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Theoretical study of positron scattering from pentane isomers
Accepted Manuscript Research paper Theoretical study of positron scattering from pentane isomers Nidhi Sinha, Bobby Antony PII: DOI: Reference:
S0009-2614(18)30830-3 https://doi.org/10.1016/j.cplett.2018.10.019 CPLETT 36005
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
30 July 2018 7 October 2018
Please cite this article as: N. Sinha, B. Antony, Theoretical study of positron scattering from pentane isomers, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.10.019
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical study of positron scattering from pentane isomers Nidhi Sinhaa , Bobby Antonya a
Atomic and Molecular Physics Lab, Department of Applied Physics, Indian Institute of Technology (Indian school of Mines), Dhanbad-826004, Jharkhand, India
Abstract This work aims to probe isomeric effect in case of positron scattering. In this pursuit, we have chosen pentane (C5 H12 ) as the target molecule. The structural isomers of pentane offers a perfect prototype to investigate isomeric effect and thus may give further knowledge into the role of the geometrical arrangements of atoms in a target molecule during the positron scattering process. The grand total, integral elastic and differential cross section are reported using the modified spherical complex optical potential (mSCOP) method. The total cross sections do not exhibit isomeric effect, which is in accordance with the experimental observation. Keywords: , Positron scattering, Isomeric effect, Total cross section, Integral elastic cross section, Differential cross section
1. Introduction The discovery of isomerism dates back to 1827, when Griedrich Woehler found that some compounds have the same molecular formula but different arrangement of the atoms in the molecules. Such group of molecules were named isomers. The different geometrical arrangement of atoms in such group of molecules infuse interest to researchers to explore the structural effect in various physio-chemical processes. We are presently interested in the scattering studies of positrons from pentane molecules. Consequently, such studies would help to gain insight on the very basic underlying physics of scattering processes. Email addresses: [email protected] (Nidhi Sinha), [email protected] (Bobby Antony) Preprint submitted to CPL
October 8, 2018
In case of electron scattering there are several work available[1, 2, 3, 4, 5, 6, 7] which primarily emphasise on the isomeric effect. These studies have established the low energy isomeric effect in electron interactions. However for positron scattering, limited studies[8, 9, 10, 11, 12, 13, 14] were found on the investigation of isomeric effect. No isomeric effect for the total cross sections was suggested from the previous literature, however differential cross sections do show some difference at higher energies[10]. However, such conclusion on DCS needs experimental validation. The pentane isomers, n-pentane, iso-pentane and neo-pentane are chosen as the targets for the present work. As can be seen from fig. 1, these isomers offers an ideal epitome to probe the role of geometrical arrangements in the scattering process. This owes to their structural difference — n-pentane being a long chain structured molecule; iso and neo-pentane being the branched one. The close values of molecular properties of these isomers (see table 1) further add to the research interest of these targets. Thus, these molecules differ mostly due to their structural arrangements; other target parameters being almost equal. Hence, the chosen isomers will aptly fulfil our aim to investigate the role of geometry in case of positron scattering.
Figure 1: Molecular structure of pentane isomers
The only available previous study for the present set of targets is due to Chiari et al.[12]. The authors have reported the measured cross sections along with their theoretical results. Their experimental total cross sections are reported from 0.1 to 50 eV for n- and iso-pentane. Further, they have used the IAM-SCAR method to theoretically estimate the various cross sections for
2
Table 1: Target properties [12]
Target
IP (eV)
Bond length (˚ A) C-H
C-C
˚3 ) Polarizability (A
∆p (eV)
n-pentane
10.35
1.118
1.531
9.98
3.55
iso-pentane
10.32
1.096
1.547
10.11
3.52
neo-pentane
10.35
1.114
1.537
10.19
3.55
these three isomers for an exhaustive energy range of 1 to 1000 eV. In their experimental work, no isomeric effect was observed. However, theoretical cross sections show slight difference in the low energy region. This is obviously due to the approximations used in the theory which dominate the formalism in the low energy region. In addition to these comparisons, total cross sections of Kimura et al. [15] is also available for n-pentane. The two experimental cross sections for this molecule is found to diverge significantly in the low energies (up to 6 eV), both in terms of magnitude and shape. This behaviour is attributed to the forward angle scattering effect according to the authors[12]. Moreover, the previous work is devoid of any isomeric effect in the positron scattering. The present work focuses on the computation of differential, integral elastic and grand total cross section for the pentane isomers. Spherical complex optical potential (SCOP) formalism [16, 17, 18] is utilised for these calculations. Interestingly, this method was developed for electron scattering. However, our group have modified the formalism to adequately solve the positron scattering problem [19, 18, 20, 21]. This is permissible since positrons are the anti-particle of electron and hence with proper modifications the theory can be applied to positron scattering. The cross sections are calculated for a comprehensive energy range of 1eV to 5keV. This attempt is the first of its kind to calculate positron scattering cross sections over such a wide range of energy for the present target molecules. The DCS are reported for energies 1, 5, 10, 15, 20, 25, 30 and 50 eV. The following section gives the brief description of the adopted theory.
2. Theoretical Methodology We have utilised modified spherical complex optical potential (mSCOP) formalism[19, 20, 18] in our previous articles exploring positron scattering systems and were able to obtain consistent
3
(a) Normal pentane
(b) Iso pentane
(c) Neo pentane
Figure 2: Electrostatic potential surface. Red circles show the identified scattering centres.
