## Introduction

Because some of the terms used to describe the phenomenon of nuclear magnetic resonance are a bit unfamiliar, a more familiar physical example illustrating these terms is presented.

To illustrate the meaning of "resonance," recall that a car driven at a certain speed, generally about 80 km/hr, will sometimes begin to vibrate due to a wheel out of balance. This vibration is an example of the phenomenon of "resonance." In a "resonant energy exchange," at a particular driving frequency (in this case the angular frequency of the wheel), a system with a "resonant state" matching the energy of the driving frequency will begin to absorb energy from an external source in a very efficient manner. The system with the resonant state in this case is the spring suspension system of the car. When the out-of-balance wheel, which is vibrating at all speeds, reaches a vibration frequency matching a "resonant frequency" of the car's suspension system, then there is a "resonant exchange" between the wheel and the remainder of the car, and the entire car begins to vibrate as the energy in the vibrating wheel becomes efficiently transferred to the remainder of the car. The energy transferred in a resonant exchange can be stored, and if not released quickly enough, can serve to destroy the system. In a car, that might mean a tire blowing out from repeated resonant bouncing. In a molecule, it could mean thermal decomposition. Alternatively, the system might relax from its high energy state by releasing energy back to the driving source in an oscillatory fashion, or by releasing energy to its surroundings in some mono-tonically decreasing manner, perhaps in an exponentially damped fashion. For example, the out-of-balance wheel on a car might simply exchange energy with the suspension system of the car and oscillate indefinitely, if the speed of the car is maintained such that resonant exchange can take place. Another possibility is that the spring suspension system of the car, receiving the vibrations of the out-of-balance wheel, might eventually be shaken apart and fail. A driver of a car with a wheel badly out of balance might automatically change the speed of the car to move away from the resonant situation, and thus allow the vibration to damp away, or relax, by energy interchange between the suspension system of the car and the surroundings; the rapid vibration might have heated the springs, and the heat could have been dissipated in the passing air.

Nuclei possessing a magnetic moment, and placed in a static magnetic field of from 5 to 10 T (for comparison, the earth's field is about .00005 T, or one-half Gauss) may be thought of as developing discrete, or quantized align ments of their magnetic moments with respect to the static magnetic field [i.e., the nuclei develop discrete energies, or states, the energy differences of which are in the radio-frequency (millions of cycles per second, or megahertz) region]. One may relate a frequency v in cycles per second, to an energy difference AE by the relation AE = hv, where h is Planck's constant, h = 6.627 x 10-27 erg sec. When such nuclei are exposed to radio-frequency radiation at the appropriate resonant driving frequency matching the energy differences of nuclear states in a magnetic field, then in much the same manner that the suspension system of a car will efficiently absorb energy from a vibrating wheel, the nuclei undergo resonant absorption of the radio-frequency energy from a driving source, which is a resonant alternating current circuit of the type used in radio broadcasting. Such excitation disturbs the nuclei from their initial equilibrium states in the static field. Changing the driving frequency or the external magnetic field will remove the resonant condition, and the nuclei will cease absorbing energy. Following, and during, a resonant absorption of energy, the nuclei can relax from their excited states back to states of energy intermediate between their initial states before resonant excitation and their excited states.

Two common relaxation processes for nuclei are termed "spin-lattice" or longitudinal relaxation, and "spin-spin" or transverse relaxation. In longitudinal relaxation, the nuclei lose their energy to the molecular framework in which they reside, and relax to their initial state, which may be thought of as alignment of their magnetic moments along the static magnetic field. In transverse relaxation, the nuclei relax to a state in which the net magnetic moment perpendicular to the static field is created by many nuclei having their spins pointing "in phase" in the same direction in the transverse plane, and the nuclei lose their phase coherence, and thus their net nuclear magnetism perpendicular to the static field. In the simplest cases (e.g., that of nuclei in molecules in many liquids), these relaxation processes are described by simple exponential forms. The time constant for longitudinal relaxation is designated by the symbol T1. After a resonant excitation, the recovery of the nuclear magnetization parallel to the static field is proportional to (1 - e-t/Tl). Note that this expression is zero at time = t = 0 (i.e., immediately following the excitation), and smoothly reaches unity at times longer than 5T1. Similarly, the time constant for relaxation of magnetization transverse to the magnetic field is called T2, and transverse relaxation is exponential in T2.

