B Spherical Harmonics

There are three broad categories of deterministic whole reactor calculation methods currently in use (a) diffusion theory (b) spherical harmonics approximations, or Pn methods, and (c) discrete ordinates or Sn methods. Diffusion theory is widely used for reactor core calculations although the faster Pn and Sn methods developed in recent years are now replacing them. 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

Expressions (3.42) and (3.43) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same /, in. This is summarized in the statement that the spherical harmonic functions are Fourier-transform invariant. It means, for example, that a dipolar density described by the function dl0, oriented along the c axis of a unit cell, will not contribute to the scattering of the (hkO) reflections, for which H is in the a b plane, which is a nodal plane of the function dU)((l, y). 

The linear relationships between the traceless moments 0 and the spherical harmonic moments lmp are obtained by use of the definitions of the functions clmp. For example, for the quadrupolar moment element xx, we obtain the equality (3x2 — l)/2 = a (x2 — y2)/2 + b(3z2 — 1). Solution for a and b for this and corresponding equations for the other moments leads to... 

To obtain the interaction energy, Eq. (9.53) is substituted into Eq. (9.50). When the charge density B is expressed in terms of one or more spherical harmonic expansions, only terms of like symmetry will integrate to nonzero values, and we obtain the expression for the interaction between two distributions as... 

Re] Reid, B.P., Spherical Harmonics, http //www.bpreid.com/applets/ poasDemo.httnl, 2004. 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

A nondegenerate irrep that is symmetric with respect to the principl axis is denoted A, while B indicates antisymmetry with respect to this axis. In point groups with a horizontal plane of reflection, primes and " respectively indicate symmetry and antisymmetry with respect to the plane, while g and u indicate symmetry and antisymmetry with respect to inversion. For doubly degenerate irreps a subscript m indicates which spherical harmonics VJ, m form basis functions for that irrep. Numerical subscripts are used on nondegenerate irreps to distinguish them where necessary the numbers indicate the first of the vertical planes or perpendicular twofold axes (in the order specified in the character table) with respect to which the irrep is antisymmetric. 

All three of ea>, e(2 e(3] can be expressed in terms of vector spherical harmonics. Thus, in addition to the nonlinear B cyclic theorem, the following linear relations occur... 

Figure 4.45 Illustration of the content of equ. (4.90) which describes the angular distribution of Auger electrons (eb) in coincidence with the preceding photoelectron (ea). The data refer to 2p3/2 ionization of magnesium by linearly polarized photons of 80 eV and subsequent L3-M1M1 Auger decay, with emission of both electrons in a plane perpendicular to the photon beam direction. The alignment tensor a,

Fig. 4.45, such a case can be found, e.g., for a = 150°. Here all components of the alignment tensor are needed, and they give positive weight to the b-dependent spherical harmonics ... 

Starting with the parametrization for two-electron emission given in equ. (4.68), and using equ. (4.67) which relates the bipolar spherical harmonics to spherical harmonics attached to the actual directions ka = a, electron emission, the triple differential cross section for two-electron emission can be rewritten as... 

In this equation the symbols X (a,b) denote the so-called nonnormalized spherical harmonics defined as... 

In Eq. (4.323) notation of the type Y(a,b) means that a spherical harmonic is built on the components of a unit vector a in the coordinate system whose polar axis points along the unit vector b. The functions. + and, S4 in Eq. (4.324) are the equilibrium parameters of the magnetic order of the particle defined, in general, by Eqs. (4.80)-(4.83). 

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