# Neutral dissociation of methane by electron impact and a complete and
consistent cross section set

Dennis Bouwman1, Andy Martinez1,2, Bastiaan J. Braams1, Ute Ebert1,2 1 Centrum
Wiskunde & Informatica (CWI), Amsterdam, The Netherlands  2 Department of
Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB
Eindhoven, The Netherlands<EMAIL_ADDRESS>
###### Abstract

We present cross sections for the neutral dissociation of methane, in a large
part obtained through analytical approximations. With these cross sections the
work of Song et al. [J. Phys. Chem. Ref. Data, 44, 023101, (2015)] can be
extended which results in a complete and consistent set of cross sections for
the collision of electrons with up to 100 eV energy with methane molecules.
Notably, the resulting cross section set does not require any data fitting to
produce bulk swarm parameters that match with experiments. Therefore
consistency can be considered an inherent trait of the set, since swarm
parameters are used exclusively for validation of the cross sections. Neutral
dissociation of methane is essential to include (1) because it is a crucial
electron energy sink in methane plasma, and (2) because it largely contributes
to the production of hydrogen radicals that can be vital for plasma-chemical
processes. Finally, we compare the production rates of hydrogen species for a
swarm-fitted data set with ours. The two consistent cross section sets predict
different production rates, with differences of $45\%$ (at $100$ Td) and
$125\%$ (at $50$ Td) for production of H2 and a similar trend for production
of H. With this comparison we underline that the swarm-fitting procedure, used
to ensure consistency of the electron swarm parameters, can possibly
deteriorate the accuracy with which chemical production rates are estimated.
This is of particular importance for applications with an emphasis on plasma-
chemical activation of the gas.

††: Plasma Sources Sci. Technol.on June 4, 2021

Keywords: Methane, Electron Collisions, Cross Sections, Neutral Dissociation,
Electric Discharges, Low-Temperature Plasma, Plasma Assisted Combustion.

##  1 Introduction

###  1.1 Methane-containing plasmas

There are many types of plasma that contain methane (CH4). Proper models of
their properties require cross sections for the collisions of electrons with
methane molecules. The present study was particularly motivated by
applications such as plasma-assisted combustion of air-methane mixtures, where
electron impact dissociation accounts for most of the plasma-produced radicals
during the discharge phase [1, 2]. Another combustion-related application is
the production of hydrogen fuel through electron impact dissociation, referred
to as low-temperature methane conversion [3]. Furthermore, methane plasmas are
found in a variety of thin film applications, such as diamond synthesis by
plasma-assisted vapour deposition [4]. Other applications range from modelling
lightning in methane-containing atmospheres (such as Titan [5, 6]) to studies
of carbon-impurities inside a tokamak [7].

###  1.2 Demands on cross sections

Theoretical and computational studies that underlie and enable the
aforementioned applications all require a set of cross sections of electron
collisions as model input. Although the requirements that are placed on a
cross section data set can vary between applications, in general a set is
required to be complete and consistent. Within the framework of low-
temperature plasma modelling these properties are often defined according to
Pitchford et al. [8] as:

  * •

Complete cross section sets accurately represent all electron energy and
momentum losses as well as the electron number changing processes such as
ionization and attachment.

  * •

Consistent cross section sets are able to reproduce measured values of swarm
parameters within an order of ten percent, when used as an input to evaluate
the electron energy distribution function from a Boltzmann solver.

Note that these definitions only apply to the behaviour of electron swarms.
Other important demands on cross sections, such as the correct approximation
of the production rates for chemical species, are not addressed.

When compiling a data set it is often found that experimental data alone is
insufficient to ensure completeness and consistency, as data for crucial
processes might be missing or measurements from different studies might
disagree. The existence of such gaps in the literature can typically be
attributed to the challenging nature of measurements for scattering processes
such as: rotational and electronic excitation, dissociative attachment and,
most notably, neutral dissociation [9]. Although theoretical cross section
calculations can be used to supplement the experiments, such results are often
constrained to specific energy ranges and are limited to simple molecules with
low atom numbers. Within the framework of low-temperature plasma modelling a
common method to overcome the limitations imposed by missing data is to fit
presumably incorrect cross sections in order to have better agreement with
measured swarm parameters [10], c.f. the IST-Lisbon data set [11]. Such data-
fitting techniques are immensely enabling for their ability to produce
consistent data sets in the absence of reliable measurements. However by
fitting cross section data the scope of applicability of a data set is limited
to describing the electron swarm behaviour of a plasma, as the rates of
individual processes may have been altered significantly and the resulting
cross section set can be non-unique [10, 12, 13]. In other words, plasma
models using such swarm-fitted data sets are not guaranteed to predict
production rates of individual chemical species with a high degree of
accuracy.

With an eye on accurately predicting the production of reactive species, it
would be a highly attractive property for a cross section set to reproduce
swarm parameters without the need for any fitting procedures. For such a set
consistency is an inherent trait, i.e. independent of the limitations imposed
by the swarm-fitting procedure. This would be especially attractive for
applications that focus on the plasma-chemical activation of the gas, since
the absence of a fitting procedure gives greater confidence that the
individual cross sections are close to their ‘true’ value. Moreover, an
unfitted and consistent cross section set could be used in any plasma-
modelling approach (e.g. hydrodynamic, multi-term Boltzmann or Monte-
Carlo/PIC).

###  1.3 Goal of the paper

The goal of this paper is to derive cross sections for the neutral
dissociation of the ground state of CH4 by electron impact. Secondly, we want
to show that these cross sections in combination with data on other relevant
scattering processes in CH4 produces a complete and consistent data set
without the need for any data fitting. Our efforts are documented in two
parts: in section 2, we will review experimental and theoretical literature on
the electron collision cross sections of CH4. We highlight a gap in the
literature corresponding to the neutral dissociation processes. In order to
fill in this gap we propose a blend of empirical and analytical cross sections
for the neutral dissociation processes in the energy range up to $100$ eV. In
section 3 we show that the addition of our cross sections to the
recommendations of Song et al. [14] produces a complete and consistent data
set without the need for any data-fitting techniques. By performing a
Boltzmann analysis in pure methane we show that the agreement between
calculated and measured swarm parameters is within error margins.

Finally, in section 4 we compare the production of hydrogen species as given
by our cross section set and the IST-Lisbon data set [11]. The observed
differences underline the issue regarding the non-uniqueness of swarm-fitted
cross section sets.

###  1.4 Relation to earlier work

An extensive data evaluation regarding electron scattering with CH4 was
published by Song et al. [14]. In their work they recommend cross section
values for most of the electron-neutral collisions: momentum-transfer [15, 16,
17, 18, 19], vibrational excitation [19], ionization [20] and dissociative
electron attachment [21]. However, recommendations for the neutral
dissociation processes have explicitly not been made due to inconsistencies in
the available data. In section 3 we demonstrate, by performing a Boltzmann
analysis in pure methane, that simply neglecting these processes results in an
ionization rate that is a factor ten larger than experimentally observed (This
behaviour has also been documented in [22]). The reason for this is that
neutral dissociation processes are an important sink of electron energy that
must be incorporated.

