The Z 1 approximation

This idea is as old as inner-shell spectroscopy itself. Already, Beutler [304], in his pioneering experiments to bridge the observational gap between optical and X-ray spectroscopy, used this comparison in a qualitative way to understand the spectral structure he observed. More recently, it has been put on a semiquantitative footing [305]. 

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The opposite case, i.e., when the band width is much larger than the hopping matrix element, can be seen in Figure 2.5 for the unoccupied As states of adsorbed on Ag. This has been measured using XAS of Ar adsorbed on Ag, since Ar using the Z + 1 approximation becomes as an effect of the final core hole state [28]. We can directly see that the As level has become a broad asymmetric resonance. The adatom resonance has a tail towards lower energies with clear cut-off at the Fermi level. The 45 level mainly interacts with the delocalized unoccupied Ag sp electrons. Most of the 45 resonance is unoccupied which indicates that charge transfer has occurred from the adatom to the substrate. 

Figure 2.5. L-edge XAS measurements of Ar adsorbed on Ag(110). The projected Ar As states becomes the 4s using the Z + 1 approximation for the core hole state.

There are, however, limitations inherent to this approximation often, the inner-shell spectrum does not appear alone, but is accompanied by double excitations in which two external electrons rather than an inner-shell electron are promoted to outer orbits, and the two modes of excitation can and do interact strongly. It is then inappropriate to use the Z + 1 approximation [189]. 

It is useful to treat the atomic absorption below the ionization (discrete line spectrum) and the continuum absorption above the ionization threshold separately. The arctan expression in eq. (1), experimentally confirmed through the K spectra, is a good approximation for the continuum absorption. The atomic absorption lines may be interpreted through optical multiplets using the Z + 1 approximation. The validity of this approximation has been successfully demonstrated in numerous cases of K and L spectra, first by Parrat (1939) in the K absorption of gaseous Ar. Here the optically determined wp Rydberg series ( > 4) of K (Z = 19 Z + 1 ) nicely fits the atomic absorption lines of 2s - 4p (K) X-ray excitation of Ar ( Z ). The Z -I-1 potential accounts for the creation of the deep core hole. The inflection point of the arctan shaped continuum absorption is located at the series limit, i.e. the first ionization potential. 

By this assumption and employing the above described (Z + 1) approximation, Johansson was able to provide further evidence for the occupation of 5f states in uranium metal, via the analysis of its XPS-measured 6p3/2 peak. 

The parabolic decrease in volume of the earlier elements in each d transition metal series with increasing atomic number (fig. 1) is closely connected to the number of valence electrons. A parabolic trend in volume means that the relative contraction, the ratio of the volume of the Z + 1 atom to that of the Z atom, decreases approximately linearly with the number of valence electrons. Furthermore, the relative volume decrease is similar for the different series, see fig. 2. 

Table 13.1 Number of qualifiers of crystal morphology in the Z(1) database (see Chapter 8) 8,519 total entries. 3D, 2D, and ID approximately label crystals grown in globnlar, plate or acicnlar form, respectively.

We first consider strain localization as discussed in Section 6.1. The material deformation action is assumed to be confined to planes that are thin in comparison to their spacing d. Let the thickness of the deformation region be given by h then the amount of local plastic shear strain in the deformation is approximately Ji djh)y, where y is the macroscale plastic shear strain in the shock process. In a planar shock wave in materials of low strength y e, where e = 1 — Po/P is the volumetric strain. On the micromechanical scale y, is accommodated by the motion of dislocations, or y, bN v(z). The average separation of mobile dislocations is simply L = Every time a disloca-... 

In these approximations for the K series the value 1 is subtracted from the atomic number Z to correct for the screening of the nuclei by the remaining K-shell electron. For the L series the screening effect of the two K-shell electrons and the seven remaining L-shell electrons must be taken into consideration by subtracting 7.4. 

Fig. 4.5. The degree of approximation for the increase of current in time for uncoupled and weakly coupled solutions for impact-loaded, x-cut quartz and z-cut lithium niobate is shown by comparison to the numerically predicted, fully coupled case. In the figure, the initial current is set to the value of 1.0 at the measured value (after Davison and Graham [79D01]).

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

Z values are obtained from Eq. (8-76) for solvents having Z in the approximate range 63-86. In more polar solvents the CT band is obscured by the pyridinium ion ring absorption, and in nonpolar solvents l-ethyl-4-carbomethoxy-pyridinium iodide is insoluble. By using the more soluble pyridine-1-oxide as a secondary standard and obtaining an empirical equation between Z and the transition energy for pyridine-1-oxide, it is possible to measure the Z values of nonpolar solvents. The value for water must be estimated indirectly from correlations with other quantities. Table 8-15 gives Z values for numerous solvents. 

This case is particularly interesting since the surface segregation energy can be directly compared to surface core level binding energy shifts (SCLS) measurements. Indeed, if we assume that the excited atom (i. e., with a core hole) is fully screened and can be considered as a (Z + 1) impurity (equivalent core approximation), then the SCLS is equal to the surface segregation energy of a (Z + 1) atom in a Z matrixi. in this approximation the SCLS is the same for all the core states of an atom. 

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