Friedel theorem

As density p is real quantity, F(q) = F(—q), the scattering pattern is always centrosymmetric (Friedel theorem)... 

We find that a layer model analysis can adequately describe the Pt NMR spectrum of nanoscale electrode materials. The shifts of the surface and sub-surface peaks of Pt NMR spectra correlate well with the electronegativity of various adsorbates, while the Knight shift of the adsorbate varies linearly with the f-LDOS of the clean metal surface. The Pt NMR response of Pt atoms from the innermost layers of the nanoparticles does not show any influence of the adsorbate present on the surface. This provides experimental evidence, which extends the applicability of the Friedel-Heine invariance theorem to the case of metal nanoparticles. Further, a spatially-resolved oscillation in the s-like E( -LDOS was observed via Pt NMR of a carbon-supported Pt catalyst sample. The data indicate that much of the observed broadening of the bulk-like peak in Pt NMR spectra of such systems can be attributed to spatial variations of the A( f). The oscillatory variation in A(A) beyond 0.4 nm indicates that the influence of the metal surface goes at least three layers inside the particles, in contrast to the predictions based on the Tellium model. 

It is important to point out that the invariance observed here is in quite a different context to that of originally proposed in the Friedel-Heine invariance theorem, which refers to the invariance of electronic properties in bulk environments of dilute alloys. The situation is quite different for nanoscale metal particles from those of bulk Pt, due to the presence of the surface and the significant reductions in the particle volume. However, the Ef-l DOSs still show a remarkable invariance towards changes in surface chemical environment, even though they vary from those of bulk Pt. These results may also be expected to lead to useful general correlations between electronic properties and more conventional chemical descriptions such as ligand electronegativity. 

Fig. 5 Superimposition of point-by-point, 8.47 T l Pt NMR spectra of a 2.5 nm, carbon-supported Pt electrocatalyst without and with different chemisorbed ligand Ru, CO, O, H, CN-, and S. The spectra were normaiized by equalizing the ampiitude at 1.131 C kHz-1 (indicated by the arrow). The invariance of signais beyond 1.131 G kHz i provides experimental confirmation of the Friedel-Heine invariance theorem. The surface peaks range over 11 000 ppm. (Reproduced with permission from Ref [10], Copyright by 2000 American Chemical Society.)...

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