Approximations for the missing cross sections of the electron impact
dissociation processes are also presented by Gadoum and Benyoucef [22]. In
essence, they employ a variation of the approximation technique formulated by
Erwin and Kunc [23, 24]. The latter is also thoroughly discussed and evaluated
in this study. The variant that Gadoum and Benyoucef [22] have used contains
more fitting parameters in their low-energy approximation. Also by reordering
the formulae, their variant requires the total (neutral and ionizing) cross
sections into CH3 as an input parameter (which they have obtained from Motlagh
and Moore [25]) instead of the total neutral dissociation. To avoid having to
discuss two variants of the same approximation technique we have chosen to
only include the original approximation technique formulated by Erwin and Kunc
[23, 24] in our analysis.

Data for a wider range of hydrocarbon collision processes in a near-wall
fusion plasma have been assembled by Janev and Reiter [26, 27]. The interest
in that work is the complete breakdown chain of methane, ethane and propane,
so including neutral and charged dissociation cross sections for electron
impact on ${\rm C}_{x}{\rm H}_{y}$ with $1\leq x\leq 3$ and $1\leq y\leq
2x+2$. Because of the paucity of data the emphasis in the work of Janev and
Reiter, especially in the more recent work [27] for the case of neutral
dissociation, is on the development of physically plausible functional forms
for the cross sections for all target hydrocarbons. The data in [26, 27] are
valuable and widely used for simulations of fusion plasma with carbon-based
wall material where collisions involving many distinct hydrocarbon radical
fragments are important. For our application to collisions with CH4 alone the
data in [26, 27] are lacking validation and uncertainty estimates, so for us
the starting point is Song et al. [14] which we supplement with neutral
dissociation cross sections validated to swarm data.

##  2 Compilation of unfitted cross sections for neutral dissociation of CH4

We will start by evaluating the literature regarding neutral dissociation. We
address the same inconsistencies as were found by Song et al. [14], but for
energies as high as $100$ eV. We then proceed by formulating our approximation
for the cross sections of this process. We will lay an emphasis on the energy
range of up to $100$ eV, relevant for low temperature plasmas. Note that some
of our proposed cross sections extend up to $500$ eV, however such high
energies are not shown because they have a negligible contribution on the
computation of swarm parameters in numerical swarm experiments, which we will
use to evaluate these approximations in section 3.

###  2.1 Neutral dissociation of CH4

The dissociation processes of methane generally occur through electronic
excitation of the molecule to an intermediate state[28]. All of the
electronically excited states of methane are short-lived and are dissociative
or subject to auto-ionization, hence the intermediate electronic excited state
can generally be omitted from consideration [14]. For the excitations that
lead to neutral dissociation several channels have to be considered:

$\displaystyle\text{e + \text{CH}${}_{4}$}\rightarrow\text{e +
\text{CH}${}_{4}^{*}$}\rightarrow$ e + CH3 \+ … (1) $\displaystyle\text{e +
\text{CH}${}_{4}$}\rightarrow\text{e + \text{CH}${}_{4}^{*}$}\rightarrow$ e +
CH2 \+ … (2) $\displaystyle\text{e + \text{CH}${}_{4}$}\rightarrow\text{e +
\text{CH}${}_{4}^{*}$}\rightarrow$ e + CH \+ … (3) $\displaystyle\text{e +
\text{CH}${}_{4}$}\rightarrow\text{e + \text{CH}${}_{4}^{*}$}\rightarrow$ e +
C + … (4)

The cross sections of these neutral dissociation processes are denoted by
$\sigma_{i}$ with $i$ representing the particular dissociated methane
fragment: CH3, CH2, etc.

Figure 1: An overview of experimental values for neutral dissociation cross
sections of each channel together with the fitted values from IST-Lisbon [11]
and our proposed values, within the considered energy range up to $100$ eV.
The shown measurements are from: Nakano et al. [29, 30], Motlagh and Moore
[25], Makochekanwa et al. [31] and Melton and Rudolph [32]. Note that these
measurements do not agree with each other.

The body of literature regarding the neutral dissociation cross sections is
sparse. For instance, no direct measurements below $100$ eV exist for
$\sigma_{\text{CH}}$ and $\sigma_{\text{C}}$. However, cross sections for the
neutral dissociation into specific excited states, e.g. CH($A^{2}\Delta$) and
CH($B^{2}\Sigma^{-}$), have been determined by Šašić et al. [33]. For the
remaining dissociation processes, the experimental observations are in
disagreement and theoretical results are only available for a narrow energy
range of $10$ eV to $16.5$ eV. In figure 1 we have shown a selection of the
experimental results evaluated by Song et al. of $\sigma_{\text{CH}_{3}}$ and
$\sigma_{\text{CH}_{2}}$ for electron energies up to $100$ eV. The relative
experimental uncertainty accompanying these measurements are $\pm 100\%$ for
Nakano et al. [29, 30], $\pm 30\%$ for Motlagh and Moore [25] and $\pm 20\%$
for Makochekanwa et al. [31]. In this figure we have also supplied the fitted
cross sections from the IST-Lisbon data set [11], our recommendations which
are derived at the end of this section and the isolated measurements of Melton
and Rudolph [32] for $\sigma_{\text{CH}_{3}}$, $\sigma_{\text{CH}_{2}}$ and
$\sigma_{\text{CH}}$ at $100$ eV. No measure of uncertainty was supplied for
the measurements by Melton and Rudolph.

Figure 2: A zoom on the low-energy range of the cross sections of neutral
dissociation into CH3 together with our recommendations (including a $\pm
25\%$ deviation). The shown values are obtained experimentally: Nakano et al.
[29, 30], Motlagh and Moore [25] and Makochekanwa et al. [31], theoretically:
Ziołkowski et al. [28], by semi-empirical approximations: Erwin and Kunc [23,
24] or by swarm fitting: IST-Lisbon [11]. Note the agreement between
Ziołkowksi and Makochekanwa.

Figure 3: A zoom on the low-energy range of the cross sections of neutral
dissociation into CH2 together with our recommendations (including a $\pm
25\%$ deviation). The shown values are obtained experimentally: Nakano et al.
[29, 30] theoretically: Ziołkowski et al. [28], by semi-empirical
approximations: Erwin and Kunc [23, 24] or by swarm fitting: IST-Lisbon [11].
Note the agreement between Ziołkowksi and Nakano.

