1 Introduction

Compounds containing carbonyl group such as aldehydes, ketones, and carboxylic acids are essential in chemical and biological mechanisms. For instance, it is known that acetaldehyde (CH\(_3\)CHO) is very relevant in atmospheric chemistry for its primary role played in the formation of peroxyacetyl nitrate (PAN) [1]. Moreover, it is one of the first polyatomic molecules detected in outer space, where it plays an essential role in the chemistry of the interstellar medium [2, 3]. In the last decades, the chemistry of such atmospheric and interstellar molecules has been subjected to intense investigations, focused mainly on the photochemical processes [4]. However, more recently, the role of electron-driven processes has been recognized as fundamental to understand the complexity of interstellar chemistry [5]. In this sense, detailed knowledge of the electron–molecule interaction cross sections in a wide range of incident electron energies may be useful for a deep description of the chemical reaction mechanisms in such environments.

Despite the importance of absolute cross sections of several electron–molecule interaction channels in physical and chemical applications, they are still very scarce for most polyatomic molecules. For instance, for acetaldehyde, only limited experimental cross section determinations were reported over the years. To our knowledge, elastic differential cross sections (DCS) were reported by Gauf et al. [6] in the energy range of 1.0 to 50 eV. Also, there are measured data for the total cross sections (TCS) in the 0.7–400 eV range reported by Szmytkowski [7]. Absolute partial (PICS) and total ionization cross sections (TICS) for electron impact were reported by Vacher et al. [8] at incident energies ranging from the threshold to 280 eV. Absolute TICS were also reported by Bull and Harland [9] in the 15–285 eV range. From the theoretical side, calculations of DCS were reported by Gauf et al. [6] in the energy range from 1 to 30 eV using the Schwinger multichannel method (SMC) and by Vinodkumar et al. [10] at 5, 10 and 15 eV using the R-matrix method. In the latter, TCS were also reported in the 0.01-5000 eV range using a combination of the R-matrix method and additivity rule.

In the present work, a joint theoretical and experimental investigation on elastic electron scattering by acetaldehyde in the low to intermediate energy range is reported. Elastic DCS in the 30-800 eV interval and scattering angle range of 10\(^{\circ }\) to 130\(^{\circ }\) were obtained in a crossed-beam experiment using the relative flow technique (RFT) [11]. Experimental integral (ICS) and momentum transfer cross sections (MTCS) were derived from these data. Moreover, theoretical DCS, ICS, MTCS, TCS and total absorption cross sections (TACS) were obtained in the energy range of 0.5 to 500 eV. A complex optical model potential was used to describe the electron–molecule interaction [12, 13], whereas the Padé approximant technique along with a single-center partial-wave expansion method was applied to solve the Lippmann–Schwinger scattering equation numerically [14, 15]. Additionally, DCS, ICS and MTCS were also obtained using the independent atom model (IAM). The IAM calculations were performed with and without including screening effects [16] in the energy range of 5 to 800 eV.

The organization of this paper is as follows: In Sec. 2, a summary of the theory and the computational details used in the present work are presented. In Sec. 3, the apparatus and experimental procedure are briefly described. Finally, in Sec. 4, our calculated and experimental results are discussed and compared with results available in the literature. Some concluding remarks are also provided.

2 Theory

The theory used in this work was detailed in our previous articles, e.g., de Souza et al. [12] and Lee et al. [13]. Briefly, the computational suite developed by Lucchese and co-workers [14, 15], and subsequently extended by our group to include absorption effects in order to describe the electron–molecule interaction above the excitation threshold [12], was employed in the calculations. In this methodology, a molecular complex optical potential (MCOP) is used to describe the electron–target interaction, viz

$$\begin{aligned} V_{opt} = V_{st} + V_{ex} + V_{cp} + iV_{ab}, \end{aligned}$$
(1)

