Pulsed Nuclear Magnetic Resonance

Transcription

1 Pulsed Nuclear Magnetic Resonance Tim Welsh, Sam Allsop, Chris Yip University of San Diego, Dept. of Physics (Dated: April 7, 013) In this experiment we use pulsed nuclear magnetic resonance (PNMR) to determine the spinlattice relaxation time T 1 and spin-spin relaxation time T for glycerine and mineral oil. T 1 for mineral oil is found to be 1.8 ±.1ms and T 1 for glycerine was found to be 41.4 ± 1.1ms. T for mineral oil was found to be 19.8 ± 3.5ms and T for glycerine was found to be 43.4 ± 3.5ms. I. INTRODUCTION Nuclear magnetic resonance was first described and measured in molecular beams by Isidor Rabi in 1938 [? ]. It wasn t until 1945, when Edward Purcell and Felix Bloch expanded the technique to solids and liquids, that modern NMR was born. Rabi, Bloch, and Purcell noticed that magnetic nuclei could absorb RF energy when placed in a magnetic field and when the RF was of a frequency specific to the identity of the nuclei. Observing this resonance allowed for fundamental measurements about the atomic and molecular structure of a substance to be made. The technique quickly became a powerful tool of chemistry and biochemistry. The most famous application of NMR is sure to be Magnetic Resonance Imagine (MRI). MRI allows noninvasive high resolution imaging of the entire human body. As a consequence, exploratory surgery is exceedingly rare in modern medicine. It is sure to be looked upon as one of the defining inventions in medical science and the greater innovative body of works as a whole. MRI is not without vices however, as of current the world faces a liquid helium shortage as 0 percent of the world supply is used by medical MRI, with most of the rest going to scientific MRI and party balloons. This issue is being combated by efforts to refit MRI machines with high temperature superconducting coils, which can be cooled with liquid nitrogen. The theory section will give a classical and quantum mechanical audit of NMR. The apparatus and experiment sections will detail the apparatus that was used as well as the experimental steps that were performed. Finally, the results and conclusion section will include a discussion of our results. II. THEORY A. Resonance The proton magnetic moment is the product of its angular momentum J and the gyromagnetic ratio γ: µ = Jγ. (1) Unlike the g-factor, γ is not unitless. The gyromagnetic ratio is a constant where γ = rad. With the s gauss nuclear angular momentum J proportional to the nuclear spin I like so: J = I. (). Protons are fermions and as such are spin 1, additionally according to the pauli exclusion principle for a single proton there is but two allowed m I s, ± 1. With m I being the nuclear spin quantum number. Thus the proton may only exist in the spin up or spin down state (or both I guess but let s not get in to that mess). Now the energy of these states is quite easily got from eq. (3): U = µb 0, (3) where B 0 is the field strength of the permanent magnet used in the apparatus. Quantifying how your apparatus effects observation is an essential part of experimentation. Now with some simple algebra one may arrive at energy states which account for field perturbations with eq. (3), eq. (1), and eq. (). We have: U = γib 0. (4) Counterintuitively (or not I guess it depends on your intuition, my intuition tells me grapefruit has a lower ground state than avocado but I can t prove it) the spin up state is of lower energy than the spin down state according to eq. (4). As it happens U is just γb 0, meaning that at zero external field the states are of the same energy, and thus equally populated, however when a field is applied the probability of being in one state or the other is not the same. This is illustrated in Fig. (1). This gap corresponds to a photon of energy ω 0 such that we can say a photon is resonant with the system if considering eq. (4): ω 0 = γb 0, (5) This shows the resonance depends on the external field, and is absent without one. This puts a limit on what systems may be interrogated with NMR.

2 FIG. 1. Energy splitting of spin states in the presence of an external magnetic field B 0. B. Quantum Interpretation A state in the x axis is a linear superposition of states in the z axis from [4] we have: χ = 1 χ z χ z, (6) Now because we have a difference in energy levels, and quantum mechanics forces quantons to behave as restless children, the relative populations will be time dependent. The total superposition of states is then expressed as: χ = 1 e ie+t. (7) e ie t In our experiment the detector coils are oriented orthogonally to the static magnetic field. So we must calculate the expectation value in the plane of the detector to have anything to talk about. The expectation value of I x is as follows: < χ I x χ > = χ I x χ [ = 1 = 4 e ie + t i Et (e = cos(γbt). ] e ie t + e i Et ) e ie t e ie + t. (8) This is quite an elegant, almost awe inspiring solution. We see in eq. (8) that the probability of finding the system in a certain spin state oscillates at a frequency ω = γb, which looks rather familiar (hint:(5)). Matching this frequency by pulsing a magnetic field orthogonal to a static field perturbs the system to a point such that it can absorb and emit radiation while the spin populations slosh back and forth [4]. Additionally this perturbation orients the spins in the x-y plane, which is essential for our experiment. If this perturbation persists for long enough one may establish a net magnetic orientation in a FIG.. Tipping of net magnetization into the x-y plane as a result of a π pulse. population. Classically this is observed as the net magnetization precessing about the z axis, while quantum mechanically it can be see as the z components of the spins settling into a superposition of ±z states. In this case the rate at which this precession appears to proceed is the resonance frequency. This situation is called a π pulse. This is explained visual in Fig. (). A π pulse is the length pulse required to fully invert the magnetization of a population, i.e. allow it to precess by 180. A π pulse is twice as long as a π as you may have guessed. A π pulse may be calculated as follows: ω = γb 1 ω = γb 1 π T π = γb 1 T π = π γb 1. C. Spin-Lattice Relaxation (9) Statistical mechanics tells us that a population with different states will be populated according to the boltzmann factor. Where the number in the spin up state is described as (N ) and the number in the spin down state is described as (N ). Such that the ratio of the two populations goes as: N = e E N kt. (10) The magnetization in the axis perpendicular to the permanent magnetic field (we ve been calling it z) is M z,

