Spherical surface harmonic

As seen above, the angle ij/ takes random values (n — 0) for structural units with cylindrical symmetry. The ODF is then defined by two angles and reduces to the following expansion in surface spherical harmonics... 

Surface Spherical Harmonics. From the two sets of orthogonal functions ITU (cos 0), cos ( up) we can form a third set of functions... 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

From an operational point of view, equations of the form (65) and (66) are not entirely satisfactory, for they are not written in a truly invariant form. Given an irregular particle, where is the axis cos 0 = 1, and how is the radius a of the undeformed sphere to be defined The difficulty stems, of course, from the description of the surface of the particle in terms of the surface spherical harmonic expansion (64). From an operational viewpoint it is more consistent with the nature of the requisite physical measurements to express the size and shape of the body in terms of an expansion involving its volumetric moments. The kih. such moment is... 

In eqn. 5.2.14 the coefficients of the series expansions depend on the variable x = /cr, and S v(0, polar angles 0 and q>. The vector coefficients 1 /Jc) are independent of the position of ion j which may be considered stationary according to eqn. 5.2.6. At infinity y) and/ vanish. Besides, Pitts assumed that for r = a the perturbations may be neglected thus the ionic potentials and distribution functions on the surface of the central ion are not affected by the external field. 

There is then a net scattering amplitude (Z-fl) /+ -fZ/ associated with l = li scattering where the neutron spin is not changed, and an amplitude (/ — /+) associated with spin flip. The non spin flip part has a factor (cos ) (surface spherical harmonic for Z = Z, m = 0) while the spin flip amplitude is multiplied by the surface spherical harmonic for Z = Zj, w==F 1 to give a factor sin times a function of cos. For 74=0 it is necessary to state more fully how Jq is composed from Ja> a> 1. The specification of the angular dependence of... 

The functions are the surface spherical harmonic functions made up... 

Fig. 2. Pattern selection on a hemisphere (approximating the growing tip of a plant). Surface spherical harmonics for index Z=3 (A) m = 2 (B) m=0 (C) equal mix of (A) and (B) produces a dichotomous branch pattern. Linear analysis predicts (A) and (B) patterns would grow equally, therefore always producing (C). Plant development shows both (B) and (C) patterns. Simulation and nonlinear analysis show that the full RD dynamics (Brusselator model) do have the capacity to produce both (B) and (C) patterns. From Holloway Harrison (2008), with permission.

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