On-line NMR sensor design

The potential for MR sensors to interrogate on-line/in-line processes in the Food Processing industry is reiterated on an almost annual basis. Nevertheless there are a number of significant technical hurdles to be 

As a first step let us expand the inhomogeneous magnetic field at a point, r, as a Taylor series  

As the spins initially at r(0) are carried along in the field gradient, they will accumulate a net phase angle given by 

The spin density p(r(t), t) of spins at r(t) at time t is clearly equal to p(r(0), 0) if we ignore diffusion and bulk motion of the sample other than the linear translation, v. The contribution d5 (r(t), t) to the total signal S(t) from spins at r(t) is therefore given by 

The phase factor exp(iyRoO i eliminated by setting the RF coil on-resonance so, neglecting the constant of proportionality, A, and writing r(0) simply as r, and p(r(0), t) simply as p(r) this becomes 

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