Dynamics of cross relaxation in modulated systems: Application to nuclear magnetic double resonance of glassy polymers

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Abstract
A statistical mechanical analysis, based on Mori’s generalized Langevin equation, is presented for the dynamics of cross relaxation in modulated systems. Relaxation rate constants are given by Onsager’s kinetic coefficients where the effects of lattice (heat bath) modulations are examined in terms of the spectral density of lattice motions as well as the transitions among the energy levels involved. Applications of the theory to nuclear magnetic double resonance relaxations of natural abundant 13C in glassy polymers are discussed. For a moderate rf field of 30–40 G (32–43 kHz), it is found that whenever lattice motions have sufficient power at these frequencies, a torsional amplitude of a few degrees suffices to yield a rotating frame longitudinal spin relaxation time T1ρ(C) of the order of milliseconds. This is much shorter than the rigid lattice value of T1ρ(C) which is equal to T12 (ADRF), the cross relaxation time under adiabatic demagnetization in rotating frame condition. The validity of the Markoffian approximation which leads to the usual time independent relaxation rate constant is examined. It seems that for proton deficient systems this approximation breaks down for the cross relaxation in matched spin–lock experiments. The origin of this breaking down is discussed and the interpretations of the cross relaxation T12(SL) in spin–lock experiments are also given.