The phenomenon of nuclei absorbing resonant radio-frequency energy in a static magnetic field is called nuclear magnetic resonance (NMR), and this phenomenon is always accompanied by nuclear relaxation. The details of resonant absorption of radio-frequency energy by nuclei in a magnetic field leads to the "spectrum" of absorption lines, and the relaxation processes are responsible for line intensities and the widths of the observed spectral lines. The study of spectra is called "spectroscopy," hence the term NMR spectroscopy. It is seen, therefore, that at the very least, an NMR experiment requires (1) a source of radio-frequency (rf) oscillation at some radial frequency m (rad sec-1) tuned to energy splittings between nuclear magnetic states, (2) a magnet developing a static field B, to produce the split nuclear magnetic states, (3) nuclei with magnetic moments M placed in a resonant radio-frequency (RF) circuit to absorb the rf energy at frequency m, and (4) some means of detecting this energy absorption. An NMR spectrometer is basically a high quality FM radio station and accompanying FM receiver. The carrier is in the (video) megahertz region, generally between 5 and 700 MHz. The information content generally comes through in the (audio) kilohertz region, but the sounds that are produced by resonating nuclei, when sent over an audio speaker, are generally fairly monotonic, and are not nearly as pleasant as those designed by a Mozart. The magnet is an expensive addendum to make a portion of the experiment possible.

The fundamental relation between the experimenter-supplied parameters, B and m, and the nuclear moment M is m a BM .

The resonant frequency of absorption of energy of magnetic nuclei in a magnetic field is proportional to both the strength of the field, and to the magnetic moment of the nucleus. The resonant condition for NMR may be achieved by varying either B or the driving frequency. As alluded to previously, the local electronic and nuclear environment abouta nucleus ina molecule, along with the external magnetic field created by a magnet, contributes to the effective value of B. Thus the resonant NMR frequency is a fingerprint of the local electronic environment of the nucleus, but depends upon the external magnetic field, which is at the control of the experimenter. The magnetic moment of a nucleus is a quantity fixed by nature, and is not an experimental variable. Table I lists all of the known magnetic nuclides, their resonant frequencies in the absence of interactions associated with the atomic or molecular environment at a field in which 1H resonates at 100 MHz, and relavent added material, which will become more meaningful as further information is developed. Note that from a quick glance at Table I, it is possible to infer that the physician, the materials scientist, the chemist, the physicist, the polymer chemist, the solid state scientist, the geologist, and the engineer all have problems that may be attacked with the help of NMR, since workers in all of these specialties deal with systems containing one or more of the nuclei listed.

Table I indicates that each magnetic nucleus has a number of fingerprints. One is its "nuclear spin quantum num ber" I, which is proportional to its magnetic moment M, the proportionality factor being the gyromagnetic ratio Y; M = y I. The values of y , and therefore the resonant frequencies for NMR at fixed field differ for each nuclide. For example, 1H and 13C have spin quantum numbers I = 1/2, whereas 6Li and 174Lu are spin 1. 27Al is spin 5/2. This spin quantum number imparts a special character to the nucleus' ability to detect its local molecular architecture, as will be seen in Section II. The NMR absorption spectra, examples of which are shown in Sections III and IV, are generally represented on an intensity (ordinate) vs frequency (abscissa) plot, and appear as a series of peaks of various widths and shapes that are a reflection of the local molecular environment of the nuclei under observation. This is to say that the local environments of nuclei in matter supply effective fields, Beff, which may be used to infer that environment.

We now inquire in more detail as to why NMR has this remarkable capability, and why this resonant spectroscopy is such a powerful tool, relative to other spectroscopies, such as ultraviolet and infrared spectroscopies.

