Incomplete Uniform Distribution Function

For the incomplete uniform distribution function, we use a global coordinate system to specify both the direction of propagation and the states of polarization of the incident and scattered waves, and the particle orientation. A special orientation with a constant orientation angle can be specified by setting Ns = 1 and (i in = max) where 6 stands for Op, (3p and 7p. For example. 

The scattering characteristics are averaged over the particle orientation by using a numerical procedure. The prescriptions for choosing the sample angles and the significance of the parameters are as in Sect. 3.1.1. For each particle orientation we compute the following quantities  

The azimuthal angles describing the positions of the scattering planes at which the phase matrix is computed are 93(1), (p 2). In each scattering 

For spherical particles, the code chooses a single orientation Opmin = pmax 0, / pmin Ppmax 0 and prnin O pmEix 9 j and Sets 

The codes perform calculations with double- or extended-precision floating point variables. The extended-precision code is slower than the doubleprecision version by a factor of 5-6, but allow scattering computations for substantially larger particles. It should also be mentioned that the 

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The scattering characteristics depend on the type of the orientation distribution function. By convention, the uniform distribution function is called complete if the Euler angles Op, / p and 7p are uniformly distributed in the intervals (0,360°), (0,180°) and (0,360°), respectively. The normalization constant is 47t for axisymmetric particles and for nonaxisymmetric particles. The uniform distribution function is called incomplete if the Euler angles Op, /3p and 7p are uniformly distributed in the intervals (apmin, pmax), (/ pmin,/ pmax), and (7pmin, 7pmax)> respectively. For axisymmetric particles, the orientational average is performed over Op and / p, and the normalization constant is... 

In order to apply the IRT method, Eq. (4.79) needs to be inverted for t, in order to extract a reaction time, which is not straightforward. The authors have devised two methods to make sampling from this distribution function possible. The first method involves fitting this probability distribution to an incomplete y-function for a range of r. A random uniform number is then generated from the appropriate y-distribution [16]. The second method generates a random number f/(0,l] uniform in the interval (0,1] and solves the equation... 

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