PROCEDURES

Getting Started:

HEATHKIT POWER SUPPLY. Flip the POWER switch to STANDBY and wait 1 minute; then switch to ON. Set the METER SWITCH to the RIGHT. Set the amplitude of the RF field by adjusting the B+ OUTPUT knob to 150 volts, as read by the voltmeter.
On the H_mod magnetic field modulation control panel. Flip POWER and SWEEP ON switches up. Turn the AMPLITUDE ADJUST knob to its maximum CW position to maximize the modulation of the field. This sends about 1.7 amp at 60 HZ to the modulation coils; this current generates a magnetic field of about 2.9 gauss.
The PHASE ADJUST knob on the H_mod panel changes the phase of the modulation signal sent to the x-input of the scope relative to the signal sent to the modulation coils, and hence relative to the detected output signal sent to the y-input of the scope. In short, it changes the phase between the x and y signals. No need to set it at this time.
PRE-AMP, Stanford Research Systems Model SR 560. Turn the unit on and plug the AUDIO OUTPUT from the back of the NMR box into INPUT A and make sure that the DC /GND/AC switch next to the A input is in the AC position. Note that if you suddenly lose your original signal while you are making various adjustments - if your scope trace goes flat - you should push the OV LD overload recovery switch down. Set the gain to 50, the LF ROLL-OFF to 3 Hz, and the HF ROLL-OFF to 10K. You will probably have to adjust these later to make your signal as clean as possible without losing any of its major features.
Turn on the RF POWER SUPPLY.
Turn on the frequency counter (Fluke PM6669) and the oscilloscope. Connect the FREQUENCY MONITOR on the back of the NMR box to the input of the frequency counter. Put the scope in the x-y mode. Then connect the PHASE ADJUST to the x-input of the scope and the PRE-AMP output to the y-input of the scope.
Now you're ready to go. Insert a glycerin sample into the NMR HEAD and find the resonance signal by doing the following. Look up in the NMR TABLE - CW Appendix II of the manual - the NMR frequency for H¹ in a 10 kilogauss field. Knowing this number and that our H_o is about 3.9 kG, calculate the approximate NMR frequency for our set-up (around 16.54xxx). Turn the TRANS FREQ knob on the NMR BOX until the counter reads this value. Now adjust the receiver coil to observe the resonance by adjusting the CR control so that the receiver frequency matches that of the transmitter. Start with it at the fiducial mark.

Figure 8: 0.1 Molar Mn++ in H2O

Figure 9: 1 Molar Mn++ in H2O

Figure 10: 1 Molar Mn++ in H2O

Slowly vary the frequency around the value you calculated above. You should be able to find the resonance, and your signal should look like Fig. 8. Note that the resonance condition is very sensitive to frequency; so go slowly. Once you have found it, "peak-up" your signal by readjusting CR on the NMR BOX for maximizing the signal. If you don't find a signal after you've fussed with everything, ask for help.

Move the NMR HEAD around gently in the gap until you have maximized the number of "wiggles" and minimized the line width. This will place the head in the most uniform or homogeneous region of the magnet, and you want to keep it there. Once you have set the position for the day, don't change it. Some of your calculations depend on the field being the same for successive measurements. Repeat this procedure each day when you first come to the lab, since other people may have moved the NMR head from where you have determined to be the best.

8. Read over the REPORT section to be sure you take all the necessary data. Read the information described in the CW Appendix IV on how to transfer data to the computer.

Taking Data

A. By using the glycerin sample you can find a narrow line (that is, a sharp resonance) in the homogeneous region of the magnet. Once the resonance is observed, tune the frequency of the RF, f, until the two peaks are symmetric about the midpoint of the H_mod trace on the scope, for both slow passage and non-adiabatic passage conditions. Measure the resonant frequency and hence (by knowing $γ$ ) the magnetic field to high precision. It is a standard technique to use NMR to determine the magnetic field strength precisely.

B. By recalling that $\left (\frac {\Delta f}{f} \right) = \left (\frac {\Delta H}{H} \right)$ , where f is the frequency and H is the magnetic field, you can devise a method of calibrating (in gauss/div) the field axis of the oscilloscope. Then you can transfer the data of the four modes (absorption/dispersion; slow passage/non-adiabatic passage) to the computer.

