Physics 440 Spring (3) 2018

Experiment QM2: NMR Measurement of Nuclear Magnetic Moments

Theoretical Background: Elementary particles are characterized by a set of properties including mass m, electric charge q, and magnetic moment µ. These same properties are also used to characterize bound collections of elementary particles such as the proton and neutron (which are built from quarks) and atomic nuclei (which are built from protons and neutrons). A particle's magnetic moment can be written in terms of spin as µ = g (q/2m) S where g is the "g-factor" for the particle and S is the particle spin. [Note 2 that for a classical rotating ball of charge S = (2mR /5) ω and g = 1]. For a nucleus, it is conventional to express the magnetic moment as µ A = gA µN IA /! where IA is the "spin", gA is the nuclear g-factor of nucleus A, and the constant µN = e!/2mp (where mp= proton mass) is known as the nuclear magneton. In a magnetic field (oriented in the z-direction) a spin-I nucleus can take on

2I+1 orientations with Iz = mI! where –I ≤ mI ≤ I. Such a nucleus experiences an orientation- dependent interaction energy of VB = –µ A•B = –(gA µN mI)Bz. Note that for a nucleus with positive g- factor the low energy state is the one in which the spin is aligned parallel to the magnetic field. For a spin-1/2 nucleus there are two allowed spin orientations, denoted spin-up and spin-down, and the up down energy difference between these states is simply given by |ΔVB| = VB −VB = 2µABz. A transition between the spin-up and spin-down states can be driven by the absorption of a photon of energy

Ephoton = hƒ = ΔVB. This induced spin-flip process is know as nuclear magnetic resonance or NMR and can be used to measure nuclear magnetic moments.€ In this lab we will measure the magnetic moments of the 1H nucleus (i.e., the proton) and of the 19F nucleus both of which have spin 1/2.

For spin-1/2 nuclei IA = !/2 and the g-factor is thus given by gA = 2 µA/µN.

Experimental Apparatus: The NMR apparatus consists of a strong (~0.3 T) permanent magnet, a pair of "scan" coils which produce a variable milli-Tesla magnetic field, and a radio frequency (MHz) oscillator which drives a small probe coil. The scan coils are mounted directly on the permanent magnet and the probe coil is positioned between the pole pieces of this magnet in a perpendicular orientation such that a sample can be inserted inside it. The probe coil is run at a fixed frequency of between 12 and 14 MHz. The exact oscillator frequency can be read from the attached digital frequency counter (set to 1MΩ input; reading is in MHz, but console outputs frequency/1000). The scan coils are driven with a linear ramp current which varies from 0 to 250 mA in a sweep time of 40 ms. This –3 produces a magnetic field at the center of the coils which varies linearly from Bo to Bo+3.2x10

Tesla (assuming the two scan coils are maximally separated) where Bo = 315±1 mT is the field of the permanent magnet. A voltage signal (0 to 1.0 V) proportional to this magnetic field is output on channel 1 of the console and should be monitored on channel 1 of the oscilloscope. An inverted

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voltage signal from the high frequency oscillator circuit is output on channel 2 of the console and should be monitored on channel 2 of the oscilloscope. A peak is observed when energy is being drawn from the oscillator (which happens when the nuclear spin-flip resonance condition is met).

Procedure: Before setting up the apparatus for NMR measurements we will measure the magnetic field produced by the "bare" scan coils. For this measurement we use a teslameter which consists of a Hall effect sensor mounted in a flat probe. The teslameter measures the field perpendicular to the probe and has an accuracy of 5%. Set the coils next to each other on the lab bench with center- center separation of 6.0 cm. (Note: In our device the coil separation ≈ the average coil diameter which is not the geometry for "Helmholtz" coils). Directly connect the positive and negative leads of the two coils and attach the other two leads to a test circuit with a power supply and ammeter. Insert the teslameter probe centered between the two coils and measure the magnetic field versus coil current, for I = 0 → 0.50 A. (Do not exceed 0.5 A). Make a plot of B vs I. For NMR measurements the strong permanent magnet and scan coils (which are directly mounted on the magnet) need to be installed on the spin-probe unit. Details of this installation are given in Sections 9.1-9.2 of the "Spin Resonance Manual". Great care needs to be exercised in this procedure!! Before rotating the magnet into its final position, be sure the scan coils are completely plugged into the base. After installing the permanent magnet, use the teslameter to directly measure the field Bo (include the telsameter uncertainty). Insert the proton probe coil oriented so that the frequency adjustment screw is accessible. Power up the console (set to MED) and check the probe frequency on the digital counter. (If the red light on the console is not illuminated, the oscillator is not running. To correct this, invert the probe and adjust the SENS screw as described below). For proton resonance this frequency needs to be around 13.4 MHz. Once the frequency is set, invert the probe coil so the oscillator sensitivity adjustment is accessible. We have three proton rich materials (glycerine, polystyrene, and water, color coded yellow, green, white) that will be used to measure the proton magnetic moment. Select one of these samples and insert it into the probe coil (from the back side of the apparatus). The two console output channels should be attached to the corresponding oscilloscope inputs and the initial oscilloscope setup should be as follows: Both channels: 200 mV/division; Timebase: 5 ms/division. Trigger: source=CH 1; type=edge; mode=normal (or auto), dc coupled. With the console and the NMR apparatus powered up you should get a linear ramp voltage on channel 1 of the scope. You may also see a peak in the channel 2 output. You may have to adjust the trigger level control to get a stable trace.

