Studies of ionization of hydrogen atoms by electron ...

Studies of ionization of hydrogen atoms by electron impact in hyperspherical partial wave theory from below 1eV excess energies to intermediate energies J. N. Das, S. Paul, and K. Chakrabarti Citation: AIP Conf. Proc. 697, 82 (2003); doi: 10.1063/1.1643682 View online: http://dx.doi.org/10.1063/1.1643682 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=697&Issue=1 Published by the American Institute of Physics.

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Studies of ionization of hydrogen atoms by electron impact in hyperspherical partial wave theory from below 1eV excess energies to intermediate energies J. N. Das∗ , S. Paul∗ and K. Chakrabarti† ∗

Department of Applied Mathematics, University College of Science, 92, A. P. C. Road, Calcutta 700 009, India. † Department of Mathematics, Scottish Church College, 1 & 3 Urquhart Square, Calcutta - 700 006, India.

Abstract. Hyperspherical partial wave theory has been applied in the study of ionization of hydrogen atoms by electron impact from very low energies (1.0eV, 0.5eV and 0.3eV above threshold) to intermediate energies (60eV). The various ionization cross sections which have been calculated are generally in good agreement with the experiments. Some times the agreements are even better than those of other high level theories, like ECS and CCC theories, requiring elaborate computations.

INTRODUCTION In recent years studies of ionization of hydrogen atoms by electron impact have concentrated in the low energy domain. Various well known approaches like the eikonal Born series (EBS) approach [1], the distorted wave Born approximation (DWBA) [2], the 3C approach [3], the continuum distorted wave with eikonal initial state (CDW-EIS) approach [4] are not generally useful in the ionization studies at low or at lower end of the intermediate energy domain (5I0 - 20I0 ), I0 being the ionization potential. The standard close coupling approaches [5,6], which have been widely used in atomic elastic and excitation scattering, have not been so widely used in ionization calculation, particularly for low energies. Bartschat and Bray [7], however have successfully used this approach for the calculation of the total ionization cross section down to low energies. A similar approach, at least in sprit, is the convergent close coupling (CCC) theory of Bray et al[8]. This approach has been applied from high to low energies [9-12] with spectacular success. Another approach which proved successful in elastic [13,14] as well as in inelastic scattering (including ionization) studies [15] begins with the solution of a coupled set of integral equations for the off-shell T-matrix elements [13]. However in their solutions some crude means were adopted [13], but yielded always moderately good results. Yet another good approach is the multiple scattering calculational approach [16]. Only first order correct (in potential) wave function has been applied with reasonable success for intermediate and high energies [17-18]. Second order correct (in potential) wave function or more accurate wave function that may be obtained by solving certain integral equation [16] in the momentum space has not yet been applied. Such an ap-

CP697, Correlation and Polarization in Photonic, Electronic, and Atomic Collisions, edited by G. F. Hanne et al. © 2003 American Institute of Physics 0-7354-0170-5/03/$20.00

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proach may be useful in the low energy domain also. Apart from the CCC approach the other successful approaches are the exterior complex scaling (ECS) approach [19,20] the hyperspherical R-matrix with semi classical outgoing wave (HRM-SOW) approach [21], the time dependent close coupling (TDCC) approach [22] and the hyperspherical partial wave (HPW) theoretic approach [23-26]. These theories are most appropriate for studies at low energies, although these may be applied to studies at intermediate energies also, requiring better efforts. For very low energies (close to threshold) there exists the well known semiclassical approach of Crothers [27]. At present there also exist several sets of experimental results for total cross sections (TICS) [28], the single differential cross section (SDCS) [29], the double differential cross sections (DDCS) [29] and the triple differential cross sections (TDCS) [30-32] for low and intermediate energies. In this article we consider calculations of various cross sections from very low (1.0eV, 0.5eV and 0.3eV above threshold) to intermediate energies (60eV) with the inclusion of a larger number of states in the hyperspherical partial wave theory compared to the calculations already published [24,25]. Since the hyperspherical partial wave theory has been described in details in [23,25] and outlined in [24,26] we only refer to these, particularly to [25,26], for the definition of the parameters and of the symbols used in the description of our results in the following section.

