A revised additivity rule for electron scattering from ethylene, propene, butene, ethane, propane and butane

Nuclear Instruments and Methods in Physics Research B 268 (2010) 1535–1539

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A revised additivity rule for electron scattering from ethylene, propene, butene, ethane, propane and butane Tan Xiao-Ming a,*, Liu Zi-Jiang b, Tian Xiao-Hong c a

School of Physics, Ludong University, Yantai 264025, China Department of Physics, Lanzhou City University, Lanzhou 730070, China c The Office of Urban Real Estate Comprehensive Development of Yantai, Yantai 264003, China b

a r t i c l e

i n f o

Article history: Received 7 February 2010 Received in revised form 14 March 2010 Available online 27 March 2010 Keywords: Electron scattering Total cross sections The revised additivity rule

a b s t r a c t Considering the difference between the bound atom in a molecule and the free atom, the original additivity rule is revised. Using the revised additivity rule, the total cross sections for electron scattering by ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) are calculated over the energy range 10–1000 eV. The results of the revised additivity rule are compared with those obtained by experiments and the revised additivity rule can give better agreement with experimental values than the original additivity rule. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) are hydrocarbon molecules. They play an important role in plasma processing in tokamak fusion devices and many other fields [1]. Their industrial importance has been established over the past few years [2]. Electron scattering cross sections for these molecules are indispensable parameters for simulating the above processes [3]. Numerous experimental investigations for electron scattering from these molecules have been made. Szmytkowski et al. [4], Sueoka and Mori [5] and Floeder et al. [6] have measured the total cross sections for electron scattering from C2H4 at energies 0.6–370 eV, 1– 400 eV and 5–400 eV, respectively. Ariyasinghe and Powers [7] have reported the total cross sections of C2H4 for the energy range 200–1400 eV and Wickramarachchi et al. [8] have measured the total cross sections for C2H4 in the energy range 200–4500 eV. Total electron cross sections for C3H6 have been measured by Szmytkowski and Kwitnewski [9], Floeder et al. [6] and Nishimura and Tawara [10] in the energy range 0.5–370 eV, 5–400 eV and 4– 500 eV, respectively. Wickramarachchi et al. [8] have obtained the experimental total cross sections for C3H6 for the energy range 200–4500 eV. For C4H8, Floeder et al. [6] have measured the total electron cross sections for 5–400 eV energy electrons and Wickramarachchi et al. [8] have reported the experimental total cross sections for the energy range 200–4500 eV. For C2H6, three * Corresponding author. E-mail address: [email protected] (T. Xiao-Ming). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.03.025

groups, Sueoka and Mori [5], Floeder et al. [6] and Nishimura and Tawara [10] have measured the total cross sections below 500 eV. Ariyasinghe et al. [11] have reported the total cross sections for electron scattering from C2H6 at energies 300–4000 eV by measurement of the electron-beam intensity attenuation through a gas cell. The experimental total cross sections for electron scattering from C3H8 have been published by Szmytkowski and Kwitnewski [12], Floeder et al. [6] and Nishimura and Tawara [10] at energies 0.5–370 eV, 5–400 eV and 4–500 eV, respectively, and by Ariyasinghe et al. [11] for 300–4000 eV. Total cross sections for C4H10 have been measured by Floeder et al. [6] at 5–400 eV and by Ariyasinghe et al. [11] at 300–4000 eV. In theory, electron-molecule scattering presents a more complex problem than corresponding electron-atom scattering due to the multi-center nature, the lack of a center of symmetry and its nuclear motion. Many approaches have been proposed and developed. Among these approaches, the additivity rule [13] is a relatively simple but effective one. The additivity rule method is based on the assumption that anisotropic electron-molecule interactions do not play an important role in shaping up the total cross sections of the intermediate- and high-energy electron-molecule collisions. Thus, the total cross section for a molecule is the sum of the total cross sections for the constituent atoms. Raj [14] made the first application of the additivity rule to obtain the elastic cross sections for electron scattering from simple molecules. Joshipura and Patel [15] and Sun et al. [16] also employed the additivity rule to obtain the total cross sections (elastic and inelastic) for electron scattering with simple diatomic and triatomic molecules and proved that the additivity rule is proper for the calculation of the

