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Pulsed Nuclear Magnetic Resonance to Obtain Characteristic Times for Mineral Oil and Gycerin Callie Fiedler,a) David Mallin,b) and Andy Vescib) (Dated: 27 March 2011) Using Pulsed Nuclear Mag thenetic spectroscopy the Spin-Lattice (T1 ) and Spin-Spin (T2 ) relaxation times of Glycerin and Mineral Oil. The T1 for Glycerin and Mineral Oil were 40.4 ± 0.1 and 20.2 ± 0.1 respectively, and T2 for Glycerin and Mineral Oil were 35.7 ± 0.1 and 15.8 ± 0.1 respectively. The Free Induction Decay (T∗2 ) was 1.2 ± 0.1 for Glycerin and 1.1 ± 0.1 for Mineral Oil. I. INTRODUCTION Nuclear Magnetic Resonance (NMR) was observed by Edward Purcell and Felix Bloch, independently, in 1946 when investigating the behavior of nuclei once placed in an external, uniform magnetic field and subject to a radio frequency (RF) magnetic field oriented perpendicular to the external field 1 . While there have been many advancements made in the sciences due to the advancements afforded by the discovery of NMR, the most notable contribution to society resides in the medical field. NMR or, in medical terms, Magnetic Resonance Imaging (MRI) is the most current and effective method used in medicine to non-invasively investigate what is physically happening within living organisms. In order to observe the inner workings of living systems one must understand the characteristic times associated with the internal molecules and their exposure to an external magnetic field. This experiment will investigate two characteristic times, relaxation times, associated with mineral oil and glycerin, two molecules with very prevalent structures to the function of not only the human body, but also our understanding of living structures in general. More specifically, the characteristic relaxation time related to an atoms interaction with surrounding molecules (T1 ) and the relaxation time associated with the internal interactions of the atom (T2 ) are determined. This paper will first address the theoretical concepts essential to this experiment(II) and discuss the physical apparatus we used to conduct research (III). I then discuss the design of our experiments (IV), first the SpinLattice Relaxation Time (T1 ) for both Glycerin and Mineral Oil (A) and the the Spin-Spin Relaxation Time (T2 ) for Glycerin and Mineral Oil (B) and the Free Induction Decay (T∗2 ) of both. The paper terminates with a discussion of the results found (V). a) Also at Physics Department, University of San Diego. Department, University of San Diego. b) Physics II. THEORY A. Nuclear Resonance To obtain Nuclear Magnetic Resonance, the nuclei in investigation must have a nonzero magnetic moment (µ) and angular momentum (J) related by µ = γJ, (1) where γ is the gyromagnetic ratio, which equals 2.657 ×104 radians per second-gauss for a proton. The angular momentum is quantized in terms of ~ and I, nuclear spin, J = ~I. (2) When exposed to an external magnetic field there is a certain energy associated to the nucleus. In particular, when a proton resides in an external magnetic field in the z-direction, the magnet energy is U = −µz B0 = −γ~Iz B0 , (3) where Iz is the nuclear spin in the z-direction and B0 is the external magnetic field. For a proton, I = 1/2 and so it follows that the allowed spins in the z-direction are mI = ±1/2 because quantum mechanics requires that these values be quantized such that mI = I, I−1, I−2, · · · −I. When nuclei with these characteristics are subjected to a uniform external magnetic field, the allowed ground energy states split in two. The energy difference between these two states, ∆U is proportional angular frequency, ∆U = ~ω0 = γ~B0 , (4) where ω0 = γB0 is the fundamental resonance condition. When a proton is exposed to a constant magnetic field and a perpendicular rf magnetic field at its resonant frequency, the energy created is sufficient to overcome ∆U and spin transitions between mI = ±1/2 can be driven, i.e. the nuclear spin of the atoms can be flipped from ”up” to ”down” or vice versa. Pulsed Nuclear Magnetic Resonance (PNMR) allows for one to determine characteristic relaxation times of certain molecular structures, e.g. mineral oil and glycerin. PNMR utilizes the same techniques of classic NMR with the addition of rf pulses with specific widths (90o or 180o ) that alter the next magnetization of the molecule 2 . 