where \(V_{st}\) and \(V_{ex}\) are the static and the exchange components, respectively, and \(V_{cp}\) is the correlation–polarization contribution. The absorption potential \(V_ {ab}\) describes the reduction in the flux of elastically scattered electrons due to the opening of inelastic scattering channels. In this framework, the many-body nature of the electron–molecule interaction is reduced to an one-particle scattering problem. In the present work, the static-exchange potential was derived exactly from a Hartree–Fock self-consistent-field (HF-SCF) target wave function, and the correlation–polarization potential was obtained in the framework of the free-electron–gas model derived from a parameter-free local density approach of Perdew and Zunger [17]. Accordingly, the correlation potential is generated in the inner region, following the prescription given in ref. [17], whereas a polarization potential asymptotically given as \({-\alpha }/{r^4}\) is used in the outer region. The crossing of the correlation and polarization potentials defines the matching distance \(r_c\). In the present study, \(r_c\) is about 4.575 a.u., slightly larger than the Van der Waals radius of 4.299 a.u. [18]. The absorption potential used was the scaled quasi-free scattering model (QFSM) absorption potential of Lee et al. [19] which is an improvement in the version 3 of the absorption model potential proposed by Staszewska et al. [20].

The HF-SCF wave function for ground state acetaldehyde was obtained using the triple-zeta valence (TZV-2d) internal basis set of Firefly QC package [21], which is partially based on the GAMESS (US) [22] source code. In the present study, the calculation was performed at the C\(_s\) point group using the optimized equilibrium geometry taken from the literature [23]. The calculated electric dipole moment is 1.221 a.u., in reasonable agreement with the experimental value of 1.0585 a.u. [23]. This difference is mainly due to the lack of electronic correlation effects in the HF calculations. The SCF total energy obtained is -152.9680112624 a.u. The calculated dipole polarizabilities are \(\alpha _{xx}\) = 32.65 a.u., \(\alpha _{yy}\) = 27.84 a.u. and \(\alpha _{zz}\) = 22.46 a.u., with result in a spherical polarizability \(\alpha _0= 27.65\) a.u., in good agreement with experimental value of 28.87 a.u. [24]. The theoretical values were used for the generation of the asymptotic form of \(V_{cp}\) [17]. For the generation of \(V_{ab}\), we used the experimental ionization potential (IP = 10.229 eV) [25] as the cutoff parameter. Therefore, no absorption effects are included in the calculations at energies under ionization threshold.

The Lippmann–Schwinger equation is solved iterative and numerically using the Padé approximant technique along with the single-center partial-wave expansion. In the present study, the single-center expansions of bound and scattering wave functions, as well as the interaction potentials and all related matrices, were truncated at \(l_{max} = 22\) for incident energies lower than 15 eV and \(l_{max} = 30\) for 15 eV and above. All the calculated cross sections were converged within 10 iterations. Additionally, in order to deal with the slow convergence of the partial-wave expansions due to the permanent dipole moment of the molecule, the Born closure formula was applied to account for the effects of higher partial-wave contributions to the scattering amplitudes, according to the procedure described in detail in refs. [26,27,28]. As described in such references, rotational excitations were taken into account in the calculations to avoid the elastic DCS divergence at the forward angles. Our theoretical value of the electric dipole moment was used in this procedure.

For energies of 5 eV and above, we have also performed DCS calculations in the IAM framework [29], in two approaches: with and without screening effects. The details of the standard IAM (without screening) calculation were described elsewhere [30, 31]. The DCS IAM calculations including screening effects were performed following the procedure described by Blanco et al. [16]. Again, a complex optical potential was used to describe the electron–atom interaction and to obtain the atomic scattering amplitudes. For that, the \(V_{st}\) atomic potentials reported by Salvat et al. [32] and the \(V_{ex}\) proposed by Furness and McCarthy [33] were used. The model potential of Perdew and Zunger [17] and Staszewska et al. [20] was used to account for the \(V_{cp}\) and \(V_{ab}\) contributions, respectively. The atomic polarizabilities, the internuclear distances and the mean excitation energy (considered as the first ionization potential) used in the calculations were taken from the literature [25]. Moreover, electron-impact TICS were calculated using the Binary-Encounter Bethe (BEB) model [34] with the same basis set used in the MCOP calculations.