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total cross sections for electron scattering from simple and smaller molecules in intermediate- and high-energy range. For complex and larger polyatomic molecules, although the additivity rule can give good results at high enough energies, larger discrepancies between the additivity rule and the measurements can still be seen at lower energies [17–19]. This is because no molecular structure is considered and the electron-molecule scattering is reduced to electron-atom scattering in the additivity rule method. In this paper, considering the difference between the bound atom in a molecule and the corresponding free atom, we shall revise the additivity rule method. The total cross sections for electron scattering from ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) in the energy range 10–1000 eV have been calculated with the revised additivity rule. The present results are compared with the available experimental and theoretical data. 2. Theoretical model In the additivity rule model [14], molecule orbits can be described by the sum of the valence orbits of all atoms present in the molecule. As a result, the total cross section of electron-molecule scattering is written as the sum of the total cross sections of atoms. Thus the total cross section Q T for molecule is given by

QT ¼

N N X X 4p 4p fj ðh ¼ 0Þ ¼ qjT ðEÞ ImfM ðh ¼ 0Þ ¼ Im k k j¼1 j¼1

ð1Þ

where qjT and fj are the total cross section due to the jth atom of the molecule and the complex scattering amplitude for constituent atoms of the molecule, respectively. The qjT of Eq. (1) for the jth atom is obtained by the method of partial-waves:

qjT ¼

lmax pX

k

2 l¼0

 2   2      ð2l þ 1Þ 1  sjl  þ 1  sjl 

ð2Þ

where sjl is the lth complex scattering matrix element of the jth atom, which is related to the partial-wave phase shift as sjl ¼ expð2idlj Þ. The limit lmax is taken, which is enough to generate the higher partial-wave contributions until a convergence of less than 0.5% is achieved in the total cross section calculation. To obtain sjl we solve the following radial equation

!

2

d

dr

2

2

þ k  2V opt 

lðl þ 1Þ ul ðrÞ ¼ 0 r2

ð3Þ

Under the boundary condition

ul ðkrÞ  kr ½jl ðkrÞ  inl ðkrÞ þ sl kr½jl ðkrÞ þ inl ðkrÞ

ð4Þ

where jl and nl are spherical Bessel and Neumann functions separately. The atom is replaced by the complex optical potential

V opt ¼ V s ðrÞ þ V e ðrÞ þ V p ðrÞ þ iV a ðrÞ

ð5Þ

It incorporates all the important physical effects. Presently the static potential V s ðrÞ for electron-atom system is calculated from the well-known Hartree–Fock atomic wave functions [20]. Exchange potential V e ðrÞ provides a semi-classical energy-dependent form of Riley and Truhlar [21]. Zhang et al. [22] gives a smooth form at all r for polarization potential V p ðrÞ, which has a correct asymptotic form at large r and approaches the free-electron-gas correlation potential [23] in the near-target region. This potential model has been proved fairly successful to total cross sections for electron-atom scattering [22]. The imaginary part of the optical potential V a ðrÞ is the absorption potential, which represents approximately the combined effect of all the inelastic channels. The absorption potential is derived from a quasifree-scattering model by Staszawska et al. [24], and then modified by Jiang et al. [25]. The absorption potential of Jiang et al. [25] has been used to calcu-

late the total cross sections for electron scattering from Ar, Kr and Xe atoms at 0.1–300 eV and obtained better agreements with experiments. Here, the absorption potential of Jiang et al. is adopted. The optical potential is dependent on the atomic charge density q0 ðrÞ. From the above equations, we can see that the original additivity rule model does not differentiate between the free atom and the bound atom in the molecule. Considering this, we present

