The Hanle effect of the two-level atom in the weak-field approximation

We apply the weak-field approximation of the line profiles, ω _B ≪ Δω (ω_B being the Larmor frequency and Δω the width of the line profile), adopting a second-order Taylor expansion in the Larmor frequency, to the polarized emission coefficients of the resonance transition in a two-level atom with values of the total angular momentum J_u and J_l, for the upper and lower levels, respectively. Using methods of Racah's algebra, we then derive rather compact algebraic expressions for the Stokes parameters of the radiation scattered locally in this transition. The advantage of using a second-order Taylor expansion of the emission coefficients is that the contributions of the Hanle effect and the Zeeman effect to the Stokes vector of the scattered radiation are easily identified and their respective roles in determining the polarization properties of the line better understood. Under the further assumption of unpolarized lower level, these expressions can be applied to derive explicit diagnostic formulae for polarized, resonance-scattering radiation, in terms of the relative geometry of the observer with respect to the solar magnetic field at the scattering center. The typical case of a spinless, two-level atom with J_u = 1 and J _l = 0 is explicitly worked out, and the corresponding diagnostic formulae are used to clarify some interesting properties of the Hanle effect. Finally, we compare these results, derived from the quantum-mechanical theory of line formation, with a recently proposed, classical formulation of the Hanle effect.