[Paper Review] Laser-Assisted Semi Relativistic Excitation of Atomic Hydrogen by Electronic Impact

S. Taj, B. Manaut+2 more|arXiv (Cornell University)|Nov 2, 2012

Atomic and Molecular Physics2 references3 citations

TL;DR

This paper presents an analytical semirelativistic treatment of electron-impact excitation of atomic hydrogen from the 1s to 2s state, both with and without a laser field, using the first Born approximation and Darwin wave functions. It demonstrates that nonrelativistic theory fails significantly beyond 2700 eV, while the semirelativistic approach agrees well with experimental data and verifies the sum rule for photon exchange up to ±50 photons in the nonrelativistic regime and ±13,945 in the relativistic regime.

ABSTRACT

The excitation of H ($1s-2s$) by electron impact in the presence and in the absence of the laser field is studied in the framework of the first Born approximation. The angular variation of the laser-assisted differential cross section (DCS) for atomic hydrogen by electronic impact is presented at various kinetic energies for the incident electron. The use of Darwin wave function as a semirelativistic state to represent the atomic hydrogen gives interesting results when the condition $z/c\ll1$ is fulfilled. A comparison with the non relativistic theory and experimental data gives good agreement. It was observed that beyond (2700 $eV$) which represents the limit between the two approaches, the non relativistic theory does not yield close agreement with our theory and that, over certain ranges of energy, it can be in error by several orders of magnitude. The sum rule given by Bunkin and Fedorov and by Kroll and Watson \cite{22} has been verified in both nonrelativistic and relativistic regimes.

Motivation & Objective

• To develop an analytical semirelativistic model for electron-impact excitation of atomic hydrogen in the presence of a laser field.
• To assess the validity of nonrelativistic approximations at high incident electron energies.
• To verify the photon number sum rule in both nonrelativistic and relativistic regimes.
• To compare semirelativistic results with nonrelativistic theory and experimental data.
• To examine angular dependence and photon exchange effects in laser-assisted scattering.

Proposed method

• Use of the first Born approximation with exact Dirac spinors for the incident and scattered electrons.
• Employment of the semirelativistic Darwin wave function for the 1s and 2s states of hydrogen, valid when z/c ≪ 1.
• Incorporation of the laser field via the Volkov wave function formalism for exact electron-laser interaction.
• Derivation of closed-form analytical expressions for the unpolarized differential cross section (DCS) in both laser-on and laser-off conditions.
• Numerical evaluation of the DCS as a function of scattering angle and photon exchange number (s).
• Verification of the sum rule by checking convergence of the DCS envelope across varying net photon numbers (s).

Experimental results

Research questions

• RQ1 How does the semirelativistic differential cross section for H(1s→2s) excitation compare with nonrelativistic predictions at high incident electron energies?
• RQ2 At what energy does the nonrelativistic approximation break down for this process, and by how much does it deviate?
• RQ3 How do laser-assisted differential cross sections vary with the number of absorbed/emitted photons, and what is the cutoff range for significant contributions?
• RQ4 Does the sum rule for photon exchange hold in both nonrelativistic and relativistic regimes, and is it numerically verified in this model?
• RQ5 What is the role of the Darwin wave function in enabling exact analytical solutions within the semirelativistic framework?

Key findings

• The semirelativistic differential cross section (SRDCS) shows excellent agreement with experimental data at 200 eV, matching nonrelativistic results in the low-energy regime.
• Beyond 2700 eV, the nonrelativistic theory diverges significantly from the semirelativistic model, with errors reaching several orders of magnitude.
• In the nonrelativistic regime (γ ≈ 1.0053), the laser-assisted SRDCS converges to the non-laser case at approximately ±50 net photons exchanged.
• In the relativistic regime (γ = 2.0), the convergence of the DCS envelope occurs at approximately ±13,945 net photons exchanged, indicating a much broader photon exchange window.
• The sum rule for photon exchange is numerically verified in both regimes, with the DCS envelope sharply dropping beyond the cutoffs at ±50 and ±13,945 photons.
• The analytical formalism reveals a direct mathematical analogy between the laser-off and laser-on differential cross sections, enabling consistent comparison across regimes.