In figure 3 we zoom in and compare the values of $\sigma_{\text{CH}_{3}}$ up
to $40$ eV from the experimental observations mentioned above with the results
from theoretical calculations by Ziołkowski et al. [28] and with the
approximations from Erwin and Kunc [23, 24]. Note the agreement between recent
experimental and theoretical results from Makochekanwa et al. [31] and
Ziołkowski et al. [28] which shows a sharp increasing cross section in the
near-threshold region. Based on this agreement and the fundamental nature of
their work, Ziołkowski et al.[28] conclude that their prediction and the
measurements of Makochekanwa et al. [31] of $\sigma_{\text{CH}_{3}}$ are more
reliable than the results of Nakano et al. [29, 30] and Erwin and Kunc [23,
24]. Furthermore, Ziołkowski et al. [28] observe that within their considered
energy range, $10$ to $16.5$ eV, their predictions match with Motlagh and
Moore [25]. However when transposing their measured relative cross sections to
an absolute scale, Motlagh and Moore only considered neutral dissociation into
CH3. This means that the contributions due to $\sigma_{\text{CH}_{2}}$,
$\sigma_{\text{CH}}$ and $\sigma_{\text{C}}$ are neglected. Although these
cross sections are not known exactly we estimate, based on the measurements of
Nakano et al. [29, 30], that the cross sections for $\sigma_{\text{CH}_{2}}$
are considerable in the region between the threshold energy (which can be
estimated to lie around $7.5$ eV) and $20$ eV. For this reason we do not use
the measured cross sections for neutral dissociation into CH3 from Motlagh and
Moore [25].

A zoomed-in view of $\sigma_{\text{CH}_{2}}$ is shown in figure 3. In this
case the only experimental observation for energies below $40$ eV are reported
by Nakano et al. [29, 30]. Contrary to their results for
$\sigma_{\text{CH}_{3}}$, the values of $\sigma_{\text{CH}_{2}}$ are in
excellent agreement with the theoretical predictions from Ziołkowski et al.
[28] in the near-threshold energy region. Moreover, the approximation by Erwin
and Kunc [24] deviates significantly from the aforementioned results, as it
does not portray the sharp rise for low energies. This qualitative difference
might be attributed to the absence of any data calibration, aside from fixing
a threshold energy, of the low-energy ($<50$ eV) approximation of Erwin and
Kunc [24]. For these reasons, Ziołkowski et al. [28] conclude that their
results and the measurements of Nakano et al. [29, 30] for the dissociation
into CH2 are more reliable.

###  2.2 Our proposed cross sections

In the previous section it was discussed, relying on the advancements
regarding neutral dissociation cross sections in the low-energy regime [31,
28], that the only available measurements for dissociation into CH3 across a
wide energy range (i.e. Motlagh and Moore [25] and Nakano et al. [29, 30]) are
unsatisfactory.

For this reason we resort to an alternative method to obtain cross sections
for neutral dissociation into CH3, following Janev and Reiter [26] and Erwin
and Kunc [23] with support from measurements of Winters [34]. Winters [34]
observed that for energies above $50$ eV the total dissociation cross section
is split equally between neutral and ionizing dissociation, suggesting a
common mechanism. Janev and Reiter [26] describe the common mechanism as: “[…]
excitation of a dissociative state which lies in the ionization continuum.
Autoionization of this state leads to dissociative ionization, while its
survival leads to dissociation to neutrals.” They conclude from this that
cross section branching ratios within the neutral dissociation channel should
match cross section branching ratios within the ionized dissociation channel.
Erwin and Kunc [23] treat these branching ratios in a similar manner.
Therefore, consistent with Janev and Reiter [26] and with Erwin and Kunc [23]
we chose to employ for electron impact energies above the threshold for
ionizing dissociation the following functional approximation for
$\sigma_{\text{CH}_{3}}$:

$\displaystyle\sigma_{\text{CH}_{3}}$
$\displaystyle=\frac{\sigma_{\text{ND}}}{\sigma_{\text{ID}}}(\sigma_{[\text{CH}_{3}^{+}+\text{H}]}+\sigma_{[\text{CH}_{3}+\text{H}^{+}]}),$
$\displaystyle\text{for }\epsilon\geq\epsilon_{c},$ (5)

with $\sigma_{\text{ID}}$ the cross section for total ionizing dissociation,
$\sigma_{\text{ND}}$ the cross section for total neutral dissociation,
$\sigma_{[\text{CH}_{3}^{+}+\text{H}]}$ and
$\sigma_{[\text{CH}_{3}+\text{H}^{+}]}$ correspond to cross sections of
specific ionizing dissociation by electron impact and $\epsilon_{c}$ is the
lowest energy at which “reliable” experimental cross sections are available.
For ionization processes we adopted the cross sections reported by Lindsay and
Mangan [20], as is recommended in Song et al. [14]. Furthermore, the value of
$\sigma_{\text{ND}}$ can be obtained by subtracting the total ionizing
dissociation from the total dissociation:

$\sigma_{\text{ND}}=\sigma_{\text{TD}}-\sigma_{\text{ID}},$ (6)

with $\sigma_{\text{TD}}$ the cross section for the total dissociation
reported by Winters [34]. Fitting functions reported in Shirai et al. [35]
were used for $\sigma_{\text{TD}}$, all ionizing dissociation cross sections,
and the dissociative electron attachment cross sections. More details on these
fitting functions and their parameters can be found in A. We can recover the
initial observation of Winters by taking
$\sigma_{\text{ND}}=\sigma_{\text{ID}}$, which generally holds for energies
above $50$ eV. Note that the approximation from equation 2.2 only holds for
energies above the threshold energies of the corresponding ionizing
dissociation reactions.

|  $\epsilon_{\text{ND}}$ (eV) |  $\epsilon_{b}$ (eV) |  $\epsilon_{c}$ (eV) |  $\sigma^{(2)}_{i}(\epsilon_{c})$ (m2) |  $a$ (m2) |  $p$
---|---|---|---|---|---|---
$\sigma_{\text{CH}_{3}}$ | $7.5$ | $10.5$ | $13.16$ | $8.8\cdot 10^{-21}$ | $1.5\cdot 10^{-20}$ | $1.5$
$\sigma_{\text{CH}}$ | $15.5$ | $18.5$ | $22.37$ | $2.9\cdot 10^{-27}$ | $1.3\cdot 10^{-26}$ | $1.6$
$\sigma_{\text{C}}$ | $15.5$ | $18.5$ | $22.37$ | $6.8\cdot 10^{-29}$ | $3.1\cdot 10^{-28}$ | $1.6$
Table 1: The parameters used for the low-energy approximations of our proposed
cross sections.

