Department of Physics, Middlebury College 1992-93
Modern Physics Laboratory

XIX. Nuclear Magnetic Resonance with a 300 MHz Fourier Transform Spectrometer


Nuclear magnetic resonance (NMR) of protons in bulk matter was first demonstrated in 1946 independently by F. Bloch and E. Purcell who were jointly awarded the Nobel prize for their discoveries in 1952. Their original motivation was simply to demonstrate that the feeble magnetism associated with atomic nuclei could be detected in ordinary matter. Their early demonstrations of proton NMR were soon followed by NMR detection of other nuclei and the use of NMR to characterize molecular and solid state environments. Today NMR spectrometers are standard research tools in physics, chemistry, biology, geology, and perhaps most prominently, in medicine for magnetic resonance imaging (MRI) of human tissue.

In this experiment you will use a modern 300 MHz Fourier transform NMR spectrometer to (a) study in detail NMR detection techniques, (b) understand the use of NMR for determining molecular structure, and (c) identify the intramolecular coupling between protons in a molecule using homonuclear decoupling techniques. As will become apparent when you browse the list of possible multi-pulse experiments listed in the software menu of the GE GN-300 Omega spectrometer, NMR spectroscopy is a mature scientific field with its own unique jargon, and you will only scratch the surface of possible NMR applications.

Excellent introductions to continuous wave (CW) and Fourier transform (FT) NMR techniques can be found in the early chapters of Refs. 1-3. A brief introduction, largely restricted to Fourier techniques, will be given here.

(1) Nuclear Magnetism of Bulk Matter.

A nucleus with a non-zero spin angular momentum I possesses a magnetic dipole moment that is conventionally written as

= g e /2mp I   (MKS units)        (1)

where e is the electronic charge, is Planck's constant divided by 2, mp is the proton mass, and g is the nuclear g-factor.4 The unit of nuclear magnetism analogous to the Bohr magneton of atomic physics is the nuclear magneton, N, defined to be

N e /2mp = 5.05 x 10-27 J/T

so that the nuclear magnetic moment can be written more simply as

= g I N .

Nuclei with I = 0 such as 12C and 16O have = 0 and therefore do not have nuclear magnetic dipole moments.The present experiment deals exclusively with protons (1H) for which I = 1/2 and p = +2.7928473386(36) N.5

In a quantum mechanical picture, a free proton at rest in a uniform, static magnetic field Bo oriented along the z direction, occupies the spin up (mI = +1/2) or spin down (mI = -1/2) state as shown in Fig. 1(a). The energies of the two spin states are given by

E = - .Bo
= - z Bo
= - g N mI Bo

so that for spin up, E = - pBo, and E = + pBo for spin down. A real liquid or solid sample contains a very large number of protons N that remain in thermal equilibrium at temperature T with their environment. For thermal equilibrium, the Boltzmann distribution gives the relative number of protons in the spin up and spin down states as

N/N = e-(E-E)/kT        (3)

where N and N are the numbers of spin up and spin down protons, respectively, and k is the Boltzmann constant. It is useful to define the nuclear polarization p as

p N - N/N + N        (4)

where p = 0 means there are equal numbers of protons in the two spin states and the sample is said to be unpolarized. For p = +1 and p = -1 all protons are in the spin up and spin down states, respectively. Substituting Eq. (3) into Eq. (4) and approximating to the high temperature limit pBo/kT<< 1 applicable to most NMR experiments, the polarization p is given by

p pBo/kT .          (5)

For the 7.05 T magnetic field of the GN-300 Spectrometer and samples kept at room temperature, p 2 x 10-5, which is an extremely small nuclear polarization. This means that of 100,000 protons in a sample at room temperature, about 50,001 are spin up and 49,999 are spin down. The bulk nuclear magnetization M produced by the N protons is given by

M = Np - Np

=  N p p

where V is the volume of the proton sample. When one considers in the above example that the magnetization M is produced by the 2 excess spin up protons in 100,000, it is remarkable that NMR succeeds at room temperature, because the magnetic contribution from the protons in a sample almost completely, but not quite, cancel each other out. Since according to Eq. (5) the polarization grows linearly with the strength of the static, uniform magnetic field Bo, there is great interest in high field NMR instrumentation.

(2) Nuclear Magnetic Resonance.

