Ionization of NH3 and CH4 by electron impact

Eur. Phys. J. D (2015) 69: 2 DOI: 10.1140/epjd/e2014-50677-1

THE EUROPEAN PHYSICAL JOURNAL D

Regular Article

Ionization of NH3 and CH4 by electron impact Istvan Tóth1,a , Radu I. Campeanu2 , and Ladislau Nagy1 1 2

Faculty of Phyics, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto M3J 1P3, Canada Received 11 September 2014 / Received in final form 22 October 2014 c EDP Sciences, Societ` Published online 8 January 2015 – a Italiana di Fisica, Springer-Verlag 2015 Abstract. Calculated triple differential cross sections are presented for the ionization of NH3 and CH4 molecules by electron impact. The cross sections are determined for symmetrical coplanar and perpendicular geometrical arrangements. The obtained results reproduce in most cases the main features observed in the experimental data for both geometries.

1 Introduction The most complete set of information about the ionization process of atoms and molecules may be obtained through kinematically complete studies. The main outcome of such studies is the TDCS (triple or fully differential cross section). The description of (e, 2e) processes is still incomplete. This is observed especially in case of molecular targets for which both experiments and theoretical treatments are very challenging. Difficulties arise for example in the separation of molecular orbitals, which may be very closely spaced in energy. On the theoretical side the description of the multi-center nature of the target or the correct nature of the interaction between the different particles involved in the ionization process are very challenging. Kinematically complete studies are performed for a growing number of molecular targets. We mention below only a few theoretical and experimental studies for simpler molecules like H2 [1–5], O2 [6], N2 [7–10] or more complex targets as CO2 [11], CH4 [12–17], NH3 [18], H2 O [19–22] and even C4 H8 O [23]. In the present study we calculate TDCSs for the ionization of NH3 and CH4 molecules in coplanar and perpendicular geometrical arrangements. The coplanar arrangement assumes that all free electrons are in the same, scattering plane. In a perpendicular geometry both outgoing electrons are detected in the same plane, perpendicular to the incident direction. One of the motivation for this study is that these molecules are iso-electronic (with equal number of electrons), therefore a comparison may reveal important information about the effect of the molecular structure on the ionization process. We investigate the ionization of the valence orbitals of NH3 designated as 3a1 , 1e1 and 2a1 . The first two orbitals are of p-character, while the a

e-mail: [email protected]

third is an s-like orbital. In case of the CH4 molecule the ionization of the 1t2 and 2a1 orbitals is studied. Here, the 1t2 orbital is of p-character. Our calculations are performed by employing the DWBA (distorted-wave Born approximation) method. The continuum states of the free electrons are approximated by distorted waves, while the initial state of the molecules is described by Gaussian type multi-centre wavefunctions. The molecular orbitals are given as linear combination of atomic-like orbitals, which are contractions of Gaussian primitives. The molecular orbital coefficients were calculated with the Gaussian package [24]. A doubly-symmetric kinematics is considered for the calculation of the TDCSs. Both the energies and the ejection angles of the outgoing electrons relative to the incident direction are equal. In addition, we consider a low-energy regime for the outgoing particles, the highest energy being equal to 20 eV. Previously, we have performed similar, low-energy calculations for H2 , but only in the perpendicular plane. Our results for H2 showed good agreement with the experimental data. Thus, another motivation for the present study is to test our calculation method also in the scattering plane and in case of more complex molecular targets like NH3 and CH4 at low electron energies. It is natural to assume, that for the above described kinematical conditions the PCI (post collision interaction) effects between the outgoing electrons are important. These effects are taken into account in the present study through the Coulomb distortion factor described in reference [25]. Nevertheless, we also perform calculations without this factor, in order to assess the consequences of the PCI effects on the cross section. Our results are compared with the experimental data and M3DW (molecular three-body distorted wave approximation) theoretical results presented in reference [18].

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Eur. Phys. J. D (2015) 69: 2

2 Theory

If we take into account the exchange effects between the ejected and scattered electrons, which are important for equal energy kinematics, the total transition matrix element may be written as:

In the present calculations we have applied the distorted wave Born approximation (DWBA) and have used atomic units. In this framework the TDCS for the ionization of a molecular orbital by electron impact may be written as: 3

d σ kf ke 2 = 2(2π)4 |t| . ˆ ˆ ki dkf dke dEe

(1)

Here kf , ke and ki are the wave-vectors of the scattered, ejected and incident electrons, respectively. The symbol t stands for the transition matrix element of the system and the energy of the ejected electron is denoted by Ee . The factor 2 on the right hand side occurs because every orbital has two electrons, and each of them can be active. For the direct ionization process, the transition matrix element of the system is td , which may be written as: Z φb (r2 , α, β, γ)φi (r1 ) . (2) td = φf (r1 )φe (r2 ) − r12 In the expression above Z = −1 is the charge of the projectile, while r12 stands for the distance between the projectile and the active target electron. We have denoted by φ with different subscripts the wavefunctions for the incident (i), bound (b), scattered (f ) and ejected (e) electrons, respectively. The bound electron wavefunction depends on the molecular orientation expressed by the Euler angles (α, β, γ). In order to perform the calculations, the wavefunctions of the scattered, ejected and incident electrons are expanded into partial wave series. The interaction potential between the projectile and the active electron is expanded into the multipole series as shown below λ 4π r< 1 ∗ = Yλμ (ˆ r1 )Yλμ (ˆ r2 ). λ+1 r12 2λ + 1 r> μ

