Tensor traceless quadrupole

In the response function terminology [47] the i, j component of the frequency-dependent dipole polarizability tensor — w) (or the ij, kl component of the traceless quadrupole polarizability tensor is defined through... 

Expressions for the other elements of the traceless quadrupole tensor are obtained by simple permutation of the indices. 

Expressions (8.43) for the elements of the traceless quadrupole tensor can be written in a slightly different form as... 

Frequently, a traceless quadrupole moment tensor is defined, which has only five independent elements given as... 

Here, D is a second-rank tensor (in dyadic notation) that describes the (traceless) quadrupole and 02 is the angle that the principal axes of D form with the coordinate axes e., e, . Without loss of generality, one can write at any point Fh... 

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). 

Nuclei with spin I > j possess a quadrupole moment eQ and may interact with electric field gradients (EFG) present in the solid. The EFG is described by the traceless symmetrical tensor (72)... 

Hm describes the hyperfine interaction with the 57Fe nucleus. A is the magnetic hyperfine tensor and Hq describes the interaction of the quadrupole moment Q of the 7=3/2 nuclear excited state with the (traceless) electric field gradient (EFG) tensor V (the nuclear ground state has 7= 1/2 and lacks a spectroscopic quadrupole moment). In the absence of magnetic effects (for instance, for S 0, or S = integer for B = 0), the Mossbauer spectrum consists of a doublet with quadrupole splitting ... 

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]... 

An electrostatic quadrupole moment is a second-rank tensor characterized by three components in its principal-axis system. Since the trace of the quadrupole moment tensor is equal to zero, and atomic nuclei have an axis of symmetry, there is only one independent principal value, the nuclear quadrupole moment, Q. This quadrupole moment interacts with the electrostatic field-gradient tensor arising from the charge distribution around the nucleus. This tensor is also traceless but it is not necessarily cylindrically symmetrical. It therefore needs in general to be characterized by two independent components. The three principal values of the field-gradient tensor are represented by the symbols qxx, qyy and qzz with the convention ... 

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... 

Nuclear quadrupolar interaction arises from the coupling between the nuclear quadrupole moment Q and the EFG at the nuclear position. The EFG varies in space and is described by a traceless second-rank tensor. The EFG tensor is diagonal and its three principal components are VXXr Vyy and Rzz with the definition of VZZ > Vyy > VX < Such a principal-axis system for the EFG tensor is defined with the direction of the external magnetic field, as illustrated in Figure 3(A). It is convenient to express such quadrupolar interactions by using the following two parameters ... 

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

One-dimensional quadrupole echo NMR lineshape analysis of powder samples is particularly informative when fast, discrete jumps occur between sites of well-defined geometry as, for example, in a phenyl group undergoing two-site exchange. In this case, the characteristic Pake-pattern is transformed into an axially asymmetric lineshape with an apparent asymmetry parameter r] 9 0 (see Equation (6.2.3)) [1-8]. The asymmetric lineshapes, shown on the left in Fig. 6.2.2, can be derived by considering how the individual components of the principal EFG tensor become averaged by the discrete jumps. Within the molecular frame, and in units of as defined by Equation (6.2.2), the static axially symmetric tensor consists of the components = 1, = — 1/2, and V y = — 112. This traceless tensor satisfies the... 

Because the EFG tensor is traceless and symmetric, two parameters are sufficient to define the magnitude of the principal components, the quadrupole coupling constant, x, and the asymmetry parameter... 

From Table 7.1, one realizes that placing MTPs up to quadrupoles on a given site will yield nine independent parameters in spherical coordinates (i.e., one for the monopole, three for the dipole, and five for the traceless second-rank tensor). However, the main computational hurdle in MD simulations is the force calculation. Although Eq. 7.6 refers to the pairwise interaction potential, it shows that the associated force (and of course energy) will consist of n x n independent terms, where n is the number of MTP coefficients. As such, the interaction between two MTP sites, described up to quadrupole, will involve 9x9 = 81 terms—to be put in perspective with the single term prescribed by the Coulomb interaction in standard PC force fields. This certainly provides one major reason why MTP force fields have not become routine in the MD community. 

