Spherical polar coordinates Fourier

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

We often need to calculate a three-dimensional Fourier transform of fir) that depends only on the radial distance r. For this purpose, we use a spherical polar coordinate for r with the polar axis running along k. Then, the transform proceeds as follows. 

The shape of such a surface may be described in a mathematical way which has several important advantages. The shape is analysed into harmonic components functions much as one might analyse a wave into a Fourier series. The spherical harmonics, as these components are called, have the necessary property of orthogonality however, their form is more complicated than the cos(njc) type of component of a Fourier series for a plane wave. The spherical harmonics are functions of the polar coordinate angle, referred to the director axis. The first four components, abbreviated Pq, P2(cosa), P4(cos Of) and PeCcos a) or simply Pq, P2, P4, Pe, are defined below and drawn in Fig. 4. 

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

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