II. THE NUCLEUS AS A PROBE OF MOLECULAR STRUCTURE; INTERNAL INTERACTIONS AND THE EFFECTS OF MOTION

A nucleus residing in a molecule, either in a solid or a liquid sample, has access to quite an intimate view of its local molecular architecture. This nucleus senses the locations and types of its nearest neighbors, and in a diffuse manner, the bulk matter around it. In addition, this nucleus is sensitive to motion of its environment. The nucleus, when properly interrogated with resonant excitations, can give detailed information about its local molecular environment when that environment is motionless. In addition, the alteration of this information caused by molecular motion is used to infer details of such motion. It is this type of information which, when properly interpreted as indicated in the introduction, can lead to the wide variety of applications described there.

The sensitivity of the nucleus to its environment and to motion are all the result of the arrangements of molecular framework electrons and nuclei about the nucleus in question. The effects of this molecular framework upon the effective magnetic fields, and thus upon the resonance frequencies of nuclei in matter are generally separated into four contributions, termed interactions: these are designated (1) "shielding," (2) "dipolar coupling," (3) electric field gradients, or "quadrupolar coupling," and (4) "scalar coupling." These interactions are all anisotropic. This means that they are directionally dependent on the relative orientations of the static magnetic

TABLE I Properties of Magnetically Active Nuclides3'6

Atomic weight/ element

(10"28 M2)

T"1 sec"1)

Resonance frequency (1H TMS 100 MHz)

