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Luminosity Integration

    Let us denote the position-velocity distribution function of j-th beam (j=1,2) at time t by tex2html_wrap_inline9704. It is normalized such that tex2html_wrap_inline9706 is the total number of particles in the j-th beam. The luminosity (per crossing) is in general given by
If all the particles in the j-th beam are ultrarelativistic and have almost the same velocity tex2html_wrap_inline9712 (tex2html_wrap_inline9714), then the expression is simplified as
where tex2html_wrap_inline8612 is the polar angle between tex2html_wrap_inline9718 and tex2html_wrap_inline9720, and tex2html_wrap_inline9722 is the number density of the j-th beam. CAIN uses this formula with tex2html_wrap_inline9726, ignoring the velocity distribution and the crossing angle.

The integration is done by introducing the time step size tex2html_wrap_inline8540, longitudinal slice width tex2html_wrap_inline9730, transverse mesh size tex2html_wrap_inline9732 and tex2html_wrap_inline9734. Summing the number of particles in each bin, the luminosity is given by
where C is an appropriate normalization factor, and tex2html_wrap_inline9738 is the number of particles of the beam j in the bin tex2html_wrap_inline9742. A problem is how to determine the transverse size  of the bin (tex2html_wrap_inline9744 and tex2html_wrap_inline9730 is mainly determined by the dynamics -- they are actually specified by the user). If the bin is too large, detail of the distribution is lost, whereas if too small, statistical error becomes large because each bin will contain only a small number of macro-particles. CAIN adopts the following way.

At first, determine the size of the whole transverse region tex2html_wrap_inline8274 such that most particles are contained there. Then, divide this region into as many bins tex2html_wrap_inline9750 as allowed by the storage requirement (n must be a power of 2. CAIN uses n=128.), and count the number of particles in each bin for both beams tex2html_wrap_inline9756 (tex2html_wrap_inline9758).

Figure 2: Bin numbering for luminosity integration. Example with n=8 and the double-sized bin n=4.

If the number of macro-particles in any of the neighbouring 4 bins are less than some number tex2html_wrap_inline9764 (CAIN adopts 5) for both beams, then sum these numbers and put the sum into a larger bin tex2html_wrap_inline9766. (For the example in Fig.2, the sum of the bins 12, 13, 14, and 15 in the figure on the left corresponds to the bin 3 on the right.) Otherwise, add tex2html_wrap_inline9768 into the luminosity sum. This doubling of the bin size is repeated so long as tex2html_wrap_inline9770. In order to make this algorithm efficient, the bin numbering system is a little complicated. Instead of using two indices tex2html_wrap_inline9772, the bins are numbered as in Fig.2. With this numbering, the sum of neighbouring bins can be simply written as
where tex2html_wrap_inline9774 is the number of particles in the k-th bin (tex2html_wrap_inline9778) in tex2html_wrap_inline9750 bin system.

next up previous contents index
Next: Beam Field Up: Physics and Numerical Methods Previous: Equation of motion under

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998