Elastic scattering of electrons by molecular oxygen

23 April 2001

Physics Letters A 282 (2001) 284–287 www.elsevier.nl/locate/pla

Elastic scattering of electrons by molecular oxygen Deo Raj, Ashok Kumar ∗ Department of Physics, C.C.S. University, Meerut 250004, India Received 11 November 2000; received in revised form 19 February 2001; accepted 21 February 2001 Communicated by B. Fricke

Abstract The absorption potential of Staszewska et al. (Phys. Rev. A 28 (1983) 2740) is known to underestimate the experimental values of the elastic differential, integral and momentum transfer cross sections. Hence, it has been modified and applied to study the elastic scattering of electrons on molecular oxygen at intermediate energies. A comparison of the elastic cross section results, obtained by using modified absorption potential, with the experimental data shows a significant improvement over the unmodified absorption potential results. The differential, integral and momentum transfer cross sections obtained using the modified absorption potential are in very good agreement with the experimental data.  2001 Elsevier Science B.V. All rights reserved. PACS: 34.80.Bm

It has been observed in a number of investigations [1–4] that the theory, which include the absorption effects through absorption potential of Staszewska et al. [1], yields low values of elastic differential (EDCS), integral (QI ) and momentum transfer (Qm ) cross sections. The reason is, the absorption potential of Staszewska et al. [1] overestimate the flux loss to the electronic excited states for large angle scattering, particularly at high incident energies. Hence, to improve the agreement between theory and experiment, contribution of the absorption potential should be reduced. In the present Letter, we have modified this absorption potential in such a way that it yielded best agreement between theory and experiment for the EDCS, QI and Qm for the elastic scattering of electrons on O2 over a wide incident energy range. For these calculations independent atom model (IAM) has been employed. In this model, the elastic differential cross section aver* Corresponding author.

E-mail address: [email protected] (A. Kumar).

aged over all orientations of internuclear axis for e− – O2 scattering is given by [5] IO2 (θ ) = 2IO (θ )(1 + sin KRe /KRe ),

(1)

where IO (θ ) is the EDCS of atomic oxygen, K (= 2k sin θ/2) is the magnitude of momentum transfer during scattering, k 2 is the energy of the incident particle (atomic units have been used where length is expressed in a0 and energy in rydbergs), Re is the equilibrium internuclear distance of oxygen molecule. To obtain IO (θ ), the following scattering equation was solved numerically under proper boundary conditions for the scattering phase shifts: 
2 d l(l + 1) 2 ul (r) = 0, (2) + k − Vopt(r) − dr 2 r2 where Vopt(r) is the optical potential appropriate to electron–oxygen atom system, which is spherically symmetric, energy dependent and complex. The real part of Vopt(r) comprises of static, exchange, and dynamic polarization potentials. The details of these

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 3 7 - 2

D. Raj, A. Kumar / Physics Letters A 282 (2001) 284–287

potentials and the method of obtaining scattering amplitude f (θ ) from phase shifts δlk may be found elsewhere [3,4]. The imaginary part of Vopt (r) is given by VImod (r) = mVI (r),

(3)

where VI (r) is the absorption potential of Staszewska et al. [1], 1/2

VI (r) =

8πρ(r)Tloc 5k 2 kf3

H (x)(A1 + A2 + A3 ),

(4)

with A1 = 5kf3 /∆,   2 2 k − kf2 , A2 = −kf3 5k 2 − 3kf2 A3 = 2H (y)y 5/2



k 2 − kf2

2

,

x = k 2 − kf2 − ∆, y = 2kf2 + ∆ − k 2 ,  1/3 kf = 3π 2 ρ(r) . Tloc and ρ(r), respectively, are the local kinetic energy and density function of the incident electron; H (x) is Heaviside unit step function and ∆ is the mean excitation energy of the atom. The m is a modification factor. To obtain the factor m the following two points were considered: 1. Eq. (4) overestimates the flux loss to the electronic excited states. 2. The effect of absorption potential should decrease with increasing energy. However, the difference between theory (broken curve) and experiment increases with increasing energy for absorption potential of Staszewska et al. (m = 1, see Figs. 1(a)– (d)). This indicates that the effect of absorption potential is increasing instead of decreasing. Keeping these points in mind, a number of modification factors were tried and the best agreement in theory and experiment was obtained for m = k −1 . This value of m is taken in the present Letter. To compare the effects of modification of absorption potential, two sets of results are obtained. In set I,