(3)

λ

In order to separate the angular and radial dependencies for the bound molecular wavefunction φb (r2 , α, β, γ), it is expanded in terms of spherical harmonics in the molecular frame as in reference [26], then these are transformed into the laboratory frame by using the Wigner D(α, β, γ) functions. By this procedure we obtain lb φb (r2 , α, β, γ) = Ylb mb (ˆ r2 ) clb ν (r2 )Dm (α, β, γ), bν ν

lb m b

(4) clb ν (r2 ) being the expansion coefficients. The direct matrix element is calculated similarly as in reference [15]. Since the above expression of the transition matrix element is given for a fixed orientation of the molecule, and in the experiments the molecules are randomly oriented, we average the cross section (1) over the Euler angles. This procedure can be performed analytically by using the expression 2π π 2π ∗l lb Dmb ν (α, β, γ)Dm (α, β, γ)dα sin βdβdγ bν 0

0

0

b

=

8π 2 δl l δm m δνν . (5) 2lb + 1 b b b b

|t|2 = |td |2 + |tex |2 − |td | |tex | .

(6)

Here the maximum interference was assumed. The exchange term is given by: Z φb (r2 , α, β, γ)φi (r1 ) , (7) tex = φe (r1 )φf (r2 ) − r12 where the coordinates of the ejected and scattered particles were exchanged relative to the direct matrix element (2). We mention here that for a doubly symmetric (equal ejection angles and energies) scattering process the direct and exchange transition matrices are identical. We have calculated the TDCS within the TS∗ (Total Screening) model, which uses distorted waves in order to describe the continuum states. In this model both outgoing electrons move in the same potential field, i.e. in the spherically averaged potential of the residual ion − Ve = Vf = Vnuclei + Velectrons ,

(8)

− where Vnuclei and Velectrons stand for the spherically averaged potential of the nuclei and residual electrons, respectively. The averaging method employed here was described elsewhere [27]. The incident projectile moves in the spherically averaged potential of the nuclei and all electrons of the target (Vi ): Vi = Vnuclei + Velectrons , (9)

Velectrons being the spherically averaged potential created by the electrons of the molecule. The PCI effects are taken into account in our calculations through the the Coulomb distortion factor (CDF) 2 CDF = G 1 F1 (iγ, 1, −2ikf erfave (10) e ) where G=

π exp(−π/kf e ) kf e [1 − exp(−π/kf e )]

(11)

is the Gamow factor. In the above equations 1 F1 is a confluent hypergeometric function, kf e = μvf e and γ = 1/vf e is the Sommerfeld parameter. Here, vf e stands for the relative velocity between the two electrons, while μ = 1/2 is the reduced mass of the electrons. In (10) rfave e is taken to be parallel to kf e and is an averaged version of the actual electron-electron separation in the final state, rf e . The average separation is given by: 2 0.627 √ π2 ave 1+ rf e =

t ln t , (12) 16 t π where t is the total energy of the scattered and ejected electrons. Using the averaged separation in (10) the TDCS (1) may be multiplied by the CDF factor. For further details see [25,28].

Eur. Phys. J. D (2015) 69: 2

3 Results In this section we show calculated TDCSs for the ionization of NH3 and CH4 by electron impact. The TDCSs were determined for two different geometrical configurations, coplanar and perpendicular, respectively. In the scattering plane we have considered the ionization of the 3a1 , 1e1 and 2a1 valence orbitals of the NH3 molecule. The first two orbitals have predominantly p-like character, while the 2a1 orbital is of s-type, being mainly spherical. The ionization energies of these orbitals are 11, 16.74 and 27.74 eV, respectively. For the CH4 molecule the 1t2 and 2a1 orbitals were studied with ionization energies of 14 and 25.7 eV. In the CH4 case the 1t2 orbital has plike character. In the perpendicular plane the TDCSs were calculated only for the outer valence orbitals 3a1 , 1e1 and 1t2 , respectively. We have determined the TDCSs for the kinematical parameters given in reference [18] and compared our results with the experimental and theoretical data presented in the same paper. The calculations were performed for a doubly symmetric case. The scattered and ejected electrons have equal energies (Ef = Ee = E) and equal emerging angles (ξf = ξe = ξ) relative to the incident direction. The energies considered here are equal or lower than 20 eV, as detailed below separately for every orbital. The calculations have been performed with and without the Coulomb distortion factor (10) in the framework of the TS∗ method. The presentation of our results is split into two parts. The first part shows the results obtained in the scattering plane, while in the second part the results for the perpendicular plane are presented. 3.1 Scattering plane TDCSs as a function of the ejection angle ξ are presented for the 3a1 , 1e1 and 2a1 orbitals of the NH3 molecule. Results are also presented for the 1t2 and 2a1 orbitals of the CH4 molecule at E = 20 eV and E = 5 eV for comparison. Comparison is also performed with the experimental data and the M3DW results from [18]. The experimental data in reference [18] were given on a relative scale, therefore all curves are normalized to unity at the higher peak location. NH3 results Figure 1 presents TDCSs for the 3a1 orbital of the NH3 molecule in case of several energy values (E = 20, 15, 10, 7.5, 5 and 2.5 eV). On every panel from (a) to (f) the experimental data, the M3DW results of reference [18] and the TDCSs obtained within our TS∗ method are plotted. We used the (TS∗ , PCI) notation to mark the case when PCI effects were taken into account through the Coulomb distortion factor. The experimental TDCS shows two distinct regions: the forward peak at angles lower than ξ = 90◦ and the

