Differential elastic scattering and electron-impact ionization cross sections of nitrous oxide

Abstract

With the aim of providing datasets for simulations of electron transport processes in the upper atmosphere, we measured singly differential elastic electron scattering and doubly differential electron-impact ionization cross sections of nitrous oxide. These measurements were conducted for primary electron energies between 30 eV and 1 keV in the angular range of 20°–150°. Secondary electron energies spanned from 4 eV to approximately half of the primary electron energy. In addition to the measurements, the differential elastic scattering cross sections of nitrous oxide were calculated using the IAM-SCAR + I model. Furthermore, the singly differential and total ionization cross sections of nitrous oxide were obtained by integrating the doubly differential ionization cross sections over emission angle and over both emission angle and secondary electron energy, respectively. These cross sections were compared to calculations performed using the BEB model and to experimental results of other groups, who determined the total ionization cross sections of nitrous oxide by collecting ions generated during electron impact.

Graphical abstract

1 Introduction

Global anthropogenic N₂O emissions are rapidly increasing and are projected to persist as the foremost anthropogenic source of an O₃-destroying substance in the foreseeable future [1]. The ozone depletion potential of nitrous oxide is heightened by its chemical inertness, enabling it to ascend the stratosphere without substantial degradation by other gases in the atmospheric layer underneath. The major process for the bulk of ozone destruction involves catalytic reactions with nitric oxides: NO + O₃ → NO₂ + O₂.

Several decades ago, a mechanism was first proposed to explain ozone depletion by cosmic ray-driven electron-induced molecular reactions. As Thorne [2] pointed out in 1977, nitric oxides can also be generated by low-energy secondary electrons that are released in the stratosphere through the interaction of cosmic ray with the atmospheric gases (CRE). Since then, there has been a growing interest in studying the influence of CRE on the Earth’s ozone layer, particularly especially in relation to anthropogenic gas emissions. Despite extensive research efforts over the years, the contribution of CRE to ozone depletion remains largely unresolved. Accurately quantifying this contribution necessitates simulation of radiation transport processes in the upper atmosphere, for which comprehensive data on electron interaction cross sections of both natural and anthropogenic gases are needed.

The first measurement of electron scattering cross sections of N₂O was reported by Ramsauer and Kollath [3], who determined the total scattering cross section (TCS) for low-energy electrons from 0.15 to 1.25 eV. About four decades later, Zecca et al. [4] extended the energy range to 10 eV, and another decade later, Kwan et al. [5] continued this investigation by measuring the TCS of N₂O for electron energies between 1 and 500 eV. Subsequent experimental work by Szmytkowski et al. [6, 7] provided data on the TCS of N₂O in the energy range of 0.8–40 eV and of 40–100 eV. Investigations at higher energies were pursued by Xing et al. [8], who determined the TCS of N₂O over electron energies ranging from 0.6 to 4.25 keV. To shed light in discrepancies in the available data, new TCS datasets were recently provided by Pyun et al. [9] for the energy range between 5 and 50 eV and by Lozano et al. [10] for impact energies ranging from 1 to 200 eV.

Several experiments [11,12,13,14,15] have been conducted to measure differential elastic scattering cross section (DCS) of N₂O. Most of these experiments concentrated on electron energies T below 100 eV, with scattering angles θ ranging from 8° to 150°. Lee et al. [15] extended the experimental study to higher energies. They published the DCS of N₂O for electron energies between 50 and 800 eV in the angular range of 15°–130°. Concerning the ionization properties of N₂O, only a limited number of measurements of the total ionization cross section (TICS) have been reported so far. Rapp and Englander-Golden [16] published the TICS of N₂O for electron energies from threshold up to 1 keV, a range also explored by Iga et al. [17]. Lopez et al. [18] further contributed to the experimental study by publishing the TICS of N₂O from threshold to 200 eV.

There are also several theoretical studies on low-energy electron scattering on N₂O [19,20,21,22,23,24,25]. Of particular interest for the present study are the theoretical works of Michelin et al. [20], da Costa and Bettega [22], and Gianturco and Stoecklin [25], who provided elastic DCS in the intermediate energy regime. They used the Born-closure Schwinger variational method, a Schwinger Multichannel Method with pseudopotentials, and an exact static-exchange formalism, respectively.

