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Pulsed NMR in Extracting Spin-Spin and Spin-Lattice Relaxation Times of Mineral Oil and Glycerol

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Pulsed NMR in Extracting Spin-Spin and Spin-Lattice Relaxation Times of Mineral Oil and Glycerol Pulsed NMR in Extracting Spin-Spin and Spin-Lattice Relaxation Times of Mineral Oil and Glycerol Kent Lee, Dean Henze, Patrick Smith, and Janet Chao University of San Diego (Dated: March 2, 2013) Using TeachSpin's PS1-A nuclear magnetic resonance (NMR) spectrometer, we preformed inver- sion recovery and spin echo experiments on samples of both glycerol and mineral oil with the goal of determining spin-lattice and spin-spin relaxation times, T1 and T2. In glycerol, T1 was found to be 41.4 ms with an uncertainty of 1.9 ms while T2 was found to be 47.6 ms with an uncertainty of 4.3 ms. In mineral oil, T1 was found to be 20.6 ms with an uncertainty of 1.0 ms while T2 was found to be 19.7 ms with an uncertainty of 1.0 ms. T2 for mineral oil agree with measurements documented by other universities, however, T1 for mineral oil and both relaxation times for glycerol do not agree with measurements documented by other universities. I. INTRODUCTION II. THEORY A. Quantum Interpretation Since Bloch and Purcell's discovery of nuclear magnetic resonance (NMR) in 1946, interest blossomed in the field of NMR theory and application. Improved understand- Protons and neutrons, the constituents of the nucleus ing and accelerated development in the field of NMR has of an atom, are much like electrons in that they are both particles with spin 1 , represented as I = 1 . Both also re- made it invaluable not only in the field of physics, but 2 2 also in the fields of chemistry and medicine. From help- side in discrete energy levels when placed in an external ing to understand the classical and quantum properties of magnetic field and both have magnetic moments. Be- nuclei to chemical structure analysis to soft-tissue imag- cause the nucleus is composed of protons and neutrons, ing, this form of non-invasive, non-destructive technology the nucleus also has angular momentum and a magnetic has contributed to the furthering of countless studies. In moment. In a given nucleus with angular momentum fact, the impact of NMR discovery is so significant that I and magnetic moment µ aligned along the spin axis, it is a widely taught subject both in undergraduate and there is a proportionality that exists, given by graduate courses. In teaching NMR theory, it is important to not only µ = γ~I; (1) explain the theory, but also show experimental applica- tions of the theory in an actual spectrometer to promote a where γ is known as the gyromagnetic ratio, with di- thorough understanding of NMR theory. As such, Teach- mensions of radians per second-tesla. For a proton, Spin has developed several NMR instruments designed γ = 2:675 × 108 s−1T−1. for teaching NMR spectroscopy and NMR theory. It is For a given nucleus of total spin I, it can exist in any of through one of TeachSpin's products that the following the (2I +1) sublevels mI , where mI = (I;I −1;I −2; :::). exploration of NMR spin-spin and spin-lattice relaxation In the absence of a magnetic field, these sublevels are times was conducted. The goal was to learn more about degenerate. However, when the nucleus is placed in a NMR theory by applying it to two sample liquids, glyc- magnetic field B0, the energies of these sublevels become erol and mineral oil. Specifically, both spin-spin and spin- different, with the energies of each sublevel being given lattice relaxation times for both glycerol and mineral oil by were determined. The significance of T1 and T2 is that they are both characteristic of the chemical being ana- E lyzed. = −γB0mI ; (2) ~ The theory behind NMR will first be discussed both from a quantum mechanical perspective and classically with the energy difference between adjacent energy sub- in Sec. II A and Sec. II B. The theory section will also levels, where ∆mI = ±1, being given by include an introduction to π and π=2 pulses in Sec. II C and spin-lattice and spin-spin relaxation times in Sec. ∆E II D and Sec. II E. After discussing the theory, the ex- = γB0 = !0; (3) perimental design of the apparatus used and the method ~ of determining both relaxation times will be introduced where !0 is known as the Larmor frequency. The splitting in Sec. III A and Sec. III B and III C respectively. The of nuclear energy levels is illustrated in Fig. 1. results of the experiment will then be presented in Sec. For simplicity, the case of a proton is considered (as in IV, followed by a discussion of the results in Sec. V. the ion H+, where the