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backward peak at angles higher than ξ = 90◦ . The first one is regarded as a consequence of a binary collision, while the backward intensity arises through a multiple collision mechanism, which involves the scattering on the nuclear core. At higher energies the forward peak shows a shoulder-like structure around 50◦ . As the energy of the outgoing particles is lowered towards 2.5 eV, the experimental backward peak becomes predominant, which may be explained by the fact that the electrons have more time to interact with the nuclear core. At higher energies the TS∗ results, which neglect PCI, show a significant shift of the forward peak towards the low angle region compared to the position of the experimental one. For energies lower than 7.5 eV, a second peak seems to emerge around 60◦ in the low angle region, which becomes dominant at the lowest energy considered here. The backward to forward peak ratio also shows an increasing tendency as approaching 2.5 eV. Nevertheless, the overall comparison with the experimental data shows significant disagreement in both regions. The (TS∗ , PCI) model, which takes into account the PCI effects, shows improvements relative to the TS∗ model. At the two highest energies (E = 20 eV and E = 15 eV), the position of the forward peak agrees well with the experimental data. It may also be observed, that our model presents a shoulder-like structure at these energy values, similar to the experimental forward peak, which may be the consequence of the mainly p-like character of the 3a1 orbital. The shoulder-like structure evolves into a second peak within our model as the energy is further lowered, but this behavior is not clearly observed in the experimental data. At even lower energies (E = 5 eV and E = 2.5 eV) the double-peak structure disappears completely, giving place to a single peak, which is shifted towards higher angles relative to the position of the forward peak at higher energies. This shift is observed also in the experimental data, and is attributed to the increased post collision interaction between the outgoing electrons, which being slower, have more time to interact with each other. It is interesting to note that our (TS∗ , PCI) model produces a double-peak structure even for the backward peak at higher energies. However, the experimental backward peak does not seem to follow this behavior, except at 15 eV, where some structure may be observed in the measured data. At 5 eV and 2.5 eV, our model shows a single peak in the high angle region, similar to the experimental data, but the position of the maximum occurs at smaller angles. Our calculated TDCS provides an increasing backward intensity as the energy of the ejected electron is lowered. However, the backward peak never becomes higher than the forward peak. This behavior of the model may be caused by a weaker electron-nuclei interaction due to the averaging of the real nuclear potential. Comparing these results with the M3DW data, one may observe significant differences. The M3DW model shows, for the whole energy range considered here, a single peak structure both in the low and high angle regions. There is also disagreement between the two models regarding the

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Eur. Phys. J. D (2015) 69: 2

(a)

E = 20 eV

Exp. M3DW TS*, PCI TS*

(d)

E = 7.5 eV


(b)

E = 15 eV


(e)

E = 5 eV


(c)

E = 10 eV


(f)

E = 2.5 eV


1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0 1.5

1.0

0.5

0.0 0

30

60

90

120

150

(deg.)

180

0

30

60

90

120

150

180

(deg.)

Fig. 1. TDCSs for the ionization of the 3a1 orbital of the NH3 molecule by electron impact in the scattering plane in case of doubly symmetric kinematics. The TDCSs are plotted for different emerging electron energies as a function of the ejection angle ξ. The experimental data and the M3DW results are from [18].

position of the peaks, similarities may be found only at the lowest energy value. The results obtained for the 1e1 orbital are presented in Figure 2 for six energy values of the ejected electron (E = 20, 15, 10, 7.5, 5 and 2.5 eV). The 1e1 orbital also has p-like character. As a consequence some similarities are displayed with the 3a1 case. For this orbital the experimental forward peak shows a more pronounced shoulder-like structure at 20 eV and 15 eV compared to the 3a1 orbital. As the energy is further lowered, at 10 eV the forward peak changes to a double-peak structure with a shallow minimum between them, which the authors of reference [18] call a “dip”. At even lower energies the forward peak seems to evolve into a single peak with some additional structure on it. The backward peak of the TDCS shows an increasing tendency relative to the forward peak as the energy is decreasing and becomes predominant below 10 eV, similarly as for the 3a1 orbital. Our (TS∗ , PCI) model shows a similar behavior as found previously for the other p-like orbital. At the two highest energies, the model seems to provide a forward