results for atoms[19], diatomic molecules[22] and complex molecules[20, 21]. In the present endeavor the same approach[20], as that of complex molecules is employed. This owes to the structure of the pentane isomers. Further, due to the large size of the molecules a multi-centre spherical complex optical potential[20, 23, 21] is employed. The cross sections for various scattering groups are calculated independently and finally summed up to get the cross sections for the target as a whole. These scattering centres are identified on the basis of bond length and atomic radii of the atoms present in the target molecule. In the present work we have used Avogadro software to identify these groups. Molecules are build and optimised and electrostatic potential surface (EPS) is then generated using the software. The scattering centres are visually selected based on the physical appearance of the EPS. EPS approach blended with mSCOP method is an efficient way to tackle the overestimation of the cross sections caused due to the approximations used in the theory and hence can be used as an alternative to screening corrections. It is noteworthy that the SCOP formalism is basically a framework which relies on the choice of potentials used to explore the scattering system. In case of electron scattering the formalism was unable to efficiently model the low energy interaction. However, for positron scattering, interestingly we found results which were in good accordance with the experimental ones even at the lowest impact scattering[19, 21, 20, 22]. Thus with the choice of proper potentials and thresholds our group was able to overcome the limitation of the present formalism in the low energy region. We have modelled the total elastic (Qel ) and total inelastic cross sections (Qinel ) using the SCOP method. In this method, the eigen value problem is solved using the partial-wave analysis method. Hence, the total positron scattering cross section Qtot is given as,
4
Qtot (Ei ) = Qel (Ei ) + Qinel (Ei )
(1)
The present calculation rests on the formulation of complex scattering potentials. The spherically averaged charge densities of the target is employed to construct this potential Vopt , given by
Vopt (r, Ei ) = VR (r, Ei ) + i Vabs (r, Ei )
(2)
The real part is the sum of static (Vst ) and polarization (Vpol ) terms, mathematically expressed as
VR (r, Ei ) = Vst (r) + Vpol (r, Ei )
(3)
We have used the Hartree-Fock wave functions of Cox and Bonham[24] to derive the analytical form of the static potential. For the polarization potential, the parameter-free model of the correlation polarization potential given by Zhang et al.[25] is used. It contains multipole nonadiabatic rectifications in the intermediate energy range and smoothly approaches the correct asymptotic form for large r. These two potentials drive the elastic part of the scattering system. However, the loss of flux due to all possible electronic inelastic processes is treated by the absorption part of the complex potential. The parameter free form proposed by Reid and Wadehra[26] is incorporated in the present work to account for the loss of incident flux.. The most intriguing feature of positron scattering when compared to other lepton interactions is the positronium formation process. Theoretically it is difficult to include this channel in the scattering process, since we cannot define it in terms of binary collisions. Now the principal factor determining the absorption potential is the choice of absorption threshold ∆. There are different variants of ∆ available in the literature which primarily ensures the energy variation of the thershold from positronium formation, ∆p to the first electronic excitation energy, ∆e . Such variation allows the inclusion of positronium formation channel in the computational codes. In the present work we have adopted the form proposed by Chiari et al.[27] given as, ∆(E) = ∆e − (∆e − ∆p ) exp
5
−(E − ∆p ) Em
(4)
Em is the energy at which the inelastic cross section attains its maximum value without considering Ps formation. These three potentials are combined to form the final optical potential, which is then used to solve the Schr¨ odinger equation using numerical method[17]. The solution are obtained in the form of complex phase shifts (δl ) carrying the signature of the scattering channel. The obtained phase shifts are employed to determine the inelasticity or the absorption factor at various energies. ηl = exp(−2Im δl )
(5)
Using this inelasticity factor, the total elastic and total inelastic cross sections are calculated through[28], ∞ π X Qel (Ei ) = 2 (2l + 1) |ηl exp(2iRe δl ) − 1|2 k
(6)
∞ π X Qinel (Ei ) = 2 (2l + 1)(1 − ηl2 ) k
(7)
l=0
l=0
The total cross section for the positron-molecule scattering is computed as the sum of these two cross sections. In the present calculation the nonspherical terms namely vibrational and rotational potentials are excluded in the full expansion of the optical potential since contributions from these terms are negligible at the intermediate and high energies.
3. Results and discussion The present work offers a comprehensive investigation of positron impact differential, integral elastic and total cross sections for pentane isomers over a wide energy range. Figures 3-7 give the pictorial representation of the cross sections along with available comparisons. The total cross sections are plotted in figs. 3 and 4 and the elastic cross sections are displayed in fig. 5. These cross sections are compared with the experimental and theoretical results of Chiari et al.[12]. Kimura et al.[15] have measured the total cross section for n-pentane and are included in our plot. In figs. 6 and 7 we have done a comparative study of the total and integral elastic and differential cross sections for the three isomers respectively. Such comparative study will help us to get a conclusive remark on the isomeric effect in positron scattering.
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Figure 3: Total cross section for positron-n-pentane scattering. solid line: Present results, dashed
line: Theoretical results of Chiari et al.[12], solid circles: Experimental results of Chiari et al.[12] and solid triangles: Experimental measurements of Kimura et al.[15]
(a) Iso pentane
(b) Neo pentane
Figure 4: Total cross section for positron-iso and neopentane scattering. solid line: Present results, dashed line: Theoretical results of Chiari et al.[12] and solid circles: Experimental results of Chiari et al.[12] for neo pentane
7
3.1. Total cross section In fig.2, the total cross section (Qtot ) for the positron impact scattering of n-pentane along with the available comparisons is graphically presented. The present result shows a similar variation with increasing energy as that of the theoretical cross sections of Chiari et al.[12]. Moreover, the two cross sections show fair agreement in the intermediate to high energy region. However, in the energy range of 5 to 30 eV, the present cross sections are observed to be significantly lower than their theoretical cross sections. This owes to the IAM-SCAR method adopted by the authors which excludes any molecular property in the calculation and the cross sections are merely the incoherent sum of the cross sections for individual atoms. It is well known that the effect of such a choice exists only upto 25-30 eV and above that the two cross sections should show good agreement with each other. This is exactly depicted in the graph. Moreover, Chiari et al.[11] have stated in the article that the dependability of the IAM-SCAR method can be extended upto 30eV. Albeit, as expected they show good agreement in the high energies. The present results are also compared to the experimental measurements of Chiari et al.[12] and Kimura et al.[15]. However, the two experimental results are found to diverge in the low energy region. This is because no corrections were made for the forward angle scattering effect by Kimura et al.[15]. Nonetheless, the present data are found to be in good accordance with the experimental results of Chiari et al.[12] in the low energies. Though the difference in the magnitudes is attributed to the exclusion of the rotational and vibrational excitation channel in the present formalism. In the intermediate energy regime the present cross sections significantly overestimate their measurements. Moreover, at energies above 100 eV, the available experimental and theoretical cross sections overlap with the present results. The total cross sections for positron-isopentane scattering is shown in fig. 3.1. As in the previous case, the present plot shows a valley structure symbolizing the point just before opening of the inelastic channel above which magnitude increases. However, when the impact energy is very high, the interaction time decreases significantly resulting in low cross sections. Such variation is well demonstrated in the present plot. The present result is compared with the theoretical and experimental data of Chiari et al.[12]. The deviations of their cross sections with the present one is similar to that in fig.3 and hence the discussion remains the same. The present total cross sections for neo pentane along with the only available theoretical esti-
8
Figure 5: Elastic cross section for positron-n-pentane scattering. Inset: integral elastic cross section
for iso and neo pentane. Solid line: present results and dashed line: theoretical results of Chiari et al.[12] mation of Chiari et al.[12] is depicted in fig. 3.1. The two cross sections vary in a similar fashion; however magnitude being higher for the present data in the intermediate energy region. Again in the low energy region, owing to the IAM-SCAR method of Chiari et al. [12], the two theoretical cross sections differ significantly in magnitude. Further, the lack of available literature for the present set of targets demand more scientific attempts to be made.
