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Screening corrections for the interference contributions to the electron and positron scattering cross sections from polyatomic molecules
Chemical Physics Letters 645 (2016) 71–75
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Screening corrections for the interference contributions to the electron and positron scattering cross sections from polyatomic molecules Francisco Blanco a , Lilian Ellis-Gibbings b , Gustavo García b,∗ a b
Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 113-bis, 28006 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 16 September 2015 In final form 29 November 2015 Available online 12 December 2015
a b s t r a c t An improvement of the screening-corrected Additivity Rule (SCAR) is proposed for calculating electron and positron scattering cross sections from polyatomic molecules within the independent atom model (IAM), following the analysis of numerical solutions to the three-dimensional Lippmann–Schwinger equation for multicenter potentials. Interference contributions affect all the considered energy range (1–300 eV); the lower energies where the atomic screening is most effective and higher energies, where interatomic distances are large compared to total cross sections and electron wavelengths. This correction to the interference terms provides a significant improvement for both total and differential elastic cross sections at these energies. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Electron and positron (e−/e+) scattering processes from atoms and molecules have been a subject of interest during the last few decades. Recently this interest has considerably increased due to the relevance of these processes in radiation interaction models for biomedical applications [1]. Scattering experiments are complicated in general and difficulties determining absolute values as well as uncertainties connected with energy and angular resolution limitations require some complementary data from theory. In this context sufficiently accurate general calculation procedures, for a wide variety of targets, over a broad energy range, are extremely useful. The lack of spherical symmetry makes ab initio calculations for electron and positron scattering cross sections by molecules almost unfeasible at intermediate and high energies, and therefore available theoretical methods rely on different approximate treatments. One of these techniques, the Independent Atom Model (IAM) [2–8], is particularly successful. The IAM treatment assumes that the molecules can be approximately substituted by their constituent atoms in their corresponding positions, which independently scatter incident electrons or positrons. One of the greatest advantages of this approach is the possibility of obtaining reliable
∗ Corresponding author. E-mail addresses: pacobr@fis.ucm.es (F. Blanco), [email protected] (G. García). http://dx.doi.org/10.1016/j.cplett.2015.11.056 0009-2614/© 2015 Elsevier B.V. All rights reserved.
results for a large number of molecular species from the data of a reduced number of atoms. With this method, total elastic, inelastic and differential elastic cross sections can be easily obtained with reasonable accuracy within its energy range of applicability, typically above 100 eV. An important limitation of the IAM treatment is that it ignores any multiple scattering of the projectile within the molecule, hence it’s application only for relatively high incident energies (>100 eV). Some years ago, we proposed an approximated method to partially account for these effects, the screening corrected additivity rule (SCAR) [9,10], which extended the applicability of the IAM method down to lower energies, typically 20–30 eV. In a recent letter [11] the IAM treatment for elastic scattering has been revisited, indicating the relevance of interference contributions arising from all the scattering centres in the molecule. These contributions were particularly important at small scattering angles, where experimental systems are unable to distinguish them. Nevertheless they contribute significantly to the integrated cross sections, and therefore it is crucial to use interference corrected values for Monte Carlo simulations [12] and for experimental data normalisation procedures [13]. However, as the AR procedure fails for energies below 100 eV by noticeably overestimating the cross section for decreasing energies, the effect of the interference terms in this range is not appreciable. In these conditions the SCAR procedure constitutes an excellent tool to evaluate the magnitude and consequences of interference terms at intermediate energies, below 100 eV.
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F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
The main objective of this study is to revise the SCAR procedure in order to include these interference contributions. For this purpose, considering only elastic scattering for representative molecular targets will be enough, and therefore, following the scheme of our previous letter [1], we will solve a three-dimensional Lippmann–Schwinger equation for some multicentre potentials. Although we have compared these results for different molecular configurations based on C and H atoms for different interatomic distances, we will present here only data for the H2 and CH4 at their equilibrium geometries as they suffice to illustrate all the relevant results. 2. Review of interference and screening corrections Assuming the IAM approximation, the molecular cross sections can be derived from the well known approximate expression [2] for multicenter dispersion F() ≈
fi ()eiq·ri
(1)
molecule, the IAM approximation fails since the atoms can no longer be considered as independent scattering centres and multiple scattering within the molecule is not negligible. Approximate methods revealed that important corrections are needed by the IAM method for these energies [8,9,19,21,24,25] and it has been shown [9] that the energy range for which these corrections are relevant depends on the size of the molecule: around 10% for N2 and CO up to 200 eV, for CO2 up to 300 eV, and for benzene up to 600 eV. Representative molecular cross-section calculations are based on a corrected form of the IAM treatment known as the SCAR (Screening Corrected Additivity Rule) procedure, which approximately accounts for these multiple scattering effects. All the details for this procedure have been extensively described elsewhere [9,10,26] and therefore they are only briefly mentioned here. Basically, within the SCAR procedure the integral molecular cross sections (both elastic and inelastic), ignoring interference contributions, are expressed as: total = molecule
where q = kf − ki is the momentum transfer, ri are the atomic positions and fi () are the atomic scattering amplitudes. As this corresponds to the independent scattering from each atom, the approximation is expected to be valid only for large interatomic distances compared to the wavelength associated to the incident projectile. From the above molecular dispersion function, the differential elastic cross section is easily obtained by averaging its modulus squared |F()|2 over all the molecule orientations’ [2,6], obtaining:
d˝
=
sin qrij ∗ fi f j
qrij
=
2 fi
i,j
+
i
sin qrij ∗ fi f j
qrij
=
i= / j
d elastic atom i
d˝
+
d interference d˝
(2)
i
where q ≡ |q| = 2k sin ⁄2, rij is the distance between i and j atoms, sin qrij /qrij = 1 when qrij = 0, and d interference /d˝ represents the ˙ i =/ j interference contribution to the molecular differential cross section. Hereafter, we will consider only elastic processes, which are enough for our purposes, and hence a real scattering potential for each atom (no imaginary inelastic part) will be used to obtain the atomic scattering amplitudes fi (). In these conditions the total cross section will be coincident with the corresponding integral elastic cross section. By integrating Eq. (2) the total molecular cross section can be written as: total = molecule