2 These pulses tip the nuclear spins into to x-y plane that then precess around the B0 magnetic field, which create a time varying voltage in a x-y plane magnetization pick-up coil. and dMy My =− , dt T2 (7) whose solutions are B. Spin-Lattice Relaxation t My (t) = Mx (t) = M0 e− dt , One of the relaxation times in interest is the spinlattice relaxation time (T1 ). When an isolated proton has its spin flipped, this magnetization can be indefinitely maintained. Because molecules have many protons which have varying magnetizations in the plane (all averaged to have a net flipped spin), the spins realign to the external, primary magnetic field (z-direction oriented). The time it takes for the spins in the molecule to realign to thermal equilibrium magnetization is defined as T1 . To determine T1 the net magnetization (M0 ) of the molecule is tipped 180o to -M0 . To ensure one is applying a true 180o pulse, the free induction decay (FID) should be minimized, corresponding to a minimized voltage amplitude within the molecule in question. A pick-up coil monitors the return of the net magnetization back to M0 , thermal equilibrium magnetization. A second pulse of 90o is introduced to determine if the net magnetization in the x-y plane has gone to zero, indicating the return to thermal equilibrium. Contrary to the 180o pulse, to ensure one is applying a true 90o pulse the FID amplitude must be maximized. By interrogating the net spin at varying increments using the pick-up coil, researchers can determine how greatly the magnetization has changed in the molecule at varying delay times. By plotting the maximized FID amplitudes, M, versus the delay times, one can observe the relaxation through the dying exponential M (t) − M0 dMz = , dt T1 (8) where t represents twice the delay time and T2 is the spin-spin relaxation time. III. A. EXPERIMENTAL DESIGN Apparatus For our experiment we utilized an oscilloscope, an RF source, and a permanent magnet system to implement and observe the effects of PNMR on different molecular samples. While the oscilloscope provides the necessary electronic feedback to observe the effects of the PNMR, the intricacies of the magnet configuration create the desired effect. As seen in Figure III A, the permanent magnet is the most external component to the magnet system, which creates a net magnetization of the molecular structure of the sample in the z-direction, e.g. mineral oil and glycerin 3 .. An rf coil configuration is positioned around the sample to allow for the necessary pulses into the x-y plane, creating a magnetic field in the x-direction. To observe the effects of PNMR, a pick-up coil is wound directly around the sample cell to create a magnetic field perpendicular to both the external, permanent magnet and the rf magnetic source in the y-direction. (5) where dMz /dt indicates the net change in magnetization in the z-direction. C. Spin-Spin Relaxation The spin-spin relaxation time (T2 ) is the time it takes for the spins to be altered from their non-thermal equilibrium magnetization in the x-y plane. Unlike the spinlattice relaxation time, to measure T2 one must first use a 90o pulse, when the FID amplitude is maximized. Because the interactions of the individual spins within an atom vary their local magnetic field and the frequency at which the spins precess back to a non-thermal equilibrium from being pulsed, the spins dephase. The dephasing of spins creates a variance in the voltage being interpreted by the rf pick-up coil. The rate at which the net magnetization in the x-y plane is represented by dMx Mx =− dt T2 (6) FIG. 1. A top-view depiction of the magnetic system used to conduct our research. The center region is the location in which the sample is placed. 3 IV. RESULTS T2 can be extracted from the exponential and was 15.8 ± 0.1. The T1 for Glycerin 40.4 ± 0.1 from our plot of the voltage detected from the pick-up coil versus the different variances of the delay time, as seen in Figure ??. The Mineral Oil spin-lattice relaxation time value was 20.2 ± 0.1 from the exponential depicted in Figure IV. FIG. 4. Graph of the spin-spin relaxation time voltage values for glycerin versus the changing delat time, displaying a decaying exponential. FIG. 2. Graph of the spin-lattice relaxation time voltage values versus the changing delay time, displaying an inverse decaying exponential, from which you can obtain the T1 value for Glycerin. FIG. 5. Graph of the spin-spin relaxation time voltage values for mineral oil versus the changing delat time, displaying a decaying exponential. V. DISCUSSION Our results were significantly similar to those of previous literature 1 . For both mineral oil and glycerin had T1 values lower than the observed T2 values, as expected. FIG. 3. Graph of the spin-lattice relaxation time voltage values for Mineral Oil versus the changing delat time, displaying a decaying exponential. VI. REFERENCES 1 Klein, The spin-spin relaxation time were in accordance with the literature for glycerin in that we obtained a T2 of 35.7 ± 0.1. Similar to the method used to determine the T1 value, this value can be expressed through the dying exponential as seen in Figure ??. As seen in ??, the W. Nuclear Magnetic Resonance: Free- Induction Decay and Spin Echoes in a 0.05-T Magnetic Field. American Journal of Physics. 1989 2 Melissinos, Adrian C. and Jim Napolitano. Mag- netic Resonance Experiments. . Experiments in Modern Physics. 2008 3 Wolff-Reichert, Barbara. A Conceptual Tour of TeachSpins Pulsed NMR. 2003.