(a) Quantum Mechanical Picture. A proton in a uniform magnetic field Bo can absorb photons when the photon energy E = h matches the energy difference between the two allowed proton spin states of Fig. 1(a). This resonant absorption leads to the nuclear magnetic resonance (NMR) frequency condition hp = 2pBo, which determines the proton NMR frequency p to be

p = 2pBo/h        (6)

The superconducting magnet of the GN-300 NMR spectrometer maintains a static magnetic field of Bo = 7.05 T that by Eq. (6) implies an NMR resonance frequency of p = 3.00 x 108 s-1 = 300 MHz. Most commercial NMR spectrometers such as the GN-300 include the proton NMR frequency, in MHz, in the instrument's model number.

(b) Classical Picture. Although the quantum mechanical approach given above leads more immediately to the nuclear magnetic resonance frequency condition of Eq. (6), it will prove useful to view the resonance process from a somewhat more naive, classical point of view. In the classical view, the magnetic moment of the spinning proton precesses about the magnetic field direction Bo as shown in Fig. 1(b). (The analogy to the precession of a spinning bicycle wheel in the earth's gravitational field should be recalled here.6) In the nuclear precessional motion the static magnetic field Bo produces a torque x B1 that produces a change in the angular momentum of the proton given by

dI/dt = x Bo.

The precessional motion of the proton magnetic moment sweeps out the surface of a cone aligned along the z direction, as shown in Fig. 1(b), and the precessional frequency can be shown to be given by

p = 2pBo/h        (7)

in agreement with the quantum mechanical result of Eq. (6). The precession frequency of Eq. (7) is a result from classical mechanics where p is known as the Larmor precession frequency.

It is also very useful to consider the precessional motion of the proton's magnetic moment from a frame of reference with axes x'y'z' that rotate at angular frequency p 2p with respect to the laboratory axes xyz, as shown in Fig. 2(a) Because the rotating frame rotates at exactly the same rate as the proton precession rate, the description of precession phenomena from the view of the rotating frame is quite simple: in the rotating frame of reference x'y'z', the proton's magnetic moment is stationary  and lies, for example, in the y'z' plane as shown in Fig. 2(b).

We now want to consider the classical analogue to the quantum mechanical transitions discussed in Section 1(a) above. To do this consider the effect of a weak, radio frequency (rf) oscillatory magnetic field B1(t) given by

B1(t) = B1 ( sin(t) + cos(t) )

that produces a magnetic field of constant magnitude B1 that rotates about the z axis of the laboratory with angular frequency = 2. (The rf frequency need not be related to and should be distinguished from the proton precession frequency p of Eqs.(6) and (7) above.) A field of this type can be produced by a pair of appropriately phased coils placed along the x and y directions of the laboratory, as shown in Fig. 3. An interesting resonance phenomenon occurs when = p, that is, when the frequency of rotation of the magnetic field B1 matches the Larmor precession frequency for protons in the static field Bo. Although this situation may appear rather complicated from the laboratory point of view, in the rotating frame x'y'z' the situation is simple: the proton magnetic moment lies in the x'z' plane, and the magnetic field B1 has constant magnitude B1 along the x' axis. But the proton magnetic moment will not remain stationary in the x'z' plane; it now finds itself in the presence of a static (relative to the x'y'z' frame) magnetic field B1 and therefore will precess about B1 as shown in Fig. 2(b). This simple precession about B1 in the rotating frame x'y'z' will of course appear very complicated from the laboratory frame, where one will observe a rapid precession about the z axis (due to Bo) combined with a slow movement of the magnetic moment vector between its up and down positions (due to B1). For our purposes it will suffice to know that when a magnetic field B1(t) rotates at the Larmor precession frequency p for protons in a static magnetic field Bo, the proton magnetic moment can be rapidly cycled between spin up and spin down configurations, in complete analogy with the quantum mechanical transitions of Section 2(a) above.