However, neutral dissociation reactions have a lower threshold energy than
their respective ionizing reactions and therefore also occur at energies below
the ionization threshold. Thus for energies below the respective ionizing
dissociation thresholds we apply the low-energy approximation method of Erwin
and Kunc. Here we only present the final result applied to
$\sigma_{\text{\text{CH}${}_{3}$}}$ used in our work, for a detailed
discussion we refer to the original work [24]. In this method the below-
ionization energy range is divided in a near-threshold range,
${\epsilon_{\text{ND}}\leq\epsilon\leq\epsilon_{b}}$, and a linear-growth
range ${\epsilon_{b}\leq\epsilon\leq\epsilon_{c}}$. Here
$\epsilon_{\text{ND}}$ represents the threshold energy for neutral
dissociation, $\epsilon_{b}$ represents the energy value separating the near-
threshold range from the linear growth range. Then the near-threshold cross
section is given by:

$\displaystyle\sigma_{\text{CH}_{3}}$
$\displaystyle=a\left(\frac{\epsilon}{\epsilon_{\text{ND}}}-1\right)^{p},$
$\displaystyle\text{ for }\epsilon_{\text{ND}}\leq\epsilon\leq\epsilon_{b},$
(7)

with $a$ and $p$ positive constants. For the linear-growth range the cross
section are blended with the relation from equation (2.2) as follows:

$\displaystyle\sigma_{\text{CH}_{3}}$ $\displaystyle=$
$\displaystyle\sigma_{\text{CH}_{3}}^{(1)}(\epsilon_{b})+\frac{\sigma_{\text{CH}_{3}}^{(2)}(\epsilon_{c})-\sigma_{\text{CH}_{3}}^{(1)}(\epsilon_{b})}{\epsilon_{c}-\epsilon_{b}}(\epsilon-\epsilon_{b}),$
(8) $\displaystyle\mbox{for }\epsilon_{b}\leq\epsilon\leq\epsilon_{c},$

with blending-parameter $\sigma_{\text{CH}_{3}}^{(1)}(\epsilon_{b})$
representing the value of the cross section evaluated at $\epsilon_{b}$ as
calculated from equation (2.2), and
$\sigma_{\text{CH}_{3}}^{(2)}(\epsilon_{c})$ the value of the experimentally-
obtained cross section at corresponding energy $\epsilon_{c}$ corresponding to
equation (2.2). As can be seen, equations (2.2)-(8) determine the cross
section for neutral dissociation into CH3 for the whole energy range.

The cross sections for the neutral dissociation into CH and C are obtained
analogously. The values for the parameters $\epsilon_{\text{ND}}$,
$\epsilon_{b}$, $\epsilon_{c}$, $\sigma_{i}^{(2)}(\epsilon_{c})$, $a$ and $p$
used in our work are given in table 1. The parameters for $\sigma_{\text{C}}$
are not covered by Erwin and Kunc [24], but here they are obtained following
the same reasoning for the parameters of CH.

Although the approximations defined in equations (2.2)-(8) can be used for any
of the neutral dissociation channels, they are only used for
$\sigma_{\text{CH}_{3}}$, $\sigma_{\text{CH}}$ and $\sigma_{\text{C}}$. For
the remaining neutral dissociation process, $\sigma_{\text{CH}_{2}}$, relying
on the experimental observations by Nakano et al. [29, 30] is preferred over
the application of a similar approximation, due to the agreement with
theoretical predictions by Ziołkowski et al. [28]. Thus, in this study we take
$\sigma_{\text{CH}_{2}}$ to be given by a fourth-order polynomial fit through
the measurements of Nakano et al.. We refer to A for the fitting parameters.

Our proposed cross sections for the neutral dissociation processes are shown
in figures 1-3. For $\sigma_{\text{\text{CH}${}_{3}$}}$ the qualitative trend
of our proposed cross section is similar to the results from Makochekanwa et
al. [31] and Ziołkowski et al. [28] in the near-threshold energy region,
although it appears shifted to higher energies by around $1.5$ eV. Our
proposed cross sections have a maximum value of ${2.29\cdot
10^{-20}\leavevmode\nobreak\ \text{m}^{2}}$ at $24$ eV, which is higher than
any of the experimental results. After attaining this maximum the value decays
and eventually agrees with the isolated measurement of Melton and Rudolph [32]
at $100$ eV. Note also that for energies above $50$ eV our proposed value
corresponds to the fitted values from the IST-Lisbon set [11]. Our proposed
values for $\sigma_{\text{\text{CH}${}_{2}$}}$, based on the measurements of
Nakano et al. [29, 30], vanish for energies above $45$ eV. This contradicts
with the measurements from Melton and Rudolph [32], which suggest that the
cross section should be around ${1.95\cdot 10^{-21}\leavevmode\nobreak\
\text{m}^{2}}$ at $100$ eV. This difference is also recognized by Nakano et
al. [29, 30]. To the best of our knowledge, there is currently no
straightforward method to reconcile these two observations. Moreover, our
proposed cross sections agree (qualitatively) with the fitted counterparts
from the IST-Lisbon set [11], although the latter appears to have shifted the
peak to lower energies by approximately $4$ eV. For
$\sigma_{\text{\text{CH}$$}}$ and $\sigma_{\text{C}}$ there is little
literature to compare with aside from observing that our approximation of
$\sigma_{\text{\text{CH}$$}}$ agrees with the isolated measurement of Melton
and Rudolph at $100$ eV. Furthermore, we can compare our values of
$\sigma_{\text{\text{CH}$$}}$ with the results for the neutral dissociation
into the excited fragments CH($A^{2}\Delta$) and CH($B^{2}\Sigma^{-}$) which
have been determined by Šašić et al. [33]. It should hold that the
dissociation into specific excited fragments is lower than
$\sigma_{\text{\text{CH}$$}}$. As shown in figure 4 this behaviour generally
holds. Only in the vicinity of the threshold, i.e. below $25$ eV, do we
observe that the cross sections for dissociation into excited fragments are
higher than our proposed value. However, this discrepancy is small compared to
the dominant inelastic scattering processes and will therefore be negligible
within the context of the swarm experiments that are presented in the
following sections.

Figure 4: The cross sections $\sigma_{\text{\text{CH}$$}}$ and
$\sigma_{\text{C}}$ used in our work alongside the experimentally-derived
cross sections into specific excited fragments by Šašić et al. [33]. The
latter should always be smaller than $\sigma_{\text{\text{CH}$$}}$. This holds
in general, aside from a small discrepancy in the vicinity of the threshold,
i.e. below $25$ eV.

##  3 Comparison of calculated and measured swarm parameters

Within the framework of low-temperature plasma modelling, a computation of
swarm parameters is performed routinely, typically for reduced electric fields
($E/N$) between $0.1$ Td and $1000$ Td, with $E$ representing the electric
field and $N$ the number density of the gas. In a fluid description of
electron swarms (e.g. [36] and references therein) the electrons are described
by their density only and this density obeys a reaction-drift-diffusion
equation governed by swarm parameters: diffusion coefficient $D$, mobility
$\mu$ and by coefficients for ionization $\alpha$ and attachment $\eta$.
Moreover, the characteristic energy $D/\mu$, reduced mobility $\mu N$ and
reduced Townsend ionization coefficient $\alpha/N$ are functions of the
reduced electric field $E/N$ only (for a not too large electric fields). These
swarm parameters can be obtained, given the gas composition and a cross
section data set, by solving the Boltzmann equation [37].

