ESR Relaxation and Lineshapes from the Generalized Cumulant and Relaxation Matrix Viewpoint

Abstract

We start with the time rate of change of the spin density matrix for a single spin system:

$$\dot \sigma \left( {\rm{t}} \right) = - {\rm{i}}\left[ {H,\sigma } \right] \equiv - {\rm{i}}{H^ \times }\sigma $$

((1))

where H = H ₀ + H ₁ (t), and we use the Kubo^2,3 notation for super-operators: A^×, such that A^×B = [A, B]. We define an interaction representation by:

$${\sigma ^\ddag }\left( {\rm{t}} \right) = {{\rm{e}}^{{\rm{i}}H_0^ \times {\rm{t}}}}\sigma = {{\rm{e}}^{{\rm{i}}{H_0}{\rm{t}}}}\sigma {{\rm{e}}^{ - {\rm{i}}{H_0}{\rm{t}}}}$$

((3))

and

$$H_1^\ddag \left( {\rm{t}} \right) = {{\rm{e}}^{{\rm{i}}H_0^ \times {{\rm{t}}_{{H_1}}}}}\left( {\rm{t}} \right) = {{\rm{e}}^{{\rm{i}}{H_0}{{\rm{t}}_{{H_1}}}}}\left( {\rm{t}} \right){{\rm{e}}^{ - {\rm{i}}{H_0}{\rm{t}}}}.$$

((4))