Complete solution for rotating frame relaxation functions during adiabatic pulses

Abstract

During adiabatic full passage (AFP) radiofrequency (RF) pulses the relaxation functions are conventionally treated in the Tilting Doubly Rotating Frame (TDRF), or the second rotating frame (SRF) of reference. Such a description is adequate when during the adiabatic passage the magnetization M is perfectly aligned with the time dependent effective magnetic field, B⁽¹⁾_eff(t), leading to T_1ρ(t) relaxation, or evolves on a plane perpendicular to B⁽¹⁾_eff(t), leading to T_2ρ(t) relaxation. Time evolution of B⁽¹⁾_eff(t) results in formation of a fictitious magnetic field, which is typically neglected during the AFP pulses operating in adiabatic regime, i.e., given that the adiabatic condition |γ⁻¹dα⁽¹⁾(t)/dt| ≪ B⁽¹⁾_eff(t) is well satisfied. Here α⁽¹⁾(t) is the angle between B⁽¹⁾_eff(t) and the axis of quantization of the first rotating frame (FRF) z′, and dα⁽¹⁾(t)/dt is the angular velocity. When the fictitious field component cannot be neglected, for the adequate description of relaxation during AFP pulses the solutions for the relaxation functions in a multi-fold rotating frame are necessary. Such a general treatment is currently unavailable for adiabatic RF pulses. Here, we obtain the solution for the relaxation functions in the Tilting Triply Rotating Frame (TTRF) during the Hyperbolic Secant (HS) pulses of the HSn family, HS1 and HS4, where n is the stretching factor. We show that the contribution to the relaxations originating from the non-negligible magnitude of the fictitious field depends on the pulse modulation functions of the AFP pulses and the parameters of the pulses. The corrections to describe the relaxations are given, which may be relevant in specific experimental setups especially for high-resolution NMR.