In effort to establish comprehensive datasets, Vinodkumar et al. [26] published theoretical TCS over a broad energy range (0.1–2000 eV), while Song et al. [27] conducted a review of available experimental and theoretical data to compile a reference dataset.

In this study, we aimed to establish a comprehensive database for cosmic ray-driven radiation transport processes in the upper atmosphere. To achieve this, we measured the DCS, ${\text{d}}\sigma_{{{\text{el}}}} /{\text{d}}\Omega$, and the doubly differential ionization cross sections (DDCS), ${\text{d}}^{2} \sigma_{i} /{\text{d}}E{\text{d}}\theta$, of N₂O as functions of scattering (emission) angle $\theta$ and secondary electron energy $E$ for primary electron energies $T$ ranging from 30 eV to 0.8 keV. The angular range spanned from 20° to 150° for the DCS and from 30° to 150° for the DDCS. The latter was experimentally determined for secondary electron energies of 4 eV to $(T-I)/2$, where $I=12.9\,\text{eV}$ [28] is the ionization threshold of the molecule. In addition to the experimental study, the DCS of N₂O was calculated using the IAM-SCAR + I model [29]. Furthermore, the singly differential (SDCS) and total ionization cross sections (TICS) of N₂O were determined based on the experimental DDCS. They were compared to calculations obtained using the Binary Encounter Bethe (BEB) model [30].

2 Experiment

2.1 Measurement method

The DCS and DDCS of N₂O were measured using a crossed-beam arrangement, where an electron beam intersects perpendicularly with a molecular beam. Electrons emerging from the interaction zone are analyzed according to their energy using an electron spectrometer. The electron beam axis and the viewing axis of the spectrometer define the scattering plane. Under the condition that the detection solid angle remains relatively constant throughout the interaction zone [31, 32], the DCS can be obtained from the count rate $\Delta {\dot{N}}_{\text{el}}(\theta )$ of elastically scattered electrons per solid angle $\Delta \Omega$ using the equation

$$ \frac{{{\text{d}}\sigma_{{{\text{el}}}} }}{{{\text{d}}\Omega }}\left( {\theta ,T} \right) = \frac{{\Delta \dot{N}_{{{\text{el}}}} }}{\Delta \Omega }\left( {\theta ,T} \right)/\left( {\frac{{\overline{{I_{0} }} }}{e}n_{F} \eta \left( T \right)} \right) , $$

(1)

where $\overline{{I }_{0}}$ is the primary electron current, $e$ is the electron charge, ${n}_{F}$ is the area number density, i.e., the number of molecules per area hit by the electron beam, and $\eta \left(T\right)$ is the detection efficiency of the electron energy analyzer. The latter is given by the transmission of the energy analyzer times the counting efficiency of the channel electron multipliers that were used as detectors.

Since direct determination of the product ${n}_{F}\times \eta \left(T\right)$ was very difficult, it was assessed using the relative flow technique [33] with N₂ as the reference gas. The DCS of N₂ had previously been measured in an independent experiment [34]. It follows from Eq. (1):

$$ \chi \equiv \hat{n}_{F} \times \eta \left( T \right) = \frac{{\Delta \widehat{{\dot{N}}}_{{{\text{el}}}} }}{\Delta \Omega }\left( {\theta ,T} \right)/\left( {\frac{{\overline{{\hat{I}_{0} }} }}{e} \times \frac{{{\text{d}}\hat{\sigma }_{{{\text{el}}}} }}{{{\text{d}}\Omega }}\left( {\theta ,T} \right)} \right) . $$

(2)

In Eq. (2), the quantities referring to the reference gas are labeled by the hat symbol. In the relative flow technique, the ratio ${n}_{F}/{\widehat{n}}_{F}$ of the area number density of the molecules of interest to that of the reference gas is determined by the relation ${n}_{F}/{\widehat{n}}_{F}=\widehat{F}/F\times \sqrt{M/\widehat{M}}=\gamma$, where $F$ is the mass flow rate through the gas effusion nozzle and $M$ is the molecular mass. This relation is assumed to be valid if the molecular beams are generated in the molecular flow regime and if the mean free paths for intermolecular collisions in both beams are equivalent.

