Knight shifts

The shift in the field thus produced can be determined by calculating the field caused by the electrons producing the susceptibility. This can be expressed in terms of the hyper-fine field produced by the conduction electrons, a contact type of interaction, 

This depends on the value of the wavefunction at the nucleus (r = 0) for electrons averaged over the Fermi surface, where II is the volume per electron. This shift of frequency in a metal was first observed in copper in 1949 by W.D. Knight and was subsequently termed the Knight shift (Townes, Herring and Knight 1950, Knight 1956), given by 

Since K depends on the wavefunction density at the nucleus, the effect is dominated by s-electrons which is certainly true in metals with unpaired s-electrons. If the Pauli susceptibility and electron density can be independently measured then the Knight shift will give an independent measure of the s-component of the conduction electron spin density. These shifts are positive and are much larger than chemical shift effects, some typical values being Li — 0.025%, Ag — 0.52% and Hg — 2.5%. In other metals the situation is more complicated when the s-electrons are paired but there are other electrons (e.g. p but especially d). As only s-electrons have significant density at the nucleus the effects of these other electrons are much smaller. The hyperfine fields of these electrons induce polarisation in the s-electrons that subsequently produce a shift, termed core polarisation. 

A major problem is that it is the net value of the shift which is measured in the experiment. Two points emerge from this. The zero of the scale needs to be known so that the contribution of the chemical shielding has to be taken into account. Also, in more complicated metals the different terms have different signs, with Ks and Kd positive whereas Kcp is negative. 

If the symmetry of the site is lower than cubic the full tensor form of the electron-nucleus interaction needs to be used, so that in addition to an isotropic term there is an anisotropic contribution. If in the PAS of the Knight shift tensor the components of the tensor are Kx, Ky and Kz, then in the laboratory frame with its orientation in the frame defined by Bo described by the Euler angles 0 and j), 

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Knight shift K s 10 -10 Interaction with conduction electrons via the contact interaction... 

This technique is the most widely used and the most useful for the characterization of molecular species in solution. Nowadays, it is also one of the most powerful techniques for solids characterizations. Solid state NMR techniques have been used for the characterization of platinum particles and CO coordination to palladium. Bradley extended it to solution C NMR studies on nanoparticles covered with C-enriched carbon monoxide [47]. In the case of ruthenium (a metal giving rise to a very small Knight shift) and for very small particles, the presence of terminal and bridging CO could be ascertained [47]. In the case of platinum and palladium colloids, indirect evidence for CO coordination was obtained by spin saturation transfer experiments [47]. 

Changes in electronic properties, such as Fermi level shifts and changes in the DOS, which still have to be confirmed experimentally. For example, Yano and co-workers used Pt NMR to probe possible changes in electronic properties induced by particle size [Yano et al., 2006]. They concluded that the surface Knight shifts as well as Ep-LDOS of surface Pt atoms showed no noticeable size dependence in the particle size range from 4.8 down to 1.6 nm hence, electronic properties are independent of size in this interval. 

Keywords Dopants Knight shifts Nanoparticles NMR Semiconductors... 

Despite this similarity with chemical shift, the Knight shift is grouped with the electron hyperfine term in (lb) to reflect the fact that both terms arise from the influence of the spin or orbital angular momentum of unpaired electrons. The distinction between the two is that for the electron hyperfine term the electron spin (or hole, as the absence of an electron can be described, e.g., in the case of d9 Cu++) is localized on a paramagnetic defect such as a deep-level transition metal ion. 

The terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

The 71Ga and 14N spectra of several of these films also showed partially-resolved shoulders shifted to higher frequency and having shorter T relaxation times that were attributed to Knight shifts in more heavily unintentionally doped regions of the film. These Knight shifts were observed in other GaN film samples [53] and will be discussed in more detail in Sects. 3.4.3 and 3.4.4, where MAS-NMR was used to improve the resolution in polycrystalline powders of h-GaN. Section 3.3.2 also shows 71Ga and 14N MAS-NMR spectra of GaN. 

Knight Shifts and Metal-Insulator Transition in Doped Silicon... 

Perhaps not surprisingly, the most thorough NMR studies of Knight shifts, Korringa relaxation, metal-insulator transitions, and the NMR of the dopant nuclei themselves have been carried out for doped silicon. Since few semiconductors other than PbTe, which presents a considerably more complicated case, have been studied in such detail, it is worthwhile here to summarize salient points from these studies. They conveniently illustrate a number of points, and can shed light on the behavior to be expected in more contemporary studies of compound semiconductors, which are often hindered by the lack of availability of a suite of samples of known and widely-varying carrier concentrations. 

Let us now consider the metallic regime, following the discussion of [18]. For nuclei coupled to a mobile system of independent electrons by an isotropic exchange interaction of the form A l S, where I is the nuclear spin and S the electron s spin, the Knight shift can be written as... 

For the metallic regime described above, when the Al S interaction is also responsible for the nuclear relaxation, Korringa [192] showed that the Knight shift and Ti are related by... 

For B-doped p-type samples of Si with an acceptor concentration nA of 8.5 x 1019 the 29Si NMR also showed a Knight shift of 33 ppm (both n-type and p-type Knight shifts were reported as unsigned, presumably positive, values see Sect. 3.4.2). Comparison with results for the n-type sample led to the conclusion that there was a nearly equal admixture of s-type wavefunction for the hole wave function near the top of the valence band. 

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