In this section we will use the difference between computed and measured swarm
parameters as an implicit metric to evaluate the cross sections for neutral
dissociation in conjunction with the recommendations from Song et al. [14]
(neglecting rotational excitations since these are already accounted for in
the elastic momentum-transfer cross section). Note that explicit evaluation of
the cross sections for neutral dissociation processes is not possible due to
disagreement in the available literature, as was shown in section 2.1. On the
other hand, the swarm parameters of a methane plasma are well-known, with the
exception of the attachment coefficient. This can be seen from a compilation
made in Alves [11] of measurements containing observations for reduced
mobility $\mu N$, characteristic energy $D/\mu$, and the reduced Townsend
ionization coefficient $\alpha/N$ [38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48]. Assuming that the recommendations by Song et al. have a sufficiently low
error margin, any disagreement between calculated and measured swarm
parameters must imply that the remaining cross sections, i.e. the neutral
dissociation processes, are inaccurate. We will compare bulk swarm parameters
as computed by a Monte-Carlo solver [49] based on the modelling framework
presented in [50]. The simulations are performed at standard temperature and
pressure. We emphasize that we show the bulk coefficients and that the
characteristic energy is based on the transversal diffusion coefficient. The
swarm parameters have been computed for four cross section data sets:

  1. 1.

the swarm-fitted IST-Lisbon database [11],

  2. 2.

the recommendations by Song et al. [14] (lacking any neutral dissociation
process),

  3. 3.

the recommendations by Song et al. in conjunction with the original
approximations by Erwin and Kunc [23, 24] for neutral dissociation,

  4. 4.

the recommendations by Song et al. in conjunction with our approximations for
neutral dissociation.

Moreover, for data set (4) we have included the effect of varying our proposed
cross sections for neutral dissociation by $\pm 25\%$. This results in an
upper and lower bound for the reproduced swarm parameters. These bounds define
a range which we will refer to as the sensitivity interval. This interval is
included to illustrate the effect that possible errors on the cross sections
for neutral dissociation might impose on the computed swarm parameters.

(a) characteristic energy $D/\mu$

(b) reduced mobility $\mu N$

(c) reduced Townsend ionization coefficient $\alpha/N$

Figure 5: Measured and calculated values of the swarm parameters in pure
methane. The shaded red region corresponds to the sensitivity interval,
obtained by including a $\pm 25\%$ deviation on the neutral dissociation
processes. Overestimating the neutral dissociation leads to underestimating
the ionization, and vice versa, hence the use of the ‘$\mp$’ sign in 5. Our
cross section set reproduces swarm parameters within a few tens of percent.

In figures 5-5 we have respectively shown the characteristic energy, mobility
and ionization from numerical and experimental studies on a double logarithmic
scale. All of the considered data sets reproduce the characteristic energy
within an error margin of $20\%$ and the mobility within an error margin of
$7.5\%$, as can be seen in figures 5 and 5. One exception to this can be found
at reduced electric fields below $10$ Td. Where data set (1) exhibits
deviations from the measured values (and the respective error margins) of the
characteristic energy ($25\%$) and the mobility ($15\%$ between $1$ Td and
$10$ Td and $30\%$ below $1$ Td). On the other hand the ionization
coefficient, in figure 6, varies strongly between different data sets. Data
set (2) overshoots the measured values by as much as a factor of ten. Clearly
the neutral dissociation of methane plays a vital role in determining the
electron number changing processes and must be incorporated. Even though
adding the cross sections for neutral dissociation from Erwin and Kunc reduces
this discrepancy, the corresponding data set (3) still exhibits notable
discrepancies with measured ionization coefficients. Given the high values of
the ionization coefficient it appears that data sets (2) and (3) critically
underestimate the sinks for electron energy. This can be explained by
considering that the underestimation of energy losses means that an electron
is more likely to obtain energies above the ionization potential and
subsequently the rate of ionization increases. By inspecting figures 3 and 3,
one can observe that for electron energies below $25$ eV the values from Erwin
and Kunc are lower than (most of) the other reported values. This behaviour is
especially pronounced for dissociation into CH2. For swarm experiments in
general, the electrons in this energy regime play a dominant role in
determining swarm parameters as the abundance of electrons typically reduces
strongly for higher electron energies. Therefore the effect of omitting or
underestimating the neutral dissociation processes as is done in data sets (2)
and (3) can be expected to introduce large discrepancies in the computed
ionization coefficient, as is also seen in figure 5.

Such an overestimation of the ionization coefficient is not present for the
other considered data sets. The swarm-fitted data set (1) reproduces the
ionization coefficient with a maximum deviation of $25\%$ in the region
between $100$ Td and $800$ Td. However below $100$ Td the deviations starts to
increase. For instance, at $90$ Td this deviation exceeds $40\%$. The large
accuracy between $100$ Td and $800$ Td is to be expected from data sets which
employ fitting procedures to ensure completeness and consistency. At $1000$ Td
the deviation is around $35\%$. The reproduction of the ionization coefficient
is also observed for our approximations in conjunction with Song et al., data
set (4), with a maximum deviation up to $35\%$ (at $100$ Td). This is somewhat
larger than observed for data set (1). For reduced electric fields below $100$
Td our reduced Townsend ionization coefficient is closer to measurements than
data set (1). Notably, up to $500$ Td it can be observed that our reduced
Townsend ionization coefficient is consistently lower than experimentally
observed ionization coefficient. This indicates, if one assumes that the
ionizing cross sections are sufficiently accurate, that the sum of all non-
ionizing inelastic cross sections used here is probably an overestimation.

Furthermore, from the sensitivity interval corresponding to data set (4) we
can conclude that the reproduction of characteristic energy and mobility is
almost completely independent of the neutral dissociation processes. In
contrast, the sensitivity interval for the ionization coefficient shows a
significant spread. This again underlies that neutral dissociation processes
are an important electron energy sink, at least within the context of low-
temperature plasmas. Moreover, the measured values of the reduced Townsend
ionization coefficient lie within the sensitivity interval, indicating that a
small adjustment ($<25\%$) of the proposed cross sections can account for the
observed deviations regarding this swarm parameter.

##  4 Production rates for hydrogen radicals

(a) Low $E/N$

(b) High $E/N$

Figure 6: The predicted reaction rates for the production of the hydrogen
species H and H2 for the (swarm-fitted) IST-Lisbon set and Ours (unfitted).
Although both cross section sets can be considered consistent (which refers to
behaviour of electron swarms only), they exhibit clear differences in the
prediction of hydrogen species production.

In the previous section we have introduced two consistent data sets: IST-
Lisbon (1) and Ours (4). The fundamental difference between these two sets is
that (4) is unfitted and consistent, whereas (1) employs a fitting procedure
to ensure reproduction of swarm parameters. The use of such data-fitting
techniques has already been discussed in section 1.2. Here we will illustrate
how both data sets predict the production of atomic and molecular hydrogen by
inspecting the sum of the reaction rates of hydrogen-producing electron
collisions as calculated by a Monte-Carlo solver [49] based on the modelling
framework presented in [50]. The simulations are performed at standard
temperature and pressure.