nucleus is simply a proton). In 2 to the Larmor frequency of the sample, with B1 B0, the probability to be in each state becomes dependent on the length of time that B1 is applied. Thus, by applying a rotating field B1 at the same frequency as the energy difference between the two states, the probability to be in each state vary in a time dependent manner and are predictable. At a time equal to nπ=2 of a cycle of time dependent perturbation (where n is an odd integer), the probability of being in the spin up state is equal to the probability of being in the spin down state; the two states are superimposed. As seen in the Stern-Gerlach exper- iment, probing a superimposed state yields polarization onto an orthogonal axis. Thus, since the spin states are in the z axis (due to the static magnetic field in the z axis), FIG. 1. When nuclei are placed in a magnetic field, splitting the superimposed states become polarized onto the +x of the energy levels occurs, forcing one energy level to become two. axis, which is the basis of the π=2 pulse discussed later. Therefore, in a nucleus with nonzero angular momen- tum (I) and nonzero magnetic moment (µ), the simulta- this experiment, we will be studying the NMR of a co- neous presence of a static magnetic field B0 and a per- valently bonded hydrogen atom. It is important to note turbing magnetic field B1 operating at the Larmor fre- that, in a free hydrogen atom, there will be no NMR due quency yields equally populated spin states if applied for to an unpaired electron. However, since the NMR of a the proper duration of time. Because B1 rotates at the covalently bonded hydrogen has paired electron spin, it Larmor frequency, it is able to promote spins from one 1 state to another, given that the energy difference is the can be treated as a proton. The spin of a proton is 2 , 1 Larmor frequency. Once the presence of B1 is removed, so the total nuclear spin (I) is I = 2 and there are only two sublevels, m = − 1 and m = + 1 . In the absense however, the population of the states will decay back I 2 I 2 to that of thermal equilibrium, with higher population of a magnetic field B0, the two sublevels are degenerate. When a B is applied to the proton, the sublevels split in the lower energy state. Since spontaneous emission 0 is forbidden by selection rules, this decay occurs due to in energy, with the state with spin parallel to B0 being lower in energy. Thus, defining +z by the direction of spin interactions with both the spins of other nuclei in- 1 side the molecule and spins of nuclei in other molecules. B0, mI = + 2 is the lower energy state. Given a popu- lation of protons in thermal equlibirum, the population The characteristic decay times due to these interactions 1 are known as spin-lattice and spin-spin relaxation times distribution in the lower energy spin up state (mI = + 2 ) will be greater than the population distribution in the respectively and will be discussed later. 1 higher energy spin down state (mI = − 2 ) because of the Boltzmann distribution −E=kT N(E) = N0e ; (4) where N(E) is the number of particles in energy state E, N0 is the normalization factor, T is the temperature in Kelvins, and k is the Boltzmann constant [1]. Since the energy of each state is known, from Eq. 2, the population distribution in this two state system is given by mI γB0 =kT N(mI ) = N0e ~ : (5) FIG. 2. The population in each state depends on the magnetic When a population of protons are at thermal equilib- field strength, with a larger population in the lower energy, rium in the presence of a magnetic field, the lower en- spin up state. ergy, spin up state will have a higher population than the higher energy spin down state by Eq. 5 above because they have different mI . The unequally occupied states' B. Classical Interpretation populations is also dependent on the magnitude of B0, as illustrated in Fig. 2. Thus, the populations of each state are time independent. However, it can be shown by As discussed above, in thermal equilibrium within a time-dependent perturbation theory that, by applying a magnetic field, a population of protons will have an un- perturbing magnetic field B1 rotating at frequency equal equal distribution of spin states, with a higher popula- 3 tion of protons being in the spin up state than the spin applied, the probabilies to be in each state are known. down state, as demonstrated by Eq. 4. This leads to a Two important pulse lengths are π=2 and π because at net magnetization in the proton population when placed these lengths of time, the population in the two states in a static magnetic field, where a net magnetic align- are equal and inverted, respectively.
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