qðrÞ ¼ f  q0 ðrÞ

ð6Þ

qðrÞ is the charge density of the bound atom in the molecule and q0 ðrÞ is the charge density of the corresponding free atom. f is a revised factor for a bound atom in the molecule. To obtain factor f, two points should be considered: (1) The total cross sections calculated from the optical potential with the additivity rule are much higher than the experimental results at lower energies. So, to get accurate total cross sections for electron scattering from molecules, the effect of the optical potential should decrease at lower energies. (2) The bound atoms in a molecule are different from the corresponding free atoms. The reason is that there exits the overlapping effect of electron cloud between two atoms which form the chemical bond in a molecule. This shows that the revision should be related to the radii of the constituent atoms and the bond length between the two atoms. Between the two atoms which form the chemical bond in a molecule, the more the electron number of the constituent atom is, the smaller the influence from the overlapping effect of electron cloud is. This indicates that the revision should also have relation with the electron number of the constituent atom. From the above two points, many f factors are tested and good agreements between the revised results and the experimental data are obtained by

f ¼1

R NZ Rþd N

ð7Þ

where d is the bond length between two bound atoms in the molecule and R is the sum of the radii of the corresponding two free atoms. Z is the electron number in the atom and N is the sum of the electron number of the two atoms which form the chemical bond. Thus, the original additivity rule is revised with Eqs. (6) and (7). Obviously, the revised additivity rule is related to the molecular structure. 3. Results and discussion The total cross sections for electron scattering from ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) have been calculated in the energy range 10–1000 eV with the above revised additivity rule. The present results are listed in Tables 1 and 2 and also compared with the experimental and other theoretical results shown in Figs. 1–6. In Fig. 1, our present results for C2H4 are compared with the measurements of Szmytkowski et al. [4], Ariyasinghe and Powers [7], Wickramarachchi et al. [8], Sueoka and Mori [5], Floeder et al. [6] and the theoretical values of Floeder et al. [6], Wickramarachchi et al. [8], Nishimura and Tawara [10], Vinodkumar et al. [26], Garcia and Manero [27], and Szmytkowski [28]. The present results are in better agreement with the experimental data than the original additivity rule results in the whole energy range. For example, the difference between the present results with the experimental data of Wickramarachchi et al. [8] is only 9% at

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T. Xiao-Ming et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1535–1539 Table 1 The present results for C2H4, C3H6 and C4H8 in unit of 1020 m2.

80

67.80 40.51 36.76 34.26 31.33 28.62 26.44 24.69 23.24 22.00 14.43 10.73 8.48 6.97 5.88 5.06 4.43 3.92 3.51

90.71 54.16 49.10 45.74 41.83 38.20 35.29 32.95 31.02 29.37 19.28 14.34 11.34 9.31 7.86 6.77 5.92 5.25 4.69

Table 2 The present results for C2H6, C3H8 and C4H10 in unit of 1020 m2. Energy (eV)

C2H6

C3H8

C4H10

10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000

51.08 30.49 27.63 25.63 23.34 21.23 19.54 18.18 17.05 16.10 10.40 7.69 6.06 4.97 4.19 3.60 3.15 2.78 2.49

74.21 44.25 40.06 37.20 33.91 30.89 28.45 26.50 24.88 23.51 15.27 11.32 8.93 7.33 6.18 5.31 4.65 4.11 3.68

97.25 57.96 52.46 48.73 44.46 40.52 37.35 34.80 32.70 30.91 20.13 14.93 11.79 9.67 8.16 7.02 6.14 5.43 4.86

Present results Additivity rule results Szmytkowski and Kwitnewski Wickramarachchi et al. Nishimura and Tawara Floeder et al. Vinodkumar et al. Wickramarachchi et al. Floeder et al. Nishimura and Tawara Szmytkowski

C3H6

2

44.82 26.82 24.38 22.73 20.80 19.00 17.56 16.39 15.43 14.61 9.58 7.12 5.62 4.62 3.90 3.36 2.94 2.60 2.33

-20

10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000

70

Total cross sections (10 m )

C4H8

60 50 40 30 20 10 0 0

200

400

600

800

1000

Electron energy (eV) Fig. 2. Total cross sections for electron scattering from C3H6. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Szmytkowski and Kwitnewski [9], Wickramarachchi et al. [8], Nishimura and Tawara [10], Floeder et al. [6]. Theoretical results (line): Vinodkumar et al. [26], empirical data of Wickramarachchi et al. [8], empirical data of Floeder et al. [6], empirical data of Nishimura and Tawara [10], empirical data of Szmytkowski [28].