total si atom i
(4)
atoms
atoms
elastic dmolecule
total atom + interference i
where the si screening coefficients reduce the contribution of each atom to the total molecular cross section (0 ≤ si ≤ 1). The calculation of these coefficients requires some simple closed expressions [9,26] based only on data about position and the total cross section total of each atom in the molecule. This procedure is applicable atom i to any arbitrary molecular geometry and size [27]. Note that only elastic processes are considered in this study and therefore the total cross section and the integral elastic cross section are coincident. As far as the differential elastic cross sections are concerned, the SCAR procedure distinguishes two contributions, one from the direct scattering cross section ( D ) which is related to the angular distribution given by the single scattering differential cross section, elastic and the re-dispersed cross section (molecule − D ) which corresponds to an assumed isotropic angular distribution. The latter contribution approximately accounts for re-dispersion processes inside the molecule. After estimating their relative weight by means of the si screening coefficients and the aforementioned angular distributions of the atomic elastic cross sections[9] the resulting expression is elastic dmolecule
d˝
∼ = (1 − XS )
elastic molecule − D
D =
+ 1 + XS
elastic molecule
D
−1
dD d˝
(5)
total si2 atom i
(6)
atoms
d elastic sin qrij dD = si2 atom i + si sj fi ()fj∗ () qrij d˝ d˝
(3)
where interference represents the integration of the above differential interference contribution. It must be noted that this contribution would not be present in (3) if this expression were directly obtained from (1) by applying the optical theorem. This latest procedure is known as the ‘Additivity Rule’ (AR) and is widely described in literature [5–7,9,14–23]. As discussed in [11], the appropriate expression is Eq. (3), and the discrepancy relies on the approximate nature of Eq. (1) which doesn’t fulfil the optical theorem. It has also been shown that interference terms are only relevant at small angles and their integrated contribution ( interference ) is non negligible even at high energies [11]. At intermediate energies (10–100 eV), where atomic crosssections are not small compared to interatomic distances in the
where D , XS and d D /d are defined by
i
atoms
4
(7)
i= / j
45◦ XS =
dD sin d d˝ 0 ◦ 180 dD sin d d˝ 0
(8)
As seen from Eq. (7), the direct contribution of the scattering is a screening version of Eq. (2). The first summation operator in (7) accounts for each atomic contribution, reduced by a si factor, whereas the second one represents the reduced interference terms. We should note that applying the AR, i.e. ignoring the integral interference contributions but including them in the differential cross section values, Eqs. (6) and (7) can be in conflict. In order to avoid this contradiction we introduced an additional reducing factor () applied to the positive values of d interference /d˝ in order to ensure that d˝(d interference /d˝) = 0. This additional condition was called the ‘normalised’ SCAR treatment. Most of the recent
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
calculations based on the SCAR method included this normalisation procedure, so we will call it here as SCAR. However, we have recently shown [11] that in order to properly account for multicentre scattering effects the integral interference contribution interference should be included in the calculation of the total cross section and therefore the AR procedure needs to be revised. This method of including interference effects into the AR procedure will be referred here as AR+I. Accordingly, the SCAR method must be reformulated in order to include these terms. This will be the purpose of next section.
3. Screening corrections to the interference contributions Previous publications on the effect of interference terms in multiple-scattering calculations are scarce. From the theoretical point of view, a complete treatment of the multiple-scattering problem is considerably difficult and hence most of the previous calculations are either limited to first order corrections, in double or triple scattering configurations [9,10,23], or they only consider small systems [24]. They generally conclude that multiple scattering effects tend to reduce both the integrated interference and the small angle differential d interference /d˝ values. However, due to their limitations and restrictions it is not possible to draw a general conclusion from them. Following the scheme we proposed in [11] for the AR method, we will compare the results of the SCAR procedure with accurate numerical solutions of the three-dimensional Lippmann Schwinger equation for some multicenter potentials. Previous applications of multiple-scattering approaches are also cited in [11]. As detailed there, the procedure we used is based on the method proposed by Polasek et al. [28] to numerically solve this equation in a discrete momentum space. In this representation the atomic model potentials do not present singularities and so discretization procedures become more efficient. Once the momen tum space is discretized by means of some appropriate mesh ki , the integration of the Lippmann–Schwinger equation turns into a system of linear equations on the k ’ i |T|kj unknown values. The numerical solution of that system provides the discretized T matrix elements, and then the total differential and integral cross sections by means of the standard relations [29] between the cross sections and the T operator. As proposed in [11], single atoms are represented for simplicity here by the Yukawa potential, V = e−r/R /r, which allows the use of simple analytical expressions for the corresponding V matrix elements. We also use the fitting of Yukawa expressions already described in [11] for H and C static atomic potentials. In these conditions we have calculated electron and positron scattering cross sections from several geometries. Here we will only present some representative results for H2 and CH4 in the energy range 1–300 eV. Analysing the general trend of interference contributions for a multicenter potential can be problematic, especially at low incident energies. Nonetheless, a systematic analysis for several geometries, interatomic distances and incident energies reveals a relatively expected regularity: the interference contributions to the AR calculated in [11] apply well when the interatomic distances are high in comparison with the geometric cross section radius and with the wavelength of the projectile. Otherwise, these contributions need to be reduced or even neglected. In order to quantify this criterion, for each i-j atom pair we compared their separation rij with a length dimensional parameter
defined as ij = max( i /, j /, 1/k), where the i and j are the
total atomic cross sections, respectively. It must be noted / corresponds to the radius of a circle of area , so that the condition rij = max(
i /,
j /) represents a situation of
73
geometrical overlap between two disks for which the centre of the smallest one approaches to the border of the other. According to this, for any pair of atoms fulfilling the simple rij > ij condition we can ensure that krij > 1 and the geometrical overlapping of their atomic cross sections is moderated. We can then propose a simple procedure by introducing a ij factor which smoothly attenuates the interference terms according to the rij > ij or rij < ij values. For practical uses we can define ij = rij 2 /(rij 2 + ij 2 ) which clearly fits the anticipated function. We will refer to this procedure to account for screened interferences as the SCAR+I treatment. Basically it differs only from the previous SCAR by the replacement of expressions (4) and (6) by the following ones: total molecule =
D =
atoms
atoms
total si atom + interference i
