Interpretation of Carbon-13 NMR Spectra [2 ed.] 0471917427

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Table of contents :
1.1 History of Carbon-13 Nuclear Magnetic Resonance
1.2 Relaxation and Nuclear Overhauser Effect
1.3 Instrumental Requirements
2.1 The Chemical Shift
2.2 Spin-spin Coupling
3.1 Summary of Methods
3.2 Proton-decoupling Techniques
3.3 Polarization Transfer and Related Experiments
3.4 Selective Excitation
3.5 Two-dimensional Experiments
3.6 Carbon-Carbon Connectivity Experiments
3.7 Lanthanide Shift Reagents
3.8 Chemical Shift Comparison
3.9 Isotope Effects
3.10 Solid State 13C-NMR
4.1 Correlation and Spectral Density Function
4.2 Dipole-Dipole Relaxation
4.3 Other Relaxation Mechanisms
4.4 Anisotropic Rotational Diffusion
4.5 Experimental Techniques for the Measurement of T1 and theNuclear Overhauser Effect (NOE)
4.6 Temperature and Concentration Dependence of the Rotational Correlation Time.
4.7 Relative Contributions from Individual Relaxation Mechanisms
5.1 Introduction
5.2 Structure Elucidation of Organic Molecules
5.3 Dynamic Processes, Conformational Analysis
5.4 Two-dimensional Techniques for the Elucidation of Exchange Networks
5.5 Rotamer equilibria
5.6 Macromolecules
5.7 Solid-state 13C-NMR Applications
5:8 Mechanistic Studies
5.9 Spin-Lattice Relaxation and Nuclear Overhauser Studies
5.10 Quantitative Analysis
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Interpretation of Carbon-13 NMR Spectra [2 ed.]

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F. W. Wehrli, A. P. Marchand, and S. Wehrli

JOHN WILEY & SONS Chichester . New York . Brisbane . Toronto . Singapore

Copyright cg 1983 Wiley Heyden Ltd; (~:) 1988 John Wiley & Sons Ltd. All rights reserved. No part of this book may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher Library of Congress Cataloging-in-Publication Data:

Wehrli. F. W. Interpretation of carbon-I 3 NMR spectra. I. Carbon-Isotopes-Spectra. 2. Nuclear magnetic resonance spectroscopy. I. Marchand, Alan P. II. Wehrli, S. III. Title. 547.3'0877 87-19020 QC462. CIW43 1988


0 471 91742 7

British Library Cataloguing in Publication Data:

Wehrli, F. W. Interpretation of carbon-13 NMR spectra -2nd ed. I. Chemistry, Physical organic 2. Nuclear magnetic resonance spectroscopy 3. Carbon -Isotopes I. Title II. Marchand, A. P. III. Wehrli, S. 547.3'0877 QD476 ISBN

0 471 91742 7

Phototypesetting at Thomson Press (India) Limited, New Delhi Printed by Anchor Brendon Ltd, Tiptree, Essex






1.1 History of Carbon-13 Nuclear Magnetic Resonance. 1.2 Relaxation and Nuclear Overhauser Effect. . 1.3 Instrumental Requirements .

1 5 7

Behaviour of the magnetization subjected to rf excitation-Time and frequency domain-Elements of a pulsed n.m.r. spectrometerChoice of instrumental parameters


2.1 The Chemical Shift .

32 32

Reference scale and solvent shifts-Theoretical aspects of carbon shieldings-Transmission of shielding effects within moleculesEmpirical relationships: substituent effects-Empirical additivity rules

2.2 Spin-spin Coupling.


Theoretical aspects-Carbon-proton coupling constants-Carboncarbon coupling constants-Coupling between carbon and other nuclei



3.1 Summary of Methods . 3.2 Proton-decoupling Techniques

98 98 100


CONTENTS Single-frequency off-resonance decoupling (SFORD)-Evaluation of off-resonance data with the aid of graphical methods-Selective decoupling-NOE-enhanced proton-coupled carbon spectraAnalysis of proton-coupled carbon spectra-Selective low-power irradiation-Stereospecificity of the three-bond coupling constant 3J CH


3.3 Polarization Transfer and Related Experiments .


Selectivepopulation inversion-INEPT (insensitivenucleus enhancement by polarization transfer)-Attached proton test-DEPT

3.4 Selective Excitation. . 3.5 Two-dimensional Experiments .

145 151

Principle-Two-dimensional J-resolved spectroscopy-Chemical shift correlation .