3 Experimental

The experimental apparatus, the sample handling system and procedures used in the present work are the same as described in our previous works [30, 35, 36], and thus, only some relevant aspects are outlined. An electron gun that produces a non monochromatic beam with a typical \(\Delta E \approx 0.5\) eV FWHM and currents around 50–150 nA were used. This beam crosses with a gas beam produced by a gas inlet to form the collision region. Three perpendicular pairs of Helmholtz coils around the vacuum chamber reduce the magnetic fields to less than 15 mG around this region. The angular distributions of the scattered electrons from 10\(^{\circ }\) to 130\(^{\circ }\) were measured using this crossed beam geometry. Before the detection, the scattered electrons were previously energy-filtered by a retarding field energy analyzer with a resolution of about 1.5 eV. This level of energy resolution is unable to distinguish the vibrationally elastic and inelastic scattered electrons. Therefore, our measured data are in fact a sum of the scattering intensities resulted from all vibrational transition channels. Further, the absolute values of elastic DCS were obtained applying the RFT [11]. The relative flow rates were obtained following the procedure already described [36]. The base pressure of the spectrometer is around \(1 \times 10^{-7}\) Torr, and the pressures during the measurement were in the range of \(\approx \) \(0.7-1 \times 10^{-6}\) Torr. The driving pressures producing the gas beam were kept below 0.3 and 0.5 Torr for acetaldehyde and the reference gas (argon), respectively. For the latter, the absolute DCS reported by Jansen et al. [37], Dubois and Rudd [38], and Williams and Willis [39] were used as reference data in the RFT.

The acetaldehyde was purchased from Sigma-Aldrich with purity better than 99.5\(\%\). The gaseous acetaldehyde was obtained from the saturated vapor above the liquid sample in a vial attached to the gas handling system [36]. The atmospheric air or other volatile contamination was eliminated by several freeze–pump–thaw cycles. The quality of the sample and the reference gas was checked during the measurements using a quadrupole mass analyzer attached to the experimental chamber, and no contaminants were observed.

The experimental ICS and MTCS were obtained by numerical integration over the DCS. The DCS in the angular regions near the forward and backward directions, not covered by the experiment, were obtained by extrapolation following the procedure previously described [40]. Essentially, DCS were obtained by extrapolation following the trend of the theoretical curves that better reproduced the shape of experimental DCS at these angular regions. Therefore, the present MCOP results were used in the extrapolation procedures from 30 to 200 eV. For 300, 400 and 500 eV, MCOP results were also used as a guide to obtain the DCS at near forward direction, whereas close to the backward direction the IAM curves were adopted. At 800 eV, the IAM curve was used to obtain DCS in both forward and backward directions.

The overall standard deviation in the present DCS was determined to be 15\(\%\). This value was estimated from the uncertainties reported for the reference gas and the contributions of the uncertainties associated with the experimental procedure [35, 36]. For the integrated cross sections, ICS and MTCS, a standard deviation around 30\(\%\) is estimated.

Table 1 Experimental DCS (in \(10^{-16}\) cm\(^{2}\)/sr), ICS and MTCS (in \(10^{-16}\) cm\(^{2}\)) for elastic e\(^-\)–CH\(_3\)CHO. The average DCS standard deviation is around 15%. The ICS and MTCS estimated standard deviation is around 30%
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Fig. 1
figure 1

DCS for elastic e\(^-\)–CH\(_3\)CHO scattering at 1, 2, 3, 5, 10, 15, 20, 30 and 50 eV. Black line presents MCOP results; black dashed line presents IAM calculations; blue dash-dotted line presents IAM calculations with screening corrections; red line presents theoretical SMC results of Gauf et al. [6]; blue dotted line presents theoretical results of Vinodkumar et al. [10]; red squares present experimental results of Gauf et al.; black circles present experimental results

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Fig. 2
figure 2

Same as Fig. 1, but at 100, 300, 500 and 800 eV

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4 Results and discussion

The present experimental DCS, ICS and MTCS for elastic e\(^-\)–CH\(_3\)CHO scattering are listed in Table 1. In Fig. 1, we compare our theoretical Born-corrected MCOP DCS in the 1–50 eV energy range with the theoretical SMC results and also the measured DCS, both reported by Gauf et al. [6]. Additionally, our experimental data at 30 and 50 eV, the IAM DCS results calculated with and without screening correction, as well as the theoretical R-matrix results of Vinodkumar et al. [10] at 5, 10 and 15 eV are also presented.