total si 2 atom + interference i
(10)
where interference is now given by:
interference =
(9)
⎛
⎝
vij si sj fi ()fj∗ ()
i= / j
⎞ sin qrij qrij
⎠ d˝,
(11)
being si the screening coefficients introduced in Eq. (4). This is a simple procedure but, as it will be shown in the next sections by comparison with accurate numerical calculation, it provides a significant improvement of the differential and integral scattering cross sections at intermediate energies, while maintaining the high energy efficiency of the IAM method. 4. Results and discussion Following the same procedure of [11], the proposed treatment has been checked by comparing results from different approximations for several molecular geometries, interatomic distances and number of atoms. For conciseness we will present here only results for H2 and CH4 species, as it is sufficient to show all the relevant results. As already mentioned, interference contributions affect to the angular distribution of the differential cross sections, particularly for small angles, and to the absolute value of the integral cross sections. To illustrate these contributions we have compared data derived from the AR, AR+I, SCAR and SCAR+I approximations with the ‘exact’ numerical multiple scattering results described in the previous section. As far as differential cross sections (DCS) are concerned, Figure 1 shows a representative example for positron scattering from H2 at 30 eV. Calculated values from all the above methods are plotted together. The AR and SCAR calculations clearly underestimate the multicentre scattering data for small angles. By introducing the interference effects to the AR calculation, the AR+I results show better agreement with the reference data but tend to overestimate them. Finally, the screening correction to the latest (SCAR+I) substantially improves the agreement for the smaller angles. As can be seen in Figure 1, maximum discrepancies between the different approaches correspond to zero scattering angles. For this reason, to find systematic trends for different incident energies, we will focus the DCS comparison on scattering in the forward direction ( = 0). Figure 2a shows the zero angle differential cross section for electron and positron scattering from H2 at different incident energies. Even for this small molecule interference effects are clearly significant. The general effect of the screening corrections (SCAR) is reducing the zero angle DCS given by the AR, especially for the
74
10
p-H2 30 eV
Integral cross section (atomic units)
Differential cross section (atomic units)
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
AR AR+I SCAR SCAR+I Multicentre
1
0.1
0.01
e/p-H2
p-H2 AR p-H2 SCAR p-H2 SCAR+I p-H2 Multicentre e-H2 AR e-H2 SCAR e-H2 SCAR+I e-H2 Multicentre
100
10
1
0.1 0
40
80
120
160
1
Scattering angle (deg) Figure 1. Differential cross sections for positron scattering from H2 at 30 eV impact , aditivity rule (AR), , additivity rule including interference terms energy. , screening corrected additivity rule (SCAR), , screening corrected (AR+I), additivity rule including interference terms (SCAR+I), , multicentre scattering calculation.
e/p-H2 θ=0
100
p-H2 AR p-H2 Multicentre p-H2 SCAR p-H2 SCAR+I e-H2 AR e-H2 Multicentre e-H2 SCAR e-H2 SCAR+I
10
1 1
1000
from energies around 10 eV. Discrepancies below this energy are a consequence of the limitations of the SCAR method to describe molecular features. In particular, the peak around 8 eV showed in Figure 2b is not a real molecular resonance but a cross section peak from the carbon atom which has been transmitted to the CH4 configuration through the SCAR procedure. The situation for positrons is similar, although in this case the good agreement between the SCAR+I calculation and the multicentre data extends along almost all the energy range considered here (0.1–300 eV). Integral electron and positron scattering cross sections from H2 are shown in Figure 3. As expected the AR electron scattering calculation tends to overestimate the integral multicentre cross section values as the energy decreases. The SCAR procedure corrects the AR values for energies below 100 eV providing good agreement with the multicentre data down to energies of around 10 eV but agreeing with the AR results for energies above 100 eV. When including interference effects in the SCAR procedure, the SCAR+I data extend this agreement down to around 3 eV, also notably improving the (b)
(a)
100
Figure 3. Integral cross sections for electron and positron scattering from H2 . , electron scattering additivity rule (AR), , electron scattering screening , electron scattering screening corrected addicorrected additivity rule (SCAR), tivity rule including interference terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering additivity rule (AR), , positron scattering , positron scattering screening corscreening corrected additivity rule (SCAR), rected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation.
10
Incident energy (eV)
100
Differential cross section (atomic units)
Differential cross section (atomic units)
lower energies, while at increasing energies both results merge as expected. Comparing these results with the multicentre calculation, there is not a clear tendency for energies below 20 eV but above this value they tend to underestimate the multicentre calculation by a factor 2. However, when interference terms are included (SCAR+I), still no significant changes are appreciated below 4 eV but for higher energies these values tend to the multicentre calculation reaching a perfect agreement for energies above 30 eV. This is an important result which confirms the efficiency of the SCAR procedure to extend the applicability of the AR calculation down to around 30 eV [9,26] but also indicates that interference effects need to be included even at higher energies. A similar behaviour can be found for positrons although in this case there is a closer agreement between the different approaches at lower energies. Figure 2b shows the same parameters for CH4 molecules. Discrepancies between the AR and multicentre calculations are even bigger, around a factor of 3, remaining relevant for increasing energies. Again, the SCAR correction improves the results down to 20–30 eV but they are still much lower that the multicentre scattering data. Including the interference terms (SCAR+I) there is a reasonable agreement with the multicentre calculation
0.1
10
Incident energy (eV)
e-CH4 AR e-CH4 SCAR e-CH4 SCAR+I e-CH4 Multicen tre p-CH4 AR p-CH4 SCAR p-CH4 SCAR+I p-CH4 Multicen tre
e/p-CH4 θ=0 100
10
1 0,1
1
10
100
Incident energy (eV)
Figure 2. (a) Differential cross sections for electron and positron scattering from H2 at zero scattering angle. (b) same as (a) for CH4 . (a) and (b): , electron scattering , electron scattering screening corrected additivity rule (SCAR), , electron scattering screening corrected additivity rule including interference additivity rule (AR), , positron scattering additivity rule (AR), , positron scattering screening corrected additivity rule terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering screening corrected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation. (SCAR),
Integral cross section (atomic units)
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
1000
e/p-CH4
100
e-CH4 AR e-CH4 SCAR e-CH4 SCAR+I e-CH4 Multicentre p-CH4 AR p-CH4 SCAR p-CH4 SCAR+I p-CH4 Multicentre
10
1 1
10
100
1000
Incident energy (eV) Figure 4. Integral cross sections for electron and positron scattering from CH4 . , electron scattering additivity rule (AR), , electron scattering screening , electron scattering screening corrected addicorrected additivity rule (SCAR), tivity rule including interference terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering additivity rule (AR), , positron scattering , positron scattering screening corscreening corrected additivity rule (SCAR), rected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation.