iH

2

99.985

5.68 x 103

26.7510

100.0000

2H

i

0.0i5

8.2 x 10-3

2.73 x 10-3

4.i064

15.351

3H

2

28.5335

i60.663

3 He

2

0.000i4

3.26 x 10-3

-20.378

76.i78

6Li

i

7.42

3.58

-8 x 10-4

3.9366

14.716

7 Li

3 2

92.58

i.54 x 103

-4.5 x 10-2

i0.3964

38.864

9Be

3 2

100

78.8

5.2 x 10-2

-3.759

i4.052

i0B

3

i9.58

22.i

7.4 x 10-2

2.8740

i0.744

iiB

3 2

80.42

7.54 x 102

3.55 x 10-2

8.5794

32.072

i3C

2

1.108

i.00

6.7283

25.i45

i4N

i

99.63

5.69

i.6 x 10-2

i.933i

7.226

i5N

2

0.37

2.i9 x 10-2

-2.7ii6

10.137

i7O

5 2

0.037

6.ii x 10-2

-2.6 x 10-2

-3.6264

i3.556

i9F

2

100

4.73 x 103

25.i8i

94.094

2i Ne

3 2

0.257

3.59 x 10-2

9 x 10-2

-2.1118

7.894

23 Na

3 2

100

5.25 x 102

0.i2

7.076i

26.452

25 Mg

5 2

10.13

i.54

0.22

- i.6375

6.i22

27 Al

5 2

100

1.17 x 103

0.i49

6.9704

26.057

29Si

2

4.70

2.09

-5.3i46

i9.867

3iP

2

100

3.77 x 102

i0.8289

40.48i

33 S

3 2

0.76

9.73 x 10-2

-5.5 x 10-2

2.0534

7.676

35 Cl

3 2

75.53

20.2

-8.0 x 10-2

2.62i0

9.798

37 Cl

3 2

24.47

3.8

-6.32 x 10-2

2.1817

8.i56

39 K

3 2

93.i

2.69

5.5 x 10-2

i.2483

4.666

40K

4

0.0i2

3.52 x 10-3

(-)c

- i.552

5.80i

41K

3 2

6.88

3.28 x 10-2

6.7 x 10-2

0.685i

2.56i

43 Ca

7 2

0.i45

5.27 x 10-2

-0.05

-i.800i

6.729

45Sc

7 2

100

i.7i x 103

-0.22

6.4982

24.292

47 Ti

5 2

7.28

0.864

0.29

±i.5084

5.639

49 Ti

7 2

5.5i

1.18

0.24

±i.5080

5.638

50V

6

0.24

0.755

±0.2i

2.649i

9.970

5iV

7 2

99.76

2.i5 x 103

-5.2 x 10-2

7.0362

26.303

53 Cr

3 2

9.55

0.49

±3 x 10-2

- 1.5120

5.652

55 Mn

5 2

100

9.94 x 102

0.55

6.6i95

24.745

57 Fe

2

2.i9

4.2 x 10-3

0.866i

3.238

59 Co

7 2

100

i.57 x 103

0.40

6.3472

23.727

6i Ni

3 2

1.19

0.24

0.i6

-2.3904

8.936

63 Cu

3 2

69.09

3.65 x 102

-0.2ii

7.0965

26.528

65 Cu

3 2

30.9i

2.0i x 102

-0.i95

7.60i8

28.4i7

67 Zn

5 2

4.ii

0.665

0.i5

i.6737

6.257

69 Ga

3 2

60.4

2.37 x 102

0.i78

6.420

24.00i

7iGa

3 2

39.6

3.i9 x 102

0.ii2

8.i58

30.497

73 Ge

9 2

7.76

0.6i7

-0.2

-9.33i

3.488

75 As

3 2

100

i.43 x 102

0.3

4.5804

17.123

77Se

2

7.58

2.98

5.1018

i9.072

79 Br

3 2

50.54

2.26 x 102

0.33

6.7023

(continues)

 Atomic weight/ element (10"28 M2) Gyromagnetic ratio (107 rad T"1 sec"1) Resonance frequency (1H TMS 100 MHz) 81Br 3 2 49.46 2.77 x 102 0.28 7.2246 27.007 83Kr 9 2 11.55 1.23 0.15 — 1.029 3.848 85Rb 5 2 72.15 43 0.25 2.5828 9.655 87Rb 3 2 27.85 2.77 x 102 0.12 8.7532 32.721 87Sr 9 2 7.02 1.07 0.36 — 1.1593 4.334 89 y 2 100 0.668 — — 1.3108 4.900 91Zr 5 2 11.23 6.04 -0.21 —2.4868 9.296 93Nb 9 2 100 2.740 x 103 -0.2 6.5476 24.476 95Mo 5 2 15.72 2.88 ±0.12 1.7433 6.517 97Mo 5 2 9.46 1.84 ±1.1 — 1.7799 6.654 99Tc 9 2 100 1.562 x 103d —0.19d 6.0211 22.508 99Ru 5 2 12.72 0.83 7.6 x 10—2 — 1.2343 4.614 101 Ru 5 2 17.07 1.56 0.44 —1.3834 5.171 103 Rh 2 100 0.177 — —0.8520 3.185 105Pd 3 2 22.23 1.41 0.8 —0.756 4.576 107Ag 2 51.82 0.195 — — 1.0828 4.048 109 Ag 2 48.18 0.276 — — 1.2448 4.654 111Cd 2 12.75 6.73 — —5.6714 21.201 113Cd 2 12.26 7.6 — —5.9328 22.178 113In 9 2 4.28 83.8 1.14 5.8493 21.866 115In 9 2 95.72 1.89 x 103 0.83 5.8618 21.913 115Sn 2 0.35 0.695 — —8.792 32.86 117Sn 2 7.61 19.54 — —9.5319 35.632 119Sn 2 8.58 25.2 — —9.9756 37.291 121 Sb 5 2 57.25 5.20 x 102 —0.53 6.4016 23.931 123Sb 7 2 42.75 1.11 x 102 —0.68 3.4668 12.959 123Te 