Second-moment tensor

In the nonzero trace definition, this equality is no longer valid. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as... 

Including the spherical component, the electric field gradient can be interpreted as the second-moment tensor of the distribution p(r)/ r — r 5. 

From the above expressions, the form birefringence and dichioism can be calculated. The form dichroism, in particular, has the simple interpretation of being the anisotropy in the second-moment tensor of the structure factor. It is also evident that the form dichroism appears at a higher order in the wave number than the form birefringence. 

Equation (7.24) predicts that the refractive index tensor is proportional to the second-moment tensor of the orientation distribution of the end-to-end vector. This expression was developed using a number of assumptions, however, and is strictly valid only for small... 

In principle, once the probability distribution function is available, bulk solution properties can be evaluated by averaging appropriate functions of conformation space and time. From the Kuhn and Grun analysis leading to equation (7.24) for the refractive index tensor, we are particularly interested in the second moment tensor,... 

Because this equation has the same form as equation (7.51), the second-moment tensor of the modes of relaxation can be solved using identical procedures to those discussed previously. 

As before, it is of interest to evaluate the second-moment tensor, (RR) = L (uu). Multiplying equation (7.62) by (uu) and integrating over the unit sphere gives the following equation of motion ... 

This equation is not closed in the unknown, second-moment tensor due to the presence of the (RRRR) term. One solution procedure often used, is to invoke a closure approximation, of which the form (RRRR) = (RR) (RR) is the simplest. This approximation, however, is only quantitatively accurate in the limit of nearly perfect orientation of the dumbbells, but is able to offer correct, qualitative responses for many purposes. 

Bulk material properties can be determined quite simply using this model. For example, consider the calculation of the second-moment tensor, Q = (u u ), which is required for the stress and refractive index tensors. Using the independent alignment approximation, we have... 

Scattering or form birefringence contributions will cause a deviation in the stress optical rule. As seen in equation (7.36), these effects do not depend on the second-moment tensor, but increase linearly with chain extension. 

This result is referred to as the Giesekus expression [62,86] and can be used to develop the form of the stress tensor for the rigid dumbbell model. Equation (7.63) for the rate of change of the second-moment tensor for this model is used to give the following result ... 

If Eq. (11-3) is multiplied by uu and integrated over the unit sphere, one obtains an evolution equation for the second moment tensor S (Doi 1980 Doi and Edwards 1986). In this evolution equation, the fourth moment tensor (uuuu) appears, but no higher moments, if one uses the Maier-Saupe potential to describe the nematic interactions. Doi suggested using a closure approximation, in which (uuuu) is replaced by (uu) (uu), thereby yielding a closed-form equation for S, namely. 

Similarly, the definition of each diagonal element of the orbital second-moment tensor is formally identical to that of the statistical central second moment., wi2 ... 

First, a coarse-grained description of the polymer melt is invoked through the definition of the conformation tensor, c, which is a global descriptor of the long-length scale conformation of polymer chains. The conformation tensor c is defined as the second moment tensor of the end-to-end distance vector of a polymer chain reduced by one-third of the unperturbed end-to-end distance and averaged over all chains in the system ... 

A point on the azimuthal scan may be represented by a unit vector, n, such that u, = cosfi and = sinjS (fi is the azimuthal angle). A weighted average of the second moment tensor of n provides a simple representation of anisotropy in the scattering pattern ... 

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 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 ... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

Note that since both /z and E are vector quantities, a is a second-rank tensor. The elements of a can be computed through differentiation of Eqs. (9.1) and (9.2). The difference between die permanent electric dipole moment and that measured in the presence of an electric field is referred to as the induced dipole moment. 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

The presence of magnetic moments /lia, b, of nuclei A,B,... in a molecule are responsible for the two observables of the NMR experiment that are most frequently utilized in chemical applications. They are physically observed in form of quantized energy differences AE that can be measured very precisely. These two observables are the nuclear shielding tensor cr for nucleus A and the so-called indirect reduced coupling tensor KAB for a pair of nuclei A,B. Both crA and Kab are second-rank tensors that are defined via the phenomenological Hamiltonians... 

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] ... 

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 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. 

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