Using χ defined by Eq. (2), (1) can be reformulated as

$$ \frac{{{\text{d}}\sigma_{\text {el}} }}{{{\text{d}}\Omega }}\left( {\theta ,T} \right) = \frac{e}{{I_{0} }} \times \frac{1}{\gamma \chi } \times \frac{{\Delta \dot{N}_{{{\text{el}}}} }}{\Delta \Omega }\left( {\theta ,T} \right) . $$

(3)

By analogy with Eq. (1), the DDCS is related to the count rate $\Delta {\dot{N}}_{i}(\theta ,E,T)$ per energy interval $\Delta E$ and solid angle $\Delta \Omega$:

$$ \frac{{{\text{d}}^{2} \sigma_{i} }}{{{\text{d}}E{\text{d}}\Omega }}\left( {\theta ,E,T} \right) = \frac{{\Delta \dot{N}_{i} }}{\Delta E\Delta \Omega }\left( {\theta ,E,T} \right)/\left[ {\frac{{\overline{{I_{0} }} }}{e}n_{F} \eta \left( E \right)} \right] = \frac{e}{{\overline{{I_{0} }} }}\frac{1}{\gamma \chi }\frac{\eta \left( T \right)}{{\eta \left( E \right)}}\frac{{\Delta \dot{N}_{i} }}{\Delta E\Delta \Omega }\left( {\theta ,E,T} \right) . $$

(4)

To determine the ratio $\eta \left(T\right)/\eta \left(E\right)$ required for calculating the DDCS with Eq. (4), we conducted measurements of the energy spectra of secondary electrons released from propane including the elastic scattering peak for various primary energies. The ratio $\eta \left(T\right)/\eta \left(E\right)$ was then determined by comparing these energy spectra with those measured in a prior experiment [35] using a different electron spectrometer, for which the ratio $\eta \left(T\right)/\eta \left(E\right)$ was known.

2.2 Experimental procedure

Figure 1 shows a schematic view of the experimental setup. The apparatus comprises an electron gun, an effusive gas source, and an electron spectrometer housed within a scattering chamber. The electron gun, purchased from Kimball Physics Inc. (type ELG-2), provided well-focused, stable electron beams from 30 eV and 2 keV with a maximum energy spread of approximately 0.5 eV at 800 eV.

Fig. 1

Schematic view of the experimental setup. The electron gun and Faraday cup (FC) were fixed in position, while the electron spectrometer was mounted on a turntable. The scattering (emission) angle θ of electrons to be detected was adjusted by rotating the turntable

The molecular beam was produced via an effusion nozzle with an exit aperture 0.3 mm in diameter. This diameter was sufficiently small to maintain a nearly constant detection solid angle across the beam. At a driving pressure of 2 mbar, the nozzle generated a molecular area number density of approximately 10¹⁴ cm⁻³. Considering that the gas kinetic diameters of N₂ and N₂O are 0.364 nm [36] and 0.33 nm [37], respectively, the mean free paths for intermolecular collisions in both gases (approximately 1 mm) are significantly larger than the nozzle diameter so that the Knudsen condition is well satisfied. This implies that the gas effusion occurs in the molecular flow regime, thus allowing for the application of the relative flow technique. To fulfill the conditions for using Eqs. (3) and (4), the driving pressure in the gas reservoir above the effusion nozzle was adjusted according to the square of the gas kinetic diameters of both gases. The mean free path of electrons in the molecular beam with the aforementioned area number density exceeded 5 cm, which was much larger than the diameter of the molecular beam. Consequently, the single collision condition was well satisfied.