3.6 Carbon-Carbon Connectivity Experiments.


13C_13C spin-spin coupling at natural abundance-Centre-band suppression techniques-Biosynthetic experiments

3.7 Lanthanide Shift Reagents. .

. 175

Dipolar (pseudocontact) shifts-Spectral assignment-Fermi contact shifts


3.8 Chemical Shift Comparison 3.9 Isotope Effects. . 3.10 Solid State 13C n.m.r. .

184 190 199



4.1 Correlation and Spectral Density Function. 4.2 Dipole-Dipole Relaxation. . 4.3 Other Relaxation Mechanisms .

212 215 219

Spin-rotation (SR)-Chemical shift anisotropy (CSA)-Scalar relaxation (SC)-Electron-nuclear relaxation

4.4 Anisotropic Rotational Diffusion . 226 4.5 Experimental Techniques for the Measurement of T 1 and the Nuclear Overhauser Effect (NOE) .


Inversion recovery-Progressive saturation-Saturation recoveryNOE measurements

4.6 Temperature and Concentration Dependence of the Rotational Correlation Time. 235 4.7 Relative Contributions from Individual Relaxation Mechanisms 236





5.1 Introduction 5.2 Structure Elucidation of Organic Molecules

241 241

Natural products-Stereochemistry, geometric isomerism-Cations, anions

5.3 Dynamic Processes, Conformational Analysis. Intramolecular rearrangements, fluxional Conformational equilibria, configurational inversion

307 molecules-

5.4 Two-dimensional Techniques for the Elucidation of Exchange Networks 319 5.5 Rotamer equilibria . 323 5.6 Macromolecules . 328 Synthetic polymers-Stereoregularity of polymers-Diene polymers-Ring-opening polymerization-Biopolymers-Proteins

5.7 Solid-state l3C n.m.r. Applications. 5:8 Mechanistic Studies .

365 370

Elucidation of reaction mechanisms-e-P'C labelling in biosynthesis

5.9 Spin-Lattice Relaxation and Nuclear Overhauser Studies


Applications to structure assignment-Anisotropic overall motionDipolar relaxation in macromolecules

5.10 Quantitative Analysis .


Applications-In vivo applications of 13C n.m.r.-Technical requirements-Applications to the study of cell metabolism in bacteria, animals, and humans







Preface Carbon-13 n.m.r. still is and remains one of the most powerful structural tools in organic chemistry, in particular when used in conjunction with its closest counterpart, proton n.m.r. After having been reprinted three times, Interpretation ofCarbon-13 NMR Spectra, which first appeared in 1976, was felt to be in need of a thorough revision, necessitated primarily by the instrumental and methodological advances made during the past 10 years. Although today n.m.r. spectroscopy is taught routinely at an undergraduate level, the practicing chemist is increasingly overwhelmed by the unusual richness of this spectroscopic technique. Contrary to other predictions and unlike any other spectroscopic method, n.m.f. continues to evolve rapidly. The technique received a new impetus with the advent of two-dimensional spectroscopy, just barely missed by the first edition. Along with this, instrumentation evolved to ever higher levels of sophistication. Whereas in the mid-70s, the superconducting spectrometer was an expensive research option, today virtually all routine spectrometers used in chemical and analytical practice are based on superconducting magnets. The concomitant increase in spectral dispersion, albeit less critical in carbon-13 than in proton spectroscopy, has greatly expanded the scope of applications. Along with the higher magnetic fields came a significant increase in signal-to-noise, making possible acquisition of carbon-13 spectra of a few milligrams of a typical organic compound in minutes rather than hours. Spurred by these events, the authors completely rewrote and significantly expanded the five chapters while leaving the basic concept intact. Chapter 1 briefly reviews the basic physics of carbon-13 n.m.r., expanding in particular on changes that occurred in instrumentation. Chapter 2 (Spectral Parameters) was in least need of refurbishing. Updates were primarily necessary in the area of carbon-proton coupling constants as well as coupling constants involving other nuclei which, owing 'to improved instrumentation, have been studied very broadly during the past 10 years. The known stereospecificity of many of these coupling constants, together with increased instrument sensitivity for obtaining proton-coupled spectra, have considerably expanded the scope of such applications.