peak slightly shifted to lower angles relative to the experimental one. However, the position of the peak is somewhat unclear due to the lack of experimental points at small angles. The model shows a shoulder-like structure, similarly to the experimental data, but the intensity of this structure is much lower compared to the experimental one. However, its position agrees well with the measurements. At lower energies (10, 7, 5 eV) our model shows a double-peak structure for the forward region. However, the second peak is much smaller than the first one, which seems to be shifted slightly towards smaller angles relative to the experimental data. At even smaller energies, the forward region evolves into a single peak. Our model reproduces the shift of the experimental forward peak toward larger angles as the energy decreases, an effect attributed to stronger PCI effects at low energies. A similar behavior may also be observed in the case of the backward peak. Our model provides a double-peak structure at high energies, which becomes a single peak at low energies. However, this behavior can not be verified at the moment, since there are not enough experimental points at high angles. The model reproduces the increasing tendency of the

Eur. Phys. J. D (2015) 69: 2

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(a)

E = 20 eV


(d)

E = 7.5 eV


(b)

E = 15 eV


(e)

E = 5 eV


(c)

E = 10 eV


(f)

E = 2.5 eV


1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0 1.5

1.0

0.5

0.0 0

30

60

90

120

150

(deg.)

180

0

30

60

90

120

150

180

(deg.)

Fig. 2. Same as Figure 1, but for the 1e1 orbital.

backward peak relative to the forward peak as the energy is decreased. However, at higher energies the intensity of the calculated backward peak underestimates the experimental one. Unlike for the 3a1 orbital, in this case the backward intensity is higher at the lowest energy than the forward peak, similarly to the experimental data. It may be possible that the enhanced backward intensity at the lowest energy is the consequence of the stronger interaction between the electron and the averaged nuclear potential, the ejected electron being, on average, closer to the nuclear core at the moment of ejection than an electron ejected from the 3a1 state. Nevertheless, the positions of the forward and backward peaks are underestimated by the model at E = 2.5 eV, as well as the position of the minimum between these peaks, which is also deeper than the experimental one. The M3DW model behaves similarly as found for the 3a1 orbital, displaying no additional structure in the forward and backward regions. The TS∗ model, with no PCI effects taken into account shows severe disagreement with the experimental data. Figure 3 shows the results for the s-like 2a1 orbital for several energy values of the ejected electron energy (E = 20, 15, 10 and 5 eV). The experimental TDCS

shows a large forward peak and a smaller backward peak at E = 20 eV. No additional shoulder-like structure is observed at this energy. At E = 15 eV the experimental data is more scattered, and there is a hint of an incipient third peak at 90◦ , which becomes much clearer at E = 10 eV. For the lowest energy considered for this orbital, the experimental data converge into a maximum around 90◦ . Unfortunately, there are no experimental points for angles higher than 130◦ , which makes it difficult to predict the behavior at higher angles for E = 5 eV. The (TS∗ , PCI) model provides TDCSs in a fair agreement with the experimental data. At E = 20 eV, the shape of the calculated cross section is very similar to the experimental one. The forward peak seems to be slightly shifted from the experimental peak towards low angles. At E = 15 eV this shift is more pronounced, but our calculated TDCS shows an incipient third peak around 65◦ . For the 10 eV case our model provides a double-peak structure of the low angle region, which is similar in shape to the experimental data, but the intensity is much smaller. Also, the position of these peaks seems to be shifted relative to the experimental data towards lower angles. At the lowest energy, our model shows a single-peak structure for the forward region around 60◦ . The experimental maximum

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Eur. Phys. J. D (2015) 69: 2

(a)

E = 20 eV


(c)

E = 10 eV


(b)

E = 15 eV


(d)

E = 5 eV


TDCS (a.u.)

1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0 0

30

60

90

120

150

180

0

30

(deg.)

60

90

120

150

180

(deg.)

Fig. 3. Same as Figure 1, but for the 2a1 orbital.

around 90◦ is not reproduced by our model, which predicts a minimum in this region. The (TS∗ , PCI) model provides a very good agreement with the experimental data at the backward peak. Both the position and the intensity of the backward peak seems to be correctly reproduced by the model for higher energies. At E = 5 eV there are no experimental points for a proper comparison. Our model also reproduces the relative increase of the backward intensity as the energy is decreased. The good agreement at the high angle region suggests that the averaging of the nuclear potential is a fair approximation in case of a mainly spherical orbital like the 2a1 . It is worth mentioning that for this orbital the theoretical backward peak does not split at higher energies, as was the case for the p-like orbitals. The M3DW model predicts a similar shape for the TDCS as for the outer orbitals. It shows no evidence of additional structures in the forward or backward region. It also predicts an increasing backward intensity as the energy is decreased. Our TS∗ model, with no PCI effects incorporated shows disagreement with the experimental data: the forward peak is shifted to very low angles at higher energies relative to the other models and the experimental data, while the backward intensity is completely missing for some energies. The different behaviour of the M3DW and our (TS∗ , PCI) models may be related to the approximations employed by each of them. In the M3DW model [18] the molecular orbitals are averaged over all possible orientations of the target instead of the TDCS. This averaging method may lead to the modification of the p-like character of the target orbitals. In our approach the TDCS is averaged analytically by employing the (5) orthogonality relations, which may preserve the character of the molecular orbitals. Further, the M3DW model in reference [18]

