Weibel instability with semirelativistic Maxwellian distribution function

A macroscopic description of the linear Weibel instability, based on semirelativistic distribution in an unmagnetized plasma is presented. In particular, analytical expressions are derived for the real and imaginary parts of the dielectric constant for the Maxwellian and semirelativistic Maxwellian distribution functions under the conditions of ξ=(ω/kparallelθparallel)>>1 and <<1. The real frequency and the growth rate of the instability for the semirelativistic case now depends upon the factor χ generated from the relativistic term in the distribution function. The presence of χ which is always greater than unity favors the Weibel instability to occur even for the small anisotropy of temperature. As we increase the value of χ large enough that it dominates over other terms, the damping changes into growth. In the limiting case, i.e., χ=1, the results approach the Maxwellian situation