EFG tensors

Here, I, I, and I are angular momentum operators, Q is the quadrupole moment of the nucleus, the z component, and r the asymmetry parameter of the electric field gradient (efg) tensor. We wish to construct the Hamiltonian for a nucleus if the efg jumps at random between HS and LS states. For this purpose, a random function of time / (f) is introduced which can assume only the two possible values +1. For convenience of presentation we assume equal... 

The trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The value of is given by the component of the EFG tensor along the main quantization axis. Therefore, in this example where the EFG is axial (77 = 0) with the main component the quadrupole shift is eQVzJ - This is just half the quadrupole splitting that would be observed in an unperturbed quadrupole spectrum without a magnetic field at the nucleus. 

Fig. 4.12 Surface plots of the EFG tensor as determined from the angular dependence of the first-order quadrupole shift, Eq of high-field magnetic Mossbauer spectra. The plots visualize the value of the function (3cos 0 - 1 + rysin d cos20) for > 0 and ry = 0 (a), r] = 0.3 (b), and 77 = 1 (c)...

For nonaxial EFG tensors, mixing of the nuclear mj basis functions occurs even for B being oriented along This results from the contributions of the shift operators and fi in the Hamiltonian described by (4.29). These contribu-... 

From a theoretical point of view, the calculation of the quadrupole splitting is relatively straightforward since it can be calculated directly from the elements of the EFG tensor at the iron nucleus (nucleus A ) as ... 

Once available, and supplemented by the nuclear contribution, the EFG tensor can be diagonalized. The numerically largest element (in atomic units) defines the value of q which is in turn used to calculate the quadrupole splitting parameter... 

By recognizing that the basis functions are attached to parent atoms, the EFG tensor can be analyzed in a transparent way by breaking it down into components that represent multicenter interactions ... 

EFG tensor. These are the ones usually held responsible for the EFG tensor since the integrals should decay fairly quickly owing to the dependence. In HE or DFT theory, the one-center terms can be further analyzed by making use of the fact that the density is a simple function of the molecular orbitals P = J2i I0( l-... 

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

Hence, the probability to meet a molecule in the Z th energy eigenstate is N[c T). During the time course of the Mossbauer measurement, thermal fluctuations will cause each individual molecule to visit all of the 15 available eigenstates with probability Nk T) and hence one obtains an averaged EFG tensor ... 

YBa2Cu30y, YBa2Cu30g EFG tensor, point charge calculation, charge states, hole on O positions... 

La2 xSrxCu04 EFG tensor, disagrees with point charge model, charge state, holes from substitution of La by Sr are localized mainly at the oxygen sites in the CU-O2 plane... 

RBa2Cu307 ceramics (R is a rare-earth metal or yttrium) EFG tensor, comparison with point charge calculation, spatial distribution of electron defects in the lattice... 

RBa2Cu307 (R = rare earth element or Y), La2 (5r,.Cu04 (0 < X < 0.3) Eu-155(Gd-155) emission Mossbauer spectroscopy, EFG tensor at R sites, in good agreement with point charge model when holes are supposed to be mainly in sublattices of the chain and at oxygen in Cu-O plane... 

Y2Ba4Cu7025 Nuclear quadrupole interaction at copper sites, EFG tensor at all sites is calculated using the point charge model, conclusion that holes in the Y2Ba4Cu70i5 lattice are localized predominantly at positions of chain oxygen... 

The structural picture that was envisaged to represent the temperature-dependent fluctuations of the EFG tensor [15] is based on the X-ray structure of MbOa that exhibits a geometric disorder of Fe02 with two different positions of the terminal O-atom [28]. Within this stmcture, the projection of the 0-0 bond on the heme plane is rotated by about 40° in position 2 compared to 1 (Fig. 9.10). Conventional Mossbauer studies of single crystals of Mb02 have shown that the principal component of the FFG tensor lies in the heme plane and is oriented along the projection of the 0-0 bond onto this plane [29]. If the terminal O-atom is located in position 2, the EFG should be of the same magnitude as in position 1, but its orientation is different. The EFG fluctuates between positions 1 and 2 with a rate that depends on temperature. 

Letting the EFG tensor fluctuate between positions 1 and 2 with a rate (O12... 

When a sodium cation jumps between two sites with the same EFG tensor components and the same residence times on both positions, but different orientations, then the two-site exchange process can be described by tensor averaging of the two positions. Depending on the jump angle of the tensor, the motionally averaged principal components, Vxx, Vyy, and Vzz, may eventually have to be redefined in ordering to fulfill the requirement in parenthesis in (4). 

Solid state 2H NMR parameters are almost exclusively governed by the quadrupole interaction with the electric field gradient (EFG) tensor at the deuteron site.1 8 The EFG is entirely intramolecular in nature. Thus molecular order and mobility are monitored through the orientation of individual C-2H bond directions. Therefore, 2H NMR is a powerful technique for studying local molecular motions. It enables us to discriminate different types of motions and their correlation times over a wide frequency range. Dynamics of numerous polymers has been examined by solid state 2H NMR.1 3,7,9 Dynamic information on polypeptides by NMR is however limited,10 26 although the main-chain secondary structures of polypeptides in the solid have been extensively evaluated by 13C and 15N CP/MAS NMR.27,28... 

It will be useful to outline the basic features of solid state 2H NMR spectra. Deuteron is a spin 7= 1 nucleus possessing an electric quadrupole moment. The EFG interacts with the electric quadrupole moment to produce a frequency shift. Frequencies of the symmetric line shape centred around the Larmor frequency >0, which depend on the relative orientation of an external magnetic field B0 and the EFG tensor, are given by1 7... 

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