uses Dyson molecular orbitals, calculated by employing DFT (density functional theory) with the standard hybrid B3LYP functional and the TZ2P (triple-zeta with two polarization function) Slater-type basis set, while we use Gaussian orbitals. However, the main source for the different behaviour of the models seems to be the averaging method employed. This is supported by a recent study on methane [16], where a proper averaging have been performed, but for different kinematical conditions compared to the present ones. By employing a proper averaging over all possible orientations, the M3DW model showed good agreement with the experimental data in reference [16]. CH4 results The results for the 1t2 orbital of the CH4 molecule are shown in Figure 4. The orbital has p-like character, similarly to the 3a1 and 1e1 orbitals of the NH3 molecule. The molecules being iso-electronic, it is interesting to compare the cross sections for the corresponding type of orbitals of these targets. TDCSs are presented for a higher (E = 20 eV) and for a lower (E = 5 eV) energy value in case of the 1t2 orbital. We compare our findings with the corresponding data of the 3a1 and 1e1 orbitals, presented above in Figures 1a, 1e and 2a, 2e, respectively. At E = 20 eV the experimental TDCS in Figure 4a shows a high forward peak and a smaller intensity at the high angle region, similarly to the cross sections of the 3a1 and 1e1 orbitals for NH3 . Again, a shoulder-like structure (according to [18] a “dip”) is observed for the forward region. The main difference relative to the ammonia case is the more pronounced nature of this structure. It also appears at a slightly lower angle than the corresponding

Eur. Phys. J. D (2015) 69: 2


E = 20 eV

TDCS (a.u.)

1.5

1.0

0.5

E = 20 eV


(b)

E = 5 eV


1.0

0.5

0.0

0.0


E = 5 eV

1.5

1.0

0.5

1.5

TDCS (a.u.)

(b)

TDCS (a.u.)

(a) 1.5

TDCS (a.u.)

(a)

Page 7 of 12

1.0

0.5

0.0

0.0 0

30

60

90

120

150

180

(deg.)

0

30

60

90

120

150

180

(deg.)

Fig. 4. Same as Figure 1, but for the 1t2 orbital of the CH4 .

Fig. 5. Same as Figure 1, but for the 2a1 orbital of the CH4 .

structure for NH3 . The backward intensities for all three orbitals appears to be very similar. Our (TS∗ , PCI) cross sections, while showing similarities for both targets, also present some differences too. The model provides a shoulder-like structure at the forward region in Figure 4a, but the ‘shoulder’ has a much lower intensity and appears to be slightly shifted relative to experimental one towards higher angles. However, the maximum of the first peak in the forward region seems to agree in position with the corresponding experimental peak. The main difference, however, comes from the backward region, where the ammonia cross sections for both the 3a1 and 1e1 orbitals have a double-peak structure (although the intensity of the peaks is very low), while for methane the model shows a single peak in this region. Nevertheless, the backward to forward peak ratio shows smaller values than the experimental one, for both molecules. The TS∗ model provides similar cross sections across the three p-like orbitals of NH3 and CH4 . The results of this model are in significant disagreement with the experimental data in all cases. The model shows a large forward peak at unrealistically small angles, while the backward peak is completely missing for all p-like orbitals at 20 eV. At E = 5 eV the experimental data behaves similarly for all orbitals considered: it shows a higher backward intensity than the forward peak (see Figs. 1e, 2e and 4b). This behavior is reproduced by the (TS∗ , PCI) cross section only in the case of the 1e1 orbital of the ammonia. The model provides a higher forward peak relative to the backward peak for the 1t2 and 3a1 orbitals and in both

cases overestimates the experimental data at this energy. The migration of the forward peak towards higher angles is observed for both target molecules as the energy is lowered, which indicates significant PCI effects. The M3DW model correctly reproduces the increased backward peak and shows similar behavior for both molecules. Our TS∗ model once again provides less satisfactory results. The results obtained in case of the 2a1 orbital of methane are depicted in Figure 5 for E = 20 eV and E = 5 eV, respectively. The orbital is of s-like character and has a nearly spherical shape. We compare our results with those obtained for the corresponding orbital and energies of the NH3 molecule shown in Figures 3a and 3d, respectively. We found that at E = 20 eV the experimental cross sections are similar in shape for both targets. However, for methane, the maximum of the forward peak is clearly defined, while in the case of ammonia it is less clear at which angle is the maximum due to the lack of experimental points. Our (TS∗ , PCI) model shows a particularly good agreement in Figure 5a at the forward peak, while for ammonia the agreement appears to be better at the backward region. The backward peak is underestimated by our model in case of the methane. The M3DW model shows a fair agreement with the experimental data for both targets. However, its forward peak appears at a higher angle than our model and the experimental data predicts. The TS∗ model shows a high

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Eur. Phys. J. D (2015) 69: 2

(a)

E = 20 eV


(d)

E = 5 eV


(b)

E = 15 eV


(e)

E = 2.5 eV


(c)

E = 10 eV


(f)

E = 1.5 eV


1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0 1.5

1.0

0.5

0.0 0

60

120

180

240

300

360

0

60

120

(deg.)

