The NMR signal is obtained by the coil, which also generate the Rabi field or a radio frequency to flip the spin.

the origin of the NMR signal is the transverse magnetization. for spin-½ system. the transverse component of the magnetization is:

M_T = ( M_x, M_y ) = A ( cos(\omega_0 t), sin(\omega_0 t))

where A is the amplitude and \omega_0 is the Larmor frquency. for consistency and cross reference in this blog, i keep the 0 with the \omega .

the magnetization is proportional a changing magnetic field. a changing magnetic field will induce an e.m.f on a coil. if the coil is perpendicular to an oscillating magnetic field a maximum e.m.f will be obtained. however, since the magnetization is rotating, the coil can be point at any direction to give the same e.m.f. . without lost of generality, the coil will define the x-axis of the system.

B = B_{NMR} ( cos (\omega_0 t ), sin ( \omega_0 t) )

and the Maxwell’s equation:

\nabla \times E = \frac { d}{dt} B

\nabla \times E = B_{NMR} \omega_0 ( - sin (\omega_0 t), cos(\omega_0 t))

we can see that the amplitude of the E field in the coil, which is the NMR signal strength, is depending on the Larmor frequency \omega_0 . That explained why NMR always looking for strong magnetic field, now can go to 22 Tesla ( earth magnetic field is just 5 \times 10^{-5} Tesla ), a higher magnetic field strength, the higher Larmor frequency, and a stronger signal.

Moreover, the magnetic field produced by the sample is proportional to number of NMR center, the polarization and a factor on how the spin ensemble to combine to be a giant single field. and also, the change of the flux of the NMR coil is depends on how the area was integrated. These all factor are not just related to the NMR coil but also on the particular sample.