3 3 FIG. 4. FID and Spin echo in a T experiment. D. Spin-Spin Relaxation FIG. 3. Magnetization as a function of time in an external field. and is just the magnetic moment multiplied by the population difference like so[3]: M z = (N N ) µ. (11) The magnetization is time depended, and depends on the strength of the imposed field. The rate of magnetization decays exponentially as members of the population line up with the imposed field. Thus a differential equation is required to properly describe the behavior of this system, and is shown by: dm z dt = M 0 M z T 1, (1) with M 0 as the equilibrium magnetization, M z is time dependent as described earlier, and T 1 is a characteristic time of the substance called the spin-lattice relaxation time, and is one of the quantities we wish to measure in this experiment. An accurate measurement of T 1 can yield important properties of a substance. Fig. (3) explains visually eq.(1). Assuming M 0 = 0 at t = 0 as initial conditions, one can solve the differential equation (1) to get an equation from which one can directly find T 1 : In equilibrium the system of protons will have certain population distribution proportional to the imposed magnetic field, because of the U of the spin states according to (3). Additionally there exists a net M z. A π pulse will succeed in aligning the net magnetization parallel to the x-y plane. There will then be an apparent precession of M z about the z axis at the resonance frequency for the material. However because no static field producible by man is without imperfection, there will exist spacial domains which feel different B 0 s. This effects local precession according to (8). It should also slightly effect the resonances frequencies for different spacial domains. Indeed I suspect this is the real cause of the inhomogeneous precession; some regions are less strongly coupled to the RF field. Anyway, Due to this inhomogeneous precession one will observe a dephasing, which is called the Free Induction Decay. The FID has a time constant associated with it called T. Perturbing the dephasing system with a π pulse which flips the net magnetization. As the moments continue to precess there is a point where the faster precessions and slower precession line up once more with the detector causing an increase in the detector voltage, this is called a Spin Echo. The spin echo is associate with our other constant of interest T, which is the spin-spin relaxation time. Like T 1, T is characteristic of the substance and revealing of certain properties of the substance. The dephasing-rephasing phenomenon is described visually in Fig. (4). This decay is described by a differential equation similar to the one that describes T 1 but does not depend on relative spin populations, as it is effected by field inhomogeneity instead: M z (t) = M 0 (1 e t T 1 ). (13) Fitting measured data to a curve of the form of (13) allows the experimenter to deduce T 1 for a given sample as stated previously. In this case it is worth while to mention that it was not the imposed field which aligned the proton spins to a particular orientation. The spin up state is of lower energy, in the presence of a field, so the probability of finding a proton in this state is greater. This means that the system is more likely to absorb a resonant photon. Returning to equilibrium the energy of this photon is relaxed into the lattice, intra-molecularly. Giving rise to the name Spin Lattice Relaxation Time. dm dt = M T. (14) Eq. (14) is easily solved in the same fashion as eq. (1) to yield the result: M(t) = M 0 e t T. (15) Fitting this to a set of measurements in the same way as T 1 allows one to derive T from experimental data. Given a perfectly homogenous magnetic field it would be possible to measure T simply from the width of the