x The most dramatic innovations, however, have occurred in the area of new experimental techniques. In the light of these developments, Chapter 3 was renamed 'Experimental Techniques for Spectral Assignment'. It is virtually impossible, and completely beyond the scope of this book, to do complete justice to the monumental advances in this area. The authors therefore attempted to make a judicious selection of the most relevant new techniques not available 10 years ago. These comprise new polarization transfer experiments like INEPT, DEPT, APT and some of their two-dimensional counterparts. Perhaps the most powerful among the latter category of experiments is the chemical shift correlation 2D technique, permitting a new and straightforward way for crossassigning proton and carbon spectra. Also new are carbon-carbon connectivity experiments in which spectra are obtained from naturally occurring isotopomers containing two C-13 isotopes, in principle making it possible to perform a complete assignment of an unknown molecule ab initio, i.e. without the need of additional structural information. Nevertheless, some ofthe classical approaches to structure determination such as chemical shift correlation and selective double resonance, will maintain their utility and are therefore covered in much detail. The increased spectral dispersion, achievable at high field, further makes it possible to study subtle isotope effects, as they are induced by deuterium and oxygen-18. Finally, Chapter 3 also accommodates a section on solid-state carbon-13 n.m.r., by providing a cursory overview of the basic experiments and a discussion of solid-state spectral phenomenology. Chapter 4 (Nuclear Spin Relaxation) was expanded, with a more detailed discussion of electron-nuclear relaxation in terms of the Bloembergen-PurcellPound model and a brief discussion of anisotropic rotational diffusion. The field dependence of relaxation is also covered in some more detail. Overall, however, it was felt that the area of relaxation measurements, in particular from the point of view of structure determination, has been relatively quiescent. Chapter 5 (Applications) reflects, by and large, the much increased scope of new techniques available to the practicing applied spectroscopist. Concomitantly, the algorithmic approach to structure determination has changed significantly during the past 10 years. Whereas at the time the first edition of this book was written, single frequency off-resonance decoupling was the only practical adjunct to proton noise-decoupled carbon-13 spectra, proton-coupled spectroscopy with polarization transfer, 2DFT (notably proton-carbon correlated 2D spectroscopy) are now used routinely. Some of the new techniques benefit a variety of applications in the macromolecular field, permitting assignment of polymer microstructure at high magnetic field. Quantitative analysis, a rapidly growing application of industrial applications, and the experimental requirements for accurate quantitation are treated in some detail. Finally, two new sections have been added, one on solid-state applications and a discussion of in-vivo spectroscopy. The latter allows us, for the first time, to study metabolic processes from intact cells, organs, animals and even humans and thus add a totally new dimension to carbon-13 n.m.r.

xi From the feedback received during the past 10 years, the authors concluded that it would be worthwhile to expatld on a particularly popular feature of the previous version of the book, i.e. the problem section. While some of the more esoteric and less useful problems were removed, a number of new problems were added, notably relating to structure determination of organic and biomolecules, as well as novel mechanistic applications. Finally, the authors would like to thank all those who have contributed to the book, in terms of advice and constructive suggestions: Professor Barry Shapiro, for critically reviewing the first four chapters of the book; professor James Cook from the University of Wisconsin in Milwaukee for providing data prior to publication and, last but not least, Dr T. Wirthlin from Varian AG, who coauthored the first edition of the book, for encouragement and advice.



Following hydrogen, carbon is the most abundant element in organic chemistry. Whereas hydrogen occurs at the periphery, carbon constitutes the backbone of molecules. On these premises the impetus to extend n.m.r. to carbon was an obvious one: in analogy to proton n.m.r. carbon chemical shifts and coupling constants could be assumed to provide information on functional groups, structure, and stereochemistry, and might ultimately give the chemist a tool to trace out the carbon skeleton of completely unknown molecules. Today we know that these farsighted anticipations have been fulfilled. It may therefore appear surprising that it took more than a decade following the first carbon-13 n.m.r. experiments in 1957, performed on molecules containing the isotope in natural abundance,l until carbon n.m.r. spectra could be obtained routinely. The reason for this long evolution phase is a technological one, and has to do with the intrinsically low receptivity of carbon-13, coupled with a low natural abundance of this only magnetic isotope of carbon. Swept spectrometers of the kind they were-and to some extent still are-utilized in proton n.m.r. therefore turned out to be inappropriate for natural-abundance carbon-13. These instrumental inadequacies were finally overcome thanks to a breakthrough in computer technology and digital data processing, along with instrumental developments such as broadbanded proton decoupling2 and pulsed excitation techniques. 3 Most problems that early spectroscopists were struggling with were in some way associated with spectrometer sensitivity. N.m.r. is known to be an intrinsically insensitive technique when compared, for example, with methods such as optical or mass spectroscopy. This is so for basically two reasons. The first is related to the energy difference ~E between ground and excited state, which is relatively small and so are therefore the Boltzmann population differences: N 21N 1 = exp (