C. Replace the glycerin sample with a 0.01M Mn⁺⁺ sample; again this signal should look like Figure 8. Download the data of this non-adiabatic rapid-passage absorption mode to the computer. (See REPORT, #3.) Now insert a more highly doped sample, 1M Mn⁺⁺. The paramagnetic Mn⁺⁺ ions relax the proton spins and the signal is closely resembled the one of a slow passage absorption mode. By further rotating the paddles you should be able to get a signal like Fig. 10, approximately a slow passage dispersion mode. You may have to increase V₁ to see this.

D. For observing the F¹⁹ resonance we use a Teflon rod. Note that the F¹⁹ resonance is a little harder to find. Using the same procedures as for H¹, find the resonance curves for F¹⁹. Accurately measure the NMR frequency ratio of F¹⁹ to H¹. To do that, measure f(H¹) and f(F¹⁹) each a number of times under identical conditions.

E. Lock-in Amplifier Operation: For fine tuning the NMR resonance, we can leave the frequency f and all the other parameters fixed and optimized, and vary the "total-DC" magnetic field by adding a small field H₂ with the output of the Wavetek 142 generator driving the wire coils. The following diagrams show what this change does to the signal. See Figure 11.

Figure 11: Lock-in amplifier signals

Read "Basic Introduction to Lock-in Amplifiers" in the CW Appendix III, and the manual of the Stanford Research Systems Model SR 830DSP Lock-in Amplifier (copied in reprints). In this apparatus, the lock-in amplifier records the derivative $dV_{ac}\left (H \right)/dH$ of the NMR signal as the frequency sweeps slowly through the resonance by ramping the field H₂ with the Wavetek generator. The signal-to-noise ratio can be 100x larger than that observed with the oscilloscope. This method is used to find and record weak signals. It also provides a convenient way to record the line width $δ H$ of a signal.

As usual, getting started is not trivial. First you need to have a good signal on the oscilloscope. Then turn on the power supply of the magnet, the lock-in amplifier, and the Wavetek generator. Set the Wavetek generator to give a .03 Hz triangular wave with maximum amplitude. If you are not familiar with the Wavetek generator, borrow another oscilloscope and look at the output signal, and the effects of turning or switching the knobs. Check out the OFFSET control.

You want to sweep the field as shown in Figure 11. Unfortunately, the power supply of the magnet does not generate a negative current and the sweep starts at the resonance field Ho. Consequently, the sweep is not symmetrical, and half the resonance curve is lost. Therefore you must change the RF frequency to a higher value to put the resonance at the middle of the triangular sweep. This means you must put the resonance curve on the scope at one end rather than in the center at a larger frequency than used in the previous part of the experiment. How much? Experiment to find out.

Connect the output of the pre-amplifier to the A-input of the lock-in amplifier. Set the lock-in controls as follows: Phase - 0; zero offset, switch off; Time Constant - 1 sec. Connect phase input (ref) to the phase adjust on the H_mod panel. Connect output of the lock-in to the Y-input of the PULSE NMR DAQ INTERFACE BOX and switch the box to CW.

Now how do you start? Turn the amplitude of H_mod way down, almost to zero. Refer to the diagrams above. For the first part of the experiment, H_mod was large; for this part it must be small. But of course once you get things working, try twisting all the knobs to peak up the signal. You should see the resonance curve sweeping across the face of the scope, and off the end. A few seconds later it should return and sweep across and off the other end. The voltage on the magnet power supply should be going from zero up and back again. If you are monitoring the Wavetek output on another oscilloscope, the horizontal line should go up and down from zero. When it is halfway, the resonance curve should be in the center of the scope.

For the H₂O samples with molarity of 3, 1, 0.3, 0.1, 0.03 molar Mn⁺⁺, adjust the mode to absorption and set H_mod small, ~1/8 the line width; this takes the derivative (see Ref. 2, p. 51). Adjust the lock-in amplifier and record the derivative with the computer.

