in a microscopic model20 to a level more easily implementable in a typical rate equation model. Ideally, such a theory should be (a) applicable to any species, (b) utilize readily available physical values as parameters, and (c) should give results within reasonable agreement with experiments. In this work, we propose a simple, astrochemically relevant formalism for estimating both the major radiolysis decomposition pathways and rate coefficients in a manner suitable for addition to standard chemical networks, which we hope satisfy these criteria. Here, we discuss the theory in Section 2, and apply our scheme to water ice in Section 3. A list of key values used in this work is given in Table 1.

Table 1 Key symbols used
Value Units Description
σ tot cm2 Total cross section
σ n cm2 Nuclear cross section
σ e cm2 Electronic cross section
σ ion cm2 Ionization cross section
σ exc cm2 Excitation cross section
S e eV cm2 Total electronic stopping cross section
S ion eV cm2 Ionization stopping cross section
S exc eV cm2 Excitation stopping cross section
W eV Mean energy per ion pair
G Species/100 eV Radiochemical yield
E ion eV Ionization potential
E 0 eV Excitation threshold
W e eV Total inelastic energy loss
W ion eV Average ionization energy loss
W exc eV Average excitation energy loss
W s eV Average sub-excitation energy loss
ε eV Secondary electron energy
ζ s−1 Cosmic ray ionization rate
ξ Excitation to ionization ratio
N e Total number of inelastic collisions
N ion Total number of ionizing collisions
N exc Total number of excitating collisions
M Number of species created or destroyed
F Total number of inelastic processes
I Total number of ionized states
J Total number of transitions between bound states
f br Branching fraction
P e Electron escape probability
P dis Dissociation probability
ϕ st Spitzer & Tomasko cosmic ray flux
ϕ ism Scaled cosmic ray flux

2 Theory

2.1 Energy loss and W values

Since the early 20th century, it has been known that when a swiftly moving charged particle of ionizing radiation, which we will call the primary ion, encounters some material, called the target, the primary ion loses velocity and physiochemical changes to the target occur.21 These changes are driven by collisions in which energy is transferred from the primary ion to the target.22,23 A convenient distinction was made by Bohr22 between energy loss by the swiftly moving charged particle to the nuclei and electrons, termed as elastic and inelastic collisions, respectively. The inelastic collisions can, in turn, be divided into collisions in which the target species are either ionized or electronically excited. Thus, the total collisional cross section can be represented by
σtot = σn + σe = σn + σion + σexc(1)
with σn and σe being the elastic and inelastic cross sections, and σion and σexc being the cross-sections for ionization and excitation. Using this distinction between nuclear and electronic collisions, we can express the “stopping power,” or energy loss per unit path length for any energetic particle, as
image file: c7cp05901a-t1.tif(2)
where n is the density of the target and Sn and Se are the stopping cross sections in units of energy times area, which are also sometimes referred to by the more descriptive term “energy loss functions”.24 The electronic stopping cross section, Se, characterizes the energy lost in inelastic collisions by a primary ion hitting some target, and is dependent on the physical properties of both.

The electronic energy loss function, Se, can be described as the sum of each possible inelastic energy loss, weighted by its cross section. For example, let F be the number of different inelastic collisional processes in which wf energy is lost by the primary ion traveling through some target. Moreover, let F be composed of (a) the ionization of I different states resulting in an energy loss of wioni and (b), J different excitations between bound and unbound states resulting in an energy loss of wexcj. The electronic energy loss function, Se, is then15

image file: c7cp05901a-t2.tif(3)
here, we have neglected higher order processes such as double ionizations.