RESULTS Low Energy Results Here we present TDCS results of this calculation for 17.6eV for equal-energy-sharing kinematics. This is the case which have been considered on several occasions both theoretically [10-12,19,20,25] and experimentally [30-32]. In Fig.1 we present results 0.3

1 o

θab=100

2 −18

cross section (10

θab=120o

0.6

−1

cm eV )

0.2

0.8

0.4

0.1

0.2 0

0 0

60

120

180

240

300

360

0

2

60

120

180

240

300

360

3 θ =150o

θab=180o

ab

1.5 2 1 1 0.5 0

0 0

60

120

180

240

300

360

0

30

60

90

120

150

180

scattering angle θ (degrees) b

FIGURE 1. TDSC in coplanar equal-energy-sharing constant Θab results for incident electron energy 17.6eV. Continuous curves, present results; dashed curves, ECS results [20], dashed dotted curve, CCC results [12]. Experiment: Röder et al [32].


for constant Θab geometries for several values of Θab and in Fig.2 for constant θa geometries. We compare our results with the most recent measurements of Röder et al [32] on the one hand and with the latest ECS results [20] and the most recent CCC results [12] on the other hand. It appears from the comparison that our present results agree best with experimental results for Θab = 150o and 180o . For Θab = 100o and 120o , cross section(a.u.)

1

1.8 θa=140o

o

θa=−20

θa=60o

0.4

1.2 0.5 0.2

0 −180 −120

−60

0

60

120

180

0.6

0 −180 −120

−60

0

60

120

180

0 −180 −120

−60

0

60

120

180

scattering angle θb (degrees)

FIGURE 2. TDSC in coplanar equal-energy-sharing constant θa results for incident electron energy 17.6eV. Continuous curves, present results; dashed curves, ECS results [20], dashed dotted curve, CCC results [12]. Experiment: Röder et al [32].

the agreement is also generally good. For Θab = 120o , the only disturbing feature is that the experimental peak value is about 25% higher than those of all these calculations. The θa constant results presented in Fig.2 also appear very good. Here only for θa = 60o the results appear little oscillatory and are not nearly converged. For better convergence inclusion of a few more L values and (l1 , l2 , n) triplets may be necessary. In our present calculation we have included L values up to 9 as in ECS calculations and included at least as many (l1 , l2 ) pairs as in those calculations. Here the range parameter R∞ has been chosen to be 1000 a.u. , much larger than in our previous calculation [25] or the ECS calculation. Considering the computational facilities used (a SUN Enterprise 450 server and a few Pentium III PCs) the present achievement is very gratifying.

Intermediate Energy DDCS and SDCS Results Here we present results of the DDCS and SDCS for 60eV energy calculated in the HPW theory for which there exist experimental results [29]. In this calculation we have taken L values upto 12 and (l1 , l2 ) pairs some what less than those in the previous section. The DDCS results are presented in Fig.3, where we have compared our results with the experimental results of Shyn [29]. Quantitatively the DDCS results for 3eV, 4eV and 5eV are generally good. For 1eV the cross sections are much large except for small angles and that the shape of the DDCS curve is somewhat different from that of experiment. It may be mentioned here that the multiple scattering calculation by Das and Seal [18] at a higher energy gives also satisfactory DDCS results. For 60eV there are no results of ECS or CCC calculation. However, for 54.4eV there exist CCC results. These results suggest that there are problems near forward directions of the CCC calculations [10]. Here it may again be stated that more experimental and theoretical results are necessary for a proper view. The SDCS results of our calculations are presented in Fig.4. Here we


cross section (10−18cm2eV−1)

5

2.5

4

2

present result (x 0.18) Shyn(1992) E =1eV

3 2

1

1

0.5

0

0

2

1.5

1.5

present result (x 0.86) E1=4eV

E =4eV Shyn(1992) 1

1

present result (x 0.98) Shyn(1992) E =3eV

1.5

1

1

E =5eV present 1

1

result (x 1.38) Shyn(1992) E1=5eV

0.5

0.5 0

0 0

60

120

180

0

60

120

180


FIGURE 3.