180

240

300

360

(deg.)

Fig. 6. TDCSs for the ionization of the 3a1 orbital of the NH3 molecule by electron impact in the perpendicular plane for a doubly symmetric kinematics. The TDSCs are plotted for different emerging electron energies as a function of the mutual angle φ between the outgoing electrons. The experimental data and the M3DW results are from [18].

forward peak at low scattering angles and a missing backward intensity for both targets. Turning to the lower energy, at E = 5 eV, we found important differences in the TDCSs of the two molecules. In Figure 5b, for methane, the experimental TDCS shows a triple-peak structure. In Figure 3d, for ammonia, the experimental TDCS shows a dominant central peak around 90◦ and some small structures at lower and higher angles, respectively. The (TS∗ , PCI) model reproduces the triple-peak structure found for methane. It shows a first peak which overestimates and a second peak that underestimates the experimental ones. The position of these peaks are well reproduced by the model. However, the backward to forward peak ratio is smaller than the experimental ratio. Our model does not reproduce the dominant central peak observed for NH3 at this energy in Figure 3d. Instead it shows a minimum at 90◦ , similarly to the M3DW model. However, we mention here that at E = 10 eV our model shows a triple-peak structure even for ammonia. The M3DW model provides a double-peak structure in

the case of CH4 too. Again, the TS∗ results are in a poor agreement with the experimental data for both targets.

3.2 Perpendicular plane Results are presented for the outer 3a1 and 1e1 orbitals of the NH3 molecule. In each case, the TDCSs are plotted as a function of the mutual angle between the outgoing electrons φ = ξf + ξe . The cross sections were studied in case of six energy values of the ejected electron: E = 20, 15, 10, 5, 2.5 and 1.5 eV. For several energies (E = 20, 10 and 1.5 eV) we present results also for the 1t2 orbital of the CH4 molecule. Comparison is performed with experimental data and M3DW results presented in reference [18]. All data are normalized to unity at the highest peak location. NH3 results First we present the results obtained for the 3a1 orbital. The TDCSs are shown in Figure 6.

Eur. Phys. J. D (2015) 69: 2

The experimental data show a double-peak structure at E = 20 eV. The peaks are located around 90◦ and 270◦, respectively. At 180◦ a deep minimum may be observed, which becomes shallower as the energy is decreased to 15 eV. For even lower energies, the minimum turns into an incipient third peak at 10 eV, while at 5 eV the distribution remains almost constant between the flanking peaks. At 2.5 and 1.5 eV this structure turns into a single-peak, which shows a wider distribution at the higher energy. Our (TS∗ , PCI) model reproduces the main features of the experimental TDCS. At the two highest energies, the model shows a double-peak structure. The position of the peaks are correctly reproduced, except maybe for a small shift towards 180◦ relative to the experimental peaks at 20 eV. The minimum at 180◦ is deeper than the experimental minimum. The positions of the flanking peaks are well reproduced at E = 10 eV. At this energy, our model shows an incipient peak around 180◦, but the intensity is smaller than in the experimental data. Decreasing the energy to 5 eV, our model shows two peaks, which are closer to 180◦ than at higher energies, and a shallow minimum between them. However, at angles lower than 100◦ and higher than 230◦ the agreement with the experimental point seems to be satisfactory. At the two lowest energies the model reproduces the experimental single-peak structure and even the wider distribution at 2.5 eV, showing good agreement with the measurements. The appearance of a single-peak structure at low energies is related to the more pronounced PCI effects at these energies. Since the electrons are slower, they have more time to interact and tend to emerge at a mutual angle of 180◦ . Similar behavior was observed also for molecular hydrogen, see for example [3,5]. Our TS∗ model predicts an unphysical intensity for the TDCS at φ = 0◦ and φ = 360◦ at all energies considered here. At the two highest energies we found a double-peak structure, with a minimum at 180◦ . The peak-positions are well reproduced. For lower energies, the peaks seem to migrate away towards lower and higher angles, respectively, leading to a disagreement with the experimental data. At 2.5 eV and 1.5 eV the TS∗ model shows some agreement with the experimental data around 180◦ , but the overall agreement is not satisfactory. The M3DW model shows for all energies a third peak at 180◦, which becomes dominant as the energy is decreased towards 1.5 eV. At higher energies, the position of the flanking peaks agrees well with the measurements. Recently performed M3DW calculations [16], which employed a proper averaging of the cross section have led to a very nice agreement with the experimental data and the three-peak structure disappeared in the perpendicular plane. These calculations have been performed for different kinematical conditions in case of methane. For the inner p-like orbital, 1e1 , the results are plotted on Figure 7. In this case for E = 20 eV there are no experimental data, but for lower energies the measured TDCSs are similar in shape with those of the outermost 3a1 orbital. Once again, the double-peak structure, with

