Unconventional high temperature superconductivity as well as three-dimensional bulk Dirac cone quantum states arising from the unique d-orbital topology has been a recent priority research area in physics. In iron pnictide compounds, although transpo
rt phenomena arisen from this multiple band Fermi surface are intriguing and scientifically important, they still do not give an adequate matching to neither experimental observations on the band picture nor theoretical calculations and a debate continues. Here we describe a new analytical approach of mobility spectrum, in which the carrier number is conveniently described as a function of mobility without any hypothesis about the number of carriers, on both longitudinal and transverse transport of high quality single crystal Ba(FeAs)$_2$ in a wide range of magnetic field. We show that the major numbers of carriers reside in large parabolic hole and electron pockets with very different topology as well as remarkably different mobility spectra, while the minor number of Dirac carriers resides in both hole- and electron- Dirac quantum states with the largest mobility as high as 70,000 cm$^2$(Vs)$^{-1}$.