In Equation (2), D and E are the axial and rhombic zero-field splitting (ZFS) parameters, respectively, and g is the electronic g tensor. The magnetic hyperfine interactions of the electronic system with the Fe nucleus are described by S-a-I, and —is the nuclear Zeeman term. The quadrupole interaction involves the traceless EFG tensor. The EFG tensor has principal components Vyy, and The asymmetry parameter t] = Vxx- VyyyiV can be confined to 0 < 7 < 1 if the convention V zl > I Vyy > V xl is adopted. A quadrupole doublet... 

Besides the magnetic dipole moment, nuclei with spin higher than 1/2 also possess an electric quadrupole moment. In a semiclassical picture, the nuclear electric quadrupole moment informs about the deviation of nuclear charge distribution from spherical symmetry. Nuclei with spin 0 or 1/2 are therefore said to be spherical, with zero electric quadrupole moment. On the other hand, if the nuclear spin is higher than 1/2, the nuclei are not spherical, assuming cylindri-cally symmetrical shapes around the symmetry axis defined by the nuclear total angular momentum [17]. Within the subspace [/,m), the nuclear electric quadrupole moment operator is a traceless tensor operator of second rank, with Cartesian components written is terms of the nuclear spin [2] ... 

The 3x3 electric field gradient tensor is traceless and symmetric, hence it contains only five independent parameters. Three of these convey the orientation of the principal axes of the efg with respect to the crystal axes, and could be obtained by single crystal experiments. The remaining two, with which we shall be principally concerned, are the quadrupole coupling constant, e qQ,a measure of the magnitude of the interaction, and the asymmetry parameter rj, which measures the departure from cylindrical symmetry of the efg tensor ... 

In this expression, C is a constant specific for each interaction, depending on gyromag-netic ratios and nuclear electric quadrupole moments. la is a Cartesian component of the nuclear spin, with the sum extended to all three Cartesian coordinates. Rap represents the components of a 3 x 3 Cartesian tensor of second rank that specifies the detailed nature of each interaction, some examples of which appeared in the expressions given previously (a, V, and J). Finally, Ap is a Cartesian component of a vector that can be the same nuclear spin vector (quadmpolar interaction), another nuclear spin vector (dipolar interaction or /-coupling), or the external magnetic field (chemical shiff interaction). Some of the tensors represented by Rap are traceless (cases of dipolar and quadmpolar interactions), whereas others (cases of chemical shift and /-coupling) posses non-vanishing trace. 

Nuclear magnetic resonance (NMR), in particular, deuterium NMR, has proven to be a valuable technique for determining the nature of molecular organization in liquid crystals. The utility of the NMR technique derives from the fact that the relevant NMR interactions are entirely intramolecular, i.e. the dominant interaction is that between the nuclear quadrupole moment of the deuteron and the local electric-field gradient (EFG) at the deuterium nucleus. The EFG tensor is a traceless, axially symmetric, second-rank tensor with its principal component along the C—D bond. In a nematic fluid rapid anisotropic reorientation incompletely averages the quadrupolar interaction tensor q, resulting in a nonzero projection similar to the result in Eq. (5.6) ... 

The quadrupole interaction (Fig. 18), a traceless tensor, is only valid when an isotope of an atom contains a nuclear spin greater than 54. This interaction only has Axial and Orthorhombic representations. The quadrupole interaction is included in the calculation if it is active and valid (red tick on the interaction tab) and may be toggled ofFon through a middle mouse click on the tick. When it is valid and inactive (blue tick), the parameters are written to the simulation file, but are not used in the ealeulation. If fliere is no tiek on the tab, then the interaction is invalid, as there are no isotopes that have a nuclear spin greater than 54. 

The uniaxial nematic phase possesses a quadrupole-type symmetry and is characterized by the order parameter Qajj which is a symmetric traceless second-rank tensor ... 

The electric field gradient (EFG) is a ground state property of solids that sensitively depends on the asymmetry of the electronic charge density near the probe nucleus. The EFG is defined as the second derivative of the electrostatic potential at the nucleus position written as a traceless tensor. A nucleus with a nuclear spin number / > 1 has a nuclear quadrupole moment (Q) that interacts with the EFG which originates from the nonspherical charge distribution surrounding this nucleus. This interaction... 

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