Theory

We consider TYPE-L CD polarization effects in the elastic scattering of a photon by an atomic target. It is assumed that the polarization properties of the target are not observed. The target is effectively treated as if it had zero total angular momentum, corresponding to a summation, weighted according to the number of electrons present, over the magnetic substates of each subshell at the level of the scattering amplitude. This approach (exact for closed-subshell atoms) is justified since most of the atomic electrons are in closed subshells, with the scattering from inner shells being dominant (except for small angles where all subshells contribute to scattering, but where the effects under consideration here are not important, in fact vanishing in the forward direction). More details of this approach can be found in [1]. Results that go beyond this approximation, performing the more correct averaging over magnetic substates at the level of the cross section, indicate that the corrections for many-electron atoms tend to be small [19].

The initial (final) photon has momentum $\bbox{k}_1$ ( $\bbox{k}_2$) and polarization $\bbox{\epsilon}_1$( $\bbox{\epsilon}_2$). Given our assumption of scattering from s-state type targets these are the only vectors in the problem. We have ${\hat{\bbox{k}}}_i = \bbox{k}_i / \vert \bbox{k}_i \vert$ and, since we are considering elastic scattering, $\vert \bbox{k}_1 \vert = \vert \bbox{k}_2 \vert = \omega / c$. The Stokes parameters of incident (scattered) photons are denoted by $\xi_1^{(1)}$, $\xi_2^{(1)}$ and $\xi_3^{(1)}$ ( $\xi_1^{(2)}$, $\xi_2^{(2)}$ and $\xi_3^{(2)}$). Note that $\xi_1^{(i)}$ corresponds to linear polarization making an angle of 45^o with respect to the scattering plane, $\xi_2^{(i)}$ corresponds to circular polarization, and $\xi_3^{(i)}$ corresponds to linear polarization parallel or perpendicular to the scattering plane, for the incident (i=1) and scattered (i=2) photons. Here we follow the notation of [8] for the Stokes parameters - this notation differs from that used in [3], as is explained in the Appendix.

The elastic scattering amplitude A, with the assumptions above, can be written in terms of two (complex) invariant amplitudes M and N [1],

$$A = (\bbox{\epsilon}_1 \cdot \bbox{\epsilon}_2^*) \; M + (\b... ...x{k}}}_2) (\bbox{\epsilon}_2^* \cdot {\hat{\bbox{k}}}_1) \; N,$$ (1)

which depend only on the photon energy $\omega$ and $\theta$, the scattering angle between $\bbox{k}_1$ and $\bbox{k}_2$. (Note that in the electric dipole approximation the N amplitude vanishes, and the M amplitude is independent of $\theta$, so that the only dependence on $\theta$ is through $\bbox{\epsilon}_1 \cdot \bbox{\epsilon}_2^*$). This expression for the scattering amplitude leads to the scattering cross section

 

$${d \sigma \over d \Omega} \vert M\vert^2 \vert\bbox{\epsilon}_1 \cdot \bbox{\epsilon}_2^*\v... ...\cdot {\hat{\bbox{k}}}_2) (\bbox{\epsilon}_2 \cdot {\hat{\bbox{k}}}_1) \right ]$$ =

$$+ \; \; 2 \; \text{Im} \;(M N^{*}) \; \text{Im} \left [ (\bbox{\e... ...cdot {\hat{\bbox{k}}}_2) (\bbox{\epsilon}_2 \cdot {\hat{\bbox{k}}}_1) \right ].$$ (2)

Alternatively the scattering cross section (again for the case of an s-state target) can be expressed in terms of the Stokes parameters and four real photon-polarization-independent parameters d_i

$${d \sigma \over d \Omega} = d_1 + d_1 \xi_3^{(1)} \xi_3^{(2)... ...} ) + d_4 (\xi_1^{(1)} \xi_2^{(2)} - \xi_2^{(1)} \xi_1^{(2)}),$$ (3)

The four real parameters d_i are given in terms of the M and N amplitudes as

 

$${1 \over 4} \left [ \vert N\vert^2 \sin^4 \theta - 2 \; \text{Re} \; (M N^*) \cos \theta \sin^2 \theta + \vert M\vert^2 (1+\cos^2 \theta) \right ],$$ d₁ =

$${1 \over 4} \left [ \vert N\vert^2 \sin^4 \theta - 2 \; \text{Re}... ...(M N^*) \cos \theta \sin^2 \theta - \vert M\vert^2 (1-\cos^2 \theta) \right ] ,$$ d₂ =

$${1 \over 2} \left [\vert M\vert^2 \cos \theta - \text{Re} \; (M N^*) \sin^2 \theta \right ] ,$$ d₃ =

$${1 \over 2} \left [\text{Im} \; (M N^*) \sin^2 \theta \right ].$$ d₄ = (4)

