A proton NMR line is not a fixed shape. As a target polarizes, the line moves away from where it sits at zero polarization and sharpens on that side, while its area keeps tracking the polarization. And when the polarization moves quickly, the shape also depends on how it got there. This page follows a 5 T target through four such episodes in 2022, two spin-ups and two reversals, and shows the model of the line that the simulator uses.
Proton event logs from three runs of a 5 T polarized target: 512-point sweeps of ±400 kHz about 212.882 MHz, each event the average of 1000–4000 sweeps taken in 20 to 80 seconds, baseline- and polynomial-subtracted as described on the simulator page. Most of each run is a plateau. The four shaded episodes are where the polarization moves; pick one in the table to follow it through the page. The sharp knock-downs and re-polarizations of 16 September are left out.
| # | Start (UTC) | Episode | Polarization | Middle 80% of the change |
|---|
About a dozen single events from the chosen episode, spread evenly along the path the polarization takes. Each is the fit-subtracted signal with its sign flipped so absorption is positive; at negative polarization the target emits, and the line as measured is negative. Offsets are from the run's line center. With equal areas the change in shape is plain; lines too small to scale, or with both signs, are left out of that view. Hover to pick out a line.
Every event of every episode, measured the same way: where the line's center of area sits, how far it extends on each side of the peak at half height, and how tall it is for its area. Each episode is a path across the plot; spin-ups run left to right, reversals right to left. The episode chosen above is drawn heavier. If the line depended only on the polarization, all the paths would lie on one curve. The dashed curve is the model below.
The two reversals pass through zero differently. In July the line has both signs at once for about 40 minutes: its low-frequency side turns negative where the negative line will peak, while the high-frequency peak of the positive line still stands, and the logged polarization, the net area, says little about the shape. In September the microwaves drive the polarization through zero in 8 minutes, and the line keeps one sign almost throughout: the positive line stretches a long low-frequency tail as it shrinks, and only the sweep nearest zero shows both signs.
A polarized sample is magnetized, for protons per unit volume, and each proton sees the applied field plus the field of all the others. Two effects follow, and both grow with . A third comes from the sample not being polarized evenly.
Averaged over the sample, the field inside differs from the applied field by times a factor set by the shape of the sample (, the demagnetizing factor) and the packing of the beads. For a proton-rich solid such as butanol or ammonia, at full polarization is about 1.2–1.6 mT, or 50–70 kHz in proton frequency, so a shift of a few tens of kHz per unit polarization is the expected size. Positive polarization (moments along the field) raises the field and the frequency.
At high polarization the spin temperature is a few millikelvin, and the dipolar fields of a proton's neighbours no longer average symmetrically: neighbours aligned with the field are more likely than not. This shifts and skews the line by amounts proportional to (Goldman; Abragam & Goldman), and is why the line sharpens on one side. Negative polarization is a negative spin temperature, and reverses both.
The coil sees the whole sample, and the measured line is the sum of the lines of all its parts, each at its own polarization. When the microwaves drive the polarization quickly, some parts lead and others lag. The sum is then broader than the line of an evenly polarized sample with the same area, with extra weight on the side toward zero polarization. Nuclear spin diffusion is far too slow to even this out across millimetres.
The first two effects expect the negative line to be the mirror image of the positive one, and it nearly is: it moves about as far from the center and its sharp edge is as sharp. Its tail is longer. Every negative line here follows a reversal, so some of that may be the same unevenness; these data cannot say.
The simulator needs one line for each polarization, so it uses the lines taken while the size of the polarization was growing: the rises, the build-up after each reversal, and the plateau before each reversal, 69 lines in all. It fits them with a shift and a linear change of shape for each sign of the polarization:
with the offset from the line center at ; and (18.6 kHz) for positive , and and (16.1 kHz) for negative. , the line at , is shared, so the shape is continuous through zero. Lines from the passage through zero, and from the start of the build-up after it (below 30%), are not fitted, because they are not the line of an evenly polarized sample. Beyond the measured range, −73% to 78%, the shape is held, and only the shift and the size follow . The dispersion that goes with each line is its exact Kramers–Kronig partner, and the area is independent of , so a TE calibration still holds.
Try it in the simulator at +78% or −73%: change the polarization, and watch the fit-subtracted line move and sharpen.