3. Numerical results and discussion

We write the forward Delbrück amplitude $D=D(\omega, \theta=0)$ in terms of a Born term D^Born, a correction term due to Coulomb and screening effects in the ordinary (electron in continuum) pair production cross section $\Delta D^{PP}$, and a correction term due to the inclusion of the bound-electron pair production cross section $\Delta D^{BPP}$:

$$D(\omega, \theta=0) = D^{Born} + \Delta D^{PP} + \Delta D^{BPP}.$$ (3)

Note that the Born term can be obtained from the ordinary pair production cross section in the Born approximation (given by [22], and in terms of simple expansions by Maximon [23]). It can also be obtained directly through the evaluation of the appropriate lowest-order Feynman graphs [9,10,11].

The corrections to the Born result obtained by using the ordinary pair production cross section, but including Coulomb and screening effects, have been investigated by Solberg et al. [20], who gave separately the corrections due to Coulomb and screening effects. This was based on previous work on the Coulomb [24,25] and screening [26,27] corrections to ordinary pair production. While the Coulomb corrections reported in these works are valid throughout the threshold regime, the screening corrections (based on screening corrections to the Born term and shifted Coulomb values to account for screening corrections to the Coulomb term) are not valid for low photon energies close to the pair production threshold ( $\omega <=2.5\;mc^2$ for Z=92) [26]. Since the threshold region is our region of interest, we will use the pair production tabulation of Hubbell et al. [28] to calculate the total correction due to Coulomb and screening effects in ordinary pair production for comparison with our results for corrections due to the bound-electron pair production cross section. At low photon energies, near the pair production threshold, the tabulation of Hubbell et al. [28] uses screening corrections based on the numerical work of Tseng and Pratt [29,30], which are then matched to the screening corrections of $\O$verb$\o$ [26,27] so as to accurately account for screening throughout the threshold regime. The Coulomb corrections [24,25] are also included, as are radiative corrections [31,32] which are small ($\approx$ 1% or less).

In addition to these effects, the Born results also do not include effects due to bound-electron pair production. As discussed in the previous section, the total cross section for bound-electron pair production into all bound states should be included. We consider the correction $\Delta D^{BPP}$, defined as the correction to the forward amplitude due to bound-electron pair production into all bound states. The imaginary part of the correction is given directly by

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

where $\sigma^{BPP}$ = $\sigma^{BPP}(\omega)$ is the total cross section for bound-electron pair production with production into all bound states, regardless of occupation. The real part of the correction is obtained by use of the dispersion relation

$$\mbox{Re} \; \Delta D^{BPP} = {\omega^2 \over 2 \pi^2} \mbox{... ...ime}) \over {\omega^{\prime}}^2 - \omega^2} d \omega^{\prime},$$ (5)

where $\mbox{P}$ indicates that the principal value of the integral should be taken. Note the lower limit of the integral is 2mc² - E_K rather than 2mc², where E_K is the binding energy of the K shell ( $\sigma^{BPP}$ vanishes for photon energies lower than 2mc² - E_K).

Though the bound-electron pair production cross section to be used in Eqs. (4) and (5) includes production into all bound states, it is well known that production into the inner shells dominates, as in the case of bound-electron pair annihilation and photoeffect at the same energies [33,34,35]. We have calculated explicitly the bound-electron pair production cross sections for production into the K and L shells. Our results for the K and L shell taken separately exhibit the expected L to K shell ratio of $\approx$ 0.18 (above the L-shell threshold) seen in photoeffect cross sections for the same Z at similar energies [35]. In [35] the ratio of the L-shell photoeffect cross section to the sum of the M- and higher-shell photoeffect cross sections is given for Z=92 as 3.105 at 1.332 MeV and as 3.090 at 0.662 MeV, quite insensitive to energy. Therefore we approximate the effect of bound-electron pair production into the M and higher shells by assuming a ratio of 0.06 to the K-shell result.

In Fig. 6.1 we show the real and imaginary parts of the forward Delbrück amplitude D, given in terms of the classical electron radius r₀, for photon energies in the range 0.5-100 MeV. This result includes both corrections due to Coulomb and screening effects in ordinary pair production, and corrections due to bound-electron pair production. We see that the real amplitude dominates the imaginary amplitude around and below the pair production threshold, with a crossover near 7 MeV. The lowest-lying threshold for (bound-electron) pair production is shown, 0.906 MeV, corresponding to production into the K shell. Below 0.906 MeV the imaginary amplitudes vanishes. (Note that without the inclusion of corrections due to bound-electron pair production the imaginary Delbrück amplitude vanishes below 1.022 MeV, the threshold for ordinary pair production.

In Figs. 6.2 and 6.3 we show the real and imaginary parts respectively of the corrections $\Delta D^{BPP}$ and $\Delta D^{PP}$, expressed as fractions of the corresponding real or imaginary parts of the full forward Delbrück amplitude D. The net correction due to beyond-Born-approximation effects, being the sum of these, is also shown. At high energies, well above threshold, the correction $\Delta D^{BPP}$ becomes unimportant, and our results are in general agreement with those of [20], where only the correction $\Delta D^{PP}$ was considered.

Below 7 MeV it is effects in the real amplitude that will most affect the scattering cross section. Figure 6.2 shows that below 2 MeV the correction Re  $\Delta D^{BPP}$ is comparable with Re  $\Delta D^{PP}$, and both need to be considered. Re  $\Delta D^{BPP}$accounts for as much as $\approx$ 11% of the real forward Delbrück amplitude in the threshold region. At low energies the corrections Re  $\Delta D^{BPP}$and Re  $\Delta D^{PP}$ cancel each other, so that the Born result is accurate at the 1% level (though the Delbrück amplitude is unimportant at low energies). At somewhat higher energies Re  $\Delta D^{PP}$is dominant, and the net correction to Born approximation is significant. It is interesting that qualitatively similar features are found in the experimental large angle scattering results. (But note that in forward scattering Re  $\Delta D^{BPP}$ is the dominant and significant correction around 1.3 MeV.)

The primary effect of the correction Im  $\Delta D^{BPP}$ on the imaginary part of the forward Delbrück amplitude is to shift the threshold below which the imaginary part of the amplitude vanishes from 1.022 MeV (ordinary pair production threshold) down to 0.906 MeV (threshold for production into the K shell for Z=92). Consequently Im  $\Delta D^{BPP}$ / Im D $\equiv$ 1 below 1.022 MeV as bound-electron pair production is then responsible for the entire contribution of the imaginary forward Delbrück amplitude. The correction Im  $\Delta D^{PP}$becomes comparable with Im  $\Delta D^{BPP}$ above threshold and dominates by 10 MeV.