high energy results. As may be seen in Figure 3, the reliability of the SCAR+I method is even clearer in the case of positrons where the agreement between the integral cross section data calculated with this procedure and those derived from the multicentre calculation is perfect from 1 to 300 eV. A similar comparison is illustrated in Figure 4 for electron and positron scattering from CH4 . As this is a more complicated molecule, screening effects are more important than for H2 and therefore the agreement between the different approaches is not as good as in the previous case. However, the general behaviour is similar to the previous one. The SCAR+I calculation provides the most reliable data over the whole energy range, both for electrons and positrons, showing a perfect agreement with the multicentre calculation from 10 to 300 eV. Below 10 eV some discrepancies appear, especially for positrons, but we should note that in the case of polyatomic molecules, the SCAR procedure is not expected to be reliable below about 30 eV. 5. Conclusions Different approaches based on the independent atom model have been compared with an ‘exact’ multicentre scattering calculation in order to study the effect of multiple scattering interferences in the differential and integral electron and positron scattering cross sections from molecules. Considering only elastic processes and selecting H2 and CH4 as representative molecular targets, we can conclude that the SCAR procedure represents a substantial improvement of the AR method by extending its applicability down to about 30 eV. Upon introducing interference terms to the SCAR procedure (SCAR+I), both the differential and integral calculated
75
cross section data are in significantly better agreement with the multicentre scattering calculation. For the differential cross sections, interference terms affect mainly small scattering angles and for the integral cross section data they extend over all the energy range considered here (1–300 eV). Additionally, including interference terms in the calculation of both integral and differential cross sections for molecular targets eliminates the inconsistency between the differential and integral cross section values which is inherent to the AR method (Eqs. (6) and (7), respectively). This means that no additional normalisation procedure is required by the SCAR+I approach in order to fulfil the optical theorem. For practical applications this new SCAR+I approach needs to be applied to a more realistic potential, including inelastic collision processes. This will be the subject of our further studies. We can finally conclude that once establishing the SCAR+I procedure over a realistic optical potential, a revision of the available electron-molecule and positron-scattering databases is required, as most of the available data were derived ignoring the interference effects, due either to avoiding theoretical complications or to limitations on the experimental detection methods. Acknowledgements This study has been partially supported by the Spanish Ministerio de Economía y Competitividad (project FIS2012-31230) and the EU COST programme (Action CM1301). L.E.-G. is supported by the EU FP7-PEOPLE-2013-ITN programme (ARGENT project). References [1] Radiation Damage in Biomolecular Systems, Springer, London, 2012. [2] N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions. Nucl. Phys., 81, Oxford University Press, 1966. [3] H.S. Massey, Electronic and Ionic Impact Phenomena, 1969. [4] S.P. Khare, D. Raj, P. Sinha, J. Phys. B Atom. Mol. Opt. Phys. 27 (1994) 2569. [5] Y. Jiang, J. Sun, L. Wan, Phys. Rev. A 52 (1995) 398. [6] Y. Jiang, J. Sun, L. Wan, Phys. Lett. A 231 (1997) 231. [7] Y. Jiang, J. Sun, L. Wan, J. Phys. B Atom. Mol. Opt. Phys. 30 (1997) 5025. [8] D. Raj, J. Phys. B Atom. Mol. Opt. Phys. 24 (1991) L431. [9] F. Blanco, G. Garcı´ıa, Phys. Lett. A 317 (2003) 458. [10] F. Blanco, G. García, Phys. Rev. A 67 (2003) 022701. [11] F. Blanco, G. García, Chem. Phys. Lett. 635 (2015) 321. [12] F. Blanco, et al., Eur. Phys. J. D 67 (2013) 199. ´ F. Blanco, G. García, B.P. Marinkovic, ´ A.R. Milosavljevic, ´ Phys. Rev. [13] J.B. Maljkovic, A 85 (2012) 042723. [14] A. Zecca, et al., J. Phys. B Atom. Mol. Opt. Phys. 43 (2010) 215204. [15] F. Blanco, J. Rosado, A. Illana, G. García, Phys. Lett. A 374 (2010) 4420. [16] A. Zecca, R. Melissa, R.S. Brusa, G.P. Karwasz, Phys. Lett. A 257 (1999) 75. [17] B.R. Miller, L.S. Bartell, J. Chem. Phys. 72 (1980) 800. [18] S. Hayashi, K. Kuchitsu, Chem. Phys. Lett. 41 (1976) 575. [19] D. Raj, S. Tomar, J. Phys. B Atom. Mol. Opt. Phys. 30 (1997) 1989. [20] A.G. Sanz, et al., J. Chem. Phys. 137 (2012) 124103. [21] P. Moejko, L. Sanche, Radiat. Environ. Biophys. 42 (2003) 201. ˙ [22] P. Mozejko, L. Sanche, Radiat. Phys. Chem. 73 (2005) 77. [23] R. Raizada, K.L. Baluja, Phys. Rev. A 55 (1997) 1533. [24] K.N. Joshipura, P.M. Patel, J. Phys. B Atom. Mol. Opt. Phys. 29 (1996) 3925. [25] Y. Jiang, J. Sun, L. Wan, Phys. Rev. A 62 (2000) 062712. [26] F. Blanco, G. García, Phys. Lett. A 330 (2004) 230. [27] F. Blanco, G. García, J. Phys. B Atom. Mol. Opt. Phys. 42 (2009) 145203. ˇ [28] M. Poláˇsek, M. Juˇrek, M. Ingr, P. Cársky, J. Horáˇcek, Phys. Rev. A 61 (2000) 032701. [29] C.J. Joachain, Quantum Collision Theory, Nort-Holland Physics Publishing, 1987.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Screening corrections for the interference contributions to the electron and positron scattering cross sections from polyatomic molecules Francisco Blanco a , Lilian Ellis-Gibbings b , Gustavo García b,∗ a b
Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 113-bis, 28006 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 16 September 2015 In final form 29 November 2015 Available online 12 December 2015
a b s t r a c t An improvement of the screening-corrected Additivity Rule (SCAR) is proposed for calculating electron and positron scattering cross sections from polyatomic molecules within the independent atom model (IAM), following the analysis of numerical solutions to the three-dimensional Lippmann–Schwinger equation for multicenter potentials. Interference contributions affect all the considered energy range (1–300 eV); the lower energies where the atomic screening is most effective and higher energies, where interatomic distances are large compared to total cross sections and electron wavelengths. This correction to the interference terms provides a significant improvement for both total and differential elastic cross sections at these energies. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Electron and positron (e−/e+) scattering processes from atoms and molecules have been a subject of interest during the last few decades. Recently this interest has considerably increased due to the relevance of these processes in radiation interaction models for biomedical applications [1]. Scattering experiments are complicated in general and difficulties determining absolute values as well as uncertainties connected with energy and angular resolution limitations require some complementary data from theory. In this context sufficiently accurate general calculation procedures, for a wide variety of targets, over a broad energy range, are extremely useful. The lack of spherical symmetry makes ab initio calculations for electron and positron scattering cross sections by molecules almost unfeasible at intermediate and high energies, and therefore available theoretical methods rely on different approximate treatments. One of these techniques, the Independent Atom Model (IAM) [2–8], is particularly successful. The IAM treatment assumes that the molecules can be approximately substituted by their constituent atoms in their corresponding positions, which independently scatter incident electrons or positrons. One of the greatest advantages of this approach is the possibility of obtaining reliable
