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A final state wave function of multiple scattering theory developed by Das and seal is utilized in the present study to calculate the triple differential cross sections (TDCS) for the ionization of metastable 3S state hydrogen atoms at incident electron energy of 250 eV with the exchange effects in the asymmetric coplanar geometry for various kinematic conditions. Our present calculation results are compared with the available hydrogenic ground state experimental data and other existing theoretical results. A good qualitative agreement is shown with those of compared results of the present study specifically with hydrogenic ground state experimental data and metastable 2S and 2P state with exchange effect results. These new results offer an extensive scope for experimental verification in such ionization process.

Bethe [

The first theoretical calculation of direct scattering amplitude of the TDCS for the coplanar asymmetric ionization of hydrogenic metastable 2S-state by electrons was calculated by Vučič et al. [

The purpose of our present work is to calculate the triple differential cross sections (TDCS) for the electron impact ionization of hydrogen atoms in the metastable 3S state for coplanar asymmetric geometry with exchange effects. The present results give an interesting good qualitative fitness with the hydrogenic ground state ionization experimental data and some other hydrogenic ground state theoretical results as well as hydrogenic metastable 2S state and 2P state results. The new observation created a new dimension in this field of research.

We have considered here the direct and exchange amplitude of the T-matrix element. The T-matrix element for ionization of hydrogen atoms by electrons [

T f i = 〈 Ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) | V i ( r ¯ 1 , r ¯ 2 ) | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (1)

Here the perturbation potential V i ( r ¯ 1 , r ¯ 2 ) is given by

V i ( r ¯ 1 , r ¯ 2 ) = 1 r 12 − Z r 2 (2)

For hydrogen atom nuclear charge (Z) = 1, r 1 and r 2 are the distance of the two electrons from the nucleus and r 12 is the distance between the two electrons. The initial channel unperturbed wave function is,

Φ i ( r ¯ 1 , r ¯ 2 ) = e i ⋅ p ¯ 2 ⋅ r ¯ 2 ( 2 π ) 3 / 2 φ 3 S ( r ¯ 1 ) = e i ⋅ p ¯ 2 ⋅ r ¯ 2 ( 2 π ) 3 / 2 ⋅ 1 81 3 π ( 27 − 18 r 1 + 2 r 1 2 ) e − λ 1 r 1 (3)

where

φ 3 S ( r ¯ 1 ) = 1 81 3 π ( 27 − 18 r 1 + 2 r 1 2 ) e − λ 1 r 1 (4)

and

λ 1 = 1 / 3

Equation (4) is the hydrogenic 3S-state wave function, p ¯ i is the incident electron momentum, Ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) is the final three-particle scattering state wave function with the electrons being in the continuum with momenta p ¯ 1 , p ¯ 2 . Co-ordinates of the two electrons taken to be r ¯ 1 and r ¯ 2 .

Here Ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) is approximate wave function and is given by,

Ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) = N ( p ¯ 1 , p ¯ 2 ) [ ϕ p ¯ 1 ( − ) ( r ¯ 1 ) e i ⋅ p ¯ 2 ⋅ r ¯ 2 + ϕ p ¯ 2 ( − ) ( r ¯ 2 ) e i ⋅ p ¯ 1 ⋅ r ¯ 1 + ϕ p ¯ ( − ) ( r ¯ ) e i P ¯ ⋅ R ¯ − 2 e i p ¯ 1 ⋅ r ¯ 1 + i p ¯ 2 ⋅ r ¯ 2 ] / ( 2π ) 3 (5)

where

r ¯ = r ¯ 2 − r ¯ 1 2 , R ¯ = ( r ¯ 2 + r ¯ 1 ) / 2 , p ¯ = ( p ¯ 2 − p ¯ 1 ) , P ¯ = ( p ¯ 2 + p ¯ 1 )

The normalization constant N ( p ¯ 1 , p ¯ 2 ) is given by

| N ( p ¯ 1 , p ¯ 2 ) | − 2 = | 7 − 2 [ λ 1 + λ 2 + λ 3 ] − [ 2 λ 1 + 2 λ 2 + 2 λ 3 ] + [ λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 1 + λ 2 λ 3 + λ 3 λ 1 + λ 3 λ 2 ] | (6)

here

λ 1 = e π α 1 2 Γ ( 1 − i α 1 ) , α 1 = 1 P 1

λ 2 = e π α 2 2 Γ ( 1 − i α 2 ) , α 2 = 1 P 2

λ 3 = e π α 2 Γ ( 1 − i α ) , α = − 1 P

The normalization constant N ( p ¯ 1 , p ¯ 2 ) is calculated numerically using Equation (6) and the approximated value of N ( p ¯ 1 , p ¯ 2 ) is 1.