It is easy to determine the frequency with which the proton magnetic moment cycles between the up and down configurations. In the rotating frame x'y'z' of Fig. 2(b), one can argue by analogy to the reasoning used to derive Eq. (6), that the frequency of the precessional motion of about B1 is given by

Rabi = 2pB1/h

where Rabi is called the flopping frequency, or Rabi frequency, for I.I. Rabi who pioneered the use of rf resonance in atomic and molecular beam experiments. For our purposes the Rabi frequency is simply the rate at which protons make transitions between the spin up and spin down states when the resonance condition = p is fulfilled. In most NMR work the Rabi frequency is much less than the Larmor precession frequency, so that in the laboratory frame of reference one thinks of the proton magnetic moment as precessing a great many times about the z direction in the time it takes for a proton's spin to make a transition from the up to the down state. The length of time T for a proton to cycle from the up state through the down state and return to the up state is given by the inverse of the Rabi frequency, that is

TRabi = 1/Rabi     (8)
= h/2pB1

As will be discussed below, when the oscillatory field B1 is turned on for a short period of time, in the form of an rf pulse, the proton magnetization can be oriented in any direction relative to the rotating frame axes x'y'z'.

(3) Detection of NMR in Bulk Matter

(a) Continuous Wave (CW) NMR. This is the traditional and perhaps simplest means for detecting nuclear magnetic resonance of protons. A sample containing hydrogen is placed between the pole pieces of a large magnet that supplies the large static magnetic field Bo. If the magnetic field Bo or the frequency of the weak rotating magnetic field B1 is slowly swept through the resonance condition of Eq. (5) there is a net absorption of energy by nuclei of the sample that manifests itself as a very small change in the current passing through the coils shown in Fig. 2(a). The magnitude of this absorption signal can be plotted against the varied parameter - Bo or - to yield an NMR absorption spectrum.7 The term CW is applied to this technique because the rf oscillator used to produce the rotating field B1 is on continuously while the frequency or the magnetic field strength Bo are swept slowly in time.

(b) Pulsed rf, Fourier Transform (FT) NMR. Most modern NMR spectrometers provide the weak rotating magnetic field B1 in the form of a short rf pulse, so that the orientation of the proton sample magnetization can be oriented for optimal detection by a pair of coils, as in Fig. 3. To understand the advantage of pulsed techniques it useful to consider the magnetization M produced by large number of protons in a sample. Although individual protons have magnetic moment components along the x and y directions of the laboratory, when the magnetic moment contributions of a large number of uncorrelated protons are summed together, the contributions along the x and y directions cancel to zero, and the resulting magnetization M lies solely along the z direction, as shown in Fig. 4(a). In the following discussion it will be convenient to consider the evolution of the net magnetization M produced by protons in a sample, rather than the magnetic moments of individual protons.

In thermal equilibrium, the magnetization M produced by the protons in a sample is constant in magnitude and direction, so that it does not produce a changing magnetic flux at the coils shown in Fig. 3, and therefore no NMR signal is detected. To detect the magnetization M by coils lying in the xy plane of the laboratory frame, it is desirable to have the magnetization M lie in the x'y' plane so that the magnetization sweeps its associated magnetic field through the area of the coils as the magnetization precesses in the laboratory. Because the magnetization precesses at the Larmor frequency, the changing flux at the coils will generate a rf signal at the Larmor precession frequency that can be detected by connecting a radio receiver tuned to 300 MHz to the coils.

The magnetization M can be tilted down to the x'y' plane by applying a rotating field B1 for a finite period of time in the form of an rf pulse. If B1 is turned on for a finite period of time T, the magnetization M will precess a corresponding finite angle about the B1 direction, and, after the rf pulse is over, once again remain stationary with respect to the rotating frame x'y'z' direction. If the period T of the rf pulse is chosen to satisfy

T = 1/4 TRabi        (9)

then the magnetization M precesses by an angle of 90o, or /2 about the x' axis, and finally lies along the y' axis. This /2 rf pulse therefore tilts the magnetization M so that it in the laboratory it precesses in the xy plane and generates an rf signal in the detection coils of Fig. 3. For the GN-300 Omega spectrometer, a /2 pulse has a duration of 10 s and therefore by Eqs. (8) and (9) the magnitude of B1 is 5.9 x 10-4 T. Note that the Rabi period in this case is 40 s, which is much longer than the precessional period of 3.3 ns for a Larmor precession frequency of 300 MHz. From these values you can verify that a proton magnetic moment precesses approximately 3000 times about the z axis as the /2 tilting process is taking place. When the resonant tilting of the magnetization M occurs so slowly compared to the time scale of the Larmor precession, the NMR process is said to be adiabatic.

