Decay of nuclear magnetization by bounded diffusion in a constant field gradient

Abstract

Transverse magnetization of spins diffusing in a bounded region in the presence of a constant field gradient is studied. We investigate the breakdown at short times of the much used formula for the Hahn echo amplitude in a constant gradient in unbounded space: script M(2τ)/script M(0)=exp(-2D 0g2τ3/3). Here D0 is the diffusion constant in unbounded space and g is the field gradient multiplied by the gyromagnetic ratio. We find that this formula is replaced by script M(2τ)/script M(0)=exp[-2Deefg2τ 3l3+script O(D05/2g4τ 13/2S/V)] with an effective diffusion coefficient D eff(2τ) = D0[1 - α√D0τ(S/V) + ⋯], where α is a constant and S/V is the surface to volume ratio of the bounded region. Breakdown is complex but we find that the interplay between a natural length scale lc=(g/D0)-1/3 and the geometry of the region governs the problem. The long-time behavior of the free induction decay and echo amplitude are then considered where pure exp[-const t] decay is expected. We consider some simple geometries and find in addition to the well-known result, ln\M(z,t)\∼-D0g 2Rp4t, valid for Rp≪lc (where Rp is the size of the confining space) that in the regime Rp≫gt;lc the decay becomes ln\M(z,t)\∼-g 2l3D01/3t. We then argue that this latter result should apply to more general geometries. We discuss implications for realistic experimental echo measurements and conclude that the g 2l3D01/3 decay regime is hard to measure. Implications for the effect of edge enhancement in NMR microscopy are also discussed. © 1994 American Institute of Physics.