~ElkT) ~




In Eqn. (1.1) N 1 and N 2 represent the spin populations on the ground and excited state, respectively, k is the Boltzmann constant, and T the absolute



temperature. The linearized approximation on the right of the exponential is justified since AE« kT. Insertion of the difference in Zeeman energy (1.2)


into Eqn. (1.1) gives (1.3) with Ii being the modified Planck constant (hI2n), y the magnetogyric ratio, -and Ho the field strength of the polarizing field. At a field of 2.3 Tesla (25.2 MHz carbon frequency) one obtains for the relative excess population (N 1 - N 2)1N 1 = 4 x 10- 6 for carbon nuclei at 25°C. Hence there is only a tiny fraction of spins available to be promoted from the ground to the excited state. The second cause for the inherent low sensitivity of the n.m.r. experiment lies in the long lifetimes of excited states (typically on the order of 10- 3 - 10 3 s). At constant excitation power this severely limits the repetition rate for successive excitations since sufficient time has to be allowed for restoration of the equilibrium populations. In contrast to optical spectroscopy, spontaneous emission is negligibly small in magnetic resonance and the only mechanism counteracting radiofrequency (d) perturbation is relaxation, more precisely, spin-lattice relaxation. The two time-dependent processes affecting the spin populations are diagrammatically illustrated in Fig. 1.1. Let us now briefly look into the nucleus-specific factors contributing to sensitivity. The three non-instrumental quantities are the magnetogyric ratio y, the spin number I, and the natural abundance a, which can be shown to ~e related to the signal strength S at a given field H 0 as follows: 4 (1.4) £

£2 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: N2

\ \ \

\ 1

I I / /




Fig. 1.1 Nuclear Zeeman levels E1 and E2 with equilibrium spin populations N 1 and N 2' Irradiation at the Larmor resonance frequency v12 promotes spins from the E1 to the E2 level whilst spin-lattice relaxation (dashed line) tends to reestablish Boltzmann, populations.



Physical properties of isotopes

Relative abundance (%) Spin I (in multiples of Ii) Magnetogyric ratio y(ins-1T- 1) Larmor frequency at 2.34 T (MHz) Relative receptivity


and 1 H

1.11 1. 2

6.741 X 107 25.16 1.78 x 10- 4

99.98 1


2.675 X 108 100.0 1

The quantity 1(1 + 1)y3 a has also been termed receptivity 5 to distinguish it from the sensitivity which involves instrumental 'factors. When we compare the receptivity of 13C with that of the most common n.m.r. isotope, 1H, we can further ignore the spin term since both have 1=!. However, because ye 3 C)/yeH) ~t and a = 1.11 x 10- 2 we realize that 13C is about 5600 times less receptive than 1H. Some of the relevant physical properties relating 13C and the proton are compiled in Table 1.1. Until major advances were made to overcome the crucial signal-to-noise (SIN) hurdle, 13C n.m.r. on molecules of practical analytical interest were not amenable to the experiment. The early work was therefore mostly concerned with lowmolecular-weight compounds, usually recorded as neat liquids in the so-called rapid-passage dispersion mode. 1 The fast sweep rates implicit to this technique .r'" resulted in fairly low-resolution spectra, as exemplified in Fig. 1.2(a).6 Another technique alleviating the sensitivity limitation is based upon heteronuclear double resonance. In the case where 13C is coupled to protons, for example, it may be possible to monitor a proton transition (13C satellite) while simultaneously sweeping through the 13C resonance region with a weakly perturbing double-resonance field. 7 ,s The concomitant alterations of the observed proton line results in what has been termed an INDOR (internuclear double resonance) spectrum. Major progress was made in the mid 1960s with the realization of proton noise decoupling2 which resulted in a sensitivity gain of one to two orders of magnitude and which henceforth permitted recording of slow-sweep high-resolution spectra. The advent of electronic storage devices falls roughly into the same period. Signal-averaging techniques 9 are based upon the stochastic properties of white noise. While coherent signals add up linearly, noise only increases with the square root of the number of passages. Signal averaging therefore led to a further substantial improvement in SIN, thereby raising the detection limit to ca 1 modejliter. 1o Figures 1.2(b) and (c) show the effect of proton noise decoupling and signal averaging on a swept spectrometer. By correcting for the different number of scans recorded for each trace one realizes that broadband proton decoupling augments SIN by a factor of ca 20. This is primarily a consequence of the collapse of the spin coupling multiplets into single lines. Paralleling such developments, instrumental research. was also aimed at