In practice, the same two coils that are used to produce the /2 rf pulse are used as receiver coils to detect the 300 MHz signals produced in the coils by the precessing magnetization of the sample. It is difficult to process the NMR signals at 300 MHz and it is standard practice to translate the NMR signals to much lower frequencies for signal processing and filtering. To do this the 300 MHz NMR signals detected by the two coils are mixed with a low power signal from the same oscillator that produces the rf /2 pulse; the beat frequency from the mixing process can be adjusted to lie in the range 0-100 kHz. When the beat frequency occurs at audio frequencies it is heard as a loud "bong" from the loudspeaker at the NMR console. These lower frequency signals are passed through a low-pass filter and digitized for recording in the computer memory. The temporal record of the digitized coil signals must be Fourier transformed to recover the spectrum of precession frequencies of protons in the sample.

An example from an NMR spectrum you will obtain in this experiment should serve to clarify the detection scheme. Fig. 5(a) shows the digitized signal from a single coil in the GN 300 Omega NMR spectrometer when a sample of acetaldehyde (CH3CHO) was irradiated by rf pulses of 8 s duration at = 300.5229000 MHz (300.5200000 MHZ base frequency + 2900.0 Hz offset frequency) and the signals received back from the coil were mixed with a much lower power signal derived from the same 300.5229000 MHz local oscillator. From the mixer, beat signals in the range 0-400 Hz were passed by the lowpass filter to the digitizer to produce Fig. 5(a). The mixer output was digitized for nearly 4 s following the /2 rf pulse. The rf coil signal shown in Fig. 5(a) has dominant frequency components, which by inspection, lie near 15 Hz and 3 Hz. The general decrease in the amplitude of the coil signal is due to the relaxation of the sample magnetization from the xy plane back to its equilibrium orientation along the Bo direction. Empirically it is found that after rf irradiation the z component of magnetization Mz(t) reestablishes itself according to the simple exponential formula

Mz(t) = Mo (1 - e-t/T1)

where T1, called the longitudinal relaxation time, has a very sensitive and useful dependence on the details of the environment in which a given proton is sited. Fig. 5(a) is known as the free induction decay (FID) of the /2-tilted magnetization in a pulsed NMR experiment.

When a Fourier transform is performed on the free induction decay signal of Fig. 5(a) one obtains the frequency spectrum of Fig. 5(b). The horizontal axis of Fig. 5(b) is the frequency shift, in Hz, of a proton NMR signal relative to the base frequency 300.520000 MHz of the NMR spectrometer. Therefore the center of the quartet of peaks near 2915 Hz in Fig. 5(b) corresponds to a true NMR resonance frequency of 300.5229150 MHz. The two beat frequencies at 3 Hz and 15 Hz in the FID signal of Fig. 5(a) correspond to the 3 Hz frequency difference between the four absorption peaks in Fig. 5(b), and the 15 Hz frequency difference between the center of the quartet and the local oscillator frequency, respectively.

(4) Chemical Shift and Spin-Spin Splitting in NMR Spectra.

After just a few moments of looking at Fig. 5(b) the observant student will surely ask why there are four absorption peaks in the spectrum of acetaldehyde and not just the single peak expected from the proton resonance condition of Eq. (6). The situation appears even more puzzling when an expanded NMR spectrum of acetaldehyde such as Fig. 6(a) is acquired that shows that there are two more absorption peaks located about 2280 Hz lower in frequency than the quartet of peaks in Fig. 5(b). Why are there six proton NMR peaks for acetaldehyde? The answer depends on two important effects that are exploited routinely to determine molecular structure by NMR techniques.

(a) Chemical shift. For a free proton there is just one NMR resonance frequency, but in liquids a proton finds itself bound within a molecule which in most cases shields the proton slightly from the external magnetic field Bo. The resulting shift of the NMR frequency is called a chemical shift and the degree of shielding depends on the exact site of a proton within a given molecule . For the spectrum of Fig. 6(a) the quartet of peaks at the left and the doublet of peaks at the right are shifted in frequency by amounts that are typical for aldehyde (CHO) and methyl (CH3) group protons, respectively. By inspecting the NMR spectra of many different organic molecules that have been identified by other techniques, it has become possible to develop empirical rules for identifying individual protons or proton groups within molecules simply on the basis of their chemical shifts.