110 100 90

Present results Additivity rule results Wickramarachchi et al. Floeder et al. Wickramarachchi et al. Floeder et al. Szmytkowski

C4H8

2

C3H6

-20

C2H4

Total cross sections (10 m )

Energy (eV)

80 70 60 50 40 30 20 10 0 0

200

400

600

800

1000

Electron energy (eV)

2

m)

C2H4 40

-20

Total cross sections (10

Fig. 3. Total cross sections for electron scattering from C4H8. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Wickramarachchi et al. [8], Floeder et al. [6]. Theoretical results (line): empirical data of Wickramarachchi et al. [8], empirical data of Floeder et al. [6], empirical data of Szmytkowski [28].

Present results Additivity rule results Szmytkowski et al. Ariyasinghe and Powers Wickramarachchi et al. Sueoka and Mori Floeder et al. Vinodkumar et al. Garcia and Manero Wickramarachchi et al. Floeder et al. Nishimura and Tawara Szmytkowski

50

30

20

10

0 0

200

400

600

800

1000

Electron energy (eV) Fig. 1. Total cross sections for electron scattering from C2H4. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Szmytkowski et al. [4], Ariyasinghe and Powers [7], Wickramarachchi et al. [8], Sueoka and Mori [5], Floeder et al. [6]. Theoretical results (line): Vinodkumar et al. [26], empirical data of Garcia and Manero [27], empirical data of Wickramarachchi et al. [8], empirical data of Floeder et al. [6], empirical data of Nishimura and Tawara [10], empirical data of Szmytkowski [28].

200 eV, while the difference for the original additivity rule reaches 51% at 200 eV. The present total cross sections for electron scattering from C3H6 are presented in Fig. 2. The measurements of Szmytkowski and Kwitnewski [9] and Wickramarachchi et al. [8] agree well with the present results at all energies, but the experimental values of Nishimura and Tawara [10] and Floeder et al. [6] are much lower than the present results below 200 eV. From Fig. 2, we can also see that large discrepancies exist between different experimental groups. Below 500 eV, the original additivity rule results are much higher than all experimental data. For example, the difference between the original additivity rule result and the experimental datum of Szmytkowski and Kwitnewski [9] reaches 45% at 60 eV, while the difference between the present result and the datum of Szmytkowski and Kwitnewski is only 5%. Fig. 3 shows the present total cross sections of C4H8 together with the available experimental and theoretical results. From

T. Xiao-Ming et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1535–1539 120 110 100 2

90

-20

Fig. 3, we can see that the present results are in better agreement with the experimental and theoretical data than the additivity rule results. For example, the additivity rule results deviate from the experimental data of Floeder et al. [6] and Wickramarachchi et al. [8] by about 27% at 60 eV and 33% at 300 eV, respectively, while the present results deviate from the experimental data of Floeder et al. only by 7% at 60 eV and show good agreement with the data of Wickramarachchi et al. at 300 eV. Fig. 4 shows the variations of the total cross sections for C2H6 over the energy range 10–1000 eV. The present results are in good agreement with the experimental data in the whole energy range, while the original additivity rule results are much higher than the experimental data below 400 eV. For example, the difference between the original additivity rule result and the experimental datum of Ariyasinghe and Powers [7] reaches 67% at 200 eV, while the difference for the present result is only 18%.