In order to make such a comparison we need ratios regarding the by-products of
dissociative electron collisions. However, such data is virtually non-
existent. For example: there are no cross sections which distinguish the
neutral dissociation processes:

$\displaystyle\text{e + \text{CH}${}_{4}$}\rightarrow$ $\displaystyle\text{e +
\text{CH}${}_{2}$ + H}_{2}$ (9) $\displaystyle\text{e +
\text{CH}${}_{4}$}\rightarrow$ e + CH2 \+ 2H (10)

It is known that the dissociation energy of the relatively strong hydrogen
bond is $4.52$ eV, therefore it can be expected that due to this additional
energy barrier the reaction rate of equation (10) will be lower than equation
(9). However, without direct observations such arguments will always remain
qualitative. For the current purpose of comparing the radical yields of the
two data sets, we will assume that the composition of hydrogen products will
always be in the lowest energy state. In other words, we assume that reactions
like equation (10), which requires additional energy for dissociation, will
not occur. The effect is that we will underestimate atomic hydrogen yield, and
subsequently overestimate the production of molecular hydrogen.

With this assumption, the reaction rates for hydrogen production have been
calculated for both data sets; they are shown in figure 6. It can be seen that
data set (1) predicts atomic hydrogen yields, approximately $35\%$ lower than
(4) above $100$ Td. Similar deviations are also observed for the molecular
hydrogen production. For instance, above $100$ Td the maximum deviation is
$45\%$. However, for reduced electric fields below $100$ Td the deviations
between the predictions of production rates for molecular hydrogen are
increasing. For instance, at $50$ Td data set (1) predicts a molecular
hydrogen yield which is $125\%$ higher than data set (4). For atomic hydrogen
we find a difference around $50\%$ at $50$ Td.

These deviations between the production rates of chemical species of two
consistent sets clearly illustrate the non-uniqueness of swarm-fitted data
sets. Whether the errors on the production rates for chemical species
introduced by relying on data-fitting are tolerable is always dependent on the
application and the extent of adjustments performed. However, given the highly
reactive nature of atomic hydrogen and the nonlinear nature of plasma-chemical
applications, such deviations have to be treated with care.

##  5 Summary and Outlook

###  5.1 Summary

The main contribution of this article are the cross sections for the neutral
dissociation of the ground state of CH4 by electron impact. Secondly, we have
used these values to arrive at a complete and consistent cross sections for
electron collisions with methane for reduced electric fields between $0.1$ Td
and $1000$ Td, without relying on any data-fitting techniques. This data set
is largely based on the recommendations of Song et al. [14], with the addition
of a blend of empirical and analytical cross sections for the remaining
neutral dissociation processes.

Furthermore this work includes a Boltzmann analysis using a Monte-Carlo
solver. We have shown that the presented set of cross sections reproduces
measured swarm parameters with maximum deviations of: $35\%$ for ionization,
$7.5\%$ for mobility and $20\%$ for characteristic energy.

The presented cross section set distinguishes itself from other data sets by
not relying on any data-fitting techniques to ensure consistency. This feature
makes our cross section set independent of the limitations imposed by the
swarm-fitting procedure. This can be especially attractive for applications
that focus on plasma-chemical activation of the gas, such as plasma-assisted
vapour deposition, low-temperature methane reforming, etc. Moreover, the
absence of any data fitting means that the presented cross section set can be
used in a variety of plasma-modelling approaches (e.g. hydrodynamic, multi-
term Boltzmann or Monte-Carlo/PIC).

###  5.2 Outlook

The validity of the cross sections proposed in this work has been considered
by comparing measured and calculated swarm parameters. In principle this is an
implicit metric, since the set of cross sections as a whole is considered as
opposed to individual cross sections. However, in section 3 we have assumed
that the recommendations of Song et al. [14] have a sufficiently low error
margin such that deviations in the reduced Townsend ionization coefficient can
be primarily attributed to the proposed cross sections for neutral
dissociation. Although this assumption enables much of the steps taken in this
work, it does not give explicit certainty. One way to improve on this is by
studying the swarm parameters of mixtures of methane with rare gases [10]. For
example, the swarm parameters in Ar-CH4 mixtures are studied by Sebastian and
Wadehra [51]. Still, benchmark experiments for individual cross sections
remain highly desirable if the difficulty of diagnosing neutral radical
fragments can be overcome.

On the side of computation it would be very desirable to see work in the style
of Ziołkowski et al. [28] (based on R-matrix calculations of electron
excitation of methane followed by quasi-classical trajectory simulations with
surface hopping) extended to higher electron collision energy than $17$ eV.
The work of Brigg et al.[52] highlights some electronic structure issues with
these computations and in particular they recommend a multi-reference
configuration interaction approach to deal with the multiply-excited target
states that are important at high impact energy. (However, the neutral
dissociation cross section calculations in Brigg et al. [52] are limited to
electron impact energies below $15$ eV and they do not supplant the results
from Ziołkowski et al. [28]). Such R-matrix calculations and trajectory
simulations would naturally predict branching ratios between $2$H and H2
channels as well, although an assessment of the importance of zero-point
energy (quantized vibrational energy in molecular fragments) should be made.
However, there are tools (such as ring polymer Molecular Dynamics [53]) to
incorporate this quantum effect into trajectory calculations.

## Acknowledgements

The authors would like to thank Grzegorz Karwasz and Jonathan Tennyson for
their expertise and valuable feedback during correspondence. D.B. acknowledges
funding through the Dutch TTW-project 16480 “Making Plasma-Assisted Combustion
Efficient”. A.M. has received funding from the European Union’s Horizon 2020
research and innovation program under project ID 722337 (SAINT).

## Availability of data

The data set will be made available on www.lxcat.net [54]. Furthermore, cross
sections obtained in this study are also available in analytical form and as
tabulated data in A and B, respectively. The code for the Monte-Carlo
Boltzmann solver can be found on www.gitlab.com/MD-CWI-NL/particle_swarm with
the version used identified by the commit hash e04a5644 made on 25th of May
2021.

##  Appendix A Fitting functions and parameters for used and reported cross
sections

The cross sections for total neutral dissociation, all dissociative ionization
processes, and dissociative electron attachment were obtained from fits
through the data points reported in tables by Song et al. [13]. The functions
used were those reported in Shirai et al. [35]. The fitting parameters were
obtained again for this paper.

