We present an experimental and theoretical study of the kinetic energy dependence of spin filtering of electrons by organized layers of DNA adsorbed on a gold substrate. When Au 4f7/2,5/2 levels are ionized by circularly polarized X-rays, the emitted electrons will be spin polarized. The spin distribution depends on the particular sublevel and is opposite for right versus left circularly polarized light. If the DNA overlayer preferentially attenuates one spin over another, then there should be a circular dichroism (CD) in the X-ray photoelectron spectroscopy (XPS) spectra observed with the different polarizations. Using synchrotron radiation excitation, XPS CD measurements were made of electrons with kinetic energies in the range 30 to 760 eV. In all cases there was no evidence of any significant dichroism. These results are explained by a model in which the longitudinal polarization is strongly dependent on the k-vector, and hence the energy or the de Broglie wavelength, which are simply connected to the magnitude of this vector of the incoming electrons. For a helix with a fixed number of turns, this dependence is due to a coherent process associated with multiple scattering. This model predicts that there is a window of energies where changes in the polarization should be expected. Two competing effects determine this window: The energy has to be small enough to allow for at least double scattering, but large enough so that the de Broglie wavelength probes the chiral structure. Also at very low energies the spin-orbit interaction weakens and no polarization results.