T2 Dependence on Time in Pulsed Nuclear Magnetic Resonance Stephen Hurley Physics Department, University of California, Santa Barbara Abstract The temperature dependencies of T2 as related to pulsed nuclear magnetic resonance are measured and analyzed. Margarine is used as the sample and one plot of data was taken from near boiling to near room temperature. There are various factors to consider including the composition of the margarine. It is concluded that T2 may or may not be affected by temperature. Introduction A brief history of pulsed nuclear magnetic resonance is in order. In 1946 nuclear magnetic resonance was discovered by Edward Purcell and Felix Bloch. A few years later, in 1950, Erwin Hahn accidentally discovered the idea of sending pulses to produce an echo (see Theory). Since then, pulsed nuclear magnetic resonance has been used in many commercial and academic institutions for understanding the structure of materials. It is also prominently used in MRI (magnetic resonance imaging) machines which are a less invasive way of scanning the human body than traditional methods such as X-rays. In the future, there will be many more applications of nuclear magnetic resonance in the future.1 Theory Pulsed nuclear magnetic resonance (PNMR) revolves around the spin of protons. The idea is to change the spin via magnetic fields and observe the decay of the net spin in a material. The x-y plane component of the average of the proton spins is observed on an oscilloscope. This can give many details about the structure of the material. The spin of the protons starts out in the +z direction. An effective magnetic field is produced in the +x direction, making the spin rotate in the y-z plane at a rate of Ω. If the magnetic field is turned on for only a moment, as in a pulse, the spin will turn specific number of radians. The trick is to rotate the spin by π/2 or π so that the spin will end up in the +y or -z directions, respectively. In the π/2 position, the spins precess about the +z axis, rotating in the x-y plane. Since there are a great number of protons, they do not all precess at the same rate. This leads to phase dealignment, and as the spins spread out throughout the x-y plane, the net spin becomes greatly diminished at a rate of (1) where T2 is called the Spin-Spin Relaxation Time. Unfortunately, it is hard to measure this directly, because the magnetic field used to create the π/2 rotation is not uniform itself. The magnetic field causes them to rotate at even greater differing rates, and so the T2 recorded is not the real T2.1 The clever trick is to rotate again by π at a time τ so that the pulses that are unaligned by the nonuniform magnetic field are rephased back. This creates an echo of Mx,y at time 2τ that can be used to find T2. Then numbers can be plugged in and T2 can be solved for giving (2) A diagram from the lab manual1 is shown here for better visualization: Figure (1) Method The main apparatus is a magnet that forces the spins of the protons of a sample that is placed in the middle (the middle is the most uniform of the magnetic field) to be pointed in an upward direction (taken as the +z direction). This sets the sample up for another magnetic field that will rotate the spins by the appropriate amounts. The second apparatus is a voltage pulse generator. The generator can handle pulses up to 15 MHz, which is sufficient for the purposes of the experiment. There are settings available to set up sequences of pulses, such as setting up a arbitrary length pulse of voltage followed by a independently arbitrary length pulse of voltage at a specific time later. The pulse generator is connected to a coil that produces a magnetic field in the +x direction. The pulse generator is set up so that the first pulse of π/2 is sent (longer pulse means greater radians). The second pulse equates to a π rotation. Then the interval between pulses (τ) is adjusted so that the echo is at its highest height. The magnetic field of the sample is converted into a voltage signal which is read on an oscilloscope. Then using the ratio of the two voltage heights of Mx,y(0)=M0 and Mx,y(2τ), the equation is used to solve for T2. Before using a sample, a test sample was used to find the location (the “sweet spot”) where the magnetic field is the greatest, since it is not uniform. This simply involved moving the sample within the magnet and finding the highest height of the oscilloscope readout. The sample used in this experiment was margarine, because it was the only semi- liquid substance within 10 feet of the experiment. It was first boiled using the only heating device within 5 feet of the experiment, a soldering iron. Then it was immediately placed in the magnet, and T2 was recorded multiple times along with the time at which the T2 measurement was made with t=0 being just after the sample was placed in the magnet. Data This is a plot of the time after boiling versus T2: Time Vs. T2 160 140 120 100 T2 (ms) Figure (2) 80 60 40 20 0 0:00 2:24 4:48 7:12 9:36 12:00 14:24 Time (min) The error bars are primarily from the inaccuracy of measuring the high points of the voltages, since the shapes tended to be somewhat jagged. Other sources of error include the inaccuracies in finding the sweet spot, because finding a maximum signal involves trial and error method, which can easily lead to error. The last variable τ was also selected to give the highest echo. Near the correct τ, the echo height plateaus, and so the τ has some leeway in each direction giving a small amount of error as well. Analysis This experiment was primarily of a qualitative nature, since no actual temperatures were recorded. The temperature, however, can be estimated from what is known about temperature dependence over time. Here is a sketch of the temperature overlaid with the data Figure (3) where the equation for the temperature is (3) This equation is roughly what the temperature graph might look like going from boiling to room temperature. The T2 graph has a similar pattern after 1:12 minutes suggesting a proportional dependency on temperature. Before 1:12, however, the graph of T2 starts near 70 ms, suggesting a completely different dependency. Discussion When margarine is heated up to boiling, the temperature is not the only thing that changes. At boiling and slightly below boiling, it is a very non viscous substance. As it cools down, however, it becomes more viscous into the form that is most commonly known as spreadable. T2 is known for being dependent on viscosity, and so this is a major factor.3 One possible solution is that T2 is dependent on both temperature and viscosity. It could be that after 1:12 the solution has reached an equilibrium viscosity, and the temperature dependence takes prominence. More tests will be needed to conclude this however. Conclusion The main conclusion is that more testing needs to be done. The margarine could be dependent on temperature, viscosity, or both. It would be good to test a sample that has viscosity independent of temperature, and therefore could be used to find a true temperature dependence of T2. If it is indeed true that T2 depends on temperature, this could affect many other fields. For example, the industries that rely on T2 would need to make sure that the temperature is stable throughout the readings. They would also need to make sure the temperature does not go outside certain bounds which would too greatly affect the T2 reading. References 1. Pulsed Nuclear Magnetic Resonance Lab Manual for Senior Lab at University of California, Santa Barbara. 2. C. Slichter, Principles of Magnetic Resonance (Harper and Row, New York, 1963) 3. J. Seymour (firstname.lastname@example.org), http://www.coe.montana.edu/che/jseymour/publicationsjds.htm, Montana State University, 11-04.
Pages to are hidden for
"T2 Dependence on Time in Pulsed Nuclear Magnetic Resonance"Please download to view full document