###  A.1 Basis functions

Shirai et al. [35] used 3 basis functions from which the fitting functions
were created:

$\displaystyle f_{1}(x)$
$\displaystyle=\sigma_{0}a_{1}\left(\frac{x}{\epsilon_{R}}\right)^{a_{2}},$
(11) $\displaystyle f_{2}(x)$
$\displaystyle=\frac{f_{1}(x)}{\left[1+\left(\frac{x}{a_{3}}\right)^{a_{2}+a_{4}}\right]},$
(12) $\displaystyle f_{3}(x)$
$\displaystyle=\frac{f_{1}(x)}{\left[1+\left(\frac{x}{a_{3}}\right)^{a_{2}+a_{4}}+\left(\frac{x}{a_{5}}\right)^{a_{2}+a_{6}}\right]},$
(13)

with $\sigma_{0}=1\cdot 10^{-20}$ m2, $\epsilon_{R}=1.361\cdot 10^{-2}$ keV
(Rydberg constant), and $a_{i}$ the parameters which will be obtained for each
specific reaction by fitting the data points.

###  A.2 Dissociative Ionization

The fitting function reported by Shirai et al. [35] to be used for the
dissociative ionization reactions is the following:

$\sigma_{i}(\epsilon)=f_{3}(\epsilon_{1}),$ (14)

with $\epsilon$ the incident electron energy in keV,
$\epsilon_{1}=\epsilon-\epsilon_{th}$, and $\epsilon_{th}$ the threshold
energy of the reaction in keV. Equation 14 was fitted through the tabulated
cross sections and threshold energies for the dissociative ionization
reactions reported in Song et al. [13]. The used data points and resulting
fits are shown up to 100 eV in figure 7. The fitting parameters are tabulated
in table 2

Figure 7: The cross sections of the dissociative ionization reactions of CH4. The solid lines are the result of fitting equation (14) to the tabulated cross sections for these reactions from Song et al. [13] which are represented by filled circles in the same color. |  $\epsilon_{th}$ (eV) | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$
---|---|---|---|---|---|---|---
CH${}_{3}^{+}$ | 12.63 | 5.5333 | 2.7119 | 0.0071 | -0.2619 | 0.0194 | 0.8917
CH${}_{2}^{+}$ | 16.20 | 0.2575 | 2.9997 | 0.0141 | -0.2828 | 0.0289 | 1.0172
CH+ | 22.20 | 0.295 | 3.4235 | 0.0207 | 0.9925 | 0.0100 | -0.5789
C+ | 22.00 | 0.0392 | 4.6413 | 0.0243 | 1.1558 | 0.0125 | -0.7372
H${}_{2}^{+}$ | 22.30 | 0.0134 | 5.0600 | 0.0147 | -0.7746 | 0.0242 | 1.0240
H+ | 21.10 | 0.0985 | 2.7831 | 0.0210 | -0.6691 | 0.0403 | 1.0503
Table 2: Parameters obtained by fitting equation (14) to the tabulated cross
sections of Song et al. [13] for dissociative ionization reactions

###  A.3 Dissociative Electron Attachment

Shirai et al. [35] use the same fitting function for dissociative ionization,
equation (14), as for fitting the dissociative electron attachment cross
sections (including the same definition for $\epsilon_{1}$). We use this
fitting function to fit the tabulated cross sections for dissociative electron
attachment reactions from Song et al. [13]. The fits and corresponding data
points are shown in figure 8 and the fitting parameters are reported in table
3.

Figure 8: The cross sections of the dissociative electron attachment reactions of CH4. The solid lines are the result of fitting equation (14) to the tabulated cross sections for these reactions from Song et al. [13] which are represented as filled circles in the same color. |  $\epsilon_{th}$ (eV) | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$
---|---|---|---|---|---|---|---
H- | 6.0 | 128.0817 | 5.0736 | 0.0024 | 0.1908 | 0.0041 | 10.1747
CH${}_{2}^{-}$ | 6.0 | 1.5496 | 3.1405$\cdot 10^{-5}$ | 0.0012 | 4.8957 | 0.0164 | -4.8826
Table 3: Parameters obtained by fitting equation (14) to the tabulated cross
sections of Song et al. [13] for dissociative electron attachment

###  A.4 Total Dissociation

The fitting function for the total dissociation used by Shirai et al. [35] is
given by:

$\sigma_{\text{TD}}(\epsilon)=f_{2}(\epsilon_{1})+a_{5}\cdot
f_{2}\left(\frac{\epsilon_{1}}{a_{6}}\right),$ (15)

where $\epsilon_{1}$ again has the same definition as for equation (14). The
total dissociation cross section was measured by Winters [34] and Perrin et
al. [55]. We have obtained data points for both measurements by extracting
them from the published graphs using WebPlotDigitizer [56]. The fits and the
data points for both measurements as well as equation (15) using the fitting
parameters reported by Shirai et al. [35] for total dissociation are shown in
figure 9. Deviations up to 20% can arise due to different fitting parameters
and fitted data points. These deviations in the total dissociation will
propagate to the cross sections of the individual neutral dissociation
reactions. Increasing the cross section of the neutral dissociation cross
sections has the effect of reducing the Townsend ionization coefficient
$\alpha$. In this paper we have used the data points of Winters [34] and the
fitting parameters as reported in table 4.

Figure 9: The total dissociation cross sections as measured by Winters [34] (filled blue circles), and by Perrin et al. [55] (filled red circles). The solid lines in the same colors as the measurement points are the fits done in this paper using equation (15). The solid line of Shirai et al. [35] is obtained by using their reported fitting parameters with equation (15). $\epsilon_{th}$ (eV) | 4.51
---|---
$a_{1}$ | 4.1200
$a_{2}$ | 3.0594
$a_{3}$ | 0.0142
$a_{4}$ | 0.3606
$a_{5}$ | 0.4630
$a_{6}$ | 3.8830
Table 4: Parameters obtained by fitting equation (15) to the measured cross
sections of Winters [34] for total dissociation.

###  A.5 Neutral Dissociation to CH2

In this paper we have taken the measured cross sections for neutral
dissociation into CH2 from Nakano et al. [29, 30]. To smooth the data we have
fitted a fourth order polynomial through the data points:

$f(\epsilon)=a_{0}+a_{1}\epsilon+a_{2}\epsilon^{2}+a_{3}\epsilon^{3}+a_{4}\epsilon^{4},$
(16)

with fitting parameters $a_{i}$, and $\epsilon$ the incident electron energy
in eV. Note that this function is only valid within the bounds of the
measurement energies i.e. $9.1$ eV $\leq\epsilon\leq 44.4$ eV. The fit and the
corresponding data points are shown in figure 10. The fitting parameters are
reported in table 5.

Figure 10: The cross sections for neutral dissociation to CH2 as measured by Nakano et al. [29, 30] and the corresponding fit obtained by equation (16) combined with the fitting parameters in table 5 $a_{0}$ | $-3.0203\cdot 10^{-20}$
---|---
$a_{1}$ | $5.4772\cdot 10^{-21}$
$a_{2}$ | $-2.8119\cdot 10^{-22}$
$a_{3}$ | $5.8213\cdot 10^{-24}$
$a_{4}$ | $-4.3221\cdot 10^{-26}$
Table 5: Parameters obtained by fitting equation (16) to the measured cross
sections of Nakano et al.[29, 30] for neutral dissociation to CH2

##  Appendix B Tabulated cross sections for neutral dissociation to CH3, CH,
and C

Calculated cross sections for neutral dissociation into CH3, CH and C are
reported in table 7, 6, and 8, respectively.