Total cross sections (10 m )

1538

80

Present results Additivity results Ariyasinghe et al. Floeder et al. Ariyasinghe et al. Floeder et al. Szmytkowski

C4H10

70 60 50 40 30 20 10 0 0

200

400

600

800

1000

Electron energy (eV) Fig. 6. Total cross sections for electron scattering from C4H10. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Ariyasinghe et al. [11], Floeder et al. [6]. Theoretical results (line): empirical data of Ariyasinghe et al. [11], empirical data of Floeder et al. [6], empirical data of Szmytkowski [28].

70

Present results Additivity rule results Ariyasinghe and Powers Ariyasinghe et al. Sueoka and Mori Floeder et al. Nishimura and Tawara Vinodkumar et al. Ariyasinghe et al. Floeder et al. Nishimura and Tawara Szmytkowski

C 2H 6

-20

2

Total cross sections (10 m )

60

50

40

30

20

10

0 0

200

400

600

800

1000

Electron energy (eV) Fig. 4. Total cross sections for electron scattering from C2H6. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Ariyasinghe and Powers [7], Ariyasinghe et al. [11], Sueoka and Mori [5], Floeder et al. [6], Nishimura and Tawara [10]. Theoretical results (line): Vinodkumar et al. [26], empirical data of Ariyasinghe et al. [11], empirical data of Floeder et al. [6], empirical data of Nishimura and Tawara [10], empirical data of Szmytkowski [28].

100 90

-20

2

Total cross sections (10 m )

Present results Additivity rule results Szmytkowski and Kwitnewski Ariyasinghe et al. Floeder et al. Nishimura and Tawara Vinodkumar et al. Ariyasinghe et al. Floeder et al. Nishimura and Tawara Szmytkowski

C3H8

80 70 60 50 40 30 20 10 0

0

200

400

600

800

1000

Electron energy (eV) Fig. 5. Total cross sections for electron scattering from C3H8. Solid line: present results, dash line: additivity rule results. Experimental data (symbol): Szmytkowski and Kwitnewski [12], Ariyasinghe et al. [11], Floeder et al. [6], Nishimura and Tawara [10]. Theoretical results (line): Vinodkumar et al. [26], empirical data of Ariyasinghe et al. [11], empirical data of Floeder et al. [6], empirical data of Nishimura and Tawara [10], empirical data of Szmytkowski [28].

In Fig. 5, the total electron scattering cross sections for C3H8 are plotted. The present results indicate good agreement with the measurements of Szmytkowski and Kwitnewski [12], Ariyasinghe et al. [11], Floeder et al. [6], Nishimura and Tawara [10], while the theoretical values of Vinodkumar et al. [26] are lower than these results below 600 eV. The additivity results are much higher than the available experimental and theoretical values below 500 eV. For example, the difference between the additivity rule result and the measurement of Szmytkowski and Kwitnewski [12] is about 60% at 40 eV, while the difference for the present result is only 14% at 40 eV. In Fig. 6, we present the total cross sections for C4H10 at 10– 1000 eV. The present results are compared the experimental data of Ariyasinghe et al. [11], Floeder et al. [6] and the empirical values of Floeder et al. [6], Ariyasinghe et al. [11] and Szmytkowski [28]. The present results give better accordance with the experimental and theoretical values than the additivity rule results. The additivity rule result deviates from the experimental value of Ariyasinghe et al. [11] by 39% at 300 eV, while very good agreement is obtained between the present result and the experimental value at 300 eV. From Figs. 1–6, we can see that the revised additivity rule considering the difference between the bound atom in a molecule and the free atom is more successful than the original additivity rule for calculating the total electron scattering cross sections from ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) in the energy range 10–1000 eV. The revised additivity rule can give better results at lower energies, especially below 600 eV. 4. Conclusions A revised additivity rule considering the difference between the bound atom in a molecule and the corresponding free atom is proposed. The revised additivity rule has been used to calculate the total cross sections for electron scattering from ethylene (C2H4), propene (C3H6), butene (C4H8), ethane (C2H6), propane (C3H8) and butane (C4H10) at 10–1000 eV. From our studies, we have the following conclusions: (1) The total cross sections calculated from the additivity rule for these molecules are not in good agreement with the experimental data at lower energies. This is because no molecular structure is considered and the electron-molecule scattering is reduced to electron-atom scattering.