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If no peak is evident on Channel 2 you need to adjust either the sensitivity or the frequency of the oscillator circuit. Try the sensitivity adjustment first. The circuit is tuned to maximum sensitivity (corresponding to "marginal" oscillation) when the signal level LED on the console begins to dim. The more marginal the oscillator, the better the peak signal to noise ratio. If the oscillator is made too marginal it will stop running and the console signal level LED will go out. If this happens turn the SENS screw on the oscillator counterclockwise until the LED is re- illuminated. If, after the above sensitivity adjustments, no peak is evident you most likely need to change the probe frequency. As you vary the probe frequency searching for a peak, make sure the signal level LED stays illuminated. Once you find the resonance peak, adjust the oscillator frequency to move the peak near the center of the field scan and adjust the sensitivity for maximum signal to noise. You may also want to "average" the displayed data over several scans which can be done from the oscilloscope's "Acquire" menu. Once you have a good signal peak, use the cursors on the oscilloscope to determine the peak location that can be correlated with a magnetic field strength. You will probably want to freeze the scope display (by pressing the "run/stop" button) to make accurate measurements.

Use both the time and voltage scales in your determination of the magnetic field at resonance BR and estimate an uncertainty for each of your BR values. For a permanent record, save the scope data to a flash drive (press the save/print button). It is most useful to record the scope output with the cursors positioned for measurement. Be sure to print and clearly label each of these saved screenshots (include the oscillator frequency!). Take data for all three of the proton rich samples and also take a data set with the blank sample (black) in place and with no sample. To measure the magnetic moment of the 19F nucleus we will use the fluorinated polymer polytetrafluoroethelyne (aka PTFE or Teflon) [color coded blue]. You will need to adjust the probe frequency to locate the fluorine resonance. The console should be set to the MID range. Once you find the resonance peak take data as above. Take an additional data set with the blank sample (black) in place and one with no sample.

Data Analysis:

Use your measurements of BR and ƒR to compute the magnetic moments (in units of µN) and g- factors for 1H and 19F (with uncertainty estimates) and compare your results with literature values.

Annotate your printouts of the resonance peaks to show how you determined BR in each case. If the magnetic field across the sample were perfectly uniform we would expect the resonance peaks to be sharp and narrow (as they are for analytic NMR instruments used in chemical analysis). The finite widths of the resonance peaks can be attributed to inhomogeneities in the external magnetic field and to the presence of internal magnetic fields due neighboring nuclear spins. Measure the widths of your resonance peaks (full-width at half max) and use these values to

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estimate the field variation for each of your samples. (Annotate your printouts of the resonance peaks to show how you define the peak width). First we examine the contribution of the scan coils to field variation across the sample. Use the Biot-Savart law to derive an equation for the magnetic field along the central axis of a current 2 2 2 3 / 2 carrying coil [Result: B(x) = (NµoI /2)R /(x + R ) ]. Use this result to construct an equation for the field along the central axis between two identical coils. For the case when coil separation equals coil diameter you should find the field at the midpoint between the two coils is given by:

NµoI /(2 2R€). Use this equation, along with your own measurements of the coil dimensions and the field produced at the center of the coils, to determine the approximate number of turns on the scan coils. Verify that your result is physically reasonable, given your measured coil dimensions. € In the experimental setup the field due to the scan coils is enhanced due to the presence of the

magnetic core such that Bw/ core = α Bno core where α is a constant. In particular, when in the setup, the scan coils produce a field at the center of the sample of 3.2 mT when I = 0. 25 A. Determine the enhancement factor α and make a plot of the magnetic field due to the scan coils (when in the setup) versus distance from the center over a distance range equal to the sample size for the coil current corresponding to your proton resonance condition. Does this field variation across the sample explain some of the width of your resonance peaks? To estimate the possible contribution to peak width due to internal fields we note that the local field produced by a nucleus with magnetic moment µ is approximately

⎛ µ ⎞ µ B ≈ ⎜ o ⎟ . local ⎝ 4π ⎠ r3

Compute the local field produced by a nucleus with µ = µN at a typical inter-atomic distance of 1.5 Å and compare this with your estimated field variation. In a liquid sample these internal fields usually average to zero due€ to rapid molecular reorientation. Do you notice a difference in peak width between the liquid and solid samples?

Other Questions: 1) Use a particle-in-a-box energy level diagram to explain why you would expect the 19F nucleus to have a net spin of 1/2. 1 2) By comparing the spin-flip energy hf to the thermal energy kBT, estimate the ratio of H nuclei having spins aligned versus anti-aligned with the applied external magnetic field at room temperature. [Recall that the probability to occupy a particular energy state ε is proportional to

exp(–ε/kBT)]. There is a digital thermometer in the lab. 3) What other sources of peak broadening might there be other than the two investigated above?

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