Consider now the case of the bombardment of a target by a single primary ion where we approximate the inelastic energy loss function to be Se = Sion + Sexc. Now, let Ne be the total number of inelastic collisions that occur between target species in the material and both the primary ion and secondary electrons. Moreover, let Nion be the total number of ionizing collisions and Nexc be the total number of exciting collisions such that the total number of inelastic collisions is Ne = Nion + Nexc. The average energy lost in exciting collisions, Wexc, is then

image file: c7cp05901a-t3.tif(4)
where wexcj is the energy lost in each excitation. One can also define an average excitation cross section, image file: c7cp05901a-t4.tif as
image file: c7cp05901a-t5.tif(5)

The remainder of the inelastic collisions, Nion = NeNexc, result in ionization of species in the target. One can similarly define the average ionization energy loss to be

image file: c7cp05901a-t6.tif(6)
and the average ionization cross section
image file: c7cp05901a-t7.tif(7)
In eqn (5) and (7), we have averaged the inelastic cross sections of both the primary ion and secondary electrons. Such an approach is physically meaningful, since the cross sections for both have similar peak values – around 10−16 cm−2 for collisions in solid O2 – and typically remain within an order of magnitude of each other at high energies.20 For every ionizing collision some energy wioni = Eion + ε is lost, where Eion is the ionization potential of the target species and ε is the energy of the newly liberated secondary electron. If ε > Eion, following Fano,25 we assume that it contributes to the total number of ionizing collisions, Nion. Unlike Fano, however, if ε is less than Eion but greater than some electronic excitation threshold of the target, E0, we will assume it contributes to the total number of exciting collisions, Nexc. Secondary electrons with energies below the excitation threshold of the target are known as sub-excitation electrons26 and contribute neither to the total number of ionization of electronic excitation collisions, though are thought to play a critical role in irradiation induced chemistry.27 We will assume (a) that all Nion secondary electrons produced in ionizing collisions either are formed with energies in the sub-excitation regime, or lose sufficient energy via subsequent ionization or excitation collisions to become sub-excitation electrons, and (b), that these sub-excitation electrons have an average energy of Ws.

As an example, consider the case of a secondary electron with energy ε in a molecular hydrogen gas, where Eion = 15.4 eV and E0 = 11.2 eV.28 Here, ε can be either the initial energy, ε0, of the secondary electron, or a subsequent value after the electron has undergone some number of inelastic collisions. If ε is greater than 15.4 eV, we assume that it further ionizes other H2 molecules. If the energy is in the range 11.2[thin space (1/6-em)] eV ≤ ε ≤ 15.4 eV, we similarly assume that the electron would cause electronic excitations in H2. Finally, if ε < 11.2 eV, we consider it a sub-excitation electron.

For every ionizing collision, there will be some number of excitation collisions. Let ξ be the average excitation to ionization ratio, i.e.

image file: c7cp05901a-t8.tif(8)
Thus, We, the total energy transferred to the target via inelastic collisions with either the primary ion or secondary electrons is
We = NionWion + ξNionWexc = Nion(Eion + Ws) + ξNionWexc.(9)
One can also express the inelastic energy loss per ionization as W, where
W = Eion + Ws + ξWexc(10)
is an important quantity in experimental nuclear physics and radiation chemistry29 and typically has a value of ∼30 eV.30,31

2.2 G values

Some of the inelastic collisions by the primary ion or secondary electrons result in the dissociation of species in the target material. The number of species created or destroyed per 100 eV of energy transferred to the system by irradiating particles is known as the G value, which is another important quantity in radiation chemistry. This radiochemical yield has a typical value between 0.1–3 molecules/100 eV for bombardment by protons.

The G value has a long history in radiation chemistry, and was proposed by Burton.32 As an example, consider the radiolysis of some species, A, described by

aAbB + cC + ⋯(11)
where the curly arrow implies bombardment by either a primary ion or a secondary electron, and the lowercase values represent the stoichiometric coefficients of the reactants and products. The corresponding G-value for the destruction of A, as it might be measured, would then be
image file: c7cp05901a-t9.tif(12)
where a is the stoichiometric coefficient of A, fbr is the branching fraction where there are multiple product channels, M(A) is the number of A molecules destroyed, and Nion is the number of ion-pairs produced. The negative value of A in G(−A) implies that A is here being destroyed. The G values for the destruction of the reactant and production of the products can then be related via
image file: c7cp05901a-t10.tif(13)
It should be noted that, strictly speaking, the measured G values characterize a system in a steady state, and are dependent on the physical conditions and nature of the target.33 Also, as will be discussed later in this work, the G values of the immediate production or destruction are not identical to the measured values due to processes that impact the actual number of species measured.