Fig. 1.2 (a)13C spectrum of pyridine recorded at 8.5 kG in the adiabatic rapid passage dispersion mode;! (b) 25.2 MHz absorption mode 13C spectrum of pyridine single passage with simultaneous noise modulated proton decoupling; (c) as (b) but with no proton irradiation; 16 scans were accumulated in a time-averaging device.

increasing basic signal strength and attenuating noise sources. The former was mainly achieved by increasing the magnetic field and sample diameters, whereas the latter can be attributed to advances in probe and receiver design. The one decisive breakthrough, however, without which 13e n.m.r. would never have attained its current analytical importance is the development of broadband excitation and Fourier transform techniques. 3 ,9 Most of the remainder of this chapter will be devoted to the description of this technique and its instrumental requirements.



Even though relaxation and its various applications to chemistry will be treated in detail in Chapter 4, it is felt that a brief introduction of the subject is necessary at this stage because of the profound impact the phenomenon has on the appearance of BC n.m.r. spectra. Phenomenologically, one differentiates between spin-lattice and spin-spin relaxation. 11 The former term implies a process involving interaction between spins and their surroundings, commonly referred to as the lattice. When transitions are induced by application of a rf field at resonance the equilibrium populations are perturbed in that spins are promoted from the lower to the higher energy levels, leading to a net increase in the spin system's energy. In order to return to thermal equilibrium the excess energy has to be dissipated to the lattice. Since the spins are isolated from the lattice the mutual interaction is weak and consequently the time constant for this process is long. Another way of visualizing spin-lattice relaxation involves the macroscopic spin magnetization M, which results from the polarization of the individual magnetic moments II = hI by the static magnetic field Ho. In the absence of a magnetic field M = 0 the spins are randomly oriented. The magnetization M can be calculated from the Boltzmann populations. In thermal equilibrium the magnetization M for a spin-t nucleus is given by12 M=

3 Nh2'}'2Ho 4kT


In Eqn. (1.5) N represents the number of spins per unit volume; the remaining quantities have previously been defined. In the absence of an external perturbation M points along the axis of Ho. The polarization of the individual magnetic moments does not take place instantaneously when the sample is placed in the field, and M evolves with a characteristic time constant equal to the spin-lattice or longitudinal relaxation , time, T 1 , as illustrated in Fig. 1.3(a). Likewise, when the spin system has been subjected to a strong saturating rffield, the return towards thermal equilibrium is governed by the same time constant T 1 • 13C spin-lattice relaxation times in isotropic liquid phase are found to be of the order of 10- 2-10 3 s. Even within the same sample they may differ by up to two orders of magnitude. Spin-spin relaxation, on the other hand, results from interactions among the spins themselves. In the absence of an oscillating rf field the spins precess about the external field axis with their phases at random; there is consequently no net magnetization perpendicular to the field (transverse magnetization). Let us suppose now that a rf field is applied at resonance with the precessing spins. This rf field, as we will see later, has the effect of tipping the magnetization away from the equilibrium position, thereby generating transverse magnetization Mxy. We could also say that the individual magnetic moments are polarized along the axis






Mxy (b)



,, I I


Fig. 1.3 (a) Recovery of the longitudinal magnetization M z towards its thermal equilibrium value MOO' (b) Decay of an initial transverse magnetization towards its equilibrium value zero.