3.2. Elastic cross section The total elastic cross sections for pentane isomers are shown in fig. 5. The only available previous cross section is due to Chiari et al.[12]. The three isomers show similar behaviour compared to the theoretical results of Chiari et al.. The present cross sections also varies as that of Chiari et al.[12] albeit with lower magnitudes in the low energies. However, above 10 eV, present results underestimate their cross sections. The approximations while deriving the potentials in their respective work and the MSCOP approach used in the present calculations might be the reason for such deviations. Further, Chairi et al.[12] have adopted the IAM-SCAR method which itself
9
(a) Total cross section
(b) Elastic cross section
Figure 6: Comparative study of integral cross sections of pentane isomers (log-log scale). Inset: linear scale. Solid line: n-pentane, dashed line: iso-pentane, dash dot dotted line: neo-pentane
has limitations in the low and intermediate energy region. Thus the cumulative effect of such limitations in the two theoretical models might have resulted in the observed discrepancy.
3.3. Isomeric effect 3.3.1. Integral cross sections Fig. 6a compares the total scattering cross sections for the three isomers of pentane. The qualitative shape of the cross sections is same for the molecules where a dip can be found at around Ps formation threshold. Above this energy the cross sections increases due to the contribution from various inelastic channels. Thus a broad peak can be seen at around 10 to 50 eV. However, as the energy increases the interaction time of the projectile and target decreases and hence the cross sections fall rapidly at higher energies. Further, comparing the cross sections, we found that they merge well with each other over the entire comparative energy range. The physio- chemical parameters of the three targets are comparable and the only difference lies in their geomterical arrangement. Thus we conclude that there is no part of geometry in case of positron scattering even at the lowest comparative energies. This observation is corroborated by the experiment of Chiari et al.[12]. The authors have measured the TCS for normal and isopentane and found the magnitude to be comparable. Further, their theoretical model also manifests a similar observation. In fig. 6b, we present the comparative plot of the elastic cross sections for the present isomers.
10
Akin to the previous case, we found no isomeric effect in the elastic cross sections. Nonetheless no experimental data is available and hence is highly reccommended to manifest the present observation.
3.3.2. Differential cross sections The differential cross sections (DCS) for elastic scattering in presence of inelastic channels are shown in figure 7. Since the present targets are polar in nature, the DCS becomes high at lower scattering angle. Moreover, polarisation effects are strongest at short range and gradually becomes weak at intermediate and long range. Hence the present DCS exhibit lower values at larger angles. Another interesting characteristic of the DCS curve is the oscillatory behaviour which arises due to the coupling of higher partial waves of the heavier C atoms in the scattering process. Comparing the DCS of the isomers, one can easily find that the cross sections overlap except at the minima for most of the energies. However, as mentioned by Chiari et al. [12] this could be due to the spherical-like symmetry of neo pentane and its manifestation of isomeric effect is still doubtful. Besides, considering the log scale, difference in the magnitudes are negligible and hence can be concluded that no isomeric effect was observed even at the DCS level.
4. Conclusions The present article reports the grand total and integral elastic cross sections for pentane isomers in the energy range of 1eV to 5keV. mSCOP formalism[19, 20, 18, 21] along with the multi-center approach[23, 20, 21] were utilised for the computation of the cross sections. The group of molecules studied here show a consistent behaviour when compared to the previous results. A good agreement was observed between the cross sections reported here and the theoretical data of Chiari et al.[12]. Moreover, in the energy span of 10 to 50 eV, significant deviations were observed. This owes to the various approximations used in these two theoretical methods. Besides, for the rest part of the comparative energy range an excellent corroboration can be seen. The inconsistency of the experimental measurements and the theoretical predictions hold in this work too. However, above 100 eV, an excellent accordance can be found between the two as seen in fig 3. The prime motivation behind the present work was to explore the isomeric effect in the positron scattering systems. Previous studies[12] show that experimentally no such effect was pronounced, but the theoretical calculations differ in magnitude for the three isomers. However, the authors did
11
not considered any role of geometry for such behaviour and attributed it to theoretical limitations. In the present work we obtained a similar result and the present set of targets, namely pentane isomers didn’t show any appreciable isomeric effect in the integral cross sections reflecting the absence of any contribution of geometry in positron scattering. Further, the determination of DCS provides a stringent test for any scattering theory, as it is sensitive to the effects which are averaged over in case of integral cross section. To a close approximation, even for the DCS we found no role of geometry in positron scattering. Experiments are much needed to validate our observations.