2.3 Decomposition pathways

The ultimately reported experimental G values characterize the steady state yields of the irradiated system. These yields are effective values that reflect the cumulative physiochemical changes that occur in the target as a result of bombardment by many primary ions. Though complex, the series of events caused by a single primary ion can be divided into several major phases. During the initial “physical” stage of irradiation, the ionizing and exciting collisions between the target and both the primary ion and secondary electrons occur. Subsequently, during the “physiochemical” stage, radiolysis products are formed through the dissociation of excited species and the charge neutralization of molecular ions.15 In a detailed microscopic model, for even a relatively simple irradiated system, i.e. a pure O2 ice, one must consider a large number of possible radiolysis products and decomposition pathways.20 This level of detail, however, is not realistically achievable in most widely used astrochemical models, which typically use a rate equation approach of some kind and are quite limited in accommodating physical complexity, particularly for grain surface and bulk chemistry.12,34,35

Therefore, it is necessary to simplify the microscopic radiochemical processes in a way that is amenable for inclusion into astrochemical models, yet captures the salient features of radiation chemistry that contribute to its astrochemical interest. In this work, we propose that essential astrochemically relevant features of irradiation chemistry are (a) the ionization and excitation of target species, (b) the destruction of the target species via charge neutralization and excitative dissociation, and (c) the formation of radiolysis products, including electronically excited suprathermal species. Thus, for molecular species, A, we propose the following basic decomposition pathways that occur after bombardment by an energetic particle

aAaA+ + e(14)
aAaA+ + e → aA* → bB* + cC*(15)
aAaA* → bB + cC(16)
where the star superscript indicates an electronically excited species. Here, we will refer to processes (14)–(17) as ion1, ion2, exc1, and exc2, respectively. If A is an atom, we assume that process (16) does not occur and that (15) is functionally identical to (17). Once they are formed, we further assume that these electronically excited, or suprathermal, species rapidly react, are quenched by the solid,15,19,20 or dissociate in the case of molecular products.36 Reactions between ground state species that have a barrier often have little or no barrier when one of the reactants is in one of these electronically excited states, which is in part why such energetic radiolysis products have been of recent astrochemical interest,20 and these suprathermal species have been implicated in driving the non-equilibrium low-temperature solid-phase chemistry seen in previous experiments.19 In a real system, the measured G values will be heavily influenced by the rate of the backwards reactions, e.g. bB + cCaA. One advantage of including reactions (14)–(17) in an astrochemical simulation is that the back reactions can be accounted for separately and explicitly, thus, we here focus on the radiolytic destruction of A and show only the forward reactions. It should be emphasized that reactions (14)–(17) most appropriately describe what occurs in the bulk of the solid, where most of the radiation chemistry is thought to occur.37 Conversely, irradiation-induced processes at the gas–surface interface, also called the selvedge, are more likely to lead to desorption, as also noted in the work by Abdulgalil et al.37

Having described the major decomposition routes for A via radiolysis, we now turn our attention to determining the products, which we have here called B and C. We first examine the products of the excitative dissociation given in (16), where A dissociates after being electronically excited due to an inelastic collision with a primary ion or secondary electrons. One can use the dissociation products from photochemistry38 as a guide in this regard,13 since there is typically no distinction whether the species absorbed wexcj eV from an inelastic collision from a particle or a photon of E = wexcj when considering a single event. Care must be taken, however, due to two key differences between photochemistry and radiation chemistry. First, in photochemistry selection rules govern the possible excitations and product channels. Secondly, since photochemical experiments often use monochromatic light, only a few, or even one, electronic transition may result, moreover, not all species in a photochemical target of mixed composition will absorb light of the utilized frequency to the same degree, and the chemical changes observed can be dominated by the dissociation of a single constituent. Conversely, for ionizing radiation, a larger number of electronic excitations are possible in the target species, and in mixed materials all components can be physio-chemically altered.39

The products of the dissociative recombination reaction given in (15) can be more difficult to estimate. For polyatomic species there exist statistical theories, such as the one proposed by us in a previous work,36 in which the excited intermediate is treated as a complex of excited atoms and the product channels are assumed to be independent of the entrance channel. This statistical method works best for molecules with ∼3 or more atoms.40 For smaller species, the dissociative recombination mechanism can be non-statistical; however, since these are typically better studied than larger species, there is likely to be more relevant experimental and theoretical work which could be used for determining the nature and energetics of the products. For both excitative dissociation and dissociative recombination, where there are multiple known product channels, the corresponding G values for the formation of the products can be multiplied by a branching fraction, as in eqn (12).