ofthe rffield. Mxy does not persist indefinitely, however. It decays to zero because of a second relaxation process, transverse or spin-spin relaxation. Even in an ideally homogeneous magnetic field there is a spread of Larmor frequencies across the sample. As a consequence ofthis the spins dephase, and they do so with a time constant given by the transverse or spin-spin relaxation time, T 2 • The exponential decay of an initial transverse magnetization M~y as a function of time is illustrated in Fig. 1.3(b). In a perfectly homogeneous field the width of a resonance line at half height is inversely proportional to T 2 : 11v1/2 = (1t T 2 ) - 1. However, the magnetic field is never perfectly homogeneous, i.e. there are always gradients across the sample, which means that the transverse magnetization decays faster than predicted by T 2 • This is usually expressed in terms' of the effective transverse relaxation time, T!, which is always shorter than T 2 • It is obvious that the full magnetization along the magnetic field axis can only be attained after Mxy has decayed to zero. From this it follows that T2 ~ T 1 • In,13C n.m.r. it is found that in the majority of cases Tl = T2 • Another phenomenon which plays an important role in 13C n.m.r. is the nuclear Overhauser effect (NOE).13 This topic will be discussed more thoroughly in connection with the treatment of dipolar relaxation in.Chapter 4. It is known that relaxation of a 1.3e nucleus in most organic molecules originates from dipolar interaction with the magnetic moments of neighbouring protons. Since the majority of 13e experiments are carried out under simulta-



neous proton decoupling, protons are saturated, which leads to a redistribution of spin populations on the 13C energy levels. Provided the 13C-proton dipolar relaxation mechanism dominates, it can be shown that the population difference between ground and excited states of the nuclear Zeeman levels increases above that at thermal equilibrium. This is equivalent to increased magnetization and therefore enhanced carbon signal. Apart from very small molecules, protonbearing carbons are known to be predominantly relaxed by this mechanism. 11 Under these conditions the NOE, as defined by the quotient MzlM o, where M z represents the 13C magnetization with and M ° without irradiation of protons, is given by (1.6) In Eqn. (1.6) ye H) and y(13C) are the magnetogyric ratios ofthe proton and 13C, respectively. Apart from the increased SjN brought about by the collapse of spin multiplets when protons are decoupled, the signals of most carbons experience an additional threefold enhancement due to the NOE. Quaternary carbons do not necessarily give the full NOE because mechanisms other than 13C_ 1H dipolar relaxation may be significant. However, in larger molecules even quaternary carbons are found to exhibit the full theoretical NOE.14 . Under standard operating techniques 13C n.m.r. therefore does not provide quantitative information because it is always possible that not all carbons within the molecule have equal NOE. A second cause for the non-quantitative behaviour lies in the differences of the spin-lattice relaxation times, causing enhanced saturation for more slowly relaxing carbons.

1.3 INSTRUMENTAL REQUIREMENTS With the exception of low-field permanent-magnet-based proton n.m.r. spectrometers, which are almost extinct, all currently available commercial n.m.r. spectrometers are based on pulsed excitation and Fourier transform (FT) methods. The authors therefore feel that any comparison between continuouswave and pulsed excitation has become redundant. For an in-depth treatment of Fourier transform n.m.r. the interested reader is referred to the pertinent specialized texts. 15 ,16 However, a basic understanding of the experimental aspects of am.r. becomes increasingly important as many of the n.m.r. facilities in academic and industrial research are open access, i.e. recording of the spectra is often done by the originator of the sample rather than by a dedicated operator. Our experience with practical 13 C n.m.r. has shown us that erroneous data interpretation is often compounded by a lack of experimental



understanding. Although today's spectrometers are easier to operate than their predecessors the operator is still not-and will never be-relieved from making certain parameter choices. Instrumental settings are related to fundamental molecular properties (such as, for example, molecular correlation times) and, of course, also to the sample state (solvent, sample quantity, temperature, etc.). They further depend on the objective of the experiment. ZI


-- -







Vi Y


(f )




" (e)

Vi Y


(g) Fig. 1.4 Magnetization in the rotating frame, subjected to rf excitation. (a) Equilibrium; (b) effect of radio frequency field Hi' causing nutation of magnetization by an angle a; (c) magnetization at the end of a nj2 rf pulse; (d) precession of the magnetization around the z axis of the laboratory frame, following n/2 pulse, induces an a.c. voltage in the receiver coil situated along the y axis of the laboratory frame; (e) out-of-phase component of the free induction signal in the laboratory frame, following n/2 rf excitation; (f) out-of-phase component of the free induction signal after phase detection, assuming an rf pulse has been applied on resonance; (g) same as (f) for the case where the rf pulse was applied off resonance by an amount /J.(f) rad/s.