The molecular beam intersected with the electron beam 1.5 mm below the exit aperture of the gas nozzle. The primary electron beam current, which was measured using the Faraday cup located beyond the molecular beam, varied between 0.2 and 0.5 μA, depending on the electron energy $T$. It was raised at higher electron energies to compensate for the decreasing cross sections with increasing energy. The electron spectrometer used to analyze the energy of electrons was a hemispherical deflection analyzer (AR65 from Omicron Nanotechnology Ltd.) with a mean radius of 65 mm and deflection angle of 180°. The distance between the molecular beam axis and the entrance aperture of the hemispherical deflection analyzer was 29 mm, corresponding to the working distance of the analyzer. While the electron gun remained fixed in position, the hemispherical deflection analyzer was mounted on a turntable, allowing adjustment of the electron detection angle $\theta$ through rotation of the turntable.

The angular resolution of the apparatus was approximately 2°, which was primarily given by the ratio of the diameter (1 mm) of the entrance aperture of the hemispherical deflection analyzer to its working distance. The overall energy resolution of the apparatus was determined by the energy width of the electron source and the resolution of the hemispherical deflection analyzer. The latter could be adjusted by varying the retardation voltage applied to the entrance slit of the hemispherical deflector [38]. For the retardation voltages used in this work, the resolution of the analyzer was 1.5 eV at $T=800\,\text{eV}$. It was better at lower electron energies.

The scattering chamber was surrounded by three orthogonal pairs of Helmholtz coils to compensate the Earth’s magnetic field. On the scattering plane, the residual magnetic field was lower than 2 μT. The deflection of electrons in this field was small compared to the angular resolution of the apparatus and therefore disregarded.

2.3 Uncertainty analysis

The standard uncertainties of the experimental DCS and DDCS were estimated following the guidelines given in the Guide to the Expression of Uncertainty in Measurement [39]. Since the measuring quantities were uncorrelated, the variances ${u}_{\text{el}}^{2}$ and ${u}_{\text{i}}^{2}$ of the DCS and DDCS, respectively, were computed from the sum of the squared sensitivity coefficients times the individual variances of the measuring quantities:

$$ u_{{{\text{el}}}}^{2} = \mathop \sum \limits_{j = 1}^{4} \left( {\frac{\partial G}{{\partial x_{j} }}} \right)^{2} u^{2} \left( {x_{j} } \right),\quad u_{i}^{2} = \mathop \sum \limits_{j = 1}^{5} \left( {\frac{\partial H}{{\partial x_{j} }}} \right)^{2} u^{2} \left( {x_{j} } \right) , $$

(5)

where $G = {\text{d}}\sigma_{{{\text{el}}}} /{\text{d}}\Omega$ and $H = {\text{d}}^{2} \sigma_{i} /{\text{d}}E{\text{d}}\Omega$ are defined by Eqs. (3) and (4), respectively. The measuring quantities ${x}_{j}$ stand for: $x_{1} = I_{0} , x_{2} = \gamma , x_{3} = \chi , x_{4} = \Delta \dot{N}_{{{\text{el}}}} /\Delta \Omega$ (or $\Delta \dot{N}_{i} /\Delta E\Delta \Omega )$, and ${x}_{5}=\eta (T)/\eta (E).$ The relative uncertainty of ${I}_{0}$, caused by temporal drift of the primary electron beam current, was 4%. The uncertainty of $\gamma$ was estimated from the uncertainties of the ratio of the flow rates of both gases, which were deduced from linear regressions with respect to the pressure decrease in the gas reservoir vs. time. It amounted to 5%. An additional uncertainty of 15% was attributed to $\chi$, primarily due to the uncertainties of the DCS of N₂. The statistical uncertainties ${x}_{4}$ in the energy spectra of N₂O did not exceed 5%. Finally, the uncertainty of ${x}_{5}$ was estimated to be 15%. The overall standard uncertainty, calculated as the positive square root of the variances, was 17% for the DCS and 23% for the DDCS.

3 Theory

The theoretical DCS was calculated using the independent atom model with screening corrected additivity rule and interference effects (IAM-SCAR + I). This method is meanwhile well-established and has been extensively documented in the literature [29, 40, 41]. Therefore, only a brief description of the concept is provided here, as the IAM-SCAR + I calculations performed in this study are analogous to our previous work on ethanol [34].

