# The Limitations of 5f Delocalization and Dispersion
## 3 Digression on ARPES  
It is useful to remember that, in the Hamiltonian of the Schrodinger Equation, it is the symmetry of the potential that determines the symmetry of the solutions, i.e., the wavefunctions [30]. Thus, a periodic lattice will give rise to periodic wavefunctions, dependent upon the linear or translational crystal momentum. Hence, in many respects, ARPES is the perfect tool with which to probe the electronic structure of an ordered lattice. Examples of this abound, but one of particular application here is the system of Ag/Cu(001) and Ag(111) [19; 20; 21; 22]. In Figure 3 is shown a summary of an extensive study that demonstrated the development from 2-D, in plane, dispersion to 3-D dispersion, as the thickness of the silver overlayer was increased from one monolayer (ML) to five monolayers, with convergence to behavior very similar to bulk Ag(111) [19; 20; 21; 22]. This case is clearly an example of strong delocalization and dispersion, evidenced by the movement of the peaks over several eV as the photon energy is increased from 6 eV to 12 eV, corresponding to a movement from near its edge (L, hv = 6 eV) to approximately halfway (hv = 12 eV) to the center of the Brillouin zone (\(\Gamma\)), see Figures 3 and 4. The dispersion of several eV is completely consistent with the most fundamental of models. If one assumes a free electron with KE = \(\frac{1}{2}\) mv\({}^{2}\), a de Broglie wavelength of \(\lambda\), and a 1-D periodic chain of atoms with a spacing of d, it can be shown that the electron states will have an energy difference of \(\sim\)10 eV for d = 2 A, \(\sim\)4 eV for d = 3 A and \(\sim\)2 eV for d = 4 A, between the center of the first Brillouin zone (BZ) and its boundary [31]. An important aspect of the analysis of the Ag(111) was the success of an interpolation scheme to model the electronic structure [20]. Here, the 4d states are treated as almost constant features, which then interact with the overlying 5s wave. The states mix and repel, based upon symmetry, giving a high level of agreement with the experimental measurements, as shown in Figure 4. This issue will be revisited below. Before returning to uranium, it is worthwhile to consider ARPES in a more complex system than Ag(111). Below, in Figure 5, are shown the spectra and dispersion curves for the \"1 eV\" state in twinned and un-twinned, single crystalline YBCO [23]. The dispersion is smaller than in Ag(111) but easily definable and with a range of several tenths of an eV, and centered upon a high symmetry line in the complicated BZ of YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{6.9}\). These observations will be utilized and applied in the analysis of the ARPES of \(\alpha\)-U that follows in the next section. Figure 3: Normal emission ARPES results for Ag/Cu(001) and Ag(111). Left panel: the Ag/Cu(001) dispersion along normal emission increases as the thickness of the over-layer increases, moving towards the limit of Ag(111)-like behavior. B\({}^{\rm F}\) is the binding energy versus the Fermi level. (**a**) Ag(111); (**b**) 5 Monolayers (ML) of Ag; (**c**) 4 ML; (**d**) 2 ML of c(10x2)Ag/Cu(001). Right panel: strong dispersion in Ag(111) over the photon energy range (hv) of 6 eV to 12 eV [19; 20; 21; 22]. Figure 4: The ARPES band-mapping of Ag(111) and the results of an interpolation scheme [20]. Left panel: Plot of valence-band binding energy with respect to Fermi energy vs. final-state wave vector for the normal emission results (open circles) and the interpolation scheme calculation (dotted lines). Right panel: Interpolation scheme calculation results showing conduction bands to 30 eV above the Fermi level [20]. Figure 5: Dispersion of the “1 eV” state in single crystalline, twinned and untwinned YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{6.9}\)[23]. Left panel: Spectra shown here were collected near the S-bar point of the Brillouin zone, using hv = 24 eV photons and a twinned crystal. At 14 degrees, the momentum is about halfway from S-bar to F-bar. Note the coalescence of the two peaks into a single peak at S-bar.