The unpolarized cross section (averaging over incident photon polarizations, summing over final photon polarizations), is given solely in terms of the parameter d₁ (all the Stokes parameters vanish in this case)

$${d \sigma \over d \Omega} = 2 d_1.$$ (5)

Thus the parameter d₁ can be determined without any polarization measurements. The parameters d₂ and d₃ can in principle be determined from measurements involving only linear polarizations - a discussion of this as well as numerical results can be found in [3], though the notations and choices of invariant amplitudes differ, as described in the Appendix.

Our interests here are the polarization effects associated with the parameter d₄, appearing in the last term in Eq. (3). These effects are (1) TYPE-L CD (when $\xi_1^{(2)}$ is non-zero), and (2) the appearance of elliptically polarized scattered photons for the case of linearly polarized incident photons (when $\xi_1^{(1)}$ is non-zero). Note that TYPE-C CD effects, involving circular polarizations of both incoming and scattered photons (thereby involving $\xi_2^{(1)} \xi_2^{(2)}$) are determined by the parameter d₃. Looking at Eq. (4) we see that for the parameter d₄ to be finite requires the beyond-dipole-approximation N amplitude, which enters physical observables modulated by a $\sin^2 \theta$ behavior. Thus TYPE-L CD is only important when retardation matters; it vanishes at forward and backward angle and has a geometric maximum at 90^o (the actual maximum shifts towards forward angle at high energy). By contrast d₃, characterizing TYPE-C CD, can still be finite even when the N amplitude vanishes or is neglected. A common approximation for Rayleigh scattering in the x-ray regime is to describe the M amplitude using form factors and (angle-independent) anomalous scattering factors, neglecting the N amplitude altogether [20]. Polarization effects involving d₄ are not present in this approximation.

We now return to the assertions in [16,17,18], mentioned earlier, that CD effects in scattering from randomly oriented systems vanish completely if one chooses to have a fixed linear polarization for the final photon. Though this is true within the approximations used in those analyses [nonrelativistic dipole approximation and A² approximation, neglecting $\bbox{p} \cdot \bbox{A}$ terms in the photon-electron interaction, $H_{\mbox{int}} = (e^2 / 2 m c^2) A^2 - (e / mc) \bbox{p} \cdot \bbox{A}$], in the more general case this condition of final linear polarization only suffices to exclude TYPE-C CD effects. TYPE-L CD effects can still be present if $\xi_1^{(2)} \ne 0$. In fact we see from Eq. (3) that CD effects in the case of randomly oriented targets vanish completely only if one has a final photon linear polarization parallel or perpendicular to the scattering plane.

We conclude this section by making some observations on the implications of the fundamental symmetry of time reversal for these effects. Consider the action of time reversal on the invariant amplitudes and the photon variables (and the Stokes parameters):

$$M,N \leftrightarrow M^{*},N^{*}; \;\;\;\; \bbox{k}_1 \leftrig... ...{(1)} \leftrightarrow \xi_1^{(2)}, \xi_2^{(2)}, \xi_3^{(2)}.$$ (6)

The combination of Stokes parameters that enters the d₄term of Eq. (3) is odd under this symmetry (T-ODD), in contrast to the combinations associated with all the other terms. Therefore the parameter d₄ must also be T-ODD in order that the cross section be invariant under the time reversal operation. (The other three parameters must be T-EVEN.) Therefore the requirement that the parameter d₄ is T-ODD (and real) implies that it should be solely the result of interference between real (T-EVEN) and imaginary (T-ODD) parts of the invariant amplitudes.

The optical theorem relates the imaginary part of the elastic scattering amplitude at forward angle to the total cross section for absorption and scattering. If one writes the optical theorem in an expansion in the fine structure constant $\alpha$, one relates the imaginary part of the 2nd-order amplitude for elastic scattering to the total cross section for absorptive scattering in lowest order (including photoeffect, bound-bound transitions and bound-electron pair production, depending on the energy involved). From this viewpoint polarization effects involving d₄ can be thought of (in leading order) as dissipation-induced effects related to the existence of dissipation channels (if they are present). A consequence of this observation is the lack of TYPE-L CD in scattering by a free electron in lowest order (as can be seen from the well-known Klein-Nishina formula; see e. g. [8]), where there is no dissipative channel - a free electron cannot absorb a photon.