In the IAM-SCAR + I model, each atomic target (N, N, O) within the molecule is described by an optical potential, where the real part accounts for elastic scattering and the imaginary part pertains to inelastic scattering. The optical potential consists of several components: (i) a static term, (ii) an exchange term, and (iii) a polarization term, which describes long-range interactions depending on the molecular polarizability. In the zeroth approximation (IAM-AR), scattering amplitudes are calculated from the optical potentials of each atom, and the molecular DCS is then derived by coherently summing the absolute squares of all relevant atomic scattering amplitudes. The IAM-SCAR + I improves the IAM-AR by including atomic screening and interference terms to account for multi-center scattering within the molecule. While the IAM-SCAR + I method is applicable for impact energies as low as 20 eV, the calculation of the atomic DCS using optical potentials and, consequently, the results obtained in this study with the IAM-SCAR + I are of limited validity below 50 eV.

4 Results and discussion

4.1 Elastic scattering

The complete experimental data of the elastic DCS are provided in Table 1 and graphically compared to other experimental data and the theoretical calculations in Fig. 2.

Table 1 Present experimental results for the elastic DCS of nitrous oxide as a function of the scattering angle $\theta$ for different electron energies in units of 10⁻¹⁶ cm²/sr

Fig. 2

Present experimental results (filled diamond) for the DCS of nitrous oxide in comparison with the experimental data reported by other groups: (open square) Marinkovic et al. [11], (open invert triangle) Johnstone and Newell [12], (open circle) Kitajima et al. [13], (open left triangle) Lee et al. [15]. The solid curves depict calculations using the IAM-SCAR + I model, the dashed curves represent calculations from da Costa and Bettega [22], and the dash-pointed curve illustrates calculations by Michelin et al. [20]

In general, the results of the present work agree satisfactorily well with the experimental data of other groups [11,12,13,14,15] within the experimental uncertainties. At small energies, the elastic DCS shows a distinct minimum around 90°, which decreases with increasing energies until it completely vanishes at energies higher than 300 eV. As expected, the DCS in the forward direction is significantly higher than in the backward direction.

While the present results agree well with the data of both Kitajima et al. [13] and Lee et al. [15], they show a significant deviation to the data of Marinkovic et al. [11], and to those of Johnstone and Newell [12], especially at 80 eV. At this energy, the data of Marinkovic et al. [11], and Johnstone and Newell [12] are consistent among each other but significantly lower than the data of this work and of Lee et al. [15]. Above 100 eV, only the data sets of Lee et al. [15] are available for comparison. They match the present results well at the energies 200 eV and 800 eV within the experimental uncertainties. At 300 eV and 400 eV, however, the present results are notably higher than the data of Lee et al. [15] in the angular range above 80°.

A satisfactory agreement was found between the calculations employing the IAM-SCAR + I model and the present results above 40 eV, with a general trend of improving agreement as the energy increases. At $T \le 60\,\text{eV}$, the IAM-SCAR + I model predicts distinct minima at scattering angles around 100°. In this angular range, the present results notably surpass the calculated values. Among the published experimental data, the present results are best reproduced by the IAM-SCAR + I model, except at 60 eV and 80 eV, where the data of Kitajima et al. [13] and Lee et al. [15], respectively, exhibit closer agreement with the calculations employing the IAM-SCAR + I model. At $T \ge 200\,\text{eV}$, the IAM-SCAR + I model excellently reproduces the experimental data of this work, while it predicts DCS diverging significantly from the data of Lee et al. [15] at scattering angles above 80°. Figure 2 also shows the theoretical calculations from da Costa and Bettega [22] for 30 eV and 80 eV, as well as the calculations from Michelin et al. [20]. Both theories significantly overestimate the DCS in the region around the minimum, which reinforces the usefulness of the IAM-SCAR + I model.

