Introduction

Advances in the use of synchrotron x-ray and $\gamma$-ray sources allow the possibility of making detailed observations of photon polarization effects in the scattering of hard photons by atomic targets. A correct theoretical analysis of this problem requires a quantum electrodynamic description with an accurate account of relativistic and retardation effects. General discussions of both theoretical and experimental results concerning elastic x-ray and $\gamma$-ray scattering by atoms can be found in [1,2] and references therein. Polarization effects in elastic photon scattering have been investigated numerically, with particular attention being paid to linear polarization effects [3]. In addition there have been more detailed investigations into the contributions of relativistic and higher multipole effects to elastic scattering [4,5], though these works mainly studied the angular distribution of scattered photons rather than polarization effects. A recent high precision experiment involving the scattering of x-rays from neon attained cross sections accurate at the level of 1% [6], though polarization effects were not considered.

The problem of photon polarization effects has been completely analyzed only for the case of photon scattering by a relativistic free electron (e. g. [7,8]). Here we concentrate on a specific beyond-dipole-approximation polarization effect that appears in the elastic scattering of photons by ground-state atoms, giving rise to finite circular dichroism (CD) when the scattered photon has a fixed suitable linear polarization. (The existence of CD in the case that the scattered photon has a fixed circular polarization is well known - see discussion below.) We understand CD as the difference between cross sections (which may generally describe any process involving an incoming photon) for different (right-handed and left-handed) circular polarizations of the incident photon, with all other observables being held fixed (including the observed polarization of the final photon, in the case of scattering).

Apparently, the possibility of CD effects in light scattering (for the case that the scattered photon has a fixed linear polarization) was first pointed out for dipole-forbidden scattering in 1980 [9], and it was studied in detail for optical frequencies in light scattering by gases in [10]. In that work a nonrelativistic treatment was used, with account of retardation effects being taken only in the first nonvanishing order. The relativistic case of CD effects in scattering of hard photons by hydrogen-like ions was first investigated numerically in 1987 [11], and later analytical results were obtained for the simplest case of atoms with closed shells [12]. Terms responsible for these effects were identified in [3] and numerical results were obtained, but the emphasis there was on linear polarization effects.

The use of circular dichroism to investigate orientations in atomic and molecular targets is longstanding [13,14,15]. The approach has typically involved measuring the differences between the cross sections for the photoabsorption of right-handed and left-handed circularly polarized photons in the optical regime. Recently the possibility of also using circular dichroism effects in inelastic scattering as a tool for investigating target orientations has been proposed [16,17]. In these works the authors point out that, even for the case of randomly oriented targets, there is still the possibility of finite CD effects. This they refer to as design-induced CD (existing in the case of a randomly oriented target), which would tend to mask the CD effects arising from the target having some definite orientation in space. The aim is to identify the conditions under which design-induced CD vanishes for a randomly oriented target, so that that any observed CD effects would then be a clear indication of target orientation.

The conclusion [16,17] (for the case of inelastic photon scattering) is that design-induced CD can exist if the scattered photon has a fixed circular polarization, but it vanishes completely if the scattered photon has a fixed linear polarization. A similar assertion has been made for elastic (coherent) photon scattering, namely that CD is not present for the case of randomly oriented targets when the scattered photon has a fixed linear polarization [18]. We wish to point out that these conclusions are limited by the approximations made in the analyses: a more general analysis reveals that design-induced CD can occur when the scattered photon has a fixed linear polarization. We examine these CD effects and determine when they will vanish. For the purposes of our discussion we distinguish between TYPE-C CD (where the scattered photon has a fixed circular polarization) and beyond-dipole-approximation TYPE-L CD (where the scattered photon has a fixed linear polarization). Of course in the general case of a fixed elliptical polarization of the scattered photon being observed, both types of CD may be present.

We point out that, given our assumptions, the parameter describing TYPE-C CD effects is also observable in measurements involving only linear polarizations, in contrast to the parameter describing TYPE-L CD effects [see Eq. (3) in next section]. Though both of these polarization effects fall under the general definition of circular dichroism (i.e. leading to differences in cross sections for left- and right-handed circularly-polarized incident radiation, with all other details of observation the same), they are nevertheless distinct effects, providing different information about the target wavefunction.

Here we specifically consider TYPE-L CD effects in the elastic scattering of x-rays by bound atomic electrons. We suppose the target atoms to be randomly oriented. We employ a fully relativistic approach for the description of atomic electrons, retaining all significant multipoles in the electron-photon interaction. We will present numerical estimates of the importance of TYPE-L CD effects for three ground-state atoms, with Z=29, Z=54, and Z=92. Attention is given to determining the regime where these effects are large and may be experimentally observable. The ideal experiment for observing such effects involves measuring the difference between the scattering cross sections for right-handed (RHC) and left-handed (LHC) circularly polarized incident photons, observing a fixed linear polarization (making an angle of 45^o with respect to the scattering plane) for the scattered photon.

The parameter determining this CD effect is also responsible for the appearance of elliptically polarized scattered photons from a linear polarized incident photon beam (even when the scattering is from an s-state target). One could therefore also measure this parameter by designing an experiment with a fixed linear polarization (making an angle of 45^o with respect to the scattering plane) for the incoming photon, measuring the difference between scattering cross sections for scattered RHC and LHC photons.

In Sec. II we discuss the general form of the scattering cross section for the case of elastic photon scattering from ground state-atoms under the assumption that polarization effects in the target are not being observed (scattering from s-state targets). A brief discussion is given regarding the consequences of the symmetry of the cross section under time reversal. Numerical results are given in Sec. III for TYPE-L CD effects in elastic photon scattering from ground-state atoms with Z=29, Z=54 and Z=92. These results suggest the situations in which the TYPE-L CD effect is most likely to be experimentally observable. General features are explained in terms of the basic scattering amplitudes. We summarize our conclusions in Sec. IV.