Numerical results

As discussed in the previous section, the polarization effects being considered here are determined by the parameter d₄, which specifies the magnitude of terms involving the Stokes parameters $\xi_2^{(i)}$, describing circular polarization, and $\xi_1^{(i)}$, describing the linear polarization making an angle of 45^o with respect to the scattering plane. Effects involving d₄ are maximized when one photon is circularly polarized and the other has a linear polarization making an angle of 45^o with respect to the scattering plane. In this case of maximum TYPE-L CD the cross section becomes

$${d \sigma \over d \Omega} = d_1 \pm d_4,$$ (7)

$\pm$ as $\xi_1^{(2)}$ and $\xi_2^{(1)}$ are of opposite (same) sign, where $\xi_1^{(2)}$ is fixed to be +1 or -1, and $\xi_2^{(1)} =$+1 (RHC incident radiation) or -1 (LHC incident radiation); all the other Stokes parameters vanish.

Since d₄ is generally fairly small compared with d₁, the cross section to be measured is always of the order of d₁. Therefore in order to observe the effect one has to be able to measure a cross section of the order of d₁, and measure it sufficiently accurately to be able to detect the influence of d₄ in Eq. (7). For this reason we have chosen to present our results by giving the magnitude of d₁, and then expressing d₄ as a percentage of d₁.

It is not in fact the case that all four parameters d_i (i=1... 4) are truly independent [3]. There exists a relation between them,

 

d₁² = d₂² + d₃² + d₄², (8)

related to the fact that the overall phase of the scattering amplitude is not an observable. In principle, if all the other parameters are known, then d₄ can be determined from Eq. (8), apart from its overall sign. That only a sign is left undetermined is a consequence of the special case of treating the target as having no angular momentum; in the general case there will be undetermined amplitudes. However, since the parameter d₄ tends to be small compared to the others (which can all be of the same order of magnitude), it is preferable to have a direct calculation, and a direct observation, of this parameter.

We present numerical results for d₁ and d₄ for ground-state atoms with Z=29, Z=54 and Z=92. We consider all scattering angles and all energies for which the effects appear to be significant. Calculations were performed using the S-matrix approach in independent particle approximation (see [1] and references therein). Electron orbitals were obtained in a Dirac-Slater type central potential. This is a fully relativistic calculation, retaining all significant multipoles in the photon-atom interaction.

Figure VIIA shows a contour plot of the magnitude of d₁ for Z=29, Z=54, and Z=92 as a function of angle and energy, given in units of r₀²( r₀ = e² / m c² being the classical electron radius). Note this parameter changes by many orders of magnitude over the range of Z, energies and angles considered. In Figs. VIIB, VIIC and VIID corresponding results are presented for d₄ (expressed as a percentage of d₁), for Z=29, Z=54, and Z=92 respectively. Results are given for all scattering angles and for photon energies ranging from 100 eV to 1 MeV. This includes the inner-shell threshold region for all but the lightest atoms (for which the effects are small).

As is seen in Fig. VIIA, as one goes to high energies, many times threshold, the elastic scattering cross section becomes very small except at forward angles (where d₄ vanishes). Thus, although d₄ can be largest relative to d₁ at high energy, the total elastic scattering cross section is becoming unimportant except at forward angles. Only for high Z is d₁ still sizeable away from forward angles at high energies. It is also the case that d₄ is largest relative to d₁ for high Z, rising to $\approx$ 20% of d₁ for Z=92. For very low Z the effect is of very little importance. From this we conclude that the TYPE-L CD effect is of experimental consequence in the regime where magnitude of the Rayleigh scattering cross section is still large (indicated by d₁), and where d₄ is significant in comparison to d₁. This is the case for high Z, intermediate scattering angles, and photon energies in the hundreds of keV range.

We now wish to discuss some of the common features of d₄ seen in Figs. VIIB, VIIC and VIID. At lower energies there is symmetry about 90^o, corresponding to the geometric maximum. At very high energies the situation has clearly changed - the maximum shifts towards forward angles and the parameter is now negative at large angles. This is due to competition between the two terms in the expression for d₄:

$$d_4 = {1 \over 2} \left [ \; \text{Im} \; (M) \; \text{Re} ... ...ext{Re} \; (M) \; \text{Im} \; (N) \; \right ] \sin^2 \theta.$$ (9)

At low energies the second term tends to dominate, since this involves the real part of the M amplitude, which contains the large (at low energy) form factor contribution. At higher energies, however, the form factor is small at finite angle, so there is more interference between the two terms, which have opposite signs, causing the d₄ parameter to pass through zero and become negative.

Since both terms involve imaginary amplitudes which (for a given subshell) will vanish below the subshell threshold, we expect d₄ to abruptly increase as the photon energy exceeds atomic thresholds (and the corresponding dissipative channels are opened). There is clear evidence of such abrupt increases in our results. For Z=29 we see such an increase occurring at the K-shell threshold ($\approx$ 9 keV). For Z=54 we see this happening at the M-shell thresholds ($\approx$ 1 keV), at the L-shell thresholds ($\approx$ 5 keV), and at the K-shell threshold ($\approx$ 34 keV). These increases are not so distinct for the case of Z=92, as there are many outer shells with binding energies in the energy range up to 10 keV, whose individual contributions are not discernible on a large scale. The inner shells have large binding energies ($\approx$ 116 keV for the K-shell) and the situation is more complicated at these high energies due to interference between the two (comparable) terms in Eq. (9).