What, though, are the G values for reactions (14)–(17)? Consider the case of Nion = 1 and Nexc = ξ. Based on processes (14)–(16), we propose the following forms for the G values describing the effects of an inelastic collision resulting in the ionization or excitation of some species, A:

image file: c7cp05901a-t11.tif(18)
image file: c7cp05901a-t12.tif(19)
image file: c7cp05901a-t13.tif(20)
image file: c7cp05901a-t14.tif(21)
where a is the stoichiometric coefficient, Pe is the electron escape probability and Pdis is the dissociation probability. The electron escape probability can be estimated using the formalism of Elkomoss and Magee,41 based on which we approximate Pe = 0 for solids, which means that Gion1 = 0, i.e. that all charged species formed in ionizations are neutralized. Regarding the dissociation probability, we here set Pdis = 0.5 in the absence of any experimental or theoretical data for A.

For each process given in eqn (14)–(17), we determine the corresponding value of M(A) from our expression for the W value given in (10). Thus, for each ionization

M(A)ion = 1(22)
while the number of excitations that occur per ion-pair is then
M(A)exc = ξ.(23)
Following Fueki and Magee42 we estimate an average value of ξ from the energy per ion-pair, W using
image file: c7cp05901a-t15.tif(24)
In solids with large band gaps, excitons, electron–hole pairs formed due to the electronic excitation of species in the material, have been noted as being both a promising link between photochemistry and radiation chemistry and an additional initiator of physiochemical changes.43 These excitons could increase the number of excited species and thereby amplify the radiochemical yield. Though a consideration of these effects is beyond the scope of the present work, they could be included via an extra factor in eqn (23) which would take into account the increased number of affected species.

Since W is typically ∼30 eV,29,30,44Gion1 and Gion2 will be similar for most species. Thus, the relative stability of some electronically excited species, A*, is reflected more by ξ. From eqn (24), one can see that ξ is inversely proportional to Wexc, and thus, the smaller the average excitation energy, the more excitation collisions will occur per ion-pair.33

To obtain values for Eion and Wexc we assume a solid composed of pure A. For solids, Eion and by extension, W will typically be lower than the corresponding gas phase values. Here, for species in interstellar ices, we take Eion to be the ionization potential from the ground state. The average excitation energy, Wexc, can be approximated as the peak UV/vis absorption energy, corresponding to the most likely electronic transition.42 Obtaining solid-phase values for Wexc can be difficult; however, gas-phase values can be easily obtained for many species from online databases such as the MPI-Mainz UV/Vis spectral atlas or the Leiden database for astrophysical molecules. The average sub-excitation energy, Ws, can be obtained from the distribution of sub-excitation electron energies given in Elkomoss and Magee,41 which is a function of Eion and E0 and is given by

image file: c7cp05901a-t16.tif(25)
where ε is the sub-excitation energy. The average sub-excitation electron energy can be obtained from
image file: c7cp05901a-t17.tif(26)
In cases where the value of E0 is not accurately known, one can approximate this energy as E0Wexc.42

2.4 Rate coefficients for radiolysis

Being thus equipped to estimate G values, we can now use them for determining rate coefficients for use in standard astrochemical models. From basic kinetic theory, one can express the first-order rate coefficient for the bombardment of some species by some particles with a flux of ϕ as k(s−1) = σϕ.

