4. Observability of zeros

We now wish to address the observability of the free-free zeros we have discussed above. For bremsstrahlung many electron angular momenta, and so many transition matrix elements, begin to contribute to cross sections at energies low compared to energies considered in current experimental and theoretical efforts; they contributed at still lower energies to angular distributions. We know from elastic scattering that higher phase shifts become comparable to s-wave phase shifts by 10 eV in neutral noble gases, so observation of zeros causing Ramsauer-Townsend minima may be confined to the region near 1-10 eV. Zeros in bremsstrahlung matrix elements from neutral atoms (but less likely for ions, for which many matrix elements generally contribute at most angles) can likewise be observable when both initial and final electrons are of low energy. However, in bremsstrahlung there is additional opportunity for observation, since as long as one electron is slow, small numbers of its partial waves contribute in bremsstrahlung even as the other electron becomes fast. Whenever dipole approximation remains valid, only a few radial matrix elements are important, and one can still expect to see effects of these zeros, as in the tip region of the spectrum of faster electrons. Note that the bremsstrahlung spectrum is fairly well described in non-relativistic dipole approximation (cancellation of relativistic, retardation and higher multipole effects [33]) up toward 100 keV, so that in the tip region the s-p dipole matrix elements continue to play a dominant role in the spectrum: effects of any zeros should be visible.

Also inverse bremsstrahlung, absorption of a photon by a slow electron scattering from an atom or ion, would be a prime candidate for observation of such zeros. In many cases experiments are conducted at electron energies $1\mbox{ eV} < E_1 < 300$ eV and photon energies near the soft photon regime (typically the external field is due to ${\rm CO}_2$ lasers with $h\nu=0.117$ eV) [34,35,36,37]. Thus we would expect that at some electron energies the scattering electrons would be transparent to the laser in the region of the zeros discussed here.

There are several previous works which might suggest the existence of observable zeros in free-free transitions. In one calculation Zon [38] observed a deep minimum in the spectrum for absorption of a photon by low energy electrons scattering from Argon. In this work, however, Zon (appropriately) included the effects of the dynamic (rather than static) polarizability of the atom in an approximate way; it is unclear what effect this treatment has on the arguments here. Zon states that his observation is unrelated to Ramsauer-Townsend minima because the ``frequency corresponding to the photoabsorption minimum is much higher in this case than the width of the Ramsauer dip.'' Our results here would indicate that zeros connected to Ramsauer-Townsend minima may be visible at energies away from the soft photon limit. In another investigation, Green [39,40] observed, but did not discuss, minima in transition cross sections obtained from non-relativistic dipole calculations (retaining all important dipole contributions) using wavefunctions corresponding to continuum electrons in finite temperature and density Thomas-Fermi potentials. In a third related work, Ashkin [41] compared various approximate theories to an ``exact calculation'' (non-relativistic but with all partial waves included) of the spectrum for absorption by electron scattering from Argon. Ashkin used the potential of Holtsmark [13], which includes a static polarizational tail and is known to produce a Ramsauer-Townsend minimum near 2 eV. His results indicated a shallow minimum in absorptivity at incident electron energies near 0.3 eV, corresponding to the outgoing electron energy of about 2 eV - far from the soft photon regime. However, this minimum is shallow enough that it is difficult to discern.