Raman scattering of photons by hydrogenlike ions I: A semirelativistic and nondipole approach

Abstract

Going beyond the usual nonrelativistic treatments of Raman scattering of light by a ground-state electron in hydrogenic ions, we propose in this work a semirelativistic and nondipole approach in the case of inelastic process under the ionization threshold. This scheme is based, on the one hand, on the discrete semirelativistic Coulomb Sturmian basis sets, and on the other hand, on the direct Foldy–Wouthuysen interaction Hamiltonian derived recently from the Dirac equation with electromagnetic potentials (Hinschberger and Hervieux 2012) and confirmed by us to seventh order in powers of $1∕m$. It includes the paramagnetic, the diamagnetic, the magnetic field–spin and the electric field–spin interactions. Moreover, the photon fields are expanded in electric and magnetic multipoles and nondipole effects are fully taken into account. On utilizing these ingredients, we build in detail the multipole expansion of the semirelativistic Raman scattering cross section when the photon energy is under the ionization threshold. For each type of the allowed multipole components, namely EE, MM and EM in the standard notations, suitable closed forms of total scattering cross sections are calculated for numerical computations. Such analytical expressions enable the evaluation of the contribution of each of these interaction operators and field multipoles in the scattering process.

Introduction

The scattering of light by atomic targets has been studied for a long time and remains a subject of strong experimental and theoretical interest [1] to make its use as a tool to answer fundamental questions about the properties of matter. Among various competing two-photon processes in this physical phenomenon is the so-called Raman effect [2], one of the primary processes responsible for the attenuation of radiation in matter. In atomic physics, it involves the excitation of the ground-state atom to a higher energy bound state. If the atomic system is initially excited, then Stokes and anti-Stokes Raman scattering (RS) may occur. In this paper we are interested in hydrogenlike targets which have been customarily investigated by many authors during the past decades. They represent the only case in which exact formulas may be obtained. To be specific the most extensive theoretical calculations have been worked out within the framework of conventional nonrelativistic perturbation theory for either ground state or $2s$ metastable state hydrogenic atoms. They stem from the chief difficulty in handling the infinite sums running over the entire atomic spectrum which enter the Schrödinger–Coulomb–Green function (SCGF) [3] involved in the second order matrix elements. Whence the analytical approximation schemes which have been suggested to accurately determine the Raman cross section (RCS) in the dipole approximation. The first attempt was made by Zernik and Klopfenstein [[4], [5], [6], [7], [8]]. They used the Kramers–Heisenberg–Waller (KHW) dispersion formula [[9], [10], [11]] in the length gauge and an implicit summation technique developed by Schwartz and Tieman [[12], [13]], and employed by Mittleman and Wolf [14], to carry out the sum over intermediate states for the metastable $2s$ atomic hydrogen. More precisely this method which was firstly introduced by Dalgarno and Lewis [[15], [16]] and reformulated by Schwartz and Tieman, consists of reducing sums over intermediate states to the integration of an inhomogeneous differential equation. It was also applied by Sadeghpour and Dalgarno [17]. Gontier and Trahin [18] extended this treatment to higher-order processes, and performed calculations for the anti-Stokes Raman scattering from the $2s$ excited state. Later on, Rapoport and Zon [19], and Zon et al. [20] studied the case in which the low lying transitions $1s\to 2s$ and $1s\to 3d$ are induced. They proposed a method based on a convenient integral representation for the radial part of the SCGF so as to avoid the problem of cumbersome sums over a complete set of intermediate states. These authors and others as Klarsfeld [21] obtained in the length gauge the resulting cross section in closed form in terms of hypergeometric functions. In order to get quantitative information concerning the magnitude of the quantity in question and its dependence on incident photon energy, Saslow and Mills [22] re-examined the work of Rapoport and Zon, and instead, they used the KHW formula in the velocity gauge to evaluate explicitly the Raman matrix elements. Another approach based on a Sturmian representation of the SCGF introduced by Hostler [23], was used by Heno [24] to evaluate Raman-like scattering processes in metastable hydrogenic atoms. This approach is known to yield, in a natural and straightforward calculation, analytical expressions of the desired amplitudes, with summation of rapidly convergent series.

However, the limitations of the aforementioned approaches became apparent especially for multiply charged ions which require to take into account relativistic effects. To the best of our knowledge, only a few works have been done so far, starting with early calculations of Manakov et al. [25]. After that pioneering work, the fully relativistic investigations conducted in this direction by Szymanowski et al. [[26], [27]], Zapryagaev [28], and Jahrsetz et al. [29] are based on the Dirac–Coulomb–Green function expressed in terms of Laguerre and Whittacker functions. On the other hand, Hinschberger and Hervieux [30] reported recently an interaction Hamiltonian derived by means of the Foldy–Wouthuysen (FW) transformation from the Dirac equation of a single electron atom coupled to a time-dependent electromagnetic field. This semirelativistic Hamiltonian includes the paramagnetic, the diamagnetic, and the direct field–spin couplings for electronic states. It appears that, on increasing the nuclear charge of ions, this expression offers the possibility of getting a detailed study of RS by isolating the contribution and the relative importance of each term. Thus, there should be a distinct advantage in using this important result over the fully relativistic approaches. It is our purpose here to base our calculation scheme on the FW photon-atom interaction Hamiltonian and to use of semirelativistic Coulomb Sturmian functions that are particularly appropriate in that context; this may give more insight into that inelastic process. It is referred to as the semirelativistic and nondipole approach (SRNDA).

The organization of this paper is as follows. Section 2 describes the essentials of semirelativistic Coulomb Sturmian functions used to construct the atomic states and the Coulomb–Green function. In Section 3, we present the Foldy–Wouthuysen interaction Hamiltonian. Section 4 describes the general formula of the multipole expansion of the semirelativistic Raman scattering cross section. Sections 5, 6, and 7 are devoted to the derivation of transition amplitudes for electric–electric, magnetic–magnetic and electric–magnetic multipoles, respectively. Details for all these sections are included in the Appendix A The sixth and seventh orders of the direct field–spin interaction, Appendix B Useful formulas for operations on the multipoles of the photon fields, Appendix C Basic relations for the computation of matrix elements of the direct field–spin interaction Hamiltonian, Appendix D Electric–electric multipole type contribution for terms involving the diamagnetic interaction in the modulus squared of the transition amplitude, Appendix E Magnetic–magnetic multipole type contribution for terms involving the diamagnetic interaction in the modulus squared of the transition amplitude, Appendix F Electric–magnetic multipole type contribution for terms involving the diamagnetic interaction in the modulus squared of the transition amplitude, Appendix G Multipole decomposition of the nonrelativistic and nondipole Raman scattering cross section. Finally, in Section 8 we make some concluding remarks. Notice that, unless otherwise stated explicitly, atomic units ($a.u.:ħ=m=e=1,c=1∕\alpha$) are used throughout, and $\alpha$ stand for the fine structure constant. $a_0$ and $a_0^2$ are the Bohr radius and the unit of cross section, respectively.