Modified spherical harmonics

The explicit forms of the first few spherical harmonics are given in table 5.1. It is sometimes more convenient to use modified spherical harmonics Cfa, defined by... 

Table 1. Tensor Operators in the Rotating Frame and Modified Spherical Harmonics for the Dipolar and CSA Interaction. The calibration has been chosen such that (j>) 0) d (cos 0)d is independent of q. ...

Tensor Operators for the Dipolar Interaction Tensor Operators for the CSA Interaction Modified Spherical Harmonics Frequency... 

Table 1. Polar forms of the modified spherical harmonic functions...

Provided, as is usually the case, the distribution function is well-behaved it can be expanded in an appropriate set of functions. Here the modified spherical harmonics, CLm(Py) [1.3], are especially... 

The other components have a particularly simple dependence on y as may be seen from the first few modified spherical harmonics... 

This result follows from the definition of the modified spherical harmonics given in EQN (24). For perfect order the molecular z axis is aligned parallel to the director so that is just unity. The other components < > with m not equal to isso vanish because th are all proportional to sinp. 

As we have come to expect, the expansion coefficients in the distribution are proportional to the order parameters for die system. These ordw parameters are obtained by exploiting the orthogonality of the modified spherical harmonics (see EQN (26)) and the Fourier fiinctions... 

The orientational order in a uniaxial phase, such as a nematic, composed of rigid particles of arbitrary shape is characterized by an infinite set of order parameters defined as the averages of modified spherical harmonics... 

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