^{1}

^{*}

^{1}

^{2}

Triple differential cross sections (TDCS) are estimated for the ionization of metastable 3d-state hydrogen atoms by electron at 250 eV for various kinematic conditions pursuing a multiple scattering theory. The present new results are compared with the theoretical results of hydrogenic different metastable states as well as the hydrogenic ground state experimental data. Obtained new finding results are in good qualitative agreement with those of compared theories. The present results give an immense opportunity for experimental trial in the field of ionization problems.

The theoretical non-relativistic studies for the atomic ionization by fast particle were first treated by Bethe [

In the field of electron, impact ionization is to develop a general theoretical framework, which will provide the accurate ionization cross sections for many atoms over a practically relevant impact energy range. Due to its perplexity, the fully quantum mechanical treatment of atomic ionization by electron is possible for the artless cases of hydrogen atom. In this work, atomic hydrogen is used as target in order to perceive the ionization mechanism of atomic system by electron impact energy.

Hydrogenic metastable 3d state is an excited state which has a relatively long lifetime than the other excited states. A metastable sate has a higher energy than the ground state. The lifetime of excited state is given by [

T i = ( ∑ A i j ) − 1

where A i j is Einstein A coefficient. The lifetime of metastable 3d state of hydrogen atom is 2.3 × 10 − 7 s .

A multiple scattering theory [

The existent new study results will present a new dimension on ionization of hydrogenic metastable states. Current results are compared with previous related theories [

Electron-impact ionization cross section is estimated by taking the ratio of the number of ionization elements per unit time and per unit target to the incident electron flux.

Ionization of atomic hydrogen by electron in most elaborate form is presently available of following type

e − + H ( 3d ) → H + + 2 e − (1)

Here 3d denotes the hydrogenic metastable state and has been attained in the coplanar geometry by examining TDCS measured in (e, 2e) coincidence experiments.

The triple differential cross section is denoted by the symbol d 3 σ d Ω 1 d Ω 2 d E 1 .

Finally, the total ionization cross section is obtained by integrating over all outgoing scattering angles and energies, and depends only on E i , the incident electron energy.

The direct T-matrix element for ionization of hydrogen atoms by electrons, following Das and Seal [

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

here, r ¯ 1 and r ¯ 2 represent the coordinates of the atomic active electron and the incident electron, ( p ¯ 1 , p ¯ 2 ) and ( E 1 , E 2 ) represent the momenta and energies of the two electrons in the final state and ( p ¯ i , E i ) are the momentum and the energy of the incident electron.

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

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

The nuclear charge of the hydrogen atom is Z = 1, r 1 and r 2 are the distance of the two electrons from the nucleus and r 12 is the distance between two electrons.

We have the initial channel unperturbed wave function is

Φ i ( r ¯ 1 , r ¯ 2 ) = e i p ¯ i ⋅ r ¯ 2 ( 2 π ) 3 / 2 ϕ 3 d ( r ¯ 1 ) .

where

ϕ 3 d ( r ¯ 1 ) = 1 81 6 π ( r 1 2 ) ( 3 cos 2 θ − 1 ) e − r 1 / 3 = 1 324 3 π 2 ( r 1 2 ) ( 3 cos 2 θ − 1 ) e − λ 1 r 1 .

Φ i ( r ¯ 1 , r ¯ 2 ) = 1 324 3 π 2 ( r 1 2 ) ( 3 cos 2 θ − 1 ) e − λ 1 r 1 (3)

here λ 1 = 1 3 , ϕ 3 d ( r ¯ 1 ) is the hydrogenic 3d-state wave function and Ψ f ( − ) ( r ¯ 1 , r ¯ 2 ) is approximate wave function 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 ] / ( 2 π ) 3 (4)

where

r ¯ = r ¯ 2 − r ¯ 1 2 , R ¯ = r ¯ 1 + r ¯ 2 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 ] | (5)

where

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

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

with

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 ¯

Now Equation (2) becomes

T f i = T B + T B ′ + T i − 2 T P B (6)

where

T B = 〈 ϕ p 1 ( − ) ( r ¯ 1 ) e i p ¯ 2 ⋅ r ¯ 2 | V i | Φ i ( r ¯ 1 , r ¯ 2 ) 〉 (7)

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

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

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

here Equation (6) is called First Born term and it may be written as

T B = 1 324 3 π 2 〈 ϕ p 1 ( − ) ( r ¯ 1 ) e i p ¯ 2 ⋅ r ¯ 2 | 1 r 12 − 1 r 2 | e i p ¯ i ⋅ r ¯ 2 ( r 1 2 ) ( 3 cos 2 θ − 1 ) e − r 1 λ 1 〉 = 1 324 3 π 2 ∫ ϕ p 1 ( − ) ∗ ( r ¯ 1 ) e − i p ¯ 2 ⋅ r ¯ 2 ( 1 r 12 − 1 r 2 ) e i p ¯ i ⋅ r ¯ 2 ( r 1 2 ) ( 3 cos 2 θ − 1 ) e − λ 1 r 1 d 3 r 1 d 3 r 2

