Intrabeam scattering growth rates for a bi - gaussian distribution

This note finds results for the intrabeam scattering growth rates for a bi-gaussian distribution. The bi-gaussian distribution is interesting for studying the possibility of using electron cooling in RHIC. Experiments and computer studies indicate that in the presence of electron cooling, the beam distribution changes so that it developes a strong core and a long tail which is not described well by a gaussian, but may be better described by a bi-gaussian. Being able to compute the effects of intrabeam scattering for a bi-gaussian distribution would be useful in computing the effects of electron cooling, which depend critically on the details of the intrabeam scattering. The calculation is done using the reformulation of intrabeam scattering theory given in [1] based on the treatments given by A. Piwinski [2] and J. Bjorken and S.K. Mtingwa [3]. The bi-gaussian distribution is defined below as the sum of two gaussians in the particle coordinates x, y, s, p x , p y , p s. The gaussian with the smaller dimensions produces most of the core of the beam, and the gaussian with the larger dimensions largely produces the long tail of the beam. The final result for the growth rates are expressed as the sum of three terms which can be interperted respectively as the contribution to the growth rates due to the scattering of the particles in the first gaussian from themselves, the scattering of the particles in the second gaussian from themselves, and the scattering of the particles in the first gaussian from the particles in the second gaussian.