2. Partitioning of the elastic scattering amplitude

The total amplitude for coherent elastic photon-atom scattering is traditionally partitioned into Rayleigh, Delbrück, and nuclear amplitudes, which can be thought of as describing scattering off the atomic bound electrons, off virtual electron-positron pairs in the atomic field, and off the nucleus, respectively. In going to a single-electron formalism [21], one has written the Rayleigh amplitude for a given bound state as a sum over all possible intermediate states, including other occupied bound states and negative energy states. By writing the Rayleigh amplitude in such a way one has defined a partitioning scheme for the Rayleigh amplitude, and for consistency the same partitioning scheme must be used for the Delbrück amplitude. This in particular implies that the Delbrück amplitude must be written so as to include the influence of all virtual electron-positron pairs, even when the electron is created in an occupied bound state of the atomic potential. Current S-matrix predictions for Rayleigh scattering [14] use this partitioning (which is regarded as attractive since the ability to sum over all intermediate states in closed form allows one to solve the inhomogeneous wave equation for a perturbed orbital instead of explicitly constructing the Green's function as a summation over the states). Hence any calculation of the Delbrück amplitude that is to be used in conjunction with the Rayleigh amplitude calculated this way must include the sum over all intermediate states, and hence it must include a sum over all bound-electron-positron virtual pairs.

This partitioning scheme implies that there will be unphysical spurious transitions present in the separate partitioned amplitudes, which violate the Pauli principle. Of course such spurious transitions must cancel when the partitioned amplitudes are added to obtain the total coherent amplitude, on which the physical observables depend. It is well known that such a phenomena occurs in the individual Rayleigh amplitudes defined for each of the bound electrons for the case of multi-electron atoms and ions. The Rayleigh amplitude for one of the occupied bound states n will contain spurious resonances corresponding to virtual transitions to another occupied bound state m. Similarly the Rayleigh amplitude for the occupied bound state m will contain spurious resonances corresponding to virtual transitions to the occupied bound state n. In summing to get the total Rayleigh amplitude the spurious resonances corresponding to transitions between occupied states cancel.

Now, write the optical theorem involving the single-particle partitioned Rayleigh amplitude for one of the occupied bound states n [21] as

$$\mbox{Im} \; R_n = {\omega \over 4 \pi c} \left ( \sigma_n^{... ... \sigma_n^{BBT+} - \sigma_n^{BBT-} - \sigma_n^{BPP} \right ),$$ (1)

where $\omega$ is the photon energy, $\sigma_n^{PE}$ is the photoeffect total cross section for the bound state n, $\sigma_n^{BBT+}$ is the total cross section for upward bound-bound transitions starting from the bound state n, $\sigma_n^{BBT-}$ is the total cross section for downward bound-bound transitions starting from the bound state n, and $\sigma_n^{BPP}$ is the bound-electron pair production total cross section with the electron being created in the bound state n. As discussed in [14] the subtraction of the bound-electron pair production cross section in Eq. (1), needed in the single-electron formalism for a complete set of intermediate states, is necessary in order that the real part of the Rayleigh amplitude, defined through the dispersion relation, will have a finite high energy limit. To get the total Rayleigh amplitude one sums over all occupied bound states n. In this partitioning scheme the optical theorem written for the Delbrück amplitude is

$$\mbox{Im} \; D = {\omega \over 4 \pi c} \left ( \sigma^{BPP} \right ),$$ (2)

where $\sigma^{BPP}$ is the bound-electron pair production total cross section, summed over production into all bound states, regardless of occupation. In summing to get the total coherent amplitude, for which we can again write the optical theorem for the imaginary part [that being just the sum of Eq. (1), summed over all occupied bound states n, and of Eq. (2)] we see that the contribution of the total bound-electron pair production cross section for production into the occupied bound states cancels, as it should.