Even-ordered spherical harmonic

Fig, 4. Sections through the even-ordered spherical harmonic functions, Pq, P2, P4 and P( which are in effect Fourier-type components of orientation distributions such as that shown in Fig. 3. ... 

The use of eqn (23) requires a knowledge of the spherical harmonics, /2),(s), for a perfectly aligned structural unit. Before proceeding to discuss values for If (s) it is appropriate to emphasize the limitations of eqn (23). It is an obvious requirement that I s, a), I (s, a) and D(a) have the appropriate axial and mirror symmetry that enables them to be expressed in terms of a series of even-order spherical harmonics. Equally the expression of the convolution of eqn (18) in the form of eqn (19) or as a deconvolution eqn (23) requires the convolution to involve the scattering from independent structural units. In other words, eqn (19) ensues from the separation of the spatial and orientational correlations. 

This time is smaller than for a linear chain of Aymonomers but much larger than rd. Since D is very small, it is difficult to measure in a simulation and has not been done. The rotational diffusion time, which should be comparable to td, can be analyzed by studying the time autocorrelation function of the center-to-end vector R or the autocorrelation of the squares of the second-order spherical harmonics of the angles at which the principal axes of the ellipsoid are oriented with respect to a fixed-coordinate system. However even for small there was no linear region in the semilog plots of C t) for the times that can presently be simulated, making it impossible at present to test eq. (9.23). 

For magnet configurations in which coils are coaxial and symmetric about the illustrated xy-plane, such as the magnet configurations in Figure 2A and C, the spherical harmonic expansion results in the elimination of all even order terms within the expansion. To further reduce computational complexity, the strategy employed here considers only one quarter of the magnet domain, and thus, the constraints in Equation (5) simplify to ... 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

Only even orders are taken into account in Eqs. 2.83 and 2.84 due to the presence of the inversion center in the diffraction pattern. The number of harmonic coefficients C and terms k(h) varies depending on lattice symmetry and desired harmonic order L. The low symmetry results in multiple terms (triclinic has 5 terms for L = 2) and therefore, low orders 2 or 4 are usually sufficient. High symmetry requires fewer terms (e.g. cubic has only 1 term for L = 4), so higher orders may be required to adequately describe preferred orientation. The spherical harmonics approach is realized in GSAS. ... 

In a diffraction experiment the crystal reflection coordinates (cj), p) are determined by the reflection index (h) while the sample coordinates (ij/, y) are determined by the orientation of the sample on the diffractometer. This formulation assumes that the probability surface is smooth and can be described by a sum of spherical harmonic terms, kfl and kjf, that depend on h and sample orientation, respectively, to some maximum harmonic order, L (ref. 25). The coefficients Cf" then determine the strength and details of the texture. Notably, only the even order, L = 2n, terms in these harmonic sums affect the intensity of Bragg reflections the odd order terms in the ODF are invisible to diffraction. 

On the other hand, the use of even higher-order DO approximations is not warranted given the current level of the accuracy and availability of radiative properties. In multidimensional systems, moment, spherical harmonics, and hybrid multiflux approximations usually do not yield results with the accuracy or efficiency of the DO approximations. 

When the theoretical magic numbers are compared with the discontinuities measured in the abundance spectrum (fig. 12.5) or in the ionisation potentials (fig. 12.6), they line up pretty well for Kre, the observed numbers are 2, 8, 18, 20, 58, 92, and this is taken to imply that the original assumption of a spherical short range potential was a reasonable one. In fact, the result is not too critically dependent on the form of the potential. Even a square well is a fair approximation, and one can also adapt a harmonic oscillator well, truncated at an appropriate radius, by adding to it a term proportional to ( , all of which generate very similar orderings of shells and magic numbers. 

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