∗ Corresponding author. E-mail addresses: pacobr@fis.ucm.es (F. Blanco), [email protected] (G. García). http://dx.doi.org/10.1016/j.cplett.2015.11.056 0009-2614/© 2015 Elsevier B.V. All rights reserved.
results for a large number of molecular species from the data of a reduced number of atoms. With this method, total elastic, inelastic and differential elastic cross sections can be easily obtained with reasonable accuracy within its energy range of applicability, typically above 100 eV. An important limitation of the IAM treatment is that it ignores any multiple scattering of the projectile within the molecule, hence it’s application only for relatively high incident energies (>100 eV). Some years ago, we proposed an approximated method to partially account for these effects, the screening corrected additivity rule (SCAR) [9,10], which extended the applicability of the IAM method down to lower energies, typically 20–30 eV. In a recent letter [11] the IAM treatment for elastic scattering has been revisited, indicating the relevance of interference contributions arising from all the scattering centres in the molecule. These contributions were particularly important at small scattering angles, where experimental systems are unable to distinguish them. Nevertheless they contribute significantly to the integrated cross sections, and therefore it is crucial to use interference corrected values for Monte Carlo simulations [12] and for experimental data normalisation procedures [13]. However, as the AR procedure fails for energies below 100 eV by noticeably overestimating the cross section for decreasing energies, the effect of the interference terms in this range is not appreciable. In these conditions the SCAR procedure constitutes an excellent tool to evaluate the magnitude and consequences of interference terms at intermediate energies, below 100 eV.
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F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
The main objective of this study is to revise the SCAR procedure in order to include these interference contributions. For this purpose, considering only elastic scattering for representative molecular targets will be enough, and therefore, following the scheme of our previous letter [1], we will solve a three-dimensional Lippmann–Schwinger equation for some multicentre potentials. Although we have compared these results for different molecular configurations based on C and H atoms for different interatomic distances, we will present here only data for the H2 and CH4 at their equilibrium geometries as they suffice to illustrate all the relevant results. 2. Review of interference and screening corrections Assuming the IAM approximation, the molecular cross sections can be derived from the well known approximate expression [2] for multicenter dispersion F() ≈
fi ()eiq·ri
(1)
molecule, the IAM approximation fails since the atoms can no longer be considered as independent scattering centres and multiple scattering within the molecule is not negligible. Approximate methods revealed that important corrections are needed by the IAM method for these energies [8,9,19,21,24,25] and it has been shown [9] that the energy range for which these corrections are relevant depends on the size of the molecule: around 10% for N2 and CO up to 200 eV, for CO2 up to 300 eV, and for benzene up to 600 eV. Representative molecular cross-section calculations are based on a corrected form of the IAM treatment known as the SCAR (Screening Corrected Additivity Rule) procedure, which approximately accounts for these multiple scattering effects. All the details for this procedure have been extensively described elsewhere [9,10,26] and therefore they are only briefly mentioned here. Basically, within the SCAR procedure the integral molecular cross sections (both elastic and inelastic), ignoring interference contributions, are expressed as: total = molecule
where q = kf − ki is the momentum transfer, ri are the atomic positions and fi () are the atomic scattering amplitudes. As this corresponds to the independent scattering from each atom, the approximation is expected to be valid only for large interatomic distances compared to the wavelength associated to the incident projectile. From the above molecular dispersion function, the differential elastic cross section is easily obtained by averaging its modulus squared |F()|2 over all the molecule orientations’ [2,6], obtaining:
d˝
=
sin qrij ∗ fi f j
qrij
=
2 fi
i,j
+
i
sin qrij ∗ fi f j
qrij
=
i= / j
d elastic atom i
d˝
+
d interference d˝
(2)
i
where q ≡ |q| = 2k sin ⁄2, rij is the distance between i and j atoms, sin qrij /qrij = 1 when qrij = 0, and d interference /d˝ represents the ˙ i =/ j interference contribution to the molecular differential cross section. Hereafter, we will consider only elastic processes, which are enough for our purposes, and hence a real scattering potential for each atom (no imaginary inelastic part) will be used to obtain the atomic scattering amplitudes fi (). In these conditions the total cross section will be coincident with the corresponding integral elastic cross section. By integrating Eq. (2) the total molecular cross section can be written as: total = molecule
total si atom i
(4)
atoms
atoms
elastic dmolecule
total atom + interference i
where the si screening coefficients reduce the contribution of each atom to the total molecular cross section (0 ≤ si ≤ 1). The calculation of these coefficients requires some simple closed expressions [9,26] based only on data about position and the total cross section total of each atom in the molecule. This procedure is applicable atom i to any arbitrary molecular geometry and size [27]. Note that only elastic processes are considered in this study and therefore the total cross section and the integral elastic cross section are coincident. As far as the differential elastic cross sections are concerned, the SCAR procedure distinguishes two contributions, one from the direct scattering cross section ( D ) which is related to the angular distribution given by the single scattering differential cross section, elastic and the re-dispersed cross section (molecule − D ) which corresponds to an assumed isotropic angular distribution. The latter contribution approximately accounts for re-dispersion processes inside the molecule. After estimating their relative weight by means of the si screening coefficients and the aforementioned angular distributions of the atomic elastic cross sections[9] the resulting expression is elastic dmolecule
d˝
∼ = (1 − XS )
elastic molecule − D
D =
+ 1 + XS
elastic molecule
D
−1
dD d˝
(5)