T. Xiao-Ming et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 1535–1539

(2) The revised additivity rule is more successful for calculating the total cross sections of these molecules in the energy range 10–1000 eV, especially below 600 eV. Of course, more experimental and theoretical investigations are needed to elucidate our present calculations.

Acknowledgements This work was supported by the Research Foundation of Ludong University of China (LY20072801), the Discipline Construction Fund of Ludong University and the Key Project of the Chinese Ministry of Education (209127). References [1] W.L. Morgan, Adv. Atom. Mol. Opt. Phys. 43 (2000) 79. [2] C. Makochekanwa, H. Kato, M. Hoshino, H. Tanaka, H. Kubo, M.H.F. Bettega, A.R. Lopes, M.A.P. Lima, L.G. Ferreira, J. Chem. Phys. 124 (2006) 024323. [3] O. Sueoka, C. Makochekanwa, H. Tanino, M. Kimura, Phys. Rev. A 72 (2005) 042705. [4] C. Szmytkowski, S. Kwitnewski, E. Ptasinska-Denga, Phys. Rev. A 68 (2003) 032715. [5] O. Sueoka, S. Mori, J. Phys. B 19 (1986) 4035.

1539

[6] K. Floeder, D. Fromme, W. Raith, A. Schuab, G. Sinapius, J. Phys. B 18 (1985) 3347. [7] W.M. Ariyasinghe, D. Powers, Phys. Rev. A 66 (2002) 052716. [8] P. Wickramarachchi, P. Palihawadana, G. Villela, W.M. Ariyasinghe, Nucl. Instrum. Methods B 267 (2009) 3391. [9] C. Szmytkowski, S. Kwitnewski, J. Phys. B 35 (2002) 2613. [10] H. Nishimura, H. Tawara, J. Phys. B 24 (1991) L363. [11] W.M. Ariyasinghe, P. Wickramarachchi, P. Palihawadana, Nucl. Instrum. Methods B 259 (2007) 841. [12] C. Szmytkowski, S. Kwitnewski, J. Phys. B 35 (2002) 3781. [13] J.W. Otvos, D.P. Stevenson, J. Am. Chem. Soc. 78 (1956) 546. [14] D. Raj, Phys. Lett. A 160 (1991) 571. [15] K.N. Joshipura, P.M. Patel, Z. Phys. D 29 (1994) 269. [16] J.F. Sun, Y.H. Jiang, L.D. Wan, Phys. Lett. A 195 (1994) 81. [17] J.F. Sun, B. Xu, Y.F. Liu, D.H. Shi, Chin. Phys. 14 (2005) 1125. [18] D.G.M. Silva, T. Tejo, J. Muse, D. Romero, M.A. Khakoo, M.C.A. Lopes, J. Phys. B 43 (2010) 015201. [19] T. Yamada, S. Ushiroda, Y. Kondo, J. Phys. B 41 (2008) 235201. [20] E. Clementi, C. Roetti, Atom. Data Nucl. Data Tables 14 (1974) 177. [21] M.E. Riley, D.G. Truhlar, J. Chem. Phys. 63 (1975) 2182. [22] X.Z. Zhang, J.F. Sun, Y.F. Liu, J. Phys. B 25 (1992) 1893. [23] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [24] G. Staszewska, D.G. Schwenke, D. Thirumalai, D.G. Truhlar, Phys. Rev. A 28 (1983) 2740. [25] Y.H. Jiang, J.F. Sun, L.D. Wan, Chin. J. Atom. Mol. Phys. 11 (1994) 418. [26] M. Vindkumar, K.N. Joshipura, C.G. Limbachiya, B.K. Antony, Eur. Phys. J. D 37 (2006) 67. [27] G. Garcia, F. Manero, Chem. Phys. Lett. 280 (1997) 419. [28] Cz Szmytkowski, Z. Phys. D 13 (1989) 69.