As observed in Fig. 1, there is a generally good agreement between our theoretical results obtained using the MCOP with both the SMC and measured data by Gauf et al. [6] except at 2 eV. At this energy, a substantial discrepancy between our calculated results and the data of Gauf et al. [6] is seen, specially at intermediate and large scattering angles. This significant difference is due to the occurrence of a shape resonance in the MCOP calculations around this energy. At the resonance region, it is known that both the shape and the magnitude of the DCS are very sensitive in respect to incident energies. At 5 eV, the R-matrix results of Vinodkumar et al. [10] are unable to reproduce the shape of both our and SMC DCS curves. However, at 10 and 15 eV, agreement between the R-matrix results and those of the MCOP and the SMC is significantly improved. It can also be noted that our MCOP results at 10 and 15 eV present a bump around 10\(^{\circ }\) which are artifacts due to the Born closure procedure, and therefore, they are not physical. At 30 and 50 eV, there is a very good agreement between our experimental DCS and the corresponding data of Gauf et al. [6]. This good agreement reinforces the reliability of both experiments.

In principle, one does not expect good performance of the IAM type of calculations at such low energies. In fact, neither the shape nor the magnitude of the DCS obtained by our standard IAM calculations was able to reproduce the experimental results, particularly for incident energies below 20 eV. However, the performance of the IAM with screening correction is significantly improved in the sense that even at energies as low as 10 eV; the screening-corrected (SC) IAM calculations were able to provide DCS in quite good agreement with those obtained using more elaborated theories such as MCOP and SMC.

Fig. 3
figure 3

a ICS and b MTCS for elastic e\(^-\)–CH\(_3\)CHO scattering. Black line presents MCOP results; black dotted line presents MCOP results without Born corrections; blue dash-dotted line presents SC-IAM calculations; red line presents theoretical SMC results of Gauf et al. [6]; red long-dashed line presents theoretical results of Gauf et al. without Born correction  [6]; red squares present experimental results of Gauf et al.; black circles present experimental results

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Fig. 4
figure 4

a Partial-channel ICS and b Eigenphase-sum for elastic e\(^-\)–CH\(_3\)CHO scattering in the 0.5–30 eV range. A\(^{\prime }\) symmetry: black line presents MCOP results; red dotted line presents SMC results of Gauf et al. [6]. A\(^{\prime \prime }\) symmetry: long-dashed line presents MCOP results; red dash-dotted line presents results of Gauf et al.

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Fig. 5
figure 5

TCS for e\(^-\)–CH\(_3\)CHO scattering. Black line presents MCOP results; blue dotted line presents theoretical results of Vinodkumar et al. [10]; long-dashed line presents TACS; red dotted line presents BEB-TICS. Solid squares present experimental data of Szmytkowski [7]

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In Fig. 2, we compare the present experimental DCS in the 100-500 eV range with our MCOP results and with the data calculated using the IAM both screening corrected and not corrected. At 800 eV, the comparison of our experimental DCS is made only with the IAM calculated data. It is seen a very good agreement between our measured DCS and those calculated using the MCOP except at 500 eV, where the MCOP calculations underestimate the DCS at large scattering angles. This behavior was already observed in our previous works [41, 42] and is due to the poor convergence in the single-center expansions of the nuclear part of the interaction potential for atoms a few angstroms away from the origin. Also, in this energy range, the IAM type of calculations reproduces quite well the experimental data, particularly at energies above 300 eV, as shown in Fig. 2. To the higher energies, the screening correction does not introduce significant improvement on the DCS calculated using the standard IAM.