total si2 atom i
(6)
atoms
d elastic sin qrij dD = si2 atom i + si sj fi ()fj∗ () qrij d˝ d˝
(3)
where interference represents the integration of the above differential interference contribution. It must be noted that this contribution would not be present in (3) if this expression were directly obtained from (1) by applying the optical theorem. This latest procedure is known as the ‘Additivity Rule’ (AR) and is widely described in literature [5–7,9,14–23]. As discussed in [11], the appropriate expression is Eq. (3), and the discrepancy relies on the approximate nature of Eq. (1) which doesn’t fulfil the optical theorem. It has also been shown that interference terms are only relevant at small angles and their integrated contribution ( interference ) is non negligible even at high energies [11]. At intermediate energies (10–100 eV), where atomic crosssections are not small compared to interatomic distances in the
where D , XS and d D /d are defined by
i
atoms
4
(7)
i= / j
45◦ XS =
dD sin d d˝ 0 ◦ 180 dD sin d d˝ 0
(8)
As seen from Eq. (7), the direct contribution of the scattering is a screening version of Eq. (2). The first summation operator in (7) accounts for each atomic contribution, reduced by a si factor, whereas the second one represents the reduced interference terms. We should note that applying the AR, i.e. ignoring the integral interference contributions but including them in the differential cross section values, Eqs. (6) and (7) can be in conflict. In order to avoid this contradiction we introduced an additional reducing factor () applied to the positive values of d interference /d˝ in order to ensure that d˝(d interference /d˝) = 0. This additional condition was called the ‘normalised’ SCAR treatment. Most of the recent
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
calculations based on the SCAR method included this normalisation procedure, so we will call it here as SCAR. However, we have recently shown [11] that in order to properly account for multicentre scattering effects the integral interference contribution interference should be included in the calculation of the total cross section and therefore the AR procedure needs to be revised. This method of including interference effects into the AR procedure will be referred here as AR+I. Accordingly, the SCAR method must be reformulated in order to include these terms. This will be the purpose of next section.
3. Screening corrections to the interference contributions Previous publications on the effect of interference terms in multiple-scattering calculations are scarce. From the theoretical point of view, a complete treatment of the multiple-scattering problem is considerably difficult and hence most of the previous calculations are either limited to first order corrections, in double or triple scattering configurations [9,10,23], or they only consider small systems [24]. They generally conclude that multiple scattering effects tend to reduce both the integrated interference and the small angle differential d interference /d˝ values. However, due to their limitations and restrictions it is not possible to draw a general conclusion from them. Following the scheme we proposed in [11] for the AR method, we will compare the results of the SCAR procedure with accurate numerical solutions of the three-dimensional Lippmann Schwinger equation for some multicenter potentials. Previous applications of multiple-scattering approaches are also cited in [11]. As detailed there, the procedure we used is based on the method proposed by Polasek et al. [28] to numerically solve this equation in a discrete momentum space. In this representation the atomic model potentials do not present singularities and so discretization procedures become more efficient. Once the momen tum space is discretized by means of some appropriate mesh ki , the integration of the Lippmann–Schwinger equation turns into a system of linear equations on the k ’ i |T|kj unknown values. The numerical solution of that system provides the discretized T matrix elements, and then the total differential and integral cross sections by means of the standard relations [29] between the cross sections and the T operator. As proposed in [11], single atoms are represented for simplicity here by the Yukawa potential, V = e−r/R /r, which allows the use of simple analytical expressions for the corresponding V matrix elements. We also use the fitting of Yukawa expressions already described in [11] for H and C static atomic potentials. In these conditions we have calculated electron and positron scattering cross sections from several geometries. Here we will only present some representative results for H2 and CH4 in the energy range 1–300 eV. Analysing the general trend of interference contributions for a multicenter potential can be problematic, especially at low incident energies. Nonetheless, a systematic analysis for several geometries, interatomic distances and incident energies reveals a relatively expected regularity: the interference contributions to the AR calculated in [11] apply well when the interatomic distances are high in comparison with the geometric cross section radius and with the wavelength of the projectile. Otherwise, these contributions need to be reduced or even neglected. In order to quantify this criterion, for each i-j atom pair we compared their separation rij with a length dimensional parameter
defined as ij = max( i /, j /, 1/k), where the i and j are the
total atomic cross sections, respectively. It must be noted / corresponds to the radius of a circle of area , so that the condition rij = max(
i /,
j /) represents a situation of
73
geometrical overlap between two disks for which the centre of the smallest one approaches to the border of the other. According to this, for any pair of atoms fulfilling the simple rij > ij condition we can ensure that krij > 1 and the geometrical overlapping of their atomic cross sections is moderated. We can then propose a simple procedure by introducing a ij factor which smoothly attenuates the interference terms according to the rij > ij or rij < ij values. For practical uses we can define ij = rij 2 /(rij 2 + ij 2 ) which clearly fits the anticipated function. We will refer to this procedure to account for screened interferences as the SCAR+I treatment. Basically it differs only from the previous SCAR by the replacement of expressions (4) and (6) by the following ones: total molecule =
D =
atoms
atoms
total si atom + interference i
total si 2 atom + interference i
(10)
where interference is now given by:
interference =
(9)
⎛
⎝
vij si sj fi ()fj∗ ()
i= / j
⎞ sin qrij qrij
⎠ d˝,
(11)
