# The Limitations of 5f Delocalization and Dispersion
## 1 Introduction  
In many respects, it is the delocalization of the 5f states that sets them apart from 4f behavior. One very powerful aspect of 5f delocalization is the size dependence of the actinides with fillings, as can be seen in Figure 1 below. In the early part of the series, the size dependence is much more like a 5d filling than a 4f filling. This observation has been known and reported on for many years [1; 2; 3]. In fact, it temporarily led to the erroneous hypothesis that the actinide series was a 6d filling series [4; 5]. It is only as the 5f states fill further that the impact of the jj-skewed intermediate coupling of 5f angular momenta becomes clear, with a dramatic change in both the atomic size (lower panel) and angular momentum coupling (upper panel) [6; 7; 8]. Interestingly, it is the nature of the 5f intermediate coupling that is the other major difference between 4f (Russell-Saunders skewed) and 5f (jj skewed) behavior. This subject will be revisited and further explored later in the discussions herein. From a consideration of the results in Figure 1, it is clear that perhaps the best example of 5f delocalization is U metal. The atomic size dependence near U is still in the 5d-like regime and the angular momentum coupling has diverged from the predictions of the intermediate coupling model [6; 7; 8; 9], which is based upon a picture assuming a high degree of localization in the 5f states. Furthermore, U metal has been the subject of many investigations and extensive data concerning its electronic structure already exist. An example of this is shown in Figure 2. Thus, the analysis in this paper is not a review of the field but rather a detailed case study with extensive prior investigations. Figure 1: A comparison of the branching ratio results (top panel) and the atomic sizes (lower panel) for a series of actinide samples. Top panel: The spin-orbit operator expectation value, \(<\)W\({}^{110}\)\(>\) versus the 5f occupation, \(n\). The top panel is based on Figure 4 of Reference [8], with additional data from [10; 11]. The experimental values at \(n=\)2 are for UF\({}_{4}\)[12] and UO\({}_{2}\). The experimental values at \(n=\) 4, 6 and 7 are for Np, Am and Cm, respectively. Three theory curves are shown for LS (Russell–Saunders), jj and intermediate coupling. Lower panel: The experimental Wigner Seitz radii for the lanthanides, actinides and 5d series are shown here. “The Wigner-Seitz radius RWS (the radius of the volume per atom in a solid) is defined as (4 \(\pi\)/3)RWS 3 = V, where V is the equilibrium volume of the primitive unit cell. The atoms of the actinides, lanthanides, and transition metals are aligned so that elements that lie on top of each other have the same number of valence electrons.” Taken from Reference [3] with permission [13]. Unless otherwise indicated, this information has been authored by an employee or employees of the Triad National Security, LLC, operator of the Los Alamos National Laboratory with the U.S. Department of Energy. The U.S. Government has rights to use, reproduce, and distribute this information. The public may copy and use this information without charge, provided that this Notice and any statement of authorship are reproduced on all copies. Neither the Government nor Triad makes any warranty, express or implied, or assumes any liability or responsibility for the use of this information. There is a remarkable level of agreement between the earlier measurements of [PERSON] and [PERSON] [14; 15] and the theory from [PERSON] et al. [8; 16; 17]. The X-ray Photoelectron Spectroscopy (XPS) and Bremsstrahlung Isochromat Spectroscopy (BIS, a type of high energy inverse photoelectron spectroscopy) provide a good measure of the occupied (ODOS) and unoccupied (UDOS) density of states, respectively. Effectively, the XPS and BIS are angle and momentum averaging results. However, to gain a further understanding of the electronic structure, more discerning measurements are desirable.