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1 Measuring the gyromagnetic ratio through continuous nuclear magnetic resonance Amy Catalano and Yash Aggarwal, Adlab, Boston University, Boston MA 02215

Abstract—In this experiment we use continuous wave nuclear magnetic resonance to measure the gyromagnetic ratios of Hydro- gen, Fluorine, the H nuclei in Water, and Water with Gadolinium. Our best result for Hydrogen was 25.4 ± 0.19 rad/Tm, compared to the previously reported value of 23 rad/Tm [1]. Our results for the gyromagnetic ratio of water and water with gadolinium contained a greater difference than expected.

I.INTRODUCTION Nuclear magnetic resonance has to do with the absorption of radiation by a nucleus in a magnetic field. The energy Fig. 1. Schematic diagram for continuous nmr of a nucleus in a B-field can be written in terms of the gyromagnetic ratio, γ the magnetic field, B, and the , f: III.DATA AND ERROR ANALYSIS ∆E = γB = 2πf (1) For Hydrogen, we plotted 16 points on a graph of frequency The gyromagnetic ratio is a fundamental nuclear constant (Mhz) and magnetic field (Gs). Using the numpy library in and is a different value for every nucleus. Every nucleus Python we calculated a linear best fit line of has a . The principle of nuclear magnetic resonance is to put the nucleus with an intrinsic nuclear y = mx + b (2) in a magnetic field and measure at which B-field and frequency where the spin of the nucleus flips. As we vary the magnetic field and the nucleus absorbs the signal from the spectrometer, the Gs m = 217.89 ± 1.95 (3) nucleus will go from a lower energy state to a higher energy MHz state. It will then re-emit the radiation and return to a lower energy state. We chose to use gadolinium, which is strongly b = −30.93 ± 26.84Gs (4) paramagnetic due to its electronic structure and 8 unpaired , because in medical diagnoses, gadolinium contrast The slope, when converted to per second and agent improves the quality of MRI images. In a pulsed nuclear plugged into eq (1) yields a gyromagnetic ratio of magnetic resonance experiment, we know that gadolinium has rad a strong effect on the time of water, so we wanted γ = 2.88 ± 0.02 × 108 (5) T s to see if gadolinium would have a strong effect on water in a continuous nuclear magnetic resonance experiment.

II.EXPERIMENTAL SET-UP We used a computer connected to a TEL-Atomic CWS 12- 50 spectrometer, which does a sweep of the magnetic field over 300 Gauss. The spectrometer is connected to an electromagnet. We put samples of HBF4, H2O, and H2O with Gadolinium in a probehead, which then goes in the electromagnet. The reading from the probehead goes back to the computer. We also used several gaussmeters to test the magnetic field within the electromagnet. We kept the frequency fixed and swept over the magnetic field, then recorded the B-field at which the signal is absorbed and the spin flips. We did this for several different , then made a plot of frequency vs magnetic field.

To be submitted to PRL 11/15/18 Fig. 2. Frequency vs for Hydrogen 2

After the experiment we did a mathematical least squares fit. We calculated chi square and got a value of: X 2 = 0.45 ± 5.29 (6) For Fluorine, the linear best fit line yields a slope and intercept of: Gs m = 242.53 ± 0.73 (7) MHz b = −177.33 ± 10.18Gs (8) This yields a gyromagnetic ratio of rad γ = 2.59 ± 0.02 × 108 (9) T s

Fig. 4. Frequency vs Magnetic Field for Water

Fig. 3. Frequency vs Magnetic Field for Fluorine

We calculated chi square and got a value of: Fig. 5. Frequency vs Magnetic Field for Water with Gadolinium X 2 = 0.001 ± 4.24 (10)

For water, the linear best fit line yields a slope and intercept We calculated chi square and got a value of: of: X 2 = 0.067 ± 5.10 (18) Gs m = 225.31 ± 1.17 (11) MHz IV. DISCUSSION b = −144.04 ± 15.92Gs (12) We compared our result for Hydrogen to the referenced This yields a gyromagnetic ratio of value for Hydrogen which was 8 rad rad γ = 2.79 ± 0.02 × 10 (13) γ = 2.68 × 108 ± 0.02 (19) T s T s [2]. We calculated chi square and got a value of: The percent difference is about 7 percent. The gyromagnetic X 2 = 0.09 ± 4.90 (14) ratios of water and water with gadolinium differ by 1.79 percent. We expected the gyromagnetic ratio of water with For water with gadolinium, the linear best fit line yields a gadolinium to be smaller and the difference to be within our slope and intercept of: error bars, so we attribute the larger difference to machine Gs errors within the set-up of the experiment. m = 220.87 ± 0.86 (15) MHz One problem was the disagreement between the reading of b = −86.00 ± 11.77Gs (16) the magnetic field on the computer and the reading of the magnetic field on four different gaussmeters. The range of This yields a gyromagnetic ratio of magnetic fields on these gaussmeters was too large to average, rad so we took the two gaussmeters that were in closest agreement γ = 2.84 ± 0.02 × 108 (17) T s and used the average of those values for the magnetic field. It is 3 highly advised that the next group to conduct this experiment test the magnetic field with more gaussmeters and more probes in order to find the true value of the magnetic field. Due to time constraints, we could not do multiple trials of the same measurement and therefore could not evaluate those statistical errors.

V. CONCLUSION This experiment attempted to measure the gyromagnetic ratios of Hydrogen, Fluorine, Water, and Water with Gadolin- ium. Our value for the gyromagnetic ratio of Hydrogen, 8 rad γ = 2.88 ± 0.02 × 10 T s was within 7 percent of the previously reported value. We expected gadolinium to have a negligible effect on the gyromagnetic ratio of the H nuclei in water, and we attribute this difference to machine errors.

REFERENCES [1] NMR and ESR Continuous Wave Spectrometer Operating and Experi- mental Manual, Tel-Atomic, Jackson MI, 2006 [2] Cohen, E. Richard, and Barry N. Taylor. The 1986 CODATA Rec- ommended Values of the Fundamental Physical Constants (1986). https://physics.nist.gov/cuu/pdf/codata86.pdf [3] https://chem.libretexts.org/Textbook Maps/Physical and Theoretical Chemistry Textbook Maps/Supplemental Modules (Physical and Theoretical Chemistry)/Spectroscopy/Magnetic Resonance Spectroscopies/Nuclear Magnetic Resonance/Nuclear Magnetic Resonance II

ACKNOWLEDGEMENTS Thank you to Yash Aggarwal, Professor Sulak, Situ, and Chris for all their help.