This is a real mapping report showing the quality of the results obtained using a Resonance Research FMU-1200 magnetic field mapping unit.

The report shows the mapping and shim convergence performed on a 500 MHz / 51 mm magnet. Such report is generated for every magnet mapped by Resonance Research.

Field Mapping, Analysis, Shim Convergence, and Lineshape Simulation

500 MHz / 51 mm Magnet

Principles of Mapping Static Magnetic Fields

The maps of the static, B0 magnetic field acquired in this study are the accumulation of measurements  at a sequence of points on a trajectory surrounding the centers of the shim coil assemblies (superconducting and room-temperature). The trajectory used is a helix on the surface of a right, circular cylinder, so that each point on the trajectory has a distinct z-axis position. The maps are then presented as a projection of the strength of the field (given as the NMR resonance frequency) at each point on the trajectory plotted against the z-position of each point.

In the absence of any inhomogeneity of the magnetic field, the map would be a flat line, with field strength independent of z-position. Real magnets are, of course, characterized by more complicated maps.  The map represents the extremes of the magnetic field on the surface of the cylinder, in this case, of 17 mm diameter.  The extremes of field internal to the cylinder are required to be smaller than at the outer surface of the cylinder (a consequence of the Laplace equation), so the map represents the extremes of the field within the volume of the cylinder.

Since the mapping computer drives the probe on the mapping trajectory, the coordinates of each point on the mapping trajectory are known. For the coordinates of each point, the spherical harmonic functions may be calculated by the computer.  This permits the map to be analyzed in terms of the contributions of the various spherical harmonic functions. Further, because the heli-cylindrical trajectory has a distinct z-position for each point, all points on the trajectory contribute to the evaluation of all harmonics.

System software permits analysis of the spherical harmonics through z11 and s8. Use of analysis with both direct maps and differences of pairs of maps permits determination of the center and orientation of the shim coil assemblies, determination of the magnetic character of individual shim coil assemblies, and iterative shimming of the magnetic field within the bore of the magnet (a process called shim convergence) by calculation of the currents in the shim coils required to cancel as many harmonics of the field as are provided by the shim coil assembly.

Since the mapping computer determines the frequency of the observed resonance at each point on the trajectory as well as driving the motion along it, the entire mapping process is automated.  When the mapping computer is put in direct control of the currents in the shim coil assemblies (as with RRI's shim systems and some others), the processes of centering, orientation, calibration, and shim convergence are all fully automated as well.

Description of the Mapping System

RRI's FMU-1200 system, used for mapping static magnetic fields in vertical-bore, NMR magnets, is based on 1H NMR and a heli-cylindrical NMR probe actuator.  The spectrometer operates from from 5 MHz to above 1.4 GHz. The heli-cylindrical probe actuator was designed for use in bore diameters between 38 and 89 mm.

The trajectory of the sample in the probe is that of a helix on the surface of a right, circular cylinder, with the parameters of radius of sample position (8.5 mm) and pitch (2.5 mm) fixed in hardware.  The transport mechanism that drives the probe is a stepper motor positioned outside the bore of the magnet under the control of the mapping system computer.  The effect is that the 0.7 mliter sample moves through a trajectory 30 mm long in 12 revolutions.  More than distinct 256 points may be acquired on a trajectory.  The angular rotation of the probe between points is commonly chosen so as to provide a distribution of azimuthal positions over the course of the mapping trajectory.

Trajectories 30 to 40 mm long are commonly used to map magnets of 50 mm bore diameter.  For magnets with 62 mm bore diameter, trajectories 35 to 45 mm long, and for 89 mm bore diameter, trajectories 40 to 50 mm long are commonly used.

MAPPING RESULTS FROM A 500 MHz / 51 mm ID MAGNET

Convergence of the RT Shims

A single map, with the SC shims set and all RT shim currents set to zero, obtained about the RT shim center, is shown in Figure 1.

Figure 1

The residual values at RT shim center for the gradients addressable by the super-conducting shims were:

The other significant gradients in the magnet are z3, z2x, z2y, zs2, s3, z3x, z3y, c4, z4x, z4y, z3c2, z2s3, zc4, zs4, and z5y (through sixth order).