2 0.89 0.89 — —7.0006 26.170 125Te 2 7.0 12.5 — —8.4398 31.550 127I 5 2 100 5.3 x 102 —0.79 5.3525 20.009 129Xe 2 26.44 31.8 — —7.4003 27.658 131 Xe 3 2 21.18 3.31 —0.12 2.1939 8.200 133 Cs 7 2 100 2.69 x 102 —3 x 10—3 3.5087 13.116 135 Ba 3 2 6.59 1.83 0.18 2.6575 9.934 137Ba 3 2 11.32 4.41 0.28 2.9728 11.113 138 La 5 0.09 0.43 —0.47 3.5295 13.194 139 La 7 2 99.91 3.36 x 102 0.21 3.7787 14.126 141 Pr 5 2 100 1.66 x 103 —5.9 x 10—2 7.836 29.291 143 Nd 7 2 12.17 2.31 —0.48 1.455 5.438 145 Nd 7 2 8.3 0.37 —0.25 0.895 3.346 147Sm 7 2 14.97 1.26 —0.21 1.104 4.128 149Sm 7 2 13.83 0.59 6 x 10—2 0.880 3.289 151 Eu 5 2 47.82 4.83 x 102 1.16 6.634 24.801 153Eu 5 2 52.18 45.3 2.9 2.930 10.952 155 Gd 3 2 14.73 0.23 1.6 1.022 3.820 157 Gd 3 2 15.68 0.48 2 1.277 4.775 159 Tb 3 2 100 3.31 x 102 1.3 6.067 22.679 {continues)
 Atomic weight/ element (10"28 M2) Gyromagnetic ratio (107 rad T"1 sec"1) Resonance frequency (1H TMS 100 MHz) 161 Dy 5 2 18.88 G.45 1.4 G.881 3.295 163 Dy 5 2 24.97 1.59 1.6 1.226 4.584 165 Ho 7 2 1GG 1.G3 x 1G3 2.82 5.487 20.513 167 Er 7 2 22.94 G.66 2.83 G.773 2.890 169 Tm 2 1GG 3.21 — -2.21 8.272 171 Yb 2 14.27 4.G5 — 4.72 17.612 173Yb 174Lu 5 2 1 16.G8 1.14 {—)b 1.31 4.852 175Lu 7 2 97.41 1.56 x 1G2 5.68 3.G5 11.407 176Lu 7 2.59 5.14 8.1 2.1G 7.872 177 Hf 7 2 18.5G G.88 4.5 G.95 4.008 179Hf 9 2 13.75 G.27 5.1 -G.6G9 2.518 181 Ta 7 2 99.988 2.G4 x 1G2 3 3.2G73 11.990 183W 2 14.28 5.89 x 1G-2 — 1.1145 4.166 185Re 5 2 37.G7 2.8 x 1G2 2.8 6.G255 22.525 187 Re 5 2 62.93 4.9G x 1G2 2.6 6.G862 22.752 187 Os 2 1.64 1.14 x 1G-3 — G.61G5 2.282 189Os 3 2 16.1 2.13 G.8 2.G773 7.765 191 Ir 3 2 37.3 2.3 x 1G-2 1.5 G.539 1.718 193Ir 3 2 62.7 5.G x 1G-2 1.4 G.391 1.871 195Pt 2 33.8 19.1 — 5.7412 21.462 197 Au 3 2 1GG 6.G x 1G-2 G.58 G.357 1.729 199Hg 2 16.84 5.42 — 4.7912 17.911 201 Hg 3 2 13.22 1.G8 G.5 -1.7686 6.612 203 Tl 2 29.5G 2.89 x 1G2 — 15.3G78 57.224 205 Tl 2 7G.5G 7.69 x 1G2 — 15.4584 57.787 207Pb 2 22.6 11.8 — 5.5797 209Po 235U 9 2 1 2 7 2 1GG G.72 7.77 x 1G2 4.9 x 1G-3 -G.4 4.1 4.2986 G.479 16.069 1.791 a Most values taken from Brevard, C., and Grager, P. (1981). "Handbook of High Resolution Multinuclear

NMR," Wiley (Interscience), New York, pp. 80-211.

b Some values taken from the Bruker NMR-NQR Periodic Table; Harris, R. K., and Mann, B. E. (1978). "NMR and the Periodic Table," Academic Press, London, pp. 5-7; Pople, J. A., Schneider, W. G., and Bernstein, H. J. (1959). "High-Resolution Nuclear Magnetic Resonance," McGraw-Hill, New York, pp. 480-485; Harris, R. K., private communication. c Poorly known or unknown.

dFranklin, K. J., Lock, C. J. L., Sayer, B. G., and Schrobilgen, G. J. (1982). J. Am. Chem. Soc. 104, 5303-5306.

field, and the three-dimensional coordinate system orienting the particular interactions. A nucleus that experiences all of the above four effects of the molecular framework will, in general, have a set of resonance frequencies that are a reflection of all of these contributions. The physical origins of each of these are now discussed in turn, and for simplicity, the effect on the resonance frequency due to each of the four contributions from the molecular framework is discussed as if that were the only contribution present. The observed spectrum of many nuclei is effectively due to a single one of the above contributions, so it makes sense to discuss them one at a time.

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