Tensor diagonalization

Fig. 4 represents the anisotropic HFSC tensor for the atoms in the cluster. The tensor diagonal elements reach their maximum values for the nearest to the vacancy carbon atoms (C5, C6 and C7), similar to the isotropic HFSC. The /C-tensor components have maximum values (up to 100 MHz). 

Unlike the isotropic hyperfine coupling constants, the anisotropic ones are mainly determined by dipole magnetic interactions. The contribution of the nearest atoms is expected to be suppressed by a distance factor. The values of AHC tensor are presented in Fig. 3. The results show that as in the previous case the tensor diagonal elements have the greatest values for the Ni atom varying from 70 to 150 MHz. But unlike to the previous situation the values for the nitrogen atoms are one order of magnitude lower as compared to the values for Ni atom. This is due to the fact that in this case the AHC constants are mainly... 

Hereafter the summation is taken from 1 to A/ for Latin indices and from 1 to Af — 1 for the Greek ones. According to Equation (6.9), tia determine the orthogonal transformation to the local frame of reference. One can check that the second condition. Equation (6.10), makes the metric tensor diagonal in the coordinates S and fo,. This condition is not essential but it simplifies the following derivations. The instanton trajectory approaches the potential minima along one of the normal modes, which we numerate as the A/th. We can also choose fa to coincide with the other N — 1 normal coordinates at the potential minimum. Due to Equation (6.10), we have... 

The values of the stress tensor components depend on the system of coordinates, in which the components are measured. The coordinate system can always be rotated in the way that the stress tensor diagonalizes ... 

Geometric Integrators for Rigid Body Simulation 355 where I = diag(/i,/2,/a) is the (diagonalized) inertial tensor,... 

FIG. 20 (a) Density profiles p(z) vs z for e = —2 and four average bulk densities (f> as indicated, (b) Surface excess vs density in the bulk for four choices of e. (c) Profiles for the diagonal components of the pressure tensor and of the total pressure for (p = l.O and e = —2. Insert in (c) shows the difference between P, and Px to show that isotropic behavior in the bulk of the film is nicely obtained, (d) Interfacial tension between the polymer film and the repulsive wall vs bulk density for all four choices of e. Curve is only a guide for the eye [18]. 

We make two additional comments. First, notice that when u 0, the momentum flux density tensor reduces to the diagonal term p5ij, where the pressure p = Cgp and Cg is the speed of sound. We thus conclude that the speed of sound in the FHP-I LG is given by... 

Fig. 2 Raman spectra of a single-crystal of orthorhombic Sg at three different polarizations in which the off-diagonal elements of the Raman scattering tensor are non-zero big, b2g, b g), after [105]. However, Raman intensities of other polarizations like flg components ( 54 cm ) penetrate in the spectra due to optical anisotropy in the crystal...

Note that for a macroscopically isotropic medium, the tensor given by Eq. (Ill) has equal elements along the diagonal, and therefore Eq. (110) is equivalent to... 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields... 

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 trace vanishes because only p- and /-electrons contribute to the EFG, which have zero probability of presence at r = 0 (i.e. Laplace s equation applies as opposed to Poisson s equation, because the nucleus is external to the EFG-generating part of the electronic charge distribution). As the EFG tensor is symmetric, it can be diagonalized by rotation to a principal axes system (PAS) for which the off-diagonal elements vanish, = 0. By convention, the principal axes are chosen such that... 

The values of the rhombicity parameters are conventionally limited to the range 0 < EjD < 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 < EID < 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships... 

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... 

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

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