In the interstellar medium, cosmic rays are a ubiquitous form of ionizing radiation. Due to the effects of the Sun, directly measuring the interstellar flux from the Earth is not possible,46 and there are a number of energy distributions in the literature which differ mainly in the abundance of cosmic rays with energies below ∼1 GeV (see Indriolo et al.47 and references therein). We here estimate j(E), the cosmic ray intensity, using the widely used energy distribution of Spitzer and Tomasko,45 given by

image file: c7cp05901a-t18.tif(27)
where EG is the cosmic ray energy in GeV. This distribution is shown in Fig. 1 and, when integrated for particle energies between 0.3 MeV to 100 GeV, corresponding to the lower limit of the Spitzer and Tomasko45 theory and a conservative upper cosmic ray energy, one obtains a flux of 8.6[thin space (1/6-em)]particles cm−2 s−1, which changes only negligibly when the upper energy limit is increased. This integrated value of the interstellar cosmic ray flux, which we shall refer to as ϕst, was used for determining what has become the canonical molecular hydrogen ionization rate of ζ = 1.36 × 10−17 s−1.45 This ionization rate is a common input parameter in astrochemical simulations, though is thought to be significantly enhanced in some sources,47–49 particularly those near the galactic center.48 Thus, we will introduce a scaling factor, relative to ϕst, that can suitably adjust the cosmic ray flux based on the chosen value of ζ for the source being simulated, i.e.
image file: c7cp05901a-t19.tif(28)
where we have assumed that increasing the ionization rate relative to the standard value increases the flux but preserves the overall energy distribution.

image file: c7cp05901a-f1.tif
Fig. 1 Cosmic ray energy distribution from Spitzer and Tomasko.45

Thus, the rate coefficients for processes (14)–(17) are

kion1 = σion1ϕism(29)
kion2 = σion2ϕism(30)
kexc1 = σexc1ϕism(31)
kexc2 = σexc2ϕism.(32)
where there are no experimental or theoretical data on the cross sections, these can be estimated using the appropriate G value from eqn (18)–(21). If one expresses Se, the inelastic stopping cross section, as the sum of the average ionization and excitation energy losses, then division by the average ionization cross section yields
image file: c7cp05901a-t20.tif(33)
and one can then solve for the ionization cross section, i.e.
image file: c7cp05901a-t21.tif(34)
By substituting one of the G values given in eqn (18)–(21) the rate coefficient for the corresponding process can be estimated.50 The first-order rate coefficients for processes n = 14–17, as can be used in astrochemical models, are therefore,
image file: c7cp05901a-t22.tif(35)
where Gn is the G value for the process (14)–(17). Eqn (35) more strongly characterizes the disappearance of the reactants than the formation of the products as a result of the correlation between the rate coefficient and the assumptions made here about dissociation and branching fractions. In order to illustrate an extreme case, we here assume a total absence of relevant theoretical and experimental data for values like Pdis and fbf; however, this will not necessarily be true, particularly for common interstellar molecules.

One advantage of eqn (34) is that there exist well known methods for calculating Se(E),15,23,51 particularly for high energy particles with energies above 1 MeV/nucleon.51 The most commonly used formula for high energy electronic stopping cross sections is the Bethe equation,23 also referred to as the Bethe–Born15 or Bethe-Bloch equation.51 For some primary ion, x with a velocity of v colliding with target atom y, the formula is

image file: c7cp05901a-t23.tif(36)
where me is the electron mass, Zx is the atomic number of x, Zy is the atomic number of y, and e is the elementary charge. The factor C/Zy is the shell correction term and accounts for cases where v is much greater than the bound electron velocities of y. The Bethe equation is also used in the popular SRIM program.52 This widely used model is particularly good at calculating energy loss, and can simulate other physical effects as well, such as lattice damage and surface sputtering; however, it neither considers the chemical changes that result from inelastic energy loss, nor does it explicitly distinguish between different kinds of electronic energy losses. We here circumvent these limitations through our use of the G values, which allows us to connect specific kinds of inelastic energy losses with specific chemical changes, i.e. radiolytic decomposition pathways. For radiolysis processes in solids, we here use the stopping cross section for water, which is typically the primary constituent of astrochemical ices.12,53 Using the PSTAR program,§ we obtained the values for Se shown in Fig. 2. A mean value of Se = 1.287 × 10−15[thin space (1/6-em)]cm−2[thin space (1/6-em)]eV was obtained using
image file: c7cp05901a-t24.tif(37)
where we have used the Spitzer and Tomasko45 cosmic ray energy distribution for weighting.

image file: c7cp05901a-f2.tif
Fig. 2 Electronic stopping cross sections for protons in water, calculated using PSTAR.