Acknowledgment BA is pleased to acknowledge the support of this research by Department of Science and Technology, Govt of India through SERB project Grant No. EMR/2016/005035. [1] C. Szmytkowski, S. Kwitnewski, Electron scattering on C3 H6 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 35 (11) (2002) 2613–2623. [2] C. Winstead, Q. Sun, V. McKoy, Low-energy electron scattering by C3 H6 isomers, The Journal of Chemical Physics 96 (6) (1992) 4246–4251. [3] F. Kossoski, T. Freitas, M. Bettega, Resonances in electron collisions with C2 H2 Cl2 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 44 (24) (2011) 245201. [4] F. Kossoski, M. Bettega, Low-energy electron scattering from the aza-derivatives of pyrrole, furan, and thiophene, The Journal of Chemical Physics 138 (23) (2013) 234311. [5] Y. Nakano, M. Hoshino, M. Kitajima, H. Tanaka, M. Kimura, Low-energy electron scattering from C3 H4 isomers: Differential cross sections for elastic scattering and vibrational excitation, Physical Review A 66 (3) (2002) 032714. [6] C. Szmytkowski, S. Kwitnewski, Isomer effects on the total cross section for electron scattering from C4 F6 molecules, Journal of Physics B: Atomic, Molecular and Optical Physics 36 (24) (2003) 4865–4873. [7] C. Makochekanwa, H. Kawate, O. Sueoka, M. Kimura, M. Kitajima, M. Hoshino, H. Tanaka, Total and elastic cross-sections of electron and positron scattering from C3 H4 molecules (allene and propyne), Chemical Physics Letters 368 (1-2) (2003) 82–86. [8] K. Floeder, D. Fromme, W. Raith, A. Schwab, G. Sinapius, Total cross section measurements for positron and electron scattering on hydrocarbons between 5 and 400 eV, Journal of Physics B: Atomic and Molecular Physics 18 (16) (1985) 3347–3359. [9] O. Sueoka, S. Mori, Total cross sections for low and intermediate energy positrons and electrons colliding with CH 4 , C2 H4 and C3 H6 molecules, Journal of Physics B: Atomic and Molecular Physics 19 (23) (1986) 4035–4050.
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[10] F. B. Nunes, M. H. Bettega, S. d. Sanchez, Positron collisions with C3 H6 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 48 (16) (2015) 165201. [11] L. Chiari, A. Zecca, S. Girardi, A. Defant, F. Wang, X. Ma, M. Perkins, M. Brunger, Positron scattering from chiral enantiomers, Physical Review A 85 (5) (2012) 052711. [12] L. Chiari, A. Zecca, F. Blanco, G. Garc´ıa, M. Brunger, Experimental and theoretical cross sections for positron scattering from the pentane isomers, The Journal of Chemical Physics 144 (8) (2016) 084301. [13] O. Sueoka, C. Makochekanwa, H. Tanino, M. Kimura, Total cross-section measurements for positrons and electrons colliding with alkane molecules: Normal hexane and cyclohexane, Physical Review A 72 (4) (2005) 042705. [14] C. Makochekanwa, M. Hoshino, H. Kato, O. Sueoka, M. Kimura, H. Tanaka, Electron and positron scattering cross sections for propene and cyclopropane, Physical Review A 77 (4) (2008) 042717. [15] M. Kimura, O. Sueoka, A. Hamada, Y. Itikawa, Vol. 111, John Wiley & Sons, Inc. [16] K. L. Baluja, A. Jain, Positron scattering from rare gases (He, N e, Ar, Kr, Xe, andRn): Total cross sections at intermediate and high energies, Phys. Rev. A 46 (1992) 1279–1290. [17] K. N. Joshipura, M. Vinodkumar, C. G. Limbachiya, B. K. Antony, Calculated total cross sections of electron-impact ionization and excitations in tetrahedral (XY 4 ) and SF 6 molecules, Phys. Rev. A 69 (2004) 022705. [18] S. Singh, S. Dutta, R. Naghma, B. Antony, Theoretical formalism to estimate the positron scattering cross section, The Journal of Physical Chemistry A 120 (28) (2016) 5685–5692. [19] N. Sinha, S. Singh, B. Antony, Positron total scattering cross-sections for alkali atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 51 (1) (2017) 015204. [20] N. Sinha, P. Modak, S. Singh, B. Antony, Positron scattering from methyl halides, The Journal of Physical Chemistry A 122 (9) (2018) 2513–2522. [21] N. Sinha, B. Antony, Electron and positron interaction with pyrimidine: A theoretical investigation, Journal of Applied Physics 123 (12) (2018) 124906. [22] S. Singh, B. Antony, Positronium formation and ionization of atoms and diatomic molecules by positron impact, EPL (Europhysics Letters) 119 (5) (2017) 50006. [23] S. Singh, R. Naghma, J. Kaur, B. Antony, Calculation of total and ionization cross sections for electron scattering by primary benzene compounds, The Journal of Chemical Physics 145 (3) (2016) 034309. [24] H. Cox Jr, R. Bonham, Elastic electron scattering amplitudes for neutral atoms calculated using the partial wave method at 10, 40, 70, and 100 kV for Z= 1 to Z= 54, The Journal of Chemical Physics 47 (8) (1967) 2599–2608. [25] X. Zhang, J. Sun, Y. Liu, A new approach to the correlation polarization potential-low-energy electron elastic scattering by He atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 25 (8)
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(1992) 1893–1897. [26] D. D. Reid, J. Wadehra, A quasifree model for the absorption effects in positron scattering by atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 29 (4) (1996) L127–L133. [27] L. Chiari, A. Zecca, S. Girardi, E. Trainotti, G. Garca, F. Blanco, R. P. McEachran, M. J. Brunger, Positron scattering from O2 , Journal of Physics B: Atomic, Molecular and Optical Physics 45 (21) (2012) 215206. [28] C. Joachain, Quantum collision theory (Amsterdam : N orth Holland) (1983).
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Figure 7: Differential cross section for pentane isomers
*Highlights (for review)
Investigation of isomeric effect in positron impact scattering. Cross sections reported for a wide energy range of 1 to 5000 eV. Inclusion of molecular structure in the computations. Efficient method to include positronium formation in the scattering channel Fair agreement with the limited experimental data. Consistent results obtained in terms of shape and magnitude with previous study.