T B = 1 324 3 π 2 ∫ ϕ p 1 ( − ) ∗ ( r ¯ 1 ) e − i p ¯ 2 ⋅ r ¯ 2 1 r 12 e i p ¯ i ⋅ r ¯ 2 r 1 2 ( 3 cos 2 θ − 1 ) e − λ 1 r 1 d 3 r 1 d 3 r 2 − 1 324 3 π 2 ∫ ϕ p 1 ( − ) ∗ ( r ¯ 1 ) e − i p ¯ 2 ⋅ r ¯ 2 1 r 2 e i p ¯ i ⋅ r ¯ 2 r 1 2 ( 3 cos 2 θ − 1 ) e − λ 1 r 1 d 3 r 1 d 3 r 2

T B = tb1 + tb2 (11)

where

tb1 = 1 324 3 π 2 ∫ ϕ p 1 ( − ) ∗ ( r ¯ 1 ) e − i p ¯ 2 ⋅ r ¯ 2 1 r 12 e i p ¯ i ⋅ r ¯ 2 r 1 2 ( 3 cos 2 θ − 1 ) e − λ 1 r 1 d 3 r 1 d 3 r 2

tb2 = − 1 324 3 π 2 ∫ ϕ p 1 ( − ) ∗ ( r ¯ 1 ) e − i p ¯ 2 ⋅ r ¯ 2 1 r 2 e i p ¯ i ⋅ r ¯ 2 r 1 2 ( 3 cos 2 θ − 1 ) e − λ 1 r 1 d 3 r 1 d 3 r 2

Using the Lewis integral [_{B} of Equation (7).

Similarly, we have calculated analytically the above Equations (8), (9) and (10) for second Born results using the Lewis integral [

d 3 σ d Ω 1 d Ω 2 d E 1 = p 1 p 2 p i | T f i | 2 (12)

In this section, We have calculated in this work the triple differential cross-sections (TDCS) at high incident energy E i = 250 eV for various ejected angles θ 1 and fixed scattering angles θ 2 . Triple differential cross sections for ionization of metastable 3d-state hydrogen atoms by incident electron are presented at different energies. The existent results are compared with the ionization of hydrogen atoms by electrons from ground state theoretical results of Dal et al. [

In this study, the ejected angle θ 1 varies from 0˚ to 360˚ considered as horizontal axis where scattering angles θ 2 is fixed and referred as vertical axis. The present results of hydrogenic metastable 3d state by electron are designed corresponding to the different scattering angles θ 2 = 30 ∘

The incident electron energy of E i = 250 is taken here. In all figures, θ 1 ( 0 ∘ - 150 ∘ ) and ϕ = 0 ∘ is considered as recoil region while θ 1 ( 150 ∘ - 360 ∘ ) and ϕ = 180 ∘ is referred as binary region.

In

In

In

In

In

In

In

In

In

Finally, Metastable 3d-state is an excited state of an atom or other system with a longer lifetime than the other excited states. The lifetime of 3d state of hydrogen atom is 2.3 × 10 − 7 . The lifetime of excited state is given by [

Ejected angle (θ_{1}) | 2S | 3S | 3P | 3d |
---|---|---|---|---|

0 | 1.6501 | 5.0059 | 11.8679 | 1.6989 |

36 | 0.9001 | 6.2634 | 0.4252 | 5.3959 |

72 | 1.3875 | 1.2531 | 2.1485 | 1.5032 |

108 | 0.1952 | 7.5195 | 3.5834 | 4.3023 |

144 | 0.5401 | 6.2167 | 9.7624 | 1.0020 |

180 | 0.8989 | 10.0276 | 0.3155 | 2.0506 |

216 | 2.3450 | 9.5268 | 0.7248 | 0.9956 |

252 | 0.3201 | 10.0284 | 8.4297 | 0.8921 |

288 | 110.00 | 12.5360 | 5.0455 | 3.1015 |

324 | 4.1569 | 6.2681 | 1.7845 | 0.8012 |

360 | 1.7890 | 1.2506 | 0.0850 | 5.4074 |

happened due to the change of the hydrogenic metastable states ionization by electrons. It is remarked that, the peak pattern of the energy spectrum as obtained from our present study is closer to the compared results [

Here a table (please see

The present estimation reveals imaginable additional formation of the cross-section curves for intermediate momentum transfer in the ionization of the hydrogen atoms in the hydrogenic metastable 3d-state at 250 eV electron impact energy. It is remarked that, the implementation of the final state wave function of Das and Seal [

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

The authors declare no conflicts of interest regarding the publication of this paper.

Banerjee, S., Dhar, S. and Hoque, A. (2018) Triple Differential Cross-Sections for Ionization of H(3d) by Incident Electron. Open Journal of Microphysics, 8, 30-41. https://doi.org/10.4236/ojm.2018.84005