Review of Published Theories, a New, Classical-Field Theory of Electron Waves as a Polarized Radiation Probe of Magnetic Surfaces, Comparison of These Theories with Published Experimental Data, and Motivation for New Experiments Presented Later in This Thesis

Theories of elastic spin-polarized electron scattering from a surface can be classified by their underlying philosophy, which may be of either a band-structure calculation or an analytical type, or by their possession or otherwise of spatial variation of the scatterer, in the plane of the surface. This gives a total of four possible types of theory:

1. band-structure theories, with spatial variation of the scatterer; Feder [24] put forward a numerical method for undertaking calculations of this type, which would make quantitative predictions about the intensities and polarizations of reflected and diffracted beams, from a sample of a precisely specified atomic-scale structure, which, along with similar methods, has been applied to particular samples by a variety of workers [1,4,25,26],
2. band-structure theories, without spatial variation of the scatterer; the method of Feder [24] can be applied in this way, but, as far as the author is aware, has not been, perhaps because ignoring the in-plane variation of the scatterer eliminates any possibility of examining diffraction orders other than the specular reflection,
3. analytical theories, with spatial variation of the scatterer; Darwin [27] produced a classical-field theory, in which an electron wave impinged on a sinusoidal electrostatic potential and magnetic flux density, whose lack of a Fourier component of zero in-plane wave-vector eliminated any possibility of examining the specular reflection, and
4. analytical theories, without spatial variation of the scatterer; the author produces a theory of this form, later in this chapter. Its lack of any in-plane variation in the scatterer means that it is valid only for the specular reflection, not for other diffraction orders.

Theories, of course, are refined (in the sense of choosing an atomic configuration for a band-structure theory, or estimating the parameters in an analytical theory,) and compared for their correspondence to reality, by their ability to match experimental results. This is primarily an experimental thesis; in part III, some experimental polarized electron reflection results, along with the means used to obtain them, are presented, and in section 5.3, they are used to estimate the parameters in the new theory presented in this chapter, and to compare, for correspondence to reality, two versions of this theory, in one of which the ferromagnetic samples exert a (statistically significant) Weiss field on the incident electrons, and in the other of which, they do not.

The only other theories mentioned above, which can be used to examine a specular reflection, are that of Feder [24], and its close relatives, and some attention should be given to the match between these theories and the experimental results of part III, and indeed the new theory of this chapter. The author has been unable to find an application of these theories to specular reflection from a copper or cobalt surface, which would allow direct, quantitative comparison with the experimental results, although one might expect, in some rather ill-defined sense, qualitative features in the specular reflections from other materials to carry over to copper and cobalt. This does not, however, prevent direct comparison of Feder-type theories with the new theory presented in this chapter, because the latter is not designed to be chemically specific.

An important link between the band-structure theories and the new theory presented in this chapter is that, in explaining the basis of the numerical band-structure calculation, Feder [24] appears to look forward to the day when the atomic-scale structure, which is fed to the numerical method, can instead be encoded in the continuous, adjustable parameters of an analytical theory, and suggests electrostatic potential and magnetic flux density as parameters for use in this fashion. Where, in a theory based on these parameters, the scatterer has mirror symmetry about the scattering plane, as does the scatterer assumed in the new theory of this chapter, this symmetry can [24] be combined with time-reversal symmetry, to show that the reflected beam polarization, for an unpolarized incident beam, is parallel to the magnetic flux density, when the latter is either in the scattering plane, or perpendicular to the scattering plane. This provides two important tests of the plausibility of any proposed emergent-phenomena theory; the new theory presented in this chapter passes both tests.

Another such link is that a band-structure calculation has been reviewed (figure 2.1) [1], which appears to agree, to approximately the same standard of precision that is possessed by relevant measurements, to be reviewed in section 2.10, with the new theory presented in this chapter, that at incident electron energies greater than $\sim{}60\,\mathrm{eV}$, where the Taylor expansion allows the latter to make quantitative predictions, the reflected beam polarization from a nickel $(001)$ surface is independent of incident electron energy and of angle of incidence. Several other band-structure theories for $Ni(001)$, with different atomic structures, have been reviewed [4], which have a lesser similarity to the new theory of this chapter, and a correspondingly less good fit to experimental data to be reviewed in section 2.10 [4].

Figure 2.1: A Figure Reproduced from ``Elastic Spin-Polarized Low Energy Electron Diffraction from Non-Magnetic Surfaces'' [1], Showing the Comparison between a Band-Structure Calculation (Lines) and Experimental Measurements (Vertical Bars,) on Graphs of Reflected Electron Beam Polarization, from $Ni(001)$, against Electron Energy, for a Variety of Angles of Incidence $\theta$. For High Energies (above $\sim{}60\,\mathrm{eV}$,) the Simpler New Theory of This Chapter Predicts an Energy-Independent Polarization. The Figure Mentions that $\phi {} = 0$. The Source Describes $\phi {}$ as the `Azimuthal Angle,' and Explains That This Is the Angle Between Some Fixed Crystallographic Axis and the Normal to the Scattering Plane; Returning to the Source's Source [2] Reveals That the Fixed Crystallographic Axis is $[110]$.

Striking differences between band-structure theories and the new theory of this chapter are:

• A band-structure theory has been reviewed [1] that predicts significant polarization (at some angles of incidence, more than $0.4$) of an $82\,\mathrm{eV}$ electron beam, by reflection from a tungsten $(001)$ surface. Since tungsten is not ferro-magnetic, the new theory of this chapter, because it ignores spin-orbit effects, predicts no polarization, yet it claims that $82\,\mathrm{eV}$ is within the region, of energies much less than the electron rest mass energy, where it is reasonable to ignore spin-orbit effects. However, experiments to be reviewed in section 2.10 suggest a way for both theories to survive this apparent contradiction between them, which will be discussed there.
• The same band-structure calculation for tungsten predicts [1] a polarization, which changes its sign several times, as the angle of incidence $I$ is varied from $0$ to $30\,\mathrm{^{\circ}}$. The new theory of this chapter predicts a polarization independent of $I$.
• A band-structure calculation has been reviewed [1] that predicts significant polarization (up to $0.8$) of the reflected beam from tungsten $(001)$ in a few narrow ( $\sim{}20\,\mathrm{eV}$) energy bands, while the new theory of this chapter predicts no polarization; however, the band-structure calculation does predict near-zero polarizations outside these bands.
• A band-structure theory has been reviewed [1] that predicts significant polarization (at some scattering plane orientations, more than $0.8$) of an $60\,\mathrm{eV}$ electron beam, by reflection from a platinum $(111)$ surface. Since platinum is not ferro-magnetic, the new theory of this chapter, because it ignores spin-orbit effects, predicts no polarization, yet it claims that $60\,\mathrm{eV}$ is within the region, of energies much less than the electron rest mass energy, where it is reasonable to ignore spin-orbit effects. However, experiments to be reviewed in section 2.10 suggest a way for both theories to survive this apparent contradiction between them, which will be discussed there.
• A band-structure theory has been reviewed [4] that predicts a reflected polarization from an iron $(110)$ surface, which oscillates as the incident electron energy is varied, with an amplitude of $\sim{}0.1$. This is rather different from the constant, possibly non-zero polarization predicted for this ferro-magnetic material by the new theory of this chapter.