The shielding effect of a molecule on its constituent protons can ultimately be traced to the phenomenon of diamagnetism. Because the diamagnetic shielding effect and associated frequency shifts increase linearly with external magnetic field Bo, chemists have adopted a definition of chemical shift that is independent of magnetic field strength Bo, so that NMR spectra obtained from spectrometers operating at different magnetic fields can be compared easily. The chemical shift of a proton peak is defined to be


where TMS is the resonance frequency of the protons in tetramethylsilane (TMS), (CH3)4Si, a compound that is easy to add to NMR samples and gives a single peak from which the frequency of other NMR peaks may be referenced. The offset frequency of 2915 Hz at the center of the aldehyde quartet in Fig. 6(b) is relative to the TMS peak so that the chemical shift of the aldehyde proton in Fig. 6(b) is given by

CHO = 2915 Hz/300.520000 MHz

= 9.70 x 10-6 = 9.7 ppm .

The center of the methyl group occurs at 632.8 Hz from TMS implying a chemical shift of CH3 = 2.1 ppm. The GN-300 Omega software will allow you to determine automatically the positions of peaks in terms of either a frequency offset, in Hz, or a chemical shift, in ppm.

It should also be mentioned the integrated areas under the methyl (CH3) and aldehyde (CHO) peaks of Fig. 6(a) occur in the ratio 3:1, a fact that agrees well with the number of protons thought responsible for the NMR signal in each case.

(b) Spin-Spin Splitting. The splitting of the aldehyde (CHO) signal into a quartet (center at 2915 Hz) and the methyl (CH3) signal into a doublet (center at 633 Hz) is due to a spin-spin interaction between these two proton groups. Although it is useful at first to think of the spin-spin interaction as a type of dipole-dipole interaction between proton magnetic moments within a molecule, the true picture of the spin-spin interaction is more subtle and involves an indirect interaction through electrons in the binding regions between the two proton spins, rather than a direct dipole-dipole interaction through the empty space between two protons.

The three protons of the methyl group (CH3) in acetaldehyde are said to be chemically equivalent because their symmetrical arrangement ensures that they experience the same chemical shift and that they respond identically in NMR experiments. Assuming the essential equivalence of the three methyl protons, it is straightforward to explain the doublet and quartet structures seen in Fig. 6(a).

(i) Methyl doublet. The aldehyde proton spin can be up or down, thus producing two possible values for the magnetic field at the CH3 protons, as shown in Fig. 7. The chemically equivalent protons resonant at two different frequencies, thus the methyl doublet.
(ii) Aldehyde quartet. The three chemically equivalent methyl protons can be found in only four different spin configurations, as shown in Fig. 4(b). The aldehyde proton therefore finds itself in four possible magnetic field environments, thus the quartet peak structure. The methyl spin configurations , , and (one spin down proton) and the configurations , , and (one spin up proton) are each three times more likely than the configurations and where all proton spins are up or down, respectively. The aldehyde proton is therefore three times more likely to find itself in a somewhat reduced magnetic interaction produced by a methyl group in a mixed spin state like , , , , , or rather than in the stronger magnetic interaction produced by pure states like or , explaining why the inner two peaks of the aldehyde quartet have three times the NMR signal of the two outer peaks. Another fact not to be overlooked in the spin-spin splitting of the acetaldehyde spectrum is the equal, 3Hz, spacing in frequency between peaks in both the aldehyde quartet and the methyl doublet, resulting from the reciprocal nature of the dipole-dipole interaction between proton spins.

(5) Homonuclear Decoupling.

You may still not be convinced that the NMR spectra of Figs. 6(a)-6(c) have been explained correctly in terms of the chemical shift and spin-spin splitting arguments given in the discussion above. If you remain unconvinced, then perhaps the following homonuclear decoupling demonstration will convince you. Consider what happens at the site of the aldehyde (CHO) proton if a second rf oscillator signal is set to continuously irradiate the sample at the center frequency (633 Hz) of the methyl doublet. This second rf oscillator is left on continuously to induce transitions between the four possible spin states of the methyl group (see Fig. 7) so that the net effect of the methyl group magnetism time averages to zero at each acetaldehyde proton. The aldehyde (CHO) proton no longer experience a spin-spin coupling with the methyl (CH3) group protons and the aldehyde quartet collapses to a single peak at the previous center of the quartet. Figs. 8(a), (b), and (c) show the aldehyde (CHO) quartet at 2915 Hz collapse to single peak as progressively greater rf power is applied at the frequency of the center of the methyl (CH3) group peak at 633 Hz. This process is known as homonuclear decoupling and it can be used to isolate and identify individual spin-spin interactions in the NMR spectra of complex molecules.