Page 9 of 12

a minimum at 180◦ , turns into a single peak as the energy approaches its lowest value. The minimum observed at E = 15 eV is gradually changing: at E = 10 eV additional structures are shown around 180◦ . For even lower energies (E = 5 eV and E = 2.5 eV) the distribution between the two flanking peaks appears to be constant, while at E = 1.5 turns into a single peak at 180◦. It may be also observed that the peaks seen first around 90◦ and 270◦ move closer to 180◦ as the energy is decreasing. Our (TS∗ , PCI) model reproduces the main features of the experimental TDCS at higher energies. Here, we observe a double-peak structure, but the minimum at 180◦ is much deeper than the experimental counterpart. However, the position of the peaks is correctly reproduced. At E = 10 eV the shape of the TDCS shows similarities with the experimental one, additional structures being observed around 180◦ . However, the position of the flanking peaks are slightly shifted relative to the position of the experimental ones. At E = 5 eV our calculated cross section is in good agreement with the experimental data, except at 180◦, where a shallow minimum appears in the calculated data. For the two lowest energy value, our model shows significant disagreement with the experiment. While the experimental data has a central peak at 180◦, our calculated cross section shows a minimum at this angle, which is surprising, since for the outermost p-like orbital we found a very good agreement at these energies. It seems that in our model the PCI effects between the outgoing electrons is not strong enough, or is modulated by some other mechanism at these energies and for the inner p-like orbital. The TS∗ cross section once again has a non-zero contribution at 0◦ and 360◦ for all the six energy values. The model shows a double-peak structure, similar to the (TS∗ , PCI) model. Between the peak, the minimum is deeper than for the (TS∗ , PCI) for all energies. The position of the peaks are shifted away relative to the experimental data towards lower and larger angles, respectively. The M3DW model behaves very similar to the 3a1 case. It shows a triple-peak structure, with an enhancement of the central peak as the energy decreases towards 1.5 eV. CH4 results Figure 8 presents the perpendicular plane result for the 1t2 orbital of the CH4 molecule. The TDCS was studied for three energies of the ejected electron (E = 20, 10 and 1.5 eV). We compare our results with the corresponding NH3 cross sections obtained for the 3a1 and 1e1 p-like orbitals. At E = 20 eV, the experimental data for NH3 and CH4 show similar cross sections in Figures 6a and 8a. Our models behave similarly to the ammonia case, showing a double-peak structure with a deep minimum between the peaks. The (TS∗ , PCI) model does not predict accurately the position of these peaks for the methane molecule, they are shifted relative to the experimental ones toward φ = 180◦. The M3DW model provides a triple-peak structure

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Eur. Phys. J. D (2015) 69: 2

(a)

E = 20 eV

M3DW TS*, PCI TS*

(d)

E = 5 eV


(b)

E = 15 eV


(e)

E = 2.5 eV


(c)

E = 10 eV


(f)

E = 1.5 eV


1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0 1.5

1.0

0.5

0.0 0

60

120

180

240

300

(deg.)

360

0

60

120

180

240

300

360

(deg.)

Fig. 7. Same as Figure 6, but for the 1e1 orbital.

for both targets, but in the case of methane the central one is dominant.

4 Conclusions

Lowering the energy to E = 10 eV, the experimental cross sections are almost identical in shape for both molecules (see Figs. 6c, 7c and 8b). The two orbitals of NH3 show, however, some variation in the angular distribution around 180◦. Our (TS∗ , PCI) model shows similar behavior for all three p-like orbitals, but the overall agreement is better for the NH3 molecule.

Calculated TDCSs for the ionization of NH3 and CH4 by electron impact have been determined in coplanar and perpendicular geometries. The calculations were performed within the DWBA formalism. For both geometrical arrangements a doubly-symmetric kinematics was studied. Both the emerging angles and energies of the outgoing electrons were considered to be equal. The ionization of the two molecular target was studied for several values of the ejection energy, which was gradually decreased from the highest value of 20 eV. For such kinematical conditions PCI effects are important to account for. The calculations have been performed for the ionization of the outer valence orbitals of NH3 : 3a1 , 1e1 (of p-like character) and 2a1 (of s-like character). In the case of CH4 the p-like 1t2 and the s-like 2a1 orbitals were studied but only for a few ejection energies. Our (TS∗ , PCI) model takes into account PCI effects by incorporating the Coulomb distortion factor of reference [25]. We present results also from the TS∗ model, which does not take into account these effects. Our results are compared with the experimental data and the M3DW calculations presented in reference [18].