being si the screening coefficients introduced in Eq. (4). This is a simple procedure but, as it will be shown in the next sections by comparison with accurate numerical calculation, it provides a significant improvement of the differential and integral scattering cross sections at intermediate energies, while maintaining the high energy efficiency of the IAM method. 4. Results and discussion Following the same procedure of [11], the proposed treatment has been checked by comparing results from different approximations for several molecular geometries, interatomic distances and number of atoms. For conciseness we will present here only results for H2 and CH4 species, as it is sufficient to show all the relevant results. As already mentioned, interference contributions affect to the angular distribution of the differential cross sections, particularly for small angles, and to the absolute value of the integral cross sections. To illustrate these contributions we have compared data derived from the AR, AR+I, SCAR and SCAR+I approximations with the ‘exact’ numerical multiple scattering results described in the previous section. As far as differential cross sections (DCS) are concerned, Figure 1 shows a representative example for positron scattering from H2 at 30 eV. Calculated values from all the above methods are plotted together. The AR and SCAR calculations clearly underestimate the multicentre scattering data for small angles. By introducing the interference effects to the AR calculation, the AR+I results show better agreement with the reference data but tend to overestimate them. Finally, the screening correction to the latest (SCAR+I) substantially improves the agreement for the smaller angles. As can be seen in Figure 1, maximum discrepancies between the different approaches correspond to zero scattering angles. For this reason, to find systematic trends for different incident energies, we will focus the DCS comparison on scattering in the forward direction ( = 0). Figure 2a shows the zero angle differential cross section for electron and positron scattering from H2 at different incident energies. Even for this small molecule interference effects are clearly significant. The general effect of the screening corrections (SCAR) is reducing the zero angle DCS given by the AR, especially for the
74
10
p-H2 30 eV
Integral cross section (atomic units)
Differential cross section (atomic units)
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
AR AR+I SCAR SCAR+I Multicentre
1
0.1
0.01
e/p-H2
p-H2 AR p-H2 SCAR p-H2 SCAR+I p-H2 Multicentre e-H2 AR e-H2 SCAR e-H2 SCAR+I e-H2 Multicentre
100
10
1
0.1 0
40
80
120
160
1
Scattering angle (deg) Figure 1. Differential cross sections for positron scattering from H2 at 30 eV impact , aditivity rule (AR), , additivity rule including interference terms energy. , screening corrected additivity rule (SCAR), , screening corrected (AR+I), additivity rule including interference terms (SCAR+I), , multicentre scattering calculation.
e/p-H2 θ=0
100
p-H2 AR p-H2 Multicentre p-H2 SCAR p-H2 SCAR+I e-H2 AR e-H2 Multicentre e-H2 SCAR e-H2 SCAR+I
10
1 1
1000
from energies around 10 eV. Discrepancies below this energy are a consequence of the limitations of the SCAR method to describe molecular features. In particular, the peak around 8 eV showed in Figure 2b is not a real molecular resonance but a cross section peak from the carbon atom which has been transmitted to the CH4 configuration through the SCAR procedure. The situation for positrons is similar, although in this case the good agreement between the SCAR+I calculation and the multicentre data extends along almost all the energy range considered here (0.1–300 eV). Integral electron and positron scattering cross sections from H2 are shown in Figure 3. As expected the AR electron scattering calculation tends to overestimate the integral multicentre cross section values as the energy decreases. The SCAR procedure corrects the AR values for energies below 100 eV providing good agreement with the multicentre data down to energies of around 10 eV but agreeing with the AR results for energies above 100 eV. When including interference effects in the SCAR procedure, the SCAR+I data extend this agreement down to around 3 eV, also notably improving the (b)
(a)
100
Figure 3. Integral cross sections for electron and positron scattering from H2 . , electron scattering additivity rule (AR), , electron scattering screening , electron scattering screening corrected addicorrected additivity rule (SCAR), tivity rule including interference terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering additivity rule (AR), , positron scattering , positron scattering screening corscreening corrected additivity rule (SCAR), rected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation.
10
Incident energy (eV)
100
Differential cross section (atomic units)
Differential cross section (atomic units)
lower energies, while at increasing energies both results merge as expected. Comparing these results with the multicentre calculation, there is not a clear tendency for energies below 20 eV but above this value they tend to underestimate the multicentre calculation by a factor 2. However, when interference terms are included (SCAR+I), still no significant changes are appreciated below 4 eV but for higher energies these values tend to the multicentre calculation reaching a perfect agreement for energies above 30 eV. This is an important result which confirms the efficiency of the SCAR procedure to extend the applicability of the AR calculation down to around 30 eV [9,26] but also indicates that interference effects need to be included even at higher energies. A similar behaviour can be found for positrons although in this case there is a closer agreement between the different approaches at lower energies. Figure 2b shows the same parameters for CH4 molecules. Discrepancies between the AR and multicentre calculations are even bigger, around a factor of 3, remaining relevant for increasing energies. Again, the SCAR correction improves the results down to 20–30 eV but they are still much lower that the multicentre scattering data. Including the interference terms (SCAR+I) there is a reasonable agreement with the multicentre calculation
0.1
10
Incident energy (eV)
e-CH4 AR e-CH4 SCAR e-CH4 SCAR+I e-CH4 Multicen tre p-CH4 AR p-CH4 SCAR p-CH4 SCAR+I p-CH4 Multicen tre
e/p-CH4 θ=0 100
10
1 0,1
1
10
100
Incident energy (eV)
Figure 2. (a) Differential cross sections for electron and positron scattering from H2 at zero scattering angle. (b) same as (a) for CH4 . (a) and (b): , electron scattering , electron scattering screening corrected additivity rule (SCAR), , electron scattering screening corrected additivity rule including interference additivity rule (AR), , positron scattering additivity rule (AR), , positron scattering screening corrected additivity rule terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering screening corrected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation. (SCAR),
Integral cross section (atomic units)
F. Blanco et al. / Chemical Physics Letters 645 (2016) 71–75
1000
e/p-CH4
100
e-CH4 AR e-CH4 SCAR e-CH4 SCAR+I e-CH4 Multicentre p-CH4 AR p-CH4 SCAR p-CH4 SCAR+I p-CH4 Multicentre
10
1 1
10
100
1000
Incident energy (eV) Figure 4. Integral cross sections for electron and positron scattering from CH4 . , electron scattering additivity rule (AR), , electron scattering screening , electron scattering screening corrected addicorrected additivity rule (SCAR), tivity rule including interference terms (SCAR+I), , electron multicentre scattering calculation; , positron scattering additivity rule (AR), , positron scattering , positron scattering screening corscreening corrected additivity rule (SCAR), rected additivity rule including interference terms (SCAR+I), , positron multicentre scattering calculation.