285

the unmodified absorption potential (Eq. (4)) was employed, and modified absorption potential (Eq. (3)) was used in the second set. The IO2 (θ ) was obtained from Eq. (1). Integral and momentum transfer cross sections were obtained by integrating IO2 (θ ) with appropriate weight factors. Iga et al. [6] have measured EDCS for e− –O2 scattering over a wide angular range (5–120◦) for incident energies varying from 300 to 1000 eV. They have also integrated their EDCS with appropriate weight factors to obtain the total elastic cross section QI and total momentum transfer cross section Qm . As they have truncated the integration at θmax = 120◦ , we have also obtained QI and Qm for set II by integrating our results of EDCS up to 120◦ for better comparison. These results of QI and Qm are given in columns 4 and 8 of Table 1, respectively. The present values of EDCS are compared with the experimental data of Iga et al. [6] in Figs. 1(a)–(d). It is evident from the Fig. 1(a) (E = 300 eV) that the EDCS results in set I (broken curve) and set II (continuous curve) are almost same for scattering angle less than 25◦ . However, for rest of the scattering angles the two results differ significantly. Set I results are much lower than the set II results. A comparison with the experimental results of Iga et al. [6] shows that set II results, obtained by using modified absorption potential, are in very good agreement with the experimental data. The curve for the set I results lie much lower than the experimental data. Similar type of agreement is obtained at other energies (see Figs. 1(b)–(d)) also. The present set II results are also in very good agreement with the other experimental data [7–9]. However, for the sake of clarity of the figure, these data are not included in the figures. From Table 1, it is clear that the present values of set II (θmax = 120◦) for QI are in very good agreement with the experiment of Iga et al. [6] at all the energies. The maximum difference 11%, which is much less than the experimental error 20%, is obtained for E = 400 eV. It is also evident from the table that the present set II results of Qm are in very good agreement with the experimental data. The present set I results of Qm are much lower than the experimental data. The difference between theory and experiment increases with the energy and at E = 1000 eV, the set I result is less than half of the experimental results. It may be noted that the set I results are obtained by integrating up to θmax = 180◦ . The integration up to θmax = 120◦ ,

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D. Raj, A. Kumar / Physics Letters A 282 (2001) 284–287

(a)

(b)

(c)

(d)

Fig. 1. Elastic differential cross sections for e− –O2 scattering at various incident energies. Present: (—) with modified absorption potential (Eq. (3)); (· · ·) with Staszewska et al. absorption potential [1] (Eq. (4)). Exp. data: () Iga et al. [6].

D. Raj, A. Kumar / Physics Letters A 282 (2001) 284–287

287

Table 1 Elastic integral (QI ) and momentum transfer (Qm ) cross sections for e− –O2 scattering (a02 ). Set I is calculated with absorption potential of Staszewska et al. [1]. Set II is calculated with modified absorption potential E (eV)

QI

Qm

Present

Exp. [6]

Present

Exp. [6]

I

II

II

I

II

II

(θmax = 180◦ )

(θmax = 180◦ )

(θmax = 120◦ )

(θmax = 180◦ )

(θmax = 180◦ )

(θmax = 120◦ )

300

9.47

10.69

10.20

10.3±2.1

1.46

2.45

1.59

1.6±0.32

400

7.18

8.30

8.05

9.08±1.82

0.84

1.59

1.15

1.31±0.26

500

5.82

6.89

6.74

7.15±1.41

0.55

1.14

0.87

0.92±0.18

1000

3.20

3.98

3.95

3.74±0.75

0.16

0.39

0.34

0.35±0.07

will further decrease these results. A comparison of the sets I and II results for QI and Qm shows that Qm results in two sets differ considerably. However, there is not much difference in QI results. This is not unexpected because a significant contribution of DCS to Qm comes from the large angle scattering due to the weight factor (1 − cos θ ), where the result of two sets differ considerably. The main contribution of DCS to QI comes from low angle scattering where the difference in two results is not large. Finally, we conclude that the modification of the absorption potential of Staszewska et al. [1] have improved the agreement between theory and experiment significantly for the e− –O2 elastic scattering. The investigations for the applicability of the modified absorption potential to other targets is in progress.

Acknowledgement Financial assistance from the University Grants Commission, New Delhi, and Physics Department,

C.C.S. University, Meerut (India), is greatfully acknowledged.

References [1] G. Staszewska, D.W. Schwenke, D. Thirumalai, D.G. Truhlar, Phys. Rev. A 28 (1983) 2740. [2] A. Jain, Phys. Rev. A 34 (1986) 3707. [3] S.P. Khare, D. Raj, P. Sinha, J. Phys. B 27 (1994) 2569. [4] D. Raj, S. Tomar, J. Phys. B 30 (1997) 1989. [5] H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody, Electronic and Ionic Impact Phenomenon, Vol. II, Clarendon, Oxford, 1969. [6] I. Iga, L.M. Tao, J.C. Nogueira, R.S. Barbieri, J. Phys. B 20 (1987) 1095. [7] J.P. Bromberg, J. Chem. Phys. 60 (1974) 1717. [8] K. Wakiya, J. Phys. B 11 (1978) 3913. [9] H. Daimon, S. Hayashi, T. Kondow, K. Kuchitsu, J. Phys. Soc. Jpn. 51 (1982) 2641.