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

摘要

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:M(2tgr;)/M(0)=exp(minus;2D0g2tgr;3/3). HereD0is the diffusion constant in unbounded space andgis the field gradient multiplied by the gyromagnetic ratio. We find that this formula is replaced byM(2tgr;)/M(0)=explsqb;minus;2Deffg2tgr;3/3 +O(D5/20g4tgr;13/2S/V)rsqb; with an effective diffusion coefficientDeff(2tgr;) =D0lsqb;1minus;agr;sqrt;D0tgr;(S/V) +...rsqb;, where agr; is a constant andS/Vis the surface to volume ratio of the bounded region. Breakdown is complex but we find that the interplay between a natural length scalelc=(g/D0)minus;1/3and the geometry of the region governs the problem. The longhyphen;time behavior of the free induction decay and echo amplitude are then considered where pure explsqb;minus;constthinsp;trsqb; decay is expected. We consider some simple geometries and find in addition to the wellhyphen;known result, lnVerbar;M(z,t)Verbar;sim;minus;D0g2R4pt, valid forRpLt;lc(whereRpis the size of the confining space) that in the regimeRpGt;lcthe decay becomes lnVerbar;M(z,t)Verbar;sim;minus;g2/3D1/30t. We then argue that this latter result should apply to more general geometries. We discuss implications for realistic experimental echo measurements and conclude that theg2/3D1/30decay regime is hard to measure. Implications for the effect of edge enhancement in NMR microscopy are also discussed.