The prediction of the converged result from the analysis of the map shown in the previous figure for the existing shims, based on the shim calibrations, correcting z – z5 and x – zs2, is shown in the following figure.  

Figure 2

The following figure shows the actual map obtained from experimental convergence using the RT shims.

Figure 3

The most significant residual gradients outside the correctable set after shimming are s3, z3x, z3y, c4, z4x, z4y, z2c3, zs4, z5x, z5y, and z4s2 (through sixth order).  

Simulation of Limiting Lineshapes

Calculations of limiting lineshape were performed for the existing shims and for RRI's 39-gradient. These calculations are for 16 mm long samples 5.0 mm in diameter and for 20 mm long samples 8.0 mm in diameter.

These calculations produce estimates of the best lineshape obtainable on this magnet with the target shims, in the limit that the analytical probe and sample induce no inhomogeneities not adjustable (and adjusted) in the control set of gradients.  The effects of the known, calibrated impurities of the control set are included in these results, for the open-bore corrections, for the RRI shims.  The RF-profile of the probe's B1 field (and so its detection sensitivity) is assumed to be flat and cylindrical over the specified sample length and to cut off sharply at the ends of the specified window.  The sample is assumed to be non-spinning, quiescent, and isothermal.

The calculations are of lineshape for natural linewidths at half-height of 0.15 Hz (in the absence of inhomogeneity; specifying T2, in effect) and, of course, depend significantly on this value.  Estimated peak heights should be compared only between simulations done with the same sample geometry.

For a 5.0 mm diameter sample, 16 mm long:

For the existing shims:

Figure 4

Displayed width:  102.4 Hz.

Linewidth at 0.55%:  54.6 1H Hz

Linewidth at 0.11%:  76.3 1H Hz

Peak intensity: 11.2

For RRI's 39-gradient shims:  

Figure 5

Displayed width:  102.4 Hz.

Linewidth at 0.55%:  2.3 1H Hz

Linewidth at 0.11%:  5.3 1H Hz

Peak intensity:  185.9

For an 8.0 mm diameter sample, 20 mm long:

For the existing shims:

Figure 6

Displayed width:  409.6 Hz.

Linewidth at 0.55%:  164 1H Hz

Linewid th at 0.11%:  265 1H Hz

Peak intensity:  11.2

For RRI's 39-gradient shims:

Figure 7

Displayed width:  409.6 Hz.

Linewidth at 0.55%:  7.2 1H Hz

Linewidth at 0.11%:  13.2 1H Hz

Peak intensity:  155.6

 
CONCLUSIONS

Convergent mapping provides a starting place for shimming on real probes and samples that is independent of the prejudices imposed by an individual probe and sample and unlikely to be found otherwise, but optimizing lineshapes with real, analytical probes and real samples still requires non-trivial efforts in managing both shims and sample.  Into this come also the behavior of the lock system, vibration isolation of the magnet, proper temperature management, and a variety of electronic details in the interactions among the spectrometer, shim, and probe subsystems.

For example, with RRI's 39-gradient shims lineshape performance for a 5 mm sample 16 mm long is commonly in the range of 2.0 to 2.5 / 4.0 to 5.5.  The prediction for this magnet is not much different, at 2.3 / 5.3.  Experimental lineshapes on real samples at installation of analytical probes in the field on such magnets with this shim system of course vary with the probe and other considerations, but most results have been in the range of 2.5 to 4 / 5 to 8, with a few results better and a few worse.  With older probes such values may be at best difficult to achieve.

The simulations shown for 5 mm samples were run for 16 mm sample columns, our usual practice.  This is realistic for modern probes but, of course, somewhat arbitrary.  Of course, for shim systems with limited gradient coverage, the lineshape will be found to be highly sensitive to how much length of the sample column the analytical probe observes.

All this b eing said, this magnet is eminently manageable, and good lineshape performance is certainly attainable with high-performance shim systems with sufficient care and attention.