Efficient calculation of Rayleigh and Raman scattering

Abstract

We present two methods for computing the Rayleigh and Raman scattering cross sections for photon scattering on atomic hydrogen, or hydrogen like systems. Both methods are applicable for incident photon energies above the ionization threshold. The first method implements the well known Gaussian quadrature approach to deal with principle value integration, and relies on evaluation of the exact eigenfunctions of hydrogen. The second, more computationally efficient approach, uses a finite L 2 basis expansion of the target, and applies complex exterior scaling methods to accurately account for the contribution of the intermediate continuum states. This method is much more general in that it does not rely on analytic solutions to the Hamiltonian, nor evaluation of any special functions, and is expected to be applicable to more complex systems where exact wavefunctions are cumbersome to evaluate. Both methods are in complete agreement with previous work based on analytical representations of the Green's function or the dipole matrix elements. Rayleigh, Raman, and photoionization cross sections for scattering on the first few excited states of atomic hydrogen are presented and compared with previous results where available.

Rayleigh and Raman Scattering from Alkali Atoms

“…If |t is a continuum state, then contour integration is used to avoid any unphysical results. McNamara et al [16] presented a more detailed discussion of the resonance behaviour. For Raman scattering, intermediate states with an energy between the initial and final state energies are not accessible via the absorption and emission of a single photon.…”

“…In this section, we summarise the two computational methods used and the changes from their previous implementation for photon scattering on atomic hydrogen [16]. Section 3.1 describes a direct approach for calculating the pole terms in the tensor expansion coefficients, while, in Section 3.2, we describe an approach that utilises analytic continuation into the complex plane to avoid dealing with the pole terms associated with incident photon energies above the ionisation threshold.…”

“…where σ T = 8πr 2 0 /3 ≈ 6.652 × 10 −29 m 2 is the Thompson cross section. The equivalent expressions with the velocity form of the dipole operator are given in [16]. We consider scattering with and without the inclusion of the spin-orbit interaction for the calculation of the target wave functions.…”

“…Previously, for hydrogen [16], this was done while using exact bound and continuum eigenstates. For the alkali atoms, box-based bound states [30] and distorted Coulomb continuum waves are used.…”

“…Recently, we have developed two computational methods for calculating Rayleigh and Raman scattering on hydrogen-like atoms that are valid for incident photon energies above and below the ionization threshold [16]. The first method involves the direct numerical calculation of KHW matrix elements and utilises principal value integration to deal with pole terms.…”