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The intrinsic contribution to the spin Hall effect in a 2DEG withnon-magnetic impurities is studied in a quantum Boltzmann approach. It is shownthat if the steady state response is perturbative in the spin-orbit couplingparameter $\lambda$, then the precession term--vital for Dyakonov-Perelrelaxation and the key to the spin Hall effect in previous similar Boltzmannstudies--must be left out to first order in spin-orbit coupling. In such a caseone would have that to lowest order in the parameters electric field,spin-orbit coupling, impurity strength and impurity concentration there is nointrinsic contribution to the spin Hall effect, not only for a Rashba couplingbut for a general spin-orbit coupling. To cover all possible lowest order termswe consider also electric field induced corrections to the collision integralin the Keldysh formalism. However, these corrections turn out to be of secondorder in $\lambda$. For comparison we derive some familiar results in the casewhen the response is not assumed to be perturbative in $\lambda$. We alsoinclude a detailed discussion of why a relaxation time approximation of thecollision integral fails. Finally we make a comment on pseudospin currents inbilayer graphene.

Boltzmann approach to the spin Hall effect revisited and electric field modified collision integrals