ϕ q ¯ ( − ) ( r ¯ ) is the Coulomb wave function and is given by

ϕ q ¯ ( − ) ( r ¯ ) = e π α 2 Γ ( 1 + i α ) e i q ¯ ⋅ r ¯ F 1 1 ( − i α , 1 , − i [ q r + q ¯ ⋅ r ¯ ] )

For the electron impact ionization the parameters α 1 , α 2 and α are given below

α 1 = 1 p 1 for q ¯ = p ¯ 1 , α 2 = 1 p 2 for q ¯ = p ¯ 2 and α = 1 p for q ¯ = p ¯

Equation (1) becomes,

T f i = N ( p ¯ 1 , p ¯ 2 ) [ T B + T B ′ + T i − 2 T P B ] (7)

where

T B = 〈 Φ p ¯ 1 ( − ) ( r ¯ 1 ) e i p ¯ 2 ⋅ r ¯ 2 | V i | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (8)

T B ′ = 〈 Φ p 2 ( − ) ( r ¯ 2 ) e i p ¯ 1 ⋅ r ¯ 1 | V i | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (9)

T i = 〈 Φ p ¯ ( − ) ( r ¯ ) e i ⋅ P ¯ ⋅ R ¯ | V i | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (10)

T P B = 〈 e i p ¯ 1 ⋅ r ¯ 1 + i p ¯ 2 ⋅ r ¯ 2 | V i | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (11)

The direct scattering amplitude f ( p ¯ 1 , p ¯ 2 ) is the determined from

f ( p ¯ 1 , p ¯ 2 ) = − ( 2 π ) 2 T f i (12)

The exchange scattering amplitude is then approximated by

g ( p ¯ 1 , p ¯ 2 ) = f ( p ¯ 2 , p ¯ 1 ) (13)

After analytical calculations using Lewis Integral [

d 3 σ d Ω 1 d Ω 2 d E 1 = p 1 p 2 p i [ 3 4 | f − g | 2 + 1 4 | f + g | 2 ] (14)

Here E 1 is the energy of the ejected electron. Hence, in our present study we have calculated the TDCS with exchange effects, given by the Equation (14) using computer programming language.

We have discussed here in this section the ionization of metastable 3S state hydrogen atoms by electrons with exchange effect. The triple differential cross-sections (TDCS) at E i = 250 eV incident energy with the ejected electron energy of 5 eV is calculated. The results of exchange effects are displayed by the

Electron-hydrogen ionization from ground state theoretical results of Dal et al. [

In

Ejected angles ( θ 1 ) | B1 (2P) | B2(3S) |
---|---|---|

0 | 5.6923 | 1.1151 |

36 | 3.625 | 5.0858 |

72 | 1.46154 | 0.0752 |

108 | 6.4231 | 5.0380 |

144 | 6.8077 | 5.0135 |

180 | 8.3077 | 7.2945 |

216 | 6.6077 | 6.1394 |

252 | 7.19231 | 5.1394 |

288 | 7.7769 | 10.0288 |

324 | 5.15 | 5.0395 |

360 | 5.38462 | 0.0250 |

In

We consider for

In a similar way

From Figures 4-9 we compared the present exchange effects results with previous theoretical results like 2P and 2S metastable states [

When we increase our scattering angle in

In

In the

The present result of 3S-state exchange effects gives a good qualitative improvement comparing with the previous 2S and 2P state exchange effect results. The present result also show a similar conduct with the hydrogenic ground state result [

Finally, the scattering mechanism for the ionization of metastable 3S state with exchange effects for 250 eV incident electron energy is presented here in this study. The scattered electrons are described by a plane wave in the first Born term of Equation (5) whereas the ejected electrons are defined by a Coulomb wave. In the second term of the Equation (5) the scattered electrons are defined by the Coulomb wave while the ejected electrons are defined by the plane wave. The projectile electron interaction appeared in the third term shows almost similar behavior in the final channel wave function. The fourth term represents two plane waves for both ejected and scattered particles. The above results gives us a strong view of peaks both in recoil region and binary region. We can conclude that the present peak values gives us a good agreement with our compared experimental results as well as the theoretical results. In Our present study the measurements of peak values gives us the encouragement for further research in this field of interest. Moreover it needs more experimental works in this field for further investigation.

Our present calculation on the triple differential cross sections for ionization of atomic hydrogen by electron impact with exchange effects in metastable 3S-state exposes a thinkable additional structure of the cross-section curves for small momentum transfer in the ionization of the hydrogen atoms. The final state wave function ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) of Das and Seal gives a good qualitative result with the hydrogenic ground state experiment as well as with the BBK model of ground state hydrogen atoms. For good qualitative agreement, the present study are very encouraging for the future experiments which may play a vital role to give interesting and significant results in this field of research.

The computational works have been performed in the Simulation Lab of the Department of Mathematics, Chittagong University of Engineering and Technology Chittagong-4349, Bangladesh.

Noor, T. and Dhar, S. (2017) The Triple Differential Cross Sections for Electron Impact Ionization of Metastable 3s State Hydrogen Atoms with Exchange Effect. Open Journal of Microphysics, 7, 53-65. http://dx.doi.org/10.4236/ojm.2017.73004