DDCS results for 60eV energy. Continuous curves, present results. Experiment, Shyn [29].

compare our results with the experimental results of Shyn [29]. The comparison shows that our calculation has only partly converged for this energy. For fully converged results a much larger size calculation may be necessary with the inclusion of more L values and (l1 , l2 , n) triplets. For energies of one of the electrons close to zero, say less than about 2 eV the cross sections are exceptionally large. For such energies, contamination with high Rydberg states is responsible for such behavior. However, beyond this energy the SDCS curve is oscillatory as in CCC calculations, but in a different way. Nature of the curve suggests a fit, not with a second order parabola, but with a fourth order parabola, symmetric about E1 = E2 = E/2 energy. The fitted curve gives a total cross section value 3.2 a.u., close to that of the measured value 3.1 a.u. of Shyn [29] and a correct spin asymmetry parameter value 0.20. In this figure we have also presented the singlet and the triplet parabolic fitted cross section curves.

20

singlet(fitted) triplet(fitted) total(fitted)

cross section(a.u.)

15

total(calculated) Shyn(1992) 10

5

0 0

9

18

27

36

45

secondary electron energy (eV)

FIGURE 4.

SDCS results for 60eV energy. Continuous curves, present results. Experiment, Shyn [29].


3

5 5

E = 1eV

cross section (10−18cm2eV−1)

E=0.5eV o θab=150

4

θ =150o

2

ab

3

1

0 0

60

120

180

240

300

360

0 0

60

120

180

240

300

360

8 E = 1eV

4

0

60

120

180

240

300

360

150

180

8 E=0.5eV

6

o

θab=180

ab

2

1 0

θ =150o

3

2

1

E=0.3eV

4

E=0.3eV

6

o

θab=180

o θab=180

4

4

2

2

2

0

0 0

30

60

90

120

150

180

0 0

30

60

90

120

150

180

0

30

60

90

120


FIGURE 5. TDSC in coplanar equal-energy-sharing constant Θab results for excess incident electron energies above threshold indicated in the figure. Continuous curves, present results; dashed curves, Deb and Crothers [33]

Very Low Energy Results Here we present TDCS results for equal-energy-sharing cases for 1.0eV, 0.5eV and 0.3eV excess energies of our present calculation. We have chosen for R∞ a value 1000 a.u. for 1.0eV, 1500 a.u. for 0.5eV and 2000 a.u. for 0.3eV for smooth convergence of the asymptotic series solutions and for smooth fittings there of these solutions with those which start from the origin. Here we have chosen a larger grid size. In these calculations we needed to retain L values up to 5 only and a small number of (l1 , l2 ) pairs. In Fig.5 we have presented results for Θab = 150o and Θab = 180o only for which there exist semiclassical calculation of Deb and Crothers [33]. Unfortunately for these low energies there exist no experimental results. However the close agreement between our results and those of Deb and Crothers is quite remarkable.

CONCLUSIONS The results presented here show that the hyperspherical partial wave theory is capable of producing reasonable cross section results from very low energies to intermediate energies. If we recall the unequal energy sharing results also presented in Ref. [24] then the capability of the hyperspherical partial wave theory may be considered to have been well demonstrated. References [1] F. W. Byron Jr. and C. J. Joachain, Phys. Rev. A8, 1267 (1973); ibid 8, 3266 (1973); J. Phys. B10, 207 (1977). [2] S. Jones, D.H. Madison, A. Franz, and P.L. Altick, Phys.Rev. A 48, R22 (1993); D. A. Konovalov, J. Phys. B 27, 5551 (1994).


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