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Pulsed Nuclear Magnetic Resonance to Explore Relaxation Times

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Pulsed Nuclear Magnetic Resonance to Explore Relaxation Times Pulsed nuclear magnetic resonance to explore relaxation times Hannah Saddler, Adam Egbert, and Will Weigand (Dated: 26 October 2015) The method of pulsed nuclear magnetic resonance was employed in this experiment to allow for measurements of the Spin Lattice relaxation time (T1), the Spin Spin relaxation time (T2), and the Free Induction Decay ∗ time (T2 ) for glycerin and mineral oil. The Spin Lattice relaxation time was measured to be 33.9 ms ± 0.2 ms for glycerin and 45.5 ms ± 0.2 ms for mineral oil. The Spin Spin relaxation time was measured to be 24.3 ms ± 0.3 ms for glycerin and 34.2 ms ± 0.3 ms. The Free Induction Decay time were measured to be 3.3 ∗ ms ± 0.1 ms for glycerin and 3.2 ms ± 0.1 ms for mineral oil. The measured values for T1, T2, and T2 were then compared to two different equations for Free Induction Decay time; it was found that the Spin Lattice Relaxation time does not have a significant influence on the Free Induction Decay time. I. INTRODUCTION used to compare two different equations for FID time to see which quantities affected it. Nuclear magnetic resonance is response of magnetic The second section of this paper will cover the theory nuclei in a uniform magnetic field to a continuous radio necessary to understand the method of PNMR. Section frequency magnetic field as the field in tuned through res- III is an overview of the experimental design and appa- onance. The magnetic nuclei in a magnetic field absorb ratus used. Section IV further details the experimental and then re-emit the electromagnetic radiation at a spe- procedure and the results of the experiment. This section cific resonance frequency. Nuclear magnetic resonance includes a discussion of the various equations presented (NMR) was discovered simultaneously in 1946 by both in the theory section and how they were manipulated Edward Purcell at Harvard and Felix Bloch at Stanford and used to derive the results from the recorded data. using different techniques and instruments. The discov- Additionally there is an explanation of experimental er- ery of NMR opened up a new form of spectroscopy that ror and uncertainty associated the results. Section V is is still used by physicists, chemists, geologists, and biol- a conclusion that ties the results of the experiment we ogists today. performed to the usefulness of the technique of pulsed nuclear magnetic resonance. A few years after the discovery of NMR, Erwin Hahn at the University of Illinois in 1950 began applying pulse bursts of RF magnetic fields and observing the response II. THEORY of the nuclei; this lead to the technique called pulsed nu- clear magnetic resonance (PNMR). Pulsed nuclear mag- Magnetic resonance is observed in systems where the netic resonance is the exploration of magnetic nuclei be- magnetic components have a magnetic moment and an- havior after being exposed to pulse burst of the radio gular momentum. In general most stable nuclei have this frequency magnetic fields. During Hahn's experiments property and from a classical perspective can be thought he observed a spin echo signal. A spin echo is a signal of as a small spinning bar magnet. For this experiment from the magnetic nuclei that occurred after a two pulse we looked at the nucleus of hydrogen which has a sin- sequence at a time equal to the delay time between the gle proton. This proton can be thought of a spinning two pulses. The method of PNMR is the choice method bar magnet that has magnetic moment µ and angular of NMR for most laboratories and commercial uses be- momentum J which are related by the equation: cause in 1966, Ernst and Anderson showed that high res- olution NMR spectroscopy could be achieved from the µ = γJ; (1) Fourier transform of the transient response. The technology of NMR has also made a huge im- where γ is called the gyromagnetic ratio. As seen by pact in medicine. Magnetic resonance imaging (MRI) has Equation 1, the gyromagnetic ratio is simply a ratio of revolutionized medical imaging technique. The imaging a particles magnetic dipole moment to its angular mo- technique is completely noninvasive and it produces three mentum. The nuclear angular momentum is quantized dimensional images which give doctors a clear picture of in units of h given by: the inner working of many living systems. The technique of MRI is still in its infancy stage and has the potential J = ~I; (2) to see many more applications in future years. where I is the nuclear spin. The magnetic energy, U, of The pulsed nuclear magnetic resonance experiment the nucleus in an external applied magnetic field is