$\epsilon$ (eV) |  $\sigma_{\text{CH}}$ (m2)
---|---
15.79 | $2.2169\cdot 10^{-29}$
17.11 | $3.4360\cdot 10^{-28}$
18.42 | $8.9545\cdot 10^{-28}$
19.74 | $1.5509\cdot 10^{-27}$
21.05 | $2.2067\cdot 10^{-27}$
22.37 | $2.8625\cdot 10^{-27}$
23.68 | $4.3996\cdot 10^{-24}$
25.00 | $3.4406\cdot 10^{-23}$
26.00 | $8.8330\cdot 10^{-23}$
28.11 | $3.0330\cdot 10^{-22}$
30.22 | $5.9931\cdot 10^{-22}$
32.33 | $8.7592\cdot 10^{-22}$
34.44 | $1.0797\cdot 10^{-21}$
36.56 | $1.2107\cdot 10^{-21}$
38.67 | $1.2892\cdot 10^{-21}$
40.78 | $1.3348\cdot 10^{-21}$
42.89 | $1.3612\cdot 10^{-21}$
45.00 | $1.3769\cdot 10^{-21}$
46.00 | $1.3820\cdot 10^{-21}$
52.00 | $1.3999\cdot 10^{-21}$
58.00 | $1.4092\cdot 10^{-21}$
64.00 | $1.4120\cdot 10^{-21}$
70.00 | $1.4063\cdot 10^{-21}$
76.00 | $1.3920\cdot 10^{-21}$
82.00 | $1.3706\cdot 10^{-21}$
88.00 | $1.3441\cdot 10^{-21}$
94.00 | $1.3146\cdot 10^{-21}$
100.0 | $1.2839\cdot 10^{-21}$
Table 6: Calculated cross sections for neutral dissociation to CH. Threshold energy is 15.5 eV. $\epsilon$ (eV) |  $\sigma_{\text{CH}_{3}}$ (m2)
---|---
7.90 | $1.7980\cdot 10^{-22}$
9.21 | $1.6220\cdot 10^{-21}$
10.53 | $3.8168\cdot 10^{-21}$
11.84 | $6.2953\cdot 10^{-21}$
13.16 | $8.7738\cdot 10^{-21}$
14.47 | $1.2135\cdot 10^{-20}$
15.79 | $1.5230\cdot 10^{-20}$
17.11 | $1.7797\cdot 10^{-20}$
18.42 | $1.9773\cdot 10^{-20}$
19.74 | $2.1205\cdot 10^{-20}$
21.05 | $2.2160\cdot 10^{-20}$
22.37 | $2.2697\cdot 10^{-20}$
23.68 | $2.2872\cdot 10^{-20}$
25.00 | $2.2732\cdot 10^{-20}$
26.00 | $2.2445\cdot 10^{-20}$
28.11 | $2.1454\cdot 10^{-20}$
30.22 | $2.0177\cdot 10^{-20}$
32.33 | $1.8871\cdot 10^{-20}$
34.44 | $1.7693\cdot 10^{-20}$
36.56 | $1.6706\cdot 10^{-20}$
38.67 | $1.5914\cdot 10^{-20}$
40.78 | $1.5296\cdot 10^{-20}$
42.89 | $1.4821\cdot 10^{-20}$
45.00 | $1.4462\cdot 10^{-20}$
46.00 | $1.4324\cdot 10^{-20}$
52.00 | $1.3817\cdot 10^{-20}$
58.00 | $1.3624\cdot 10^{-20}$
64.00 | $1.3547\cdot 10^{-20}$
70.00 | $1.3482\cdot 10^{-20}$
76.00 | $1.3388\cdot 10^{-20}$
82.00 | $1.3259\cdot 10^{-20}$
88.00 | $1.3102\cdot 10^{-20}$
94.00 | $1.2926\cdot 10^{-20}$
100.0 | $1.2743\cdot 10^{-20}$
Table 7: Calculated cross sections for neutral dissociation to CH3. Threshold energy is 7.5 eV. $\epsilon$ (eV) |  $\sigma_{\text{C}}$ (m2)
---|---
15.79 | $5.2967\cdot 10^{-31}$
17.11 | $8.2093\cdot 10^{-30}$
18.42 | $2.1394\cdot 10^{-29}$
19.74 | $3.7055\cdot 10^{-29}$
21.05 | $5.2723\cdot 10^{-29}$
22.37 | $6.8390\cdot 10^{-29}$
23.68 | $7.1010\cdot 10^{-26}$
25.00 | $9.3899\cdot 10^{-25}$
26.00 | $3.3122\cdot 10^{-24}$
28.11 | $1.9703\cdot 10^{-23}$
30.22 | $6.0694\cdot 10^{-23}$
32.33 | $1.2475\cdot 10^{-22}$
34.44 | $1.9488\cdot 10^{-22}$
36.56 | $2.5467\cdot 10^{-22}$
38.67 | $2.9881\cdot 10^{-22}$
40.78 | $3.2963\cdot 10^{-22}$
42.89 | $3.5130\cdot 10^{-22}$
45.00 | $3.6723\cdot 10^{-22}$
46.00 | $3.7348\cdot 10^{-22}$
52.00 | $4.0210\cdot 10^{-22}$
58.00 | $4.2381\cdot 10^{-22}$
64.00 | $4.4125\cdot 10^{-22}$
70.00 | $4.5416\cdot 10^{-22}$
76.00 | $4.6243\cdot 10^{-22}$
82.00 | $4.6645\cdot 10^{-22}$
88.00 | $4.6697\cdot 10^{-22}$
94.00 | $4.6478\cdot 10^{-22}$
100.0 | $4.6063\cdot 10^{-22}$
Table 8: Calculated cross sections for neutral dissociation to C. Threshold
energy is 15.5 eV.

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examples of possible applications of this framework to the philosophical
literature on determinism, modal structure and the past hypothesis.

The examples we have offered here are intended only as a preview of the ways
in which this framework might be applied; all of the discussion in section 6
could be significantly expanded, and we have not yet touched on the status of
laws in accounts of causation, explanation, prediction and confirmation.
Moreover, once we take seriously the possibility of non-dynamical laws, there
are important scientific questions about how researchers might furnish
evidence for non-standard laws which can’t straightforwardly be tested in
ordinary laboratory experiments. My hope is that the constraint approach will
furnish a clear and precise language in which some of these questions can be
articulated and answered.

##  9 Note

Shortly before publication the author became aware of a similar paper which
also analyses lawhood in terms of constraints, though within a more explicitly
primitivist framework: see ref [118]. There is no connection between these two
papers, but it is interesting to see convergence on these ideas from different
sources.

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