Electric quadrupole tensor

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

Solid-phase 27A1 n.m.r. spectra have been obtained for epidote, Ca2Al2-(Fe,Al)Si3012(0H).452 Two different nuclear electric quadrupole tensors... 

If we depart from pure classical electrostatics, things become slightly more complicated. The deviation from spherical symmetry of an atomic nucleus can be described by a nuclear electric quadrupole tensor Q. A natural axis system for this tensor is one where one axis is colinear with the nuclear spin axis. By virtue of the effective cylindrical symmetry of a rapidly spinning deformed nucleus, the components of the quadrupole tensor are determined by a single scalar quantity Q and in addition by the components of the nuclear spin vector i. 

The term Q isa component of the electric quadrupole tensor written in the PAS of the tensor V. The PAS of the tensor Q is characterized by the symmetry axis of the distribution (axis z ), which makes an angle 9 with z-In this system, we have ... 

We observe that it has the form of the scalar product of a nuclear electric quadrupole tensor and an electric field gradient tensor, each of rank two. 

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... 

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 ... 

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] ... 

The spin interactions, dipole-dipole (D), nuclear electric quadrupole (Q) and chemical shielding (C.S), may be formally written in terms of irreducible tensors of rank l34 in Hz ... 

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... 

JJor chemists interested in modem theories of chemical bonding, the most useful data obtainable by the Mossbauer technique are the magnitude and sign of the electric quadrupole field gradient tensor and the magnitude of the shift, 8, (which we prefer to call the chemical isomeric. Cl, shift), of the center of the Mossbauer spectrum relative to some standard absorber. Although a considerable amount of chemical and structural information is potentially available from quadrupole data on iron compounds, relatively little use has been made of such data in the literature, and we will not discuss this parameter here. We will instead restrict ourselves to two main points review of the explanations put forth to explain Cl shift data in iron compounds, and a survey of some of the correlations and generalizations which have been found. 

Electric-field-gradient tensor, 230 Electric potential, 230 Electric quadrupole moment, 229-231, 365-366... 

When a nucleus with spin I is placed in a static magnetic field, the energy splits into 21 + 1 equally spaced levels, and a 2J-fold degenerate resonance line can be observed in an NMR experiment (16). Nuclei with spin I > 1/2 possess an electric quadrupole moment Q which may interact with the gradient of the electric crystal field at the site of the nucleus. This field gradient is a traceless tensor (17). 

In this model, the structural symmetry of the boundary region is reflected in the form and magnitude of the tensor elements of the surface nonlinear susceptibility, xf and the bulk anisotropic susceptibility, . For 1.06/tm excitation, the penetration depth of E(co) is about 100 A. The surface electric dipole contribution is thought to arise from the first 10 A. The electric quadrupole allowed contribution to E(2to) from the decaying incident field is attenuated by e-2 relative to the surface dipole contribution. Consequently, the symmetry of the SH response should reflect the symmetry of at least the first two topmost layers. For a perfectly terminated (111) surface, the observed symmetry should be reduced from the 6 m symmetry of the topmost layer to 3 m symmetry as additional layers are included. This is consistent with the observations for the centrosymmetric Si(lll) surface response shown in Fig. 3.2 [67, 68]. 

For oriented samples, the rotation of the plane-polarized light becomes a tensor - that is, the optical rotation becomes directionally dependent - and includes a contribution from the electric dipole-electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30], The term arising from these quadrupolar interactions can be expressed as [30]... 

The predominant interaction for a 2H spin system is the quadrupolar interaction, which couples the electric quadrupole moment of the 2H nucleus to its electronic surrounding. This interaction is a second-rank tensor Hq which lies approximately along the C-2H bond in organic molecules. Thus, in practice, 2H nuclei may be considered to be isolated. It shows that the 2H NMR formalism is similar to that of an isolated proton pair [8] ... 

The spacing of the different energy levels studied by NQR is due to the interaction of the nuclear quadrupole moment and the electric field gradient at the site of the nucleus considered. Usually the electric quadrupole moment of the nucleus is written eQ, where e is the elementary charge Q has the dimension of an area and is of the order of 10 24 cm2. More exactly, the electric quadrupole moment of the nucleus is described by a second order tensor. However, because of its symmetry and the validity of the Laplace equation, the scalar quantity eQ is sufficient to describe this tensor. 

Nuclear Quadrupole Resonance (NQR) [50-54], Nuclear (electric) quadrupole resonance (NQR) was invented in 1950 [55] and is applicable to nuclei with nonzero nuclear electric quadrupoles eQ, which are 3x3 tensors, whose significant components are the quadrupole coupling constant QCC ... 

The potential energy due to the electrical quadrupole in a local principal-axis system, where the eQ tensor is diagonal and qzz > qyy > qxx is... 

The Qjk are components of a second-rank cartesian tensor called the nuclear quadrupole tensor, and the Vjk are components of the electric field gradient tensor. Both Q and V are traceless, symmetric tensors of the second rank. The electric field gradient tensor components can be written in a more compact form by noting that... 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

The nuclear electric quadrupole interaction was described in detail in chapter 4, where we showed that it could be represented by the scalar product of two second-rank tensors ... 

For nuclei with 7 1, the nuclear electric quadrupole moment interacts with the electric field gradient in a nonsymmetric environment of electrons around the nucleus. The electric field gradient is described by a tensor that can be expressed in diagonal form (components Vx, VY, and IQ) in a principal axis system fixed in the molecule. By Laplace s equation the sum of the three components is zero, so there are two independent parameters, usually taken as the largest of the three components, V7 = d2V/dz2, and the asymmetry parameter tj = (Vx — IQ)/ Vz, with the convention that VZ VX. The electrical interaction leads to... 

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