performed focuses on an exploration of the Spin Lat- tice relaxation time, the Spin Spin relaxation time, and U = −µ • B: (3) the Free Indiction Decay time. The design of this ex- periment allowed us to measure these three times using Therefore if the magnetic field is applied in the z- PMNR techniques. The measured quantities were then direction, then according to Equation 3 the magnetic en- 2 ergy is and the thermal equilibrium magnetization per unit vol- ume for total number of N magnetic moments is U = −µzB0 = −γ~lzB0: (4) µB µ2B M = Nµtanh( ) = N : (11) Since quantum mechanics requires that the allowed val- 0 KT KT ues of lz and ml be quantized as seen below The magnetization does not appear instantaneously ml = l; l − 1; l − 2; l − 3; ::: − l; (5) when the sample is placed into the magnetic field. It takes a measurable amount of time for the magnetization our proton with spin one half leaves us with allowed val- to build up to its equilibrium value along the direction ues of lz of of the external magnetic field. For this experiment, the m = ±1=2: (6) samples are placed along the external magnetic field that l is called the z-direction. The external magnetic field is This means that there are only two allowed magnetic applied in the form of the radio frequency magnetic field. energy states for a proton residing in a constant magnetic For most systems the magnetization along the z-axis is observed to grow exponentially. This is depicted in field of B0. A substance in thermal equilibrium generally lacks an Figure 2. The differential equation that describes this external magnetic field. In the presence of an external magnetic field it is possible to create two different energy states according to Equation 4 and Equation 6. This idea is demonstrated in Figure 1. When B=B0, the lower en- FIG. 2. This depiction shows the exponential growth in mag- FIG. 1. Diagram represents the energy states at zero magnetic netization of the sample along the z-direction as a function of field and an external magnetic field of B=B0 time. ergy state is the magnetic dipoles that are aligned with process assumes that the rate of approaching equilibrium the applied field and the higher energy state is the mag- is proportional to the separation from equilibrium and is netic dipoles that are antialigned with the applied field. given by The difference in energy between the two states is given dM M − M by z = 0 z ; (12) dt T1 ∆U = ~!0 = γ~B0: (7) where T1 is called the Spin Lattice relaxation time. Since From Equation 7, the fundamental resonance condition the unmagnetized sample is placed in a magnetic field can be written as such that it has initial conditions of t=0 and Mz=0, the differential equation can be solved by direct integration !0 = γB0: (8) to give If a sample of a substance is placed in a magnetic M (t) = M (1 − e(−t=T1)): (13) field in the z-direction, a nuclear magnetization in the z 0 z-direction eventually becomes established. This nuclear The rate at which the magnetization approaches thermal magnetization occurs because there is an unequal popu- equilibrium is specific to the particular sample. The dif- lation of the two possible states. If N1 and N2 are the ference in relaxation times for different substances is a spins per unit volume then the ratio of populations is heavily studied topic. This characteristic time is called given by the Boltzmann factor as the Spin Lattice relaxation time because during the mag- netization process energy flows from the systems to the N2 ∆U − h!0 = e KT = e : (9) surroundings and the surroundings absorbing this energy N KT 1 are referred to as lattices. The nuclear magnetization that is established is given by The process of magnetization involves more that just energy flow. Since each proton has angular momentum Mz = (N1 − N2)µ, (10) and that angular momentum must also be transferred to 3 the lattice during magnetization, the spins must experi- system which also rotates at the precessional frequency, ence a torque that is capable of changing their angular the magnetization appears stationary. For a magnetiza- momentum. In terms of quantum mechanics, the lattice tion to remain stationary there must be no torque on it. must have angular momentum states available to allow If the magnetic field is zero at the reference frame, then the spins to go from ml=-1/2 to ml=+1/2. The existence the torque is always zero regardless of the orientation of of these states is the critical component to understanding M. This implies that the magnetic field is zero if we add the large variations in Spin Lattice relaxation times for the effective field, −!/γk. different substances. The PNMR technique gives direct The classical equation of motion of the magnetization measurements for T1.
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