Multiexponential attenuation of the CPMG spin echoes due to a geometrical confinement

The CPMG multi-echo technique is often used to investigate the translational motion of diffusing nuclei in a confining medium. Henceforth, periodically repeated RF pulses with a diffusion-sensitizing gradient yield a formation of spin echoes of gradually decreasing amplitudes. The parameters of their exponential fits may characterize the structure of porous materials or biological tissue. In this paper, a multiexponential character of the CPMG measurements is rigorously demonstrated, once a geometrical confinement is present. Based on the multiple propagator approach, we derived a spectral representation for the echo amplitudes under external magnetic field of an arbitrary gradient profile. The multiple relaxation times and their spectral weights were found in a general form. The study of simple restrictive media allowed to obtain a quantitative condition under which the multiexponential attenuation is reduced to a monoexponential one.