3 Results & discussion

We will now apply our theory to the radiolysis of water ice. The resulting G values we obtain using our method are listed in Table 2. It should be noted that water is not an understudied species by any means in the context of radiation chemistry,33,56 rather, it is the existence of such data that allows us to estimate the utility of the approximate treatment we propose here. Again, since we are considering solid-phase species, we will approximate Pe = 0, i.e. that electron recapture occurs with unit probability. For the remaining radiolysis pathways we will consider the following product channels:
H2O ⇝ H2O+ + e → OH* + H*(38a)
→O + H + H*(38b)
H2O ⇝ H2O* → OH + H(39)
H2O ⇝ H2O*(40)
where products for reaction (39) are based on photo products57 and those shown in (38) are taken from previous water ice radiolysis studies.33 We further denote processes (38a)–(40) as ion2a, ion2b, exc1, and exc2, respectively. For the sake of illustration, we let f(ion2a)br = f(ion2b)br = 0.5. Thus, the cosmic ray driven dissociation rate for water, image file: c7cp05901a-t25.tif, is
image file: c7cp05901a-t26.tif(41)
and the time dependent abundances of the suprathermal species are
image file: c7cp05901a-t27.tif(42)
image file: c7cp05901a-t28.tif(43)
image file: c7cp05901a-t29.tif(44)
where the sums are over all destruction reactions. In solids the lifetime of these suprathermal species is very short19,20 since they will either rapidly react with a nearby species or will be quenched by the solid. Thus, the abundance of these suprathermal species at any time will be very low.
Table 2 Ionization, excitation, and sub-excitation energies as well as the resulting G values and rate coefficients for watera
Parameter Value Source
a Rate coefficients were calculated assuming a cosmic ray ionization rate of ζ = 1.3 × 10−17 s−1.
W 27 eV Johnson33
E ion 12.62 eV Lias54
W exc 11.12 eV Keller-Rudek et al.55
W s 3.85 eV See text
G ion2a(–H2O) 1.85 species/100 eV See text
G ion2b(–H2O) 1.85 species/100 eV See text
G exc1(–H2O) 1.75 species/100 eV See text
G exc2(–H2O) 1.75 species/100 eV See text
k ion2a 2.05 × 10−16 s−1 See text
k ion2b 2.05 × 10−16 s−1 See text
k exc1 1.93 × 10−16 s−1 See text
k exc2 1.93 × 10−16 s−1 See text
P e 0.0 See text
P dis 0.5 See text

In order to calculate the G values, and by extension, the rate coefficients, we use an ionization energy of 12.62 eV54 and an excitation energy of 11.12 eV, corresponding to the strong VUV absorption peak at 111.5 nm.55 These values were then used in eqn (25) to determine the sub-excitation energy distribution, shown in Fig. 3 and a mean sub-excitation electron energy of Ws = 3.85 eV using (26). Since, W, the mean energy per ion-pair is typically around ∼30 eV, the total ionization G value is 100/W ≈ 3 molecules/100 eV. For water ice, we used a value of W = 27 eV33 and, assuming complete charge-neutralization, the resulting value we obtain for Gion2 = Gion2a + Gion2b is 3.70 molecules/100 eV. Assuming Pdis = 0.5 means that Gexc1 = Gexc2, which for water ice results in G values of 1.75 molecules/100 eV.

image file: c7cp05901a-f3.tif
Fig. 3 Sub-excitation electron energy distribution for water, calculated using eqn (25).