Behaviour of the magnetization subjected to rf excitation iS -


The resonance phenomenon is most conveniently described in terms of the previously introduced macroscopic spin magnetization M. We have seen that in thermal equilibrium M (0,0, M 0)' i.e. the magnetization has no transverse components. It is straightforward to show that rf absorption can only ensue following generation of transverse magnetization. For this purpose a rffield Hi has to be applied perpendicular to Ho in such a way that it rotates at a frequency near the Larmor frequency of the precessing magnetic moments. If the two are in synchronism, 1\1 experiences a continued torque. It is readily seen that the motion of M is quite complex, as it is composed of a precession around Ho and one around Hi (this latter is also denoted nutation). In order to simplify the description of this motion it is practical to convert the static co-ordinate system into a rotating one, such that the x and y axes rotate in phase with Hi' In this socalled rotating frame of reference the Hi field obviously becomes static. In the special case, where the excitation frequency is exactly equal to the resonance frequency of the precessing spins, the only motion occurring is a nutation at angular frequency roN 2n'vN = yH i' The nutation persists as long as Hi is acting on the spin system. The resulting nutation or pulse flip angle IX is given by (1.7) where '1: stands for the duration or width of the rf pulse. The behaviour ofM before and during the pulse is shown in Figs 1.4(a) and (b). The voltage induced in the receiver coil which is oriented perpendicular to Ho is maximum when all initial magnetization has become transverse following the pulse. This is the case for IX = n/2 (Fig. 1.4(c)). It is appropriate at this stage to return to the fixed reference frame in which the transverse magnetization precesses at the Larmor frequency roo around Ho, thus inducing a voltage vY ex [M0 cos ro 0 t] e - tlT~ (1.8) in the receiver coil, as shown in Figs 1.4(d) and (e). In the rotating frame, however, the signal is detected as a simple exponential (Fig. 1.4(f)). In effect, it turns out that the rotating frame representation has some physical justification as well, since in the receiver coil the signal is mixed with the transmitter frequency so that actually a difference frequency is detected which, at resonance, is zero. This means that for a transmitter operating slightly off resonance the detected signal V~ is a lowfrequency damped cosine V~ ex [M°cos Arot]e -tIT~


with Aro being the difference between transmitter and· nuclear resonance frequency (Fig. l.4(g)). Because of the spread of resonance frequencies caused by the chemical shift the transmitter frequency will always differ from the nuclear resonance frequencies.



In a real situation the detected signal therefore is a complex interference pattern resulting from the superposition of the indjvidual precession signals. It is therefore sometimes termed 'interferogram' in analogy to infra-red interferometry. The generally adopted term, however, is free induction decay (FID), implying the origin of the signal (induction) as well as its transient character. The FID contains all relevant information, i.e. frequency, linewidth, and intensity for each of the spectral lines. However, instead of relating intensity to frequency, as is common in spectroscopy, the FID can be considered the time evolution of the magnetization.

Time and frequency domain 9 ,15,17 Although the FID contains the desired spectral information the individual frequencies cannot directly be extracted save in very simple systems. Figure 1.5, for example, shows an expanded section of an experimental FID for a two-line system. The two frequencies present are clearly discernable, a high-frequency (VI + v2)j2 and a low-frequency (VI - v2)/2.* . It has beeD: known for some time that FID and frequency-domain spectrum form a pair of Fourier transforms, i.e. F(w)~$i ~



or, explicitly for the sine and cosine transforms: F(OJ) =

!oro f(t)cosOJtdt


F(OJ) =

!oro f(t) sin OJt dt


It is readily verified that the cosine transform for a decaying exponential

exp( - tiT!), as shown in Fig. 1.4(e), is equal to a Lorentzian F(w) = T~/[l

+ w2(T~)2]


of half-width IIT~ (in rad s -1) and a maximum at w = O. The FID in Fig. 1.4(g) likewise transforms into a Lorentzian with the sole difference that its position is displaced by an amount Aw. The sine transform (Eqn. (l.lib)) yields the respective dispersion spectrum. It is important to realize that only ifthe FID is sampled until it has completely decayed (theoretically until t = 00), it transforms into a Lorentzian. There are clearly practical impediments to this, but it turns out that the line distortions caused by the finite sampling period are normally not too serious. For the real case of a multicomponent spectrum, Fourier transformation is


12 (0)


I 0-2









I 1-2

Time / s


carrier I