Graphical Abstract
S0009-2614(18)30830-3 https://doi.org/10.1016/j.cplett.2018.10.019 CPLETT 36005
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
30 July 2018 7 October 2018
Please cite this article as: N. Sinha, B. Antony, Theoretical study of positron scattering from pentane isomers, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.10.019
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Theoretical study of positron scattering from pentane isomers Nidhi Sinhaa , Bobby Antonya a
Atomic and Molecular Physics Lab, Department of Applied Physics, Indian Institute of Technology (Indian school of Mines), Dhanbad-826004, Jharkhand, India
Abstract This work aims to probe isomeric effect in case of positron scattering. In this pursuit, we have chosen pentane (C5 H12 ) as the target molecule. The structural isomers of pentane offers a perfect prototype to investigate isomeric effect and thus may give further knowledge into the role of the geometrical arrangements of atoms in a target molecule during the positron scattering process. The grand total, integral elastic and differential cross section are reported using the modified spherical complex optical potential (mSCOP) method. The total cross sections do not exhibit isomeric effect, which is in accordance with the experimental observation. Keywords: , Positron scattering, Isomeric effect, Total cross section, Integral elastic cross section, Differential cross section
1. Introduction The discovery of isomerism dates back to 1827, when Griedrich Woehler found that some compounds have the same molecular formula but different arrangement of the atoms in the molecules. Such group of molecules were named isomers. The different geometrical arrangement of atoms in such group of molecules infuse interest to researchers to explore the structural effect in various physio-chemical processes. We are presently interested in the scattering studies of positrons from pentane molecules. Consequently, such studies would help to gain insight on the very basic underlying physics of scattering processes. Email addresses: [email protected] (Nidhi Sinha), [email protected] (Bobby Antony) Preprint submitted to CPL
October 8, 2018
In case of electron scattering there are several work available[1, 2, 3, 4, 5, 6, 7] which primarily emphasise on the isomeric effect. These studies have established the low energy isomeric effect in electron interactions. However for positron scattering, limited studies[8, 9, 10, 11, 12, 13, 14] were found on the investigation of isomeric effect. No isomeric effect for the total cross sections was suggested from the previous literature, however differential cross sections do show some difference at higher energies[10]. However, such conclusion on DCS needs experimental validation. The pentane isomers, n-pentane, iso-pentane and neo-pentane are chosen as the targets for the present work. As can be seen from fig. 1, these isomers offers an ideal epitome to probe the role of geometrical arrangements in the scattering process. This owes to their structural difference — n-pentane being a long chain structured molecule; iso and neo-pentane being the branched one. The close values of molecular properties of these isomers (see table 1) further add to the research interest of these targets. Thus, these molecules differ mostly due to their structural arrangements; other target parameters being almost equal. Hence, the chosen isomers will aptly fulfil our aim to investigate the role of geometry in case of positron scattering.
Figure 1: Molecular structure of pentane isomers
The only available previous study for the present set of targets is due to Chiari et al.[12]. The authors have reported the measured cross sections along with their theoretical results. Their experimental total cross sections are reported from 0.1 to 50 eV for n- and iso-pentane. Further, they have used the IAM-SCAR method to theoretically estimate the various cross sections for
2
Table 1: Target properties [12]
Target
IP (eV)
Bond length (˚ A) C-H
C-C
˚3 ) Polarizability (A
∆p (eV)
n-pentane
10.35
1.118
1.531
9.98
3.55
iso-pentane
10.32
1.096
1.547
10.11
3.52
neo-pentane
10.35
1.114
1.537
10.19
3.55
these three isomers for an exhaustive energy range of 1 to 1000 eV. In their experimental work, no isomeric effect was observed. However, theoretical cross sections show slight difference in the low energy region. This is obviously due to the approximations used in the theory which dominate the formalism in the low energy region. In addition to these comparisons, total cross sections of Kimura et al. [15] is also available for n-pentane. The two experimental cross sections for this molecule is found to diverge significantly in the low energies (up to 6 eV), both in terms of magnitude and shape. This behaviour is attributed to the forward angle scattering effect according to the authors[12]. Moreover, the previous work is devoid of any isomeric effect in the positron scattering. The present work focuses on the computation of differential, integral elastic and grand total cross section for the pentane isomers. Spherical complex optical potential (SCOP) formalism [16, 17, 18] is utilised for these calculations. Interestingly, this method was developed for electron scattering. However, our group have modified the formalism to adequately solve the positron scattering problem [19, 18, 20, 21]. This is permissible since positrons are the anti-particle of electron and hence with proper modifications the theory can be applied to positron scattering. The cross sections are calculated for a comprehensive energy range of 1eV to 5keV. This attempt is the first of its kind to calculate positron scattering cross sections over such a wide range of energy for the present target molecules. The DCS are reported for energies 1, 5, 10, 15, 20, 25, 30 and 50 eV. The following section gives the brief description of the adopted theory.
2. Theoretical Methodology We have utilised modified spherical complex optical potential (mSCOP) formalism[19, 20, 18] in our previous articles exploring positron scattering systems and were able to obtain consistent
3
(a) Normal pentane
(b) Iso pentane
(c) Neo pentane
Figure 2: Electrostatic potential surface. Red circles show the identified scattering centres.