Our theoretical MCOP and experimental results for the ICS and MTCS are shown in Fig. 3 along with those of Gauf et al. [6]. Our SC-IAM calculations above 5 eV are also shown for comparison. In Figs. 3(a), our Born-corrected (BC) MCOP ICS shows a reasonable agreement with the corresponding SMC results of Gauf et al. at the overlapping energy range. Both calculations show a broad shoulder at about 10 eV. However, a resonance-like feature centered at about 2 eV shown in our BC MCOP ICS curve is not clearly seen in the BC SMC curve. In order to understand the origin of these resonances, a partial-channel ICS and eigenphase-sum analyses were made and the results are presented in Fig. 4. These analyses reveal that the peak located at near 2 eV corresponds to a C=O \(^2\)A\(^{\prime \prime }\) (\(\pi ^*\)) resonance. The broad shoulder at near 10 eV is identified as a C-H \(^2\)A\(^{\prime }\) (\(\sigma ^*\)) resonance. These resonances and assignments agree with those reported by Gauf et al. However, these authors found that the energy position of the \(\pi ^*\) resonance is about 1.6 eV. In comparison with experimental results of 1.3 and 1.2 eV reported by van Venn et al. [43] and Dressler & Allan [44], respectively, the energy positions from both theoretical results are shifted to higher energies. The difference between our calculated energy position and the experimental values could be due to the lack of many-body effects, since a model potential is used in our calculations. In addition, the discrepancy between the energy positions of our calculations and that of Gauf et al. is probably due to different approaches used to represent the polarization effects. Apparently, our \(V_{cp}\) causes weaker stabilization than theirs. The present resonance width was determined to be 0.68 eV from the eigenphase-sum by fitting them to the Breit–Wigner profile [45]. In addition, all the theoretical results of ICS agree well with the experimental data.

In Fig. 3(b), we show the comparison of the BC MCOP MTCS, the SMC MTCS without BC and SC IAM MTCS with the experiments. It is seen that the resonance at near 2 eV is more visible in both the MCOP and SMC curves. However, the resonance calculated using the SMC is slightly shifted toward lower energies relative to ours, probably due to different ways to account for polarization effects in the calculations. Indeed, this shift in the position of the resonance is the origin of the discrepancy of the calculated DCS at 2 eV shown in Fig. 1, where the MCOP DCS are significantly larger than the SMC data. On the other hand, there is no evidence of the existence of shape resonances in the experimental ICS and MTCS, probably due to the wide energy gap of the measurements and maybe also due to the substantial experimental uncertainties. In addition, there is a good agreement between the MCOP MTCS and SMC MTCS with the measured data. However, the SC IAM calculations overestimate the experimental data and the present MCOP results.

The theoretical TCS and TACS are presented in Fig. 5 in comparison with the experimental TCS of Szmytkowski [7] and with the theoretical results of Vinodkumar et al. [10]. The present calculated TICS using the BEB model are also shown to compare with our theoretical TACS. In Fig. 5, both our TCS and the theoretical TCS of Vinodkumar et al. [10] show a resonance at about 2 eV, nevertheless not shown in the experiment. At 10 eV and above, our MCOP TCS agree very well with the experimental data of Szmytkowski [7]. At lower energies, the MCOP calculations overestimate the experimental data. In contrast, the calculation of Vinodkumar et al. [10] underestimates the TCS in the entire energy range except near the resonance. A possible explanation for the disagreement between our data and those of Szmytkowski was already presented in some of our previous works, see, e.g, ref. [13]: electron transmission experiments are unable to distinguish the transmitted electrons from those elastically scattered at very small angles. For polar molecules, the strongly peaked DCS near the forward direction, particularly in the low energy region, may lead to an underestimation of experimental TCS.

Moreover, the comparison of MCOP TACS and BEB TICS shows a qualitative agreement between them. The MCOP results lie above the TICS in the entire energy range studied. This difference may be due to the fact that the calculated TACS account for all inelastic-scattering channels, including both the excitation and ionization processes, whereas TICS include only the ionization channels. Therefore, the observed difference is plausible.

In summary, we present an experimental and theoretical study of elastic e\(^-\)–CH\(_3\)CHO scattering in the low- and intermediate-energy ranges. Our theoretical MCOP method provided DCS in reasonably good agreement with our experimental results and with other theoretical and experimental data available in the literature. Our calculations predict two resonances located at around 2 eV and 10 eV in agreement with the previous theoretical results of Gauf et al.