Relativistic Corrections to the Sunyaev-Zel'dovich Effect for Clusters   of Galaxies. V. Multiple Scattering
  We extend the formalism for the calculation of the relativistic corrections to the Sunyaev-Zel'dovich effect for clusters of galaxies and include the multiple scattering effects. We present a systematic method for the inclusion of the multiple scattering effects. The multiple scattering contribution is found to be very small compared with the single scattering contribution. For high-temperature galaxy clusters of k_{B}T_{e} = 15keV, the ratio of the both contributions is -0.3% in the Wien region. In the Rayleigh--Jeans region the ratio is -0.03%. Therefore the multiple scattering contribution is safely neglected for the observed galaxy clusters. 
Naoki Itoh, Youhei Kawana, Satoshi Nozawa and Yasuharu Kohyama