Quadrupole tensors

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

Quadrupole interaction energy tensor X Residual resistivity (solid state) Pr... 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

In principle, the electronic quadrupole can also be extracted from the calculated valence charge distribution. This is a tensor quantity and its components are defined by the matrix... 

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... 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

The tensor of the nuclear quadrupole moment Q has nine elements... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear... 

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)...

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

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... 

RBa2Cu40g (R = Sm, Y, Er) Nuclear-quadrupole coupling parameters at the rare-earth metal and copper sites from Cu ( Zn) and Ga( Zn) Mossbauer emission spectroscopy, EEG tensor in comparison with point charge model, shows that holes in lattices are localized primarily at chain-oxygen sites... 

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... 

A t)tpical feature of the Mossbauer spectra of five- or six-coordinate iron(IV) with an axial oxo group (or a OCH3, a nitrido or a imido group) is a low isomer shift (+0.1 0.15 mm s ), a large and positive quadrupole splitting (1-2 mm s ), an anisotropic hyperfine coupling tensor with moderately large values for A x/gNl N and (—16 to —23 T) and a rather small value for A Jg i (0 to —10 T)... 

The electric quadrupole Q 2co) involves both the gradient of the electromagnetic incident electric field E u)) and the gradient of the electric quadrupole susceptibility tensor Xq 2o), CO, co). This problem is nonetheless solved by the mere addition of supplementary terms in the surface susceptibility tensor. As a result, the surface susceptibility tensor becomes an effective tensor instead of a purely surface specific one [27,38] ... 

Oxirane is an important Lewis base in the present context. The O atom carries two equivalent n-pairs of electrons, as it does in H20, but oxirane has the advantage over water in that it is possible to determine both angles 0 and 9 for its complexes with HC1 and ClF because the non-zero off-diagonal element Xab(Cl) of the Cl nuclear quadrupole coupling tensor is available. The corresponding Lewis base in which an S atom carries two equivalent n-pairs is thiirane. Each of the pair of complexes (CL S- -HC1 and (CL S- -ClF has Cs symmetry and here it is the off-diagonal element Xac(Cl) that is non-zero... 

The five second-moment spherical tensor components can also be calculated and are defined as the quadrupolar polarization terms. They can be seen as the ELF basin equivalents to the atomic quadrupole moments introduced by Popelier [32] in the case of an AIM analysis ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

Quadrupole coupling, isomer shift Quadrupole tensor, nuclear Zeeman splitting, g values, coupling constants, relaxation times... 

It has recently been demonstrated that the analysis of MAS sidebands patterns can be used to study molecular dynamics in the solid state [85-88]. Indeed, the line narrowing effect of MAS can be partly offset, or completely eliminated, if the 2H quadrupole tensor is reoriented due to motion on a time scale comparable to (first-order quadrupolar broadening, such motion-induced effects should be less evident in the DQMAS spectrum, as has indeed been observed by Wimperis and colleagues in several deuterated solids [87, 88]. For example, the simulation of the SQ spectrum of tetrathionate dihydrate-cfi yielded the same reorientational rate constant as the previously described quadrupolar echo approach (Fig. 6). 

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