The roles of the aldehyde and methyl protons in the decoupling process can be reversed. The methyl doublet is collapsed to a single line in the series of spectra in Fig. 8 by irradiating at the center of the aldehyde quartet at 2915 Hz.8

(6) Final Thoughts.

By this point you probably realize that NMR is a valuable tool for elucidating the structure of complex organic molecules. It is worth pausing though to appreciate two facts that conspire to make NMR relatively easy to use in organic chemistry. The first fact is that the bonding rules for organic chemistry inevitably lead to molecules with an even number of electrons so that the electron magnetic moments cancel exactly within each molecule. This condition is crucial because the electron has a magnetic moment that is 658 times greater than the proton's and the resulting additional electron-proton splittings in NMR spectra would complicate interpretation significantly. A second fact is that many "connector" atoms in organic chemistry such as carbon and oxygen have nuclei (12C and 16O) with spin I = 0. Although there is a small abundance of the nonzero spin isotopes 13C and 17O in ordinary carbon and oxygen, their effect on NMR spectra is often undetectable and the proton NMR spectra can be interpreted solely in terms of chemical shifts of protons and proton-proton interactions. Spectral interpretation would be significantly more difficult if additional spin-spin couplings were present.


(1) Reproduce the acetaldehyde spectra of Figs. 6(a)-6(c) by carrying out the following steps.
(a) Lower the 5 mm acetaldehyde sample tube into the NMR magnet and adjust the spinning rate to 20 Hz.
(b) Shim the NMR magnet to obtain a strong deuterium lock signal from the d6-acetone solvent of the acetaldehyde sample.
(c) Run the data acquisition program "1puls" and choose appropriate frequency offsets and spectral widths to acquire spectra like those in Figs. 6(a)-6(c). Save your spectra to disk and print them out on the laser printer.

(2) Reproduce the decoupling spectra of Fig. 8(c) and Fig. 9(c) by carrying out the following steps.
(a) Set the decoupling frequency to the center of the methyl doublet. Be sure the decoupler is turned on in the homonuclear mode, with 70 dB power, and observe the decoupling of the aldehyde peak. Save your spectra to disk and print them out on the laser printer.
(b) Set the decoupling frequency to the center of the aldehyde doublet. Be sure the decoupler is turned on in the homonuclear mode, with 70 dB power, and observe the decoupling of the methyl peak. Save your spectra to disk and print them out on the laser printer.

(3) The instructor will give you an NMR tube containing a sample of an unknown organic liquid. Obtain NMR spectra for your sample and perform homonuclear decoupling measurements on any multiplets you observe in your spectra. Identify the unknown organic molecule responsible for your spectra. Explain how your NMR spectra provide evidence to support your identification.

The instructor is well aware that you may not have much background in organic chemistry. The unknown you will identify is one of the simplest organic molecules and anyone with a little knowledge of organic chemistry will be able to give you suggestions as to what it might be.

1. A.E. Derome, Modern NMR Techniques for Chemistry Research  (Pergamon Press, New York, 1987).
2. J.K.M. Saunders and B.K. Hunter, Modern NMR Spectroscopy  (Oxford University Press, New York, 1987).
3. J.D. Roberts, Nuclear Magnetic Resonance: Applications to Organic Chemistry (McGraw-Hill, New York, 1959).
4. R.A. Serway, C.J. Moses, and C.A. Moyer, Modern Physics  (Saunders College Publishing, Philadelphia, 1989), pp. 267-293.
5. P. Raghavan, Atomic Data and Nuclear Data Tables 42 (1989) p. 189.
6. R. Wolfson and Jay M. Pasachoff, Physics  (Little, Brown and Co., Boston, 1987) p. 297.
7. A. Kenyon, Spin-Spin Split and Spin Decoupled Nuclear Magnetic Resonance in Acetaldehyde, B.A. thesis, Middlebury College, 1984.
8. K. Hutchinson, 1H, 31P, 15N Nuclear Magnetic Resonance Spectroscopy, B.A. thesis, Middlebury College, 1991.

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