A direct comparison of our calculated G values with experimentally determined ones is complicated by the fact that distinguishing between different radiolysis pathways can be difficult, since excited molecular species either relax, dissociate, or react very quickly upon formation19,20 and it is more common to infer these G values from the measured abundances of product species.13 Hart and Platzman58 determined a value of G(H2) ≈ 0.7 molecules/100 eV based on out-gassing of H2 from water ice irradiated with α particles. In the water decomposition pathways given in (38a)–(40), hydrogen is produced in (38) and (39). Since the canonical value of Gion is typically 3, and since the suprathermal products of the dissociative recombination are likely to be very short lived due to their increased reactivity,19 we will focus our comparison on Gexc1. If one rewrites process (39) as H2O ⇝ OH + (1/2)H2, then, following the relation given in (13), we estimate a G value for the formation of H2 of Gexc2(H2) = 0.875 molecules/100 eV, which is ∼20% larger than the measured value of 0.7 molecules/100 eV. The difference between the measured and calculated G values underscores the difference between the G values of immediate formation and destruction and the measured quantities, which in this case are likely lower due to some combination of (a) outgassing of unreacted atomic hydrogen (b) competing reaction pathways, or (c) trapping in the ice. In a standard gas-grain astrochemical model, (a) and (b) at least are usually accounted for in some manner.12 Using eqn (35) with the previously given values for ϕism and Se results in rate coefficients of kion2a = kion2b = 2.05 × 10−16 s−1 and kexc1 = kexc2 = 1.93 × 10−16 s−1.

How important are these processes, though, in interstellar chemistry? In dense molecular clouds, where external UV photons are quickly attenuated, a major dissociation mechanism for gas and grain species is typically bombardment by internal UV photons, which are formed as a result of cosmic ray ionization of molecular hydrogen.11 Gredel et al.59 express the cosmic ray induced photodissociation rate, R(CRP)A, for some species, A, as

image file: c7cp05901a-t30.tif(45)
where ω is the grain albedo and pA is an efficiency factor for species A. From Gredel et al.,59 we obtain ω = 0.5 and pH2O = 971. Comparing image file: c7cp05901a-t31.tif, where image file: c7cp05901a-t32.tif is given in eqn (41), implies that the “direct” cosmic ray driven process of water radiolysis occurs with ∼3% of the rate of “indirect” internal UV photodissociation. By way of further comparison, image file: c7cp05901a-t33.tif is ∼109 times greater than the rate of photodissociation caused by external UV photons60 from the interstellar radiation field61 in a source with visual extinction of Av = 10, typical of dense molecular clouds. As indicated by previous experimental work,18,19 radiolysis is astrochemically attractive, in part, because of the resulting suprathermal products, which can drive the formation of even complex molecules in low temperature ices, in addition to enhancing diffusion and desorption.8–10 Thus, our calculated rate for the cosmic ray driven radiation chemistry of water is suggestive of the potential importance of these processes for other interstellar species. This conclusion is in line with our preliminary modeling results in which experimental G values were used.19 The simulation results presented in Abplanalp et al.19 indicated that the importance of direct cosmic ray ice processing is on par with, and in that case even more important than, indirect cosmic ray VUV photon processing.

As previously alluded to in this work, the underlying processes which drive the radiation chemistry are indeed complex,20 and interpreting experimental data may require significantly more speculation than is needed in photochemistry;13 however, given the level of detail of standard astrochemical models, reducing the complexity to a few key processes, as described here, can at least allow workers to assess the impact of ionizing radiation on the overall abundances predicted with astrochemical models.

4 Conclusion

In conclusion, in this work we have presented a general theory for estimating the radiolysis pathways and rate coefficients for any arbitrary species, A, which can be utilized by non-experts in irradiation chemistry. This formalism uses readily available physical values as input, and yields G values which are within an astrochemically satisfactory level of agreement with experimental data. The size of the calculated rate coefficients compared with the standard interstellar ionization rate points to the potential importance of radiolysis in grain-surface chemical networks, as does the growing body of laboratory astrophysics data showing that non-equilibrium irradiation-induced solid-phase chemistry can drive low-temperature complex organic molecule formation.18,19

Conflicts of interest

There are no conflicts to declare.


E. H. wishes to thank the National Science Foundation for continuing to support the astrochemistry program at the University of Virginia. This research has made use of NASA's Astrophysics Data System Bibliographic Services.


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