Averaged tensor

The authors also reconsidered radical 20, which, in the current work, was formed by x-irradiation and studied at 77 K. The experimentally determined hyperfine coupling tensors for the two methylene protons have the same principal values and agreed with the tensors found in the earlier work.16 These tensors are presumed to be the average of the two individual proton tensors. By assuming that C5 is sp2 hybridized the authors were able to calculate, from the observed average tensors, that Ami = 1.06 mT and Amai = 3.45 mT for the methylene protons. Using the individual tensors for the methylene protons and a 3.6 mT isotropic coupling for the C4(H) proton, a satisfactory simulation of the... 

For the isotropic average of the first-order hyperpolarizability fi only elements with all three spatial indices different from one another are non zero. sgn(o) is the sign of the permutation o(xyz)=ijk of the spatial indices. For the nonlinear polarization in z-direction for example, the two elements (P)zxy= +Pchiral and (P)Zyx= Pchiral are different from zero. Far from any resonance Kleinman symmetry is valid (Pxyz=Pyzx=PZXy=PyxZ=Pzyx=PXZy) and the terms in the numerator of rra/ cancel each other resulting in (/J)p=0 for all averaged tensor elements of the first-order hyperpolarizability. 

The averaged tensor elements of the second-order hyperpolarizability yare only different from zero if at least twice the same index appears. In most experiments the polarizations of the electric fields are always kept parallel to the same direction (e.g. x). The tensor element of interest is then always (y)xxxx=yro(. 

For the Pd compound [46], since gPd is rhombic whereas gPer is isotropic, a rhombic g tensor is the result. The principal values of the averaged tensor are g, = gPer + x(g, - gPcr), where x is the fraction of interacting Pd spins in the sample and i = 1,2,3 refer to the principal values of isolated [Pd(mnt)2] species. The combined analysis of the ESR powder spectra (with each of the three g, values) and the static susceptibility made possible the separation of the contributions to Xper and Xpspin susceptibility Xper was considered to be the same. 

For many decades the principal aim of strain/stress analysis by diffraction was the determination of the average strain and stress tensors e and s in materials. The determination was based on the supposition that the elastic strain and stress tensors e(g), s(g) in a crystallite are connected to the average tensors s, e as follows ... 

In such a measuring geometry the relation between the orientationally averaged tensor components and the molecular tensor components are expressed as follows ... 

While such data is sufficient to define the motionally-averaged tensors and interpret the structural constraints in light of such tensors, it has not defined the frequency of the molecular motions. Dynamics are gaining increasing attention in the polymer field for characterizing the flexibility, noncrystalline domains and phase boundaries. In biological polymers, the characterization of dynamics can lead to insights about the kinetic rates for catalysis and ion conductance. 

In the presence of rapid molecular motion, the NMR line shape is modified by the reduced EFG tensor depending on the rate of reorientation and the orientations of the principal axes of the EFG tensor relative to the rotation axis. A calculation of the averaged tensor is performed by transformation of the EFG tensor in motion to a final reference frame in terms of Euler angles a, and y [151,154,155]. The spectral frequency in the fast... 

Let us note that this expression has two parts. The first one is position-independent and represents the dielectric tensor y averaged on a volume with dimensions much larger than the pitch / . It is worthwhile to emphasize that this averaged tensor is uniaxial with the symmetry axis parallel to the helix. Moreover, its global anisotropy is opposite to the local anisotropy. For example, if the local dielectric anisotropy is negative, that is to say, if l < x, then... 

Rotation of whole, rigid molecules on lattice sites in crystals Rotation about an n-fold axis, or hopping between n equivalent sites, produces an averaged tensor, one of whose principal axes, r, lies along the rotation axis. The location of the other two principal axes, s and will be in the plane orthogonal to r, and they are fixed with respect to the lattice and not the molecules. Thus,... 

When n> > the averaged tensor is axially symmetric about r, so that Equation [11] simplifies to... 

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