Figure 12 shows a typical result for a 1 Molar sample. The two sharp glitches occur at the ramp voltage rapid reset. (Chart speed = 5 cm/minute; Wavetek: q15V at 0.01 Hz; lock-in time contact = 0.3 sec; sensitivity = 25 mV on channel A input; DC pre-filter = 100 ms; input filters: hi-pass = 50 Hz, lo-pass = 0.1 kHz; mode = fund f; lock-in reference phase adjusted to give maximum peak-to-peak signal on lock-in meter; output zero offset adjusted to center trace on paper.) By scanning more slowly, one can get a good measure of the line width H between the peaks. To measure the signal-to-noise ratio (S/N), one can increase the input gain by, say, 10x or more to record the RMS noise off resonance.

When the NMR paddle is set to the dispersion mode for a 0.33 Molar sample, the lock-in signal is shown in Fig. 13 under very similar conditions to Fig. 12.

Figure 12: Protons in Mn++ in H2O Absorption Mode. Output from the Lock-In amplifier.

Figure 13: ~0.3 Molar Mn++ in H2O Dispersion Mode. Output from the Lock-In amplifier.

REPORT

Include the following data, analysis, and calculations.

The Bloch Equations are the key to understanding this experiment:
1. $\frac {d}{dt} M_x = \gamma \left ( \overrightarrow {M} \times \overrightarrow {H} \right)_x - \frac {M_x}{T_2} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\, \left (1 \right )$
2. $\frac {d}{dt} M_y = \gamma \left ( \overrightarrow {M} \times \overrightarrow {H} \right)_y - \frac {M_y}{T_2} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\, \left (2 \right )$
3. $\frac {d}{dt} M_z = \gamma \left ( \overrightarrow {M} \times \overrightarrow {H} \right)_z + \frac {\left (M_0 - M_z \right)}{T_1} \qquad\qquad\qquad\qquad\qquad\quad\, \left (3 \right )$
4. What is M? Is H a magnetic field inherent in your sample, or is it an applied field? Derive equations (1) to (3) for the case of no damping (none of the T₁ or T₂ terms). Hint: consider the classical torque equation N = dL/dt.
5. The equations you derived are applicable to a set of identical magnetic moments (spins), i.e., all the spins see the same magnetic field. The damping terms in (1) to (3) are added to take the different environment of each spin into account.
6. In equations (1) and (2), T₂ is often called a dephasing time, the time required for (M)_x,y to decay to zero after the resonance condition is removed. What is getting out of phase? How can this arise from inhomogeneities in H?
7. T₁ is the relaxation time for the z-component of M to come to an equilibrium value M_o when the resonance condition (applied RF) is removed. (You can see this by setting H to zero in (3).) What determines this equilibrium value, and what is it? [Hint: look at the classical Boltzmann distribution.]
What (very general) physical factor of your sample accounts for the magnitude of T₁? [Hint: Mo is a thermal equilibrium magnetization.] More specifically, how might the addition of paramagnetic ions Mn⁺⁺ affect T₁ (qualitatively)?
Measure the magnetic field H₀ of the magnet as precisely as you can in Gauss (average value and standard deviation). Are paramagnetic corrections significant? Measure H₀ on several different days: does it vary? Why?
Include best computer plot with calibrated H axes for proton signals in glycerin, showing the "best wiggles." Estimate the magnetic field inhomogeneity from the damping rate of the wiggles [see Ref. 1, pp. 84-86; and Ref. 2, p. 53; and problem 2-7, p. 57].
Computer plots with calibrated H axes for absorption and for dispersion NMR signals in H₂O under slow passage and under non-adiabatic passage.
Measure as precisely as you can the ratio of the magnetic moment of F¹⁹ to H¹ , with standard deviation. Show best scope photo of F¹⁹ resonance.
Record and show sequence of absorption lock-in trace from the computer for protons in the presence of M = 3, 1, 0.3, 0.1, 0.03 Molar Mn⁺⁺. Measure the line width (H in Gauss and plot (H vs. Molarity; explain this graph; what is the residual line width (in Gauss) due to field non-uniformity?
From your best lock-in data for resonance in 1 Molar Mn^++M in H₂O, compute the expected S/N ratio for the deuteron resonance in a 1 Molar Mn⁺⁺ in D₂O sample under two different assumptions:
1. The field H_o is kept at 3900 G and the oscillator frequency is adjusted to f(H²).
2. The oscillator frequency is kept at 16.5 MHz and the DC field is adjusted for resonance.
Under condition 2 above, compute the S/N ratio expected for O¹⁷ NMR resonance in naturally abundant water; compare to Ref. 5, Fig. 4.