results for atoms[19], diatomic molecules[22] and complex molecules[20, 21]. In the present endeavor the same approach[20], as that of complex molecules is employed. This owes to the structure of the pentane isomers. Further, due to the large size of the molecules a multi-centre spherical complex optical potential[20, 23, 21] is employed. The cross sections for various scattering groups are calculated independently and finally summed up to get the cross sections for the target as a whole. These scattering centres are identified on the basis of bond length and atomic radii of the atoms present in the target molecule. In the present work we have used Avogadro software to identify these groups. Molecules are build and optimised and electrostatic potential surface (EPS) is then generated using the software. The scattering centres are visually selected based on the physical appearance of the EPS. EPS approach blended with mSCOP method is an efficient way to tackle the overestimation of the cross sections caused due to the approximations used in the theory and hence can be used as an alternative to screening corrections. It is noteworthy that the SCOP formalism is basically a framework which relies on the choice of potentials used to explore the scattering system. In case of electron scattering the formalism was unable to efficiently model the low energy interaction. However, for positron scattering, interestingly we found results which were in good accordance with the experimental ones even at the lowest impact scattering[19, 21, 20, 22]. Thus with the choice of proper potentials and thresholds our group was able to overcome the limitation of the present formalism in the low energy region. We have modelled the total elastic (Qel ) and total inelastic cross sections (Qinel ) using the SCOP method. In this method, the eigen value problem is solved using the partial-wave analysis method. Hence, the total positron scattering cross section Qtot is given as,
4
Qtot (Ei ) = Qel (Ei ) + Qinel (Ei )
(1)
The present calculation rests on the formulation of complex scattering potentials. The spherically averaged charge densities of the target is employed to construct this potential Vopt , given by
Vopt (r, Ei ) = VR (r, Ei ) + i Vabs (r, Ei )
(2)
The real part is the sum of static (Vst ) and polarization (Vpol ) terms, mathematically expressed as
VR (r, Ei ) = Vst (r) + Vpol (r, Ei )
(3)
We have used the Hartree-Fock wave functions of Cox and Bonham[24] to derive the analytical form of the static potential. For the polarization potential, the parameter-free model of the correlation polarization potential given by Zhang et al.[25] is used. It contains multipole nonadiabatic rectifications in the intermediate energy range and smoothly approaches the correct asymptotic form for large r. These two potentials drive the elastic part of the scattering system. However, the loss of flux due to all possible electronic inelastic processes is treated by the absorption part of the complex potential. The parameter free form proposed by Reid and Wadehra[26] is incorporated in the present work to account for the loss of incident flux.. The most intriguing feature of positron scattering when compared to other lepton interactions is the positronium formation process. Theoretically it is difficult to include this channel in the scattering process, since we cannot define it in terms of binary collisions. Now the principal factor determining the absorption potential is the choice of absorption threshold ∆. There are different variants of ∆ available in the literature which primarily ensures the energy variation of the thershold from positronium formation, ∆p to the first electronic excitation energy, ∆e . Such variation allows the inclusion of positronium formation channel in the computational codes. In the present work we have adopted the form proposed by Chiari et al.[27] given as, ∆(E) = ∆e − (∆e − ∆p ) exp
5
−(E − ∆p ) Em
(4)
Em is the energy at which the inelastic cross section attains its maximum value without considering Ps formation. These three potentials are combined to form the final optical potential, which is then used to solve the Schr¨ odinger equation using numerical method[17]. The solution are obtained in the form of complex phase shifts (δl ) carrying the signature of the scattering channel. The obtained phase shifts are employed to determine the inelasticity or the absorption factor at various energies. ηl = exp(−2Im δl )
(5)
Using this inelasticity factor, the total elastic and total inelastic cross sections are calculated through[28], ∞ π X Qel (Ei ) = 2 (2l + 1) |ηl exp(2iRe δl ) − 1|2 k
(6)
∞ π X Qinel (Ei ) = 2 (2l + 1)(1 − ηl2 ) k
(7)
l=0
l=0
The total cross section for the positron-molecule scattering is computed as the sum of these two cross sections. In the present calculation the nonspherical terms namely vibrational and rotational potentials are excluded in the full expansion of the optical potential since contributions from these terms are negligible at the intermediate and high energies.
3. Results and discussion The present work offers a comprehensive investigation of positron impact differential, integral elastic and total cross sections for pentane isomers over a wide energy range. Figures 3-7 give the pictorial representation of the cross sections along with available comparisons. The total cross sections are plotted in figs. 3 and 4 and the elastic cross sections are displayed in fig. 5. These cross sections are compared with the experimental and theoretical results of Chiari et al.[12]. Kimura et al.[15] have measured the total cross section for n-pentane and are included in our plot. In figs. 6 and 7 we have done a comparative study of the total and integral elastic and differential cross sections for the three isomers respectively. Such comparative study will help us to get a conclusive remark on the isomeric effect in positron scattering.
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Figure 3: Total cross section for positron-n-pentane scattering. solid line: Present results, dashed
line: Theoretical results of Chiari et al.[12], solid circles: Experimental results of Chiari et al.[12] and solid triangles: Experimental measurements of Kimura et al.[15]
(a) Iso pentane
(b) Neo pentane
Figure 4: Total cross section for positron-iso and neopentane scattering. solid line: Present results, dashed line: Theoretical results of Chiari et al.[12] and solid circles: Experimental results of Chiari et al.[12] for neo pentane
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3.1. Total cross section In fig.2, the total cross section (Qtot ) for the positron impact scattering of n-pentane along with the available comparisons is graphically presented. The present result shows a similar variation with increasing energy as that of the theoretical cross sections of Chiari et al.[12]. Moreover, the two cross sections show fair agreement in the intermediate to high energy region. However, in the energy range of 5 to 30 eV, the present cross sections are observed to be significantly lower than their theoretical cross sections. This owes to the IAM-SCAR method adopted by the authors which excludes any molecular property in the calculation and the cross sections are merely the incoherent sum of the cross sections for individual atoms. It is well known that the effect of such a choice exists only upto 25-30 eV and above that the two cross sections should show good agreement with each other. This is exactly depicted in the graph. Moreover, Chiari et al.[11] have stated in the article that the dependability of the IAM-SCAR method can be extended upto 30eV. Albeit, as expected they show good agreement in the high energies. The present results are also compared to the experimental measurements of Chiari et al.[12] and Kimura et al.[15]. However, the two experimental results are found to diverge in the low energy region. This is because no corrections were made for the forward angle scattering effect by Kimura et al.[15]. Nonetheless, the present data are found to be in good accordance with the experimental results of Chiari et al.[12] in the low energies. Though the difference in the magnitudes is attributed to the exclusion of the rotational and vibrational excitation channel in the present formalism. In the intermediate energy regime the present cross sections significantly overestimate their measurements. Moreover, at energies above 100 eV, the available experimental and theoretical cross sections overlap with the present results. The total cross sections for positron-isopentane scattering is shown in fig. 3.1. As in the previous case, the present plot shows a valley structure symbolizing the point just before opening of the inelastic channel above which magnitude increases. However, when the impact energy is very high, the interaction time decreases significantly resulting in low cross sections. Such variation is well demonstrated in the present plot. The present result is compared with the theoretical and experimental data of Chiari et al.[12]. The deviations of their cross sections with the present one is similar to that in fig.3 and hence the discussion remains the same. The present total cross sections for neo pentane along with the only available theoretical esti-
8
Figure 5: Elastic cross section for positron-n-pentane scattering. Inset: integral elastic cross section
for iso and neo pentane. Solid line: present results and dashed line: theoretical results of Chiari et al.[12] mation of Chiari et al.[12] is depicted in fig. 3.1. The two cross sections vary in a similar fashion; however magnitude being higher for the present data in the intermediate energy region. Again in the low energy region, owing to the IAM-SCAR method of Chiari et al. [12], the two theoretical cross sections differ significantly in magnitude. Further, the lack of available literature for the present set of targets demand more scientific attempts to be made.