At E = 1.5 eV, the experimental TDCS are slightly different for the two molecules (see Figs. 6f, 7f and 8c). While for NH3 the angular distribution shows a single peak around φ = 180◦ , the CH4 cross section has an almost flat shape between φ = 120◦ and φ = 240◦ . The (TS∗ , PCI) shows similar behavior as the experimental one, except for the 1e1 orbital of the NH3 molecule, where provides a minimum instead of the maximum observed in the experimental data. Once again, the M3DW cross section shows a triple peak structure, with a dominant central peak, for all cases considered here. The TS∗ model, which does not take into account PCI effects, provides unsatisfactory agreement with the experimental data.

Eur. Phys. J. D (2015) 69: 2

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(a)

E = 20 eV


(b)

E = 10 eV


1.5

1.0

0.5

0.0

TDCS (a.u.)

1.5

1.0

0.5

0.0

(c)


E = 1.5 eV

1.5

1.0

0.5

0.0 0

60

120

180

240

300

360

(deg.) Fig. 8. Same as Figure 6, but for the 1t2 orbital of the CH4 .

In the scattering plane our (TS∗ , PCI) model shows very similar cross sections for the p-like outermost orbitals of NH3 . The model reproduces qualitatively the behavior of the experimental TDCS. For both orbitals it shows an increasing backward peak relative to the forward peak as the energy is decreasing. This behavior may be the consequence of an enhanced electron-nuclei interaction at low energies. Although, the backward intensity predicted by the model is increasing as the energy is lowered, it becomes dominant at low energies relative to the forward peak only for the 1e1 orbital in our calculations. The experimental data shows higher backward than forward peaks for both orbitals at low energies. It may be possible that for the 3a1 orbital, the electron-nuclei interaction is not strong enough due to the averaging of the nuclear potential in our model. The model predicts a shift of the forward peak towards higher ejection angles (a consequence of PCI effects) as the energy decreases, similarly to the experimental data. For energies higher than 10 eV,

the experimental data predict a shoulder-like structure or a “dip” in the forward region. Our model reproduces this structure, although its intensity is smaller than the experimental counterpart. In fact, the model predicts a doublepeak structure even for the backward region at energies higher than 7.5 eV in case of both orbitals. However, this behavior can not be verified at the moment due to the lack of experimental points at high ejection angles. At the two lowest energy, our model predicts no additional structure in the forward or backward peak. Similar observation apply in case of the 1t2 orbital of the CH4 molecule, except that the high angle region has a single-peak structure in our calculated TDCS. Our (TS∗ , PCI) model shows a good agreement with the experimental data in case of the s-like orbital of NH3 for higher ejection energies. In this case the forward peak does not show the shoulder-like structure observed for the outermost orbitals. As the energy is lowered, the experiment shows an additional peak emerging at an ejection angle of 90◦ . Our model reproduces qualitatively this behavior of the experimental TDCS, however, the position and the intensity of this new structure is underestimated. At the lowest energy our model shows a less satisfactory behavior as it can not account for the large central peak observed in the experimental data at 90◦ . In case of the CH4 molecule we found a similar behavior as for the NH3 at the highest energy. The agreement with the experiment is especially good at the forward peak location. At the lowest energy, our model shows a triple-peak structure for CH4 , similarly to the experimental data. This is in contrast with the corresponding results for NH3 , where our model predicted a double-peak structure for the TDCS, while the experimental one showed a dominant peak around 90◦ . Although the M3DW model shows several features of the experimental data, it does not accounts for the additional structures observed in case of the different orbitals. In the perpendicular plane both targets show similar cross sections. In this case the 3a1 , 1e1 and 1t2 orbitals of NH3 and CH4 were studied. As the energy is decreasing, the double-peak structure of the experimental TDCS turns into a single peak around 180◦ at the lowest energy, which again may be the consequence of PCI effects. This peak is not so clear for the 1t2 orbital of the CH4 molecule, where the angular distribution shows a rather flat structure between the flanking peaks. Our (TS∗ , PCI) model reproduces well this behavior observed for the experimental TDCS, except for the two lowest energy in case of the 1e1 orbital. Our model describes better the TDCS in the perpendicular plane relative to the M3DW cross sections, which show a triple-peak structure for all energies and orbitals studied. We mention here that in a recent study [16] the M3DW model showed good agreement with the experimental data even in the perpendicular plane by employing a proper averaging of the cross section. Our TS∗ model (with no PCI included) shows a less satisfactory behavior in case of both the scattering and perpendicular plane results and for both molecules, which

Page 12 of 12

reflects the importance of the PCI effects at the kinematical conditions employed in the present study. As a final conclusion we may say that we found similar TDCSs for the two iso-electronic molecules both in the coplanar and perpendicular geometrical arrangements. As stated in reference [18], the TDCS seem to be more sensitive to the type of molecular orbitals than the structural differences between the NH3 and CH4 molecules. For most energies and orbitals our (TS∗ , PCI) model leads to fair agreement with experiment, and clearly is the best description of the ionization process for these targets and kinematical conditions so far. However, the existing discrepancies suggest that further improvements of the theoretical description are needed. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project No. PN-II-ID-PCE-2011-3-0192 (L. Nagy) and by a grant of the Babe¸s-Bolyai University, project No. GTC34032/2013 (I. T´ oth).

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