4 4 FIG. 5. Schematic of the control panel. FID, unfortunately this is a practical impossibility. Hahn shows how the spin echo effect is used to determine the true value of T fortunately [3]. III. APPARATUS FIG. 6. Diagram of the apparatus. For our experiment, we used a PS1-A spectrometer developed by TeachSpin. The device consists of a receiver coil wrapped around a sample vial containing the isotopes we are examining at a volume of ml. There is a permanent magnet with an associated field B 0 that is perpendicular to the length of the receiver coil, and a transmitter coil on either side of the vial thats length is likewise perpendicular to the static field produced by the permanent magnet. The transmitter coils are used to send Π and Π pulses into the sample. The signal from the receiver coil as well as the transmitter coils are connected to a control panel with two output wires. The control panel consists of a pulse programmer, a receiver, an amplifier, a receiver and an oscilloscope. The pulse programmer provides the pulse that is amplified and sent to the transmitter coils. The resulting precession frequency signal is picked up by the receiver coil which sends the signal to the receiver. The receiver displays the magnitude of the amplitude of the signal out on one channel of the oscilloscope, and the mixer mixes this signal and the main pulse programmer signal and outputs it on a separate channel on the oscilloscope. The first signal is used to determine spin-echo and the free induction decay, whilst the second mixed signal is used to determine the proper resonance frequency. A schematic of the control panel can be seen in Fig. (5), while a sketch of the apparatus can be seen in Fig. (6) [3]. IV. EXPERIMENT What follows is a description of the techniques used to determined T 1 and T of a material. The first step in this process is to find the resonance frequency for the substance in question. This is done by observing the mixer channel, when a smooth curve is attained, one has arrived at his resonance frequency. Next one must maximize the FID by adjusting the A pulse. The results of this analysis are shown in the results section. A. Measuring T 1 To measure the minimum time for a nonequilibrium magnetization to decay to equilibrium T 1, the substance is perturbed such that all moments are aligned in the z axis with a π pulse. Some time is allowed to elapse before perturbing the substance again, but this time with a π pulse to align all the moments with the x axis, which is the axis of the detector. Finally the substance is allowed to completely dephase by waiting 100ms. On the next iteration a longer period is allowed to elapse between each pulse. After each iteration the height of the FID is recorded along with the delay time. The expected result is the solution to (1), which is the exponential (13). B. Measuring T Due to the inhomogeneity of the permanent magnet field, different molecules may undergo transverse dephaseing at different rates. If one had a perfectly homogenous field one could simply measure the width of the FID to determine the spin-spin relaxation time T. To overcome this the assumption is made that regions which dephase faster in one direction, will rephase just as fast in the other direction, such that if the magnetization is flipped, the net magnetization can be recovered. To measure this, we administer a π pulse to establish equal numbers of spin up and spin down protons. A small amount of time is waited, then a π pulse is administed to flip all the spins 180. This rephasing causes an echo signal to appear, the echo magnitude and delay times are plotted to reveal T according to the solution to the spin echo differential equation (14), which is the exponential (15),

5 5 described in section (II D). The controller may be programmed to pulse multiple times such that T can be recorded in one go, as seen in fig. (7). FIG. 7. Multipulse measurement of T in mineral oil. FIG. 9. Glycerine T 1 measurements. V. RESULTS Overall we show very strong results. The measured results for T 1 in mineral oil are shown in Fig.(8). The measured results for T 1 of glycerine can be found in Fig. (9). All measurements for the T 1 s before the zero crossing have been made negative in accordance with the theory. The trend lines are fit using the MATLAB fit() function with a.95 confidence interval. TABLE I. Summary of Results Sample T 1 T Mineral Oil 1.8 ±.1ms 19.8 ± 3.5ms Glycerine 41.4 ± 1.1ms 43.4 ± 3.5ms T 1 for glycerine to be 47.8 ± 4.5ms. T 1 for glycerine is thus not corroborated by both methods. The graphs for T are arrived at in a similar fashion, using the same trendline fitting method. The graph showing T for mineral oil is shown in Fig. (10), and the graph for the T of glycerine is shown in Fig. (11). FIG. 8. Mineral Oil T 1 measurements. From these graphs one can observe T 1 for mineral oil to be 1.8 ±.1ms with the T 1 for glycerine shown to be 41.4 ± 1.1ms. Alternatively T 1 can be found from the relation: T 1 = t n (16) ln() Where t n is the time of the zero crossing. With this method we find T 1 of mineral oil to be 3.9 ±.ms and FIG. 10. Mineral Oil T measurements. From these graphs one can observe T for mineral oil to be 19.8 ± 3.5ms and T for glycerine is found to be 43.4 ± 3.5ms. A summary of results is shown in Table 1.

6 6 VI. CONCLUSION FIG. 11. Glycerine T measurements. We expected T 1 for both materials to be greater than T however this was only shown for mineral oil, what s worse is the uncertainties overlap so there is no assertible correlation. This is probably due to the fact that measurements were taken over multiple days at different resonance frequencies and different temperatures. One improvement which could be made in future experiments is to more tightly control the test environment. Be that as it may we found T 1 for mineral oil is found to be 1.8±.1ms and T 1 for glycerine was found to be 41.4 ± 1.1ms. T for mineral oil was found to be 19.8 ± 3.5ms and T for glycerine was found to be 43.4 ± 3.5ms. Additionally we show the resonance frequency for mineral oil to be ±.001Mhz and the resonance frequency for glycerine to be ±.001M hz. The uncertainties in the resonances are guestimated from the observed resonances due to temperature drift. [1] A.C. Melissinos and J. Napolitano Experiments in Modern Physics, Academic Press nd Edition, 003. [] What is NMR?. University of Colorado, n.d. Web. Apr [3] USD Physics Faculty. NMR Spin Echo Lab. TeachSpin, 009. [4] Greg Severn. Notes on Π pulses. University of San Diego, 013.