3.2. Elastic cross section The total elastic cross sections for pentane isomers are shown in fig. 5. The only available previous cross section is due to Chiari et al.[12]. The three isomers show similar behaviour compared to the theoretical results of Chiari et al.. The present cross sections also varies as that of Chiari et al.[12] albeit with lower magnitudes in the low energies. However, above 10 eV, present results underestimate their cross sections. The approximations while deriving the potentials in their respective work and the MSCOP approach used in the present calculations might be the reason for such deviations. Further, Chairi et al.[12] have adopted the IAM-SCAR method which itself
9
(a) Total cross section
(b) Elastic cross section
Figure 6: Comparative study of integral cross sections of pentane isomers (log-log scale). Inset: linear scale. Solid line: n-pentane, dashed line: iso-pentane, dash dot dotted line: neo-pentane
has limitations in the low and intermediate energy region. Thus the cumulative effect of such limitations in the two theoretical models might have resulted in the observed discrepancy.
3.3. Isomeric effect 3.3.1. Integral cross sections Fig. 6a compares the total scattering cross sections for the three isomers of pentane. The qualitative shape of the cross sections is same for the molecules where a dip can be found at around Ps formation threshold. Above this energy the cross sections increases due to the contribution from various inelastic channels. Thus a broad peak can be seen at around 10 to 50 eV. However, as the energy increases the interaction time of the projectile and target decreases and hence the cross sections fall rapidly at higher energies. Further, comparing the cross sections, we found that they merge well with each other over the entire comparative energy range. The physio- chemical parameters of the three targets are comparable and the only difference lies in their geomterical arrangement. Thus we conclude that there is no part of geometry in case of positron scattering even at the lowest comparative energies. This observation is corroborated by the experiment of Chiari et al.[12]. The authors have measured the TCS for normal and isopentane and found the magnitude to be comparable. Further, their theoretical model also manifests a similar observation. In fig. 6b, we present the comparative plot of the elastic cross sections for the present isomers.
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Akin to the previous case, we found no isomeric effect in the elastic cross sections. Nonetheless no experimental data is available and hence is highly reccommended to manifest the present observation.
3.3.2. Differential cross sections The differential cross sections (DCS) for elastic scattering in presence of inelastic channels are shown in figure 7. Since the present targets are polar in nature, the DCS becomes high at lower scattering angle. Moreover, polarisation effects are strongest at short range and gradually becomes weak at intermediate and long range. Hence the present DCS exhibit lower values at larger angles. Another interesting characteristic of the DCS curve is the oscillatory behaviour which arises due to the coupling of higher partial waves of the heavier C atoms in the scattering process. Comparing the DCS of the isomers, one can easily find that the cross sections overlap except at the minima for most of the energies. However, as mentioned by Chiari et al. [12] this could be due to the spherical-like symmetry of neo pentane and its manifestation of isomeric effect is still doubtful. Besides, considering the log scale, difference in the magnitudes are negligible and hence can be concluded that no isomeric effect was observed even at the DCS level.
4. Conclusions The present article reports the grand total and integral elastic cross sections for pentane isomers in the energy range of 1eV to 5keV. mSCOP formalism[19, 20, 18, 21] along with the multi-center approach[23, 20, 21] were utilised for the computation of the cross sections. The group of molecules studied here show a consistent behaviour when compared to the previous results. A good agreement was observed between the cross sections reported here and the theoretical data of Chiari et al.[12]. Moreover, in the energy span of 10 to 50 eV, significant deviations were observed. This owes to the various approximations used in these two theoretical methods. Besides, for the rest part of the comparative energy range an excellent corroboration can be seen. The inconsistency of the experimental measurements and the theoretical predictions hold in this work too. However, above 100 eV, an excellent accordance can be found between the two as seen in fig 3. The prime motivation behind the present work was to explore the isomeric effect in the positron scattering systems. Previous studies[12] show that experimentally no such effect was pronounced, but the theoretical calculations differ in magnitude for the three isomers. However, the authors did
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not considered any role of geometry for such behaviour and attributed it to theoretical limitations. In the present work we obtained a similar result and the present set of targets, namely pentane isomers didn’t show any appreciable isomeric effect in the integral cross sections reflecting the absence of any contribution of geometry in positron scattering. Further, the determination of DCS provides a stringent test for any scattering theory, as it is sensitive to the effects which are averaged over in case of integral cross section. To a close approximation, even for the DCS we found no role of geometry in positron scattering. Experiments are much needed to validate our observations.
