  In the presence of a thermal medium or an external electro-magnetic field,
neutrinos can interact with photon, mediated by the corresponding charged
leptons (real or virtual). The effect of a medium or an electromagnetic field
is two-fold - to induce an effective $\nu \gamma$ vertex and to modify the
dispersion relations of all the particles involved to render the processes
kinematically viable. It has already been noted that in a medium neutrinos
acquire an effective charge, which in the standard model of electroweak
interaction comes from the vector type vertex of weak interaction. On the other
hand in a magnetized plasma, the axial vector part also start contributing to
the effective charge of a neutrino. This contribution corresponding to the
axial vector part in the interaction Lagrangian is denoted as the axial
polarisation tensor. In an earlier paper we explicitly calculated the form of
the axial polarisation tensor to all odd orders in external magnetic field. In
this note we complete that investigation by computing the same, to all even
orders in external magnetic field. We further show its gauge invarience
properties. Finally we infer upon the zero external momentum limit of this
axial polarisation tensor.