To determine the integral elastic (ICS) and momentum transfer cross sections (MTCS), the elastic DCS was integrated over the solid angle $\Omega$. The MTCS ${\sigma }_{\text{m}}$ is given by

$$ \sigma_{\text {m}} = \smallint \left( {1 - \cos \theta } \right)\frac{{{\text{d}}\sigma_{\text {el}} }}{{{\text{d}}\Omega }}{\text{d}}\Omega . $$

(6)

The DCS beyond the measured angular range was obtained by extrapolating the measured data using the IAM-SCAR + I model. The exact extrapolation procedure is explained in “Appendix A1”. As is evident from Fig. 3, the determined ICS and MTCS agree well with the literature data within the experimental uncertainties. It should be noted that, due to the extrapolation of the measured data to $\theta < 20^\circ ,$ where the DCS steeply rises with decreasing scattering angle, the present experimental results contributed only approximately 32% to the ICS at 800 eV. This fraction rises to 71% as energy decreases. Consequently, the integral cross sections are highly susceptible to inaccuracies in the IAM-SCAR + I model.

Fig. 3

Integral elastic scattering cross section (left panel) and the momentum transfer cross section (right panel) of nitrous oxide as a function of electron energy T. The present results (filled diamond) are compared to the data obtained by other groups: (filled triangle) Lozano et al. [10], (filled star) Marinkovic et al. [11], (open square) Johnstone and Newell [12], (multiply) Lee et al. [15], (filled circle) Michelin et al. [20]

It is important to mention that the energy resolution (1.6 eV) of the present apparatus was insufficient to resolve rotational excitation from elastic scattering. Therefore, the present results incorporate rotational excitation cross sections. According to the estimate using the equation of Collins and Norcross [42], they make up maximally 5% within the measured angular range and maximally 10% for angles smaller than 20°. They are deemed negligibly small compared to other uncertainties for electron energies above 100 eV. The uncertainties of the ICS and MTCS, derived from the estimated uncertainties of the extrapolation of the DCS to $\theta < 20^\circ$ and $\theta >150^\circ$ as well as the uncertainties of the experimental DCS, amounted to 24% and 23%, respectively. The theoretical values are pure elastic scattering cross sections, with no rotational excitation correction applied. Therefore, also the uncertainty due to the rotational excitations was considered during the extrapolation to small angles.

4.2 Ionization

Figure 4 shows the present experimental DDCS of N₂O as a function of the emission angle θ for several selected primary electron energies. Different symbols are used to represent the data at various secondary electron energies. The complete dataset is available in Table 2 in Online resource 1.

Fig. 4

DDCS of nitrous oxide as a function of emission angle θ for various primary electron energies T. Experimental data points corresponding to different secondary electron energies are indicated by distinct symbols: (open square) 5 eV, (open circle) 10 eV, (open triangle) 20 eV, (filled triangle) 30 eV, (invert triangle) 50 eV, (open diamond) 100 eV, (asterisk) 200 eV. The dashed curves represent the best fits by a semi-empirical equation (see below)

As can be seen from Fig. 4, the DDCS decreases with increasing secondary electron energy, which is in line with theoretical predictions. At low secondary electron energies, electron emission tends to be nearly isotropic. However, as secondary electron energies increase, binary collision peaks become more pronounced. This behavior is anticipated due to the finite momentum distribution of bound electrons, which causes a broadening of the binary collision peaks [43]. The influence of the finite momentum distribution on the emission angle distribution increases with decreasing secondary electron energy.

It is further evident in Fig. 4 that the angle position ${\theta }_{0}$ of the peak changes with $E$. This is also anticipated by the kinematics of binary collisions between the incident and bound electrons. The cosine of θ₀ is approximately given by [43]

$$\text{cos}{\theta }_{0}\cong {\left(\frac{\varepsilon +1}{t}\right)}^{1/2},$$

(7)

with $\varepsilon =E/I$ and $t=T/I.$ According to Eq. (7), the binary collision peaks are expected to occur in the angular interval between 45° and 90°, which is consistent with the observation made in Fig. 4.