Acknowledgment BA is pleased to acknowledge the support of this research by Department of Science and Technology, Govt of India through SERB project Grant No. EMR/2016/005035. [1] C. Szmytkowski, S. Kwitnewski, Electron scattering on C3 H6 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 35 (11) (2002) 2613–2623. [2] C. Winstead, Q. Sun, V. McKoy, Low-energy electron scattering by C3 H6 isomers, The Journal of Chemical Physics 96 (6) (1992) 4246–4251. [3] F. Kossoski, T. Freitas, M. Bettega, Resonances in electron collisions with C2 H2 Cl2 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 44 (24) (2011) 245201. [4] F. Kossoski, M. Bettega, Low-energy electron scattering from the aza-derivatives of pyrrole, furan, and thiophene, The Journal of Chemical Physics 138 (23) (2013) 234311. [5] Y. Nakano, M. Hoshino, M. Kitajima, H. Tanaka, M. Kimura, Low-energy electron scattering from C3 H4 isomers: Differential cross sections for elastic scattering and vibrational excitation, Physical Review A 66 (3) (2002) 032714. [6] C. Szmytkowski, S. Kwitnewski, Isomer effects on the total cross section for electron scattering from C4 F6 molecules, Journal of Physics B: Atomic, Molecular and Optical Physics 36 (24) (2003) 4865–4873. [7] C. Makochekanwa, H. Kawate, O. Sueoka, M. Kimura, M. Kitajima, M. Hoshino, H. Tanaka, Total and elastic cross-sections of electron and positron scattering from C3 H4 molecules (allene and propyne), Chemical Physics Letters 368 (1-2) (2003) 82–86. [8] K. Floeder, D. Fromme, W. Raith, A. Schwab, G. Sinapius, Total cross section measurements for positron and electron scattering on hydrocarbons between 5 and 400 eV, Journal of Physics B: Atomic and Molecular Physics 18 (16) (1985) 3347–3359. [9] O. Sueoka, S. Mori, Total cross sections for low and intermediate energy positrons and electrons colliding with CH 4 , C2 H4 and C3 H6 molecules, Journal of Physics B: Atomic and Molecular Physics 19 (23) (1986) 4035–4050.
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[10] F. B. Nunes, M. H. Bettega, S. d. Sanchez, Positron collisions with C3 H6 isomers, Journal of Physics B: Atomic, Molecular and Optical Physics 48 (16) (2015) 165201. [11] L. Chiari, A. Zecca, S. Girardi, A. Defant, F. Wang, X. Ma, M. Perkins, M. Brunger, Positron scattering from chiral enantiomers, Physical Review A 85 (5) (2012) 052711. [12] L. Chiari, A. Zecca, F. Blanco, G. Garc´ıa, M. Brunger, Experimental and theoretical cross sections for positron scattering from the pentane isomers, The Journal of Chemical Physics 144 (8) (2016) 084301. [13] O. Sueoka, C. Makochekanwa, H. Tanino, M. Kimura, Total cross-section measurements for positrons and electrons colliding with alkane molecules: Normal hexane and cyclohexane, Physical Review A 72 (4) (2005) 042705. [14] C. Makochekanwa, M. Hoshino, H. Kato, O. Sueoka, M. Kimura, H. Tanaka, Electron and positron scattering cross sections for propene and cyclopropane, Physical Review A 77 (4) (2008) 042717. [15] M. Kimura, O. Sueoka, A. Hamada, Y. Itikawa, Vol. 111, John Wiley & Sons, Inc. [16] K. L. Baluja, A. Jain, Positron scattering from rare gases (He, N e, Ar, Kr, Xe, andRn): Total cross sections at intermediate and high energies, Phys. Rev. A 46 (1992) 1279–1290. [17] K. N. Joshipura, M. Vinodkumar, C. G. Limbachiya, B. K. Antony, Calculated total cross sections of electron-impact ionization and excitations in tetrahedral (XY 4 ) and SF 6 molecules, Phys. Rev. A 69 (2004) 022705. [18] S. Singh, S. Dutta, R. Naghma, B. Antony, Theoretical formalism to estimate the positron scattering cross section, The Journal of Physical Chemistry A 120 (28) (2016) 5685–5692. [19] N. Sinha, S. Singh, B. Antony, Positron total scattering cross-sections for alkali atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 51 (1) (2017) 015204. [20] N. Sinha, P. Modak, S. Singh, B. Antony, Positron scattering from methyl halides, The Journal of Physical Chemistry A 122 (9) (2018) 2513–2522. [21] N. Sinha, B. Antony, Electron and positron interaction with pyrimidine: A theoretical investigation, Journal of Applied Physics 123 (12) (2018) 124906. [22] S. Singh, B. Antony, Positronium formation and ionization of atoms and diatomic molecules by positron impact, EPL (Europhysics Letters) 119 (5) (2017) 50006. [23] S. Singh, R. Naghma, J. Kaur, B. Antony, Calculation of total and ionization cross sections for electron scattering by primary benzene compounds, The Journal of Chemical Physics 145 (3) (2016) 034309. [24] H. Cox Jr, R. Bonham, Elastic electron scattering amplitudes for neutral atoms calculated using the partial wave method at 10, 40, 70, and 100 kV for Z= 1 to Z= 54, The Journal of Chemical Physics 47 (8) (1967) 2599–2608. [25] X. Zhang, J. Sun, Y. Liu, A new approach to the correlation polarization potential-low-energy electron elastic scattering by He atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 25 (8)
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(1992) 1893–1897. [26] D. D. Reid, J. Wadehra, A quasifree model for the absorption effects in positron scattering by atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 29 (4) (1996) L127–L133. [27] L. Chiari, A. Zecca, S. Girardi, E. Trainotti, G. Garca, F. Blanco, R. P. McEachran, M. J. Brunger, Positron scattering from O2 , Journal of Physics B: Atomic, Molecular and Optical Physics 45 (21) (2012) 215206. [28] C. Joachain, Quantum collision theory (Amsterdam : N orth Holland) (1983).
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Figure 7: Differential cross section for pentane isomers
*Highlights (for review)
Investigation of isomeric effect in positron impact scattering. Cross sections reported for a wide energy range of 1 to 5000 eV. Inclusion of molecular structure in the computations. Efficient method to include positronium formation in the scattering channel Fair agreement with the limited experimental data. Consistent results obtained in terms of shape and magnitude with previous study.
Graphical Abstract