high energy results. As may be seen in Figure 3, the reliability of the SCAR+I method is even clearer in the case of positrons where the agreement between the integral cross section data calculated with this procedure and those derived from the multicentre calculation is perfect from 1 to 300 eV. A similar comparison is illustrated in Figure 4 for electron and positron scattering from CH4 . As this is a more complicated molecule, screening effects are more important than for H2 and therefore the agreement between the different approaches is not as good as in the previous case. However, the general behaviour is similar to the previous one. The SCAR+I calculation provides the most reliable data over the whole energy range, both for electrons and positrons, showing a perfect agreement with the multicentre calculation from 10 to 300 eV. Below 10 eV some discrepancies appear, especially for positrons, but we should note that in the case of polyatomic molecules, the SCAR procedure is not expected to be reliable below about 30 eV. 5. Conclusions Different approaches based on the independent atom model have been compared with an ‘exact’ multicentre scattering calculation in order to study the effect of multiple scattering interferences in the differential and integral electron and positron scattering cross sections from molecules. Considering only elastic processes and selecting H2 and CH4 as representative molecular targets, we can conclude that the SCAR procedure represents a substantial improvement of the AR method by extending its applicability down to about 30 eV. Upon introducing interference terms to the SCAR procedure (SCAR+I), both the differential and integral calculated
75
cross section data are in significantly better agreement with the multicentre scattering calculation. For the differential cross sections, interference terms affect mainly small scattering angles and for the integral cross section data they extend over all the energy range considered here (1–300 eV). Additionally, including interference terms in the calculation of both integral and differential cross sections for molecular targets eliminates the inconsistency between the differential and integral cross section values which is inherent to the AR method (Eqs. (6) and (7), respectively). This means that no additional normalisation procedure is required by the SCAR+I approach in order to fulfil the optical theorem. For practical applications this new SCAR+I approach needs to be applied to a more realistic potential, including inelastic collision processes. This will be the subject of our further studies. We can finally conclude that once establishing the SCAR+I procedure over a realistic optical potential, a revision of the available electron-molecule and positron-scattering databases is required, as most of the available data were derived ignoring the interference effects, due either to avoiding theoretical complications or to limitations on the experimental detection methods. Acknowledgements This study has been partially supported by the Spanish Ministerio de Economía y Competitividad (project FIS2012-31230) and the EU COST programme (Action CM1301). L.E.-G. is supported by the EU FP7-PEOPLE-2013-ITN programme (ARGENT project). References [1] Radiation Damage in Biomolecular Systems, Springer, London, 2012. [2] N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions. Nucl. Phys., 81, Oxford University Press, 1966. [3] H.S. Massey, Electronic and Ionic Impact Phenomena, 1969. [4] S.P. Khare, D. Raj, P. Sinha, J. Phys. B Atom. Mol. Opt. Phys. 27 (1994) 2569. [5] Y. Jiang, J. Sun, L. Wan, Phys. Rev. A 52 (1995) 398. [6] Y. Jiang, J. Sun, L. Wan, Phys. Lett. A 231 (1997) 231. [7] Y. Jiang, J. Sun, L. Wan, J. Phys. B Atom. Mol. Opt. Phys. 30 (1997) 5025. [8] D. Raj, J. Phys. B Atom. Mol. Opt. Phys. 24 (1991) L431. [9] F. Blanco, G. Garcı´ıa, Phys. Lett. A 317 (2003) 458. [10] F. Blanco, G. García, Phys. Rev. A 67 (2003) 022701. [11] F. Blanco, G. García, Chem. Phys. Lett. 635 (2015) 321. [12] F. Blanco, et al., Eur. Phys. J. D 67 (2013) 199. ´ F. Blanco, G. García, B.P. Marinkovic, ´ A.R. Milosavljevic, ´ Phys. Rev. [13] J.B. Maljkovic, A 85 (2012) 042723. [14] A. Zecca, et al., J. Phys. B Atom. Mol. Opt. Phys. 43 (2010) 215204. [15] F. Blanco, J. Rosado, A. Illana, G. García, Phys. Lett. A 374 (2010) 4420. [16] A. Zecca, R. Melissa, R.S. Brusa, G.P. Karwasz, Phys. Lett. A 257 (1999) 75. [17] B.R. Miller, L.S. Bartell, J. Chem. Phys. 72 (1980) 800. [18] S. Hayashi, K. Kuchitsu, Chem. Phys. Lett. 41 (1976) 575. [19] D. Raj, S. Tomar, J. Phys. B Atom. Mol. Opt. Phys. 30 (1997) 1989. [20] A.G. Sanz, et al., J. Chem. Phys. 137 (2012) 124103. [21] P. Moejko, L. Sanche, Radiat. Environ. Biophys. 42 (2003) 201. ˙ [22] P. Mozejko, L. Sanche, Radiat. Phys. Chem. 73 (2005) 77. [23] R. Raizada, K.L. Baluja, Phys. Rev. A 55 (1997) 1533. [24] K.N. Joshipura, P.M. Patel, J. Phys. B Atom. Mol. Opt. Phys. 29 (1996) 3925. [25] Y. Jiang, J. Sun, L. Wan, Phys. Rev. A 62 (2000) 062712. [26] F. Blanco, G. García, Phys. Lett. A 330 (2004) 230. [27] F. Blanco, G. García, J. Phys. B Atom. Mol. Opt. Phys. 42 (2009) 145203. ˇ [28] M. Poláˇsek, M. Juˇrek, M. Ingr, P. Cársky, J. Horáˇcek, Phys. Rev. A 61 (2000) 032701. [29] C.J. Joachain, Quantum Collision Theory, Nort-Holland Physics Publishing, 1987.