The presence of the copper solenoid RF coil in a NMR magnetometry field probe creates a susceptibility boundary which induces a broadening of the field in the region of interest.¹ Susceptibility matching the material that encapsulates the solenoid, in this case epoxy, and sample vial has been shown to reduce this field broadening.² Additionally the presence of air bubbles that can occur in the epoxy is potentially a source of both field broadening and offset. The effects of such air bubbles have yet to be discussed in the literature. This work seeks to investigate the systematic errors due to the susceptibility of epoxy and presence of air bubbles in NMR field probes, as a guide to construction of such probes for accurate quantification of time varying fields in MR systems.

Magnetic field NMR probes were modeled using COMSOL, solved using the ‘Magnetic Fields, No Current’ physics solver. Four separate models were created. The first was a realistic NMR field probe which consisted of a glass cylinder of radius 1.4mm and wall thickness 0.2mm, containing water, surrounded by a 4 turn copper solenoid with wire thickness 0.225mm. Encapsulating this assembly was an epoxy ellipsoid with major and minor radii 15 and 10mm respectively. (Fig. 1) The second and third models were identical to the first with the exception that the second model did not contain the solenoid domain and the third model while containing the solenoid domain did not contain the water domain. The fourth model differed from the first only by the addition of a 1mm sphere of susceptibility 0 placed at various positions w.r.t. the center of the vial. In this model the susceptibility of the epoxy was kept at a constant value of -9.5658 ppm.

All models were simulated in a static homogenous magnetic field.

To determine the optimal susceptibility for the epoxy, the first 3 models were run using a range of susceptibilities of epoxy from -20 to 1.2 ppm. To investigate the systematic errors caused by the presence of an air bubble, model 4 was used. It was solved with the sphere at various angles while remaining fixed in the Z=0 plane at a radius of 5mm in addition to the sphere at different z positions while remaining fixed in the XY plane.

Values of the magnitude of the B field from the water domain were taken and the weighted variance of those field points was calculated using the sensitivity profile of the solenoid as a weighting factor. The field broadening is defined as the square root of the weighted variance. The field offset was calculated by subtracting the weighted mean of the B field data from the applied field.

The results from varying the susceptibility of the epoxy are shown in figure 2. The resulting field offset for the NMR field probe as a function of epoxy susceptibility is plotted in figure 3. Results from introducing an air bubble at various angles of a circle of radius 5mm in the Z=0 plane are shown in figure 4. Figure 5 shows the results from the model of the NMR field probe with an air bubble at a constant (X,Y) value but at varying positions in Z.

For the case of susceptibility matching the epoxy, figure 2 shows that the optimal susceptibility (vertical black line) lies somewhere between the susceptibility of water (vertical blue line) and copper (vertical red line) and is not that of the susceptibility of copper, as might have been supposed. The best matched case is an almost tenfold improvement in field broadening compared to having the probe in air alone, as expected. Although field offset is shown to decrease as susceptibility increases, offset is inherent and can be calibrated for. Broadening cannot be calibrated for, thus the main concern should be to susceptibility match.

Figure 4 shows that the effect of the bubble on broadening is symmetric as expected. Figure 5 shows that the broadening is particularly severe when the bubble is close to the solenoid.

It was shown that using susceptibility matched epoxy improves field broadening by a factor of almost 10 and that the optimal susceptibility occurs when the epoxy is matched closely to that of the water. In the future, epoxy susceptibility should be matched for highest accuracy. Additionally as the presence of air bubbles is shown to increase field broadening and is particularly severe at certain critical regions within proximity to the solenoid, it is necessary to eliminate them in construction though the use of a vacuum pot method.