The dashed curves in Fig. 4 represent the best fits of a semi-empirical equation to the experimental data. The semi-empirical equation comprises two Lorentzian proposed by Rudd [43] and a constant term accounting for increased electron emission in the forward direction:

$$ \frac{{{\text{d}}^{2} \sigma_{{\text{i}}} }}{{{\text{d}}\varepsilon {\text{d}}\Omega }}\left( {\theta ,\varepsilon ,t} \right) = p_{1} \left[ {f_{{{\text{BE}}}} \left( {\theta ,\varepsilon ,t} \right) + p_{2} f_{{\text{b}}} \left( {\theta ,\varepsilon ,t} \right) + {\Lambda }} \right] , $$

(8)

where ${f}_{\text{BE}}$ and ${f}_{\text{b}}$ are the Lorentzian functions representing the binary collision peak and electron ejection in the backward direction, respectively. The parameters ${p}_{1}$ and ${p}_{2}$ are fitting parameters, and ${f}_{\text{BE}}$ and ${f}_{\text{b}}$ are given by

$${f}_{\text{BE}}(\theta ,w,t)=\frac{1}{1+{\left[(\text{cos}\theta - \text{cos}{\theta }_{0})/\left({a}_{1}\sqrt{{\text{sin}}^{2}{\theta }_{0}/\varepsilon }\right)\right]}^{2}}$$

(9)

and

$${f}_{\text{b}}(\theta ,w,t)=\frac{1}{1+{\left[(\text{cos}\theta +1)/{a}_{2}\right]}^{2}} ,$$

(10)

where $\text{cos}{ \theta }_{0}$ is given by Eq. (7), and the coefficients ${a}_{1}$ and ${a}_{2}$ are related to the width of the binary collision peak and backward emission peak, respectively. In this work, the DDCS was fitted with $\Lambda =0$ for $T >60$ eV and $\Lambda =0.25$ for $T \le 60$ eV. The values of ${a}_{1}$ and ${a}_{2}$ were $0.95$ and $0.36$, respectively. The confidence interval for the best-fit parameter values, estimated using the Chi-square test, was 95%. It is evident in Fig. 4 that the results of the best fits well reproduce the experimental DDCS within the experimental uncertainties, except for a discrepancy at $T=100\,\text{eV}$ in the angular range from 105° to 120°.

Equation (8) with the best-fit parameter values is numerically integrated over the solid angle to obtain singly differential ionization cross section (SDCS) of N₂O for secondary electron energies between 4 eV and $(T-I)/2$. To determine the SDCS at $E < 4\,\text{eV}$, the SDCS was extrapolated to lower energies. For this purpose, the SDCS above 4 eV was fitted by an equation based on the BEB model [30]:

$$g\left(w,t\right)=\sum_{k=1}^{3}{z}_{k}\left(t\right)\left[{u}_{k}\left(w+1\right)+{u}_{k}\left(t-w\right)\right] ,$$

(11)

with ${u}_{k}\left(w\right)={(w+1)}^{-k}$ and ${u}_{k}\left(t-w\right)={(t-w)}^{-k}$ and fit parameters ${z}_{k}$. In Fig. 5, the SDCS obtained in this way was compared to calculations with the BEB model [30]. The required binding energies and average kinetic energies of electrons in the 11 molecular orbitals of nitrous oxide were calculated using the Gaussian09 software [44] with the 6–311++G basis sets. It should be noted that the binding energy of the HOMO was replaced by the experimental ionization threshold $I=12.9\,\text{eV}$ [28] for better comparison to Eqs. (7)–(10). The kinetic energy of the HOMO was scaled accordingly. It can be seen in Fig. 5 that the former is well reproduced by the latter within the experimental uncertainties.

Fig. 5

SDCS of nitrous oxide versus secondary energy $E$. The SDCS is given for the primary energies 60 eV, 100 eV, 400 eV, and 800 eV as indicated in the figure. The dashed and solid curves represent the best fits of Eq. (12) to the experimental SDCS and calculations with the BEB model [30], respectively

The SDCS determined above was integrated over secondary electron energies to obtain the total ionization cross section (TICS) of N₂O. Figure 6 shows a comparison between the TICS determined through this integration, calculation using the BEB model [30] as well as the BEB model with acceleration correction (GBEB) [45], and experimental data from other groups [10, 16, 17, 46]. The dashed line represents the best fit of the present TICS by an equation proposed by Kim and Rudd [30]

Fig. 6

Present results (filled diamond) for the TICS of nitrous oxide vs. $T$ in comparison with the literature data: (filled triangle) Lozano et al. [10], (plus) Rapp et al. [16] and (filled star) Iga et al. [17]. The dot-dashed curve represents the data recommended by Lindsay and Mangan [46], while the solid and dotted curves depict calculations with the BEB and GBEB models, respectively. The dashed curve represents the best fit to the present experimental data utilizing the formula from Ref. [30]

$${\sigma }_{i}\left(t\right)=\frac{4\pi {a}_{0}^{2}}{t}\left[{b}_{1}\text{ln}t+{b}_{2}\left(1-\frac{1}{t}\right)+{b}_{3}\frac{\text{ln}t}{t+1}\right] .$$

(12)

The best-fit parameter values were ${b}_{1}=6.11\times {10}^{-2}, {b}_{2}=-8.89\times {10}^{-2},$ and ${b}_{3}=2.65\times {10}^{-2}.$ It can be seen in Fig. 6 that the present results agree with other data within the experimental uncertainties. However, the former appear to be higher than the latter below 100 eV. While the BEB model aligns more closely with the other experimental data, the GBEB model agrees well with the best fit to the present experimental data.

5 Conclusion

The DCS of nitrous oxide measured in the present work generally aligns well with the experimental data reported by other groups [10,11,12,13,14,15,16,17] within the experimental uncertainties. Additionally, satisfactory agreement was observed between the calculations employing the IAM-SCAR + I model and the present results above 40 eV. At electron energies above 200 eV, the IAM-SCAR + I model excellently reproduced the experimental results of this work. As expected, the agreement between the experimental data and theoretical calculations based on the independent atom model improves with increasing electron energy.

Unfortunately, there are no existing literature data available for the DDCS of nitrous oxide that could be directly compared to the results obtained in this work. Nevertheless, the angular and secondary energy dependence of the present DDCS aligned with the semi-empirical formalism based on the experimental data for other atomic and molecular targets. Furthermore, the SDCS obtained by integrating the DDCS over the solid angle exhibited excellent agreement with calculations using the BEB model for primary electron energies down to 60 eV. Additionally, the TICS of nitrous oxide determined in this work was well reproduced by the BEB model within the experimental uncertainties. Our results for the TICS tended to be higher than the experimental data of other groups that were determined by collecting the ions created by electron impact. This was particularly noticeable at primary electron energies below 100 eV.

Appendix A1: Theoretical extrapolation of the experimental DCS

Since the present experimental DCS was only measured for angles between 20° and 150°, the experimental results are extrapolated beyond the angular range to obtain integral elastic (ICS) and momentum transfer cross sections (MTCS) using the IAM-SCAR + I. We first normalized the DCS obtained with the IAM-SCAR + I to the experimental data in the 20°–150° angular range. The normalization was performed using the integral DCS of both datasets over the 20°–150° angular range. Subsequently, the normalized IAM-SCAR + I values below 20° and above 150° were used as extrapolated values. These extrapolated values are detailed in Table 1.

For the calculation of the uncertainties of the extrapolation, we first estimated the relative mean deviation between the normalized IAM-SCAR + I data and experimental results for the 10 measured angles, which did not exceed 13% above $T = 40\,\text{eV}$. For the lower energies, where the agreement between the IAM-SCAR + I and experiment is worse, the relative uncertainty of extrapolation was estimated to be 35% for 30 eV and 21% for 40 eV. The overall standard uncertainties of the ICS and MTCS were calculated as the square root of the quadratic sums of the uncertainties of both the experimental and the extrapolated values, as well as the uncertainty arising due to the rotational excitations mentioned in the main text.

The uncertainties amounted to maximally 24% for the MTCS and 23% for the ICS.