New comprehensive biochemistry vol 11 modern physical methods in biochemistry part a

MODERN PHYSICAL METHODS
IN BIOCHEMISTRY, PART A


New Comprehensive Biochemistry

Volume 11A

General Editors

A. NEUBERGER
London

L.L.M. van DEENEN
Utrecht

ELSEVIER
AMSTERDAMeNEW YORK*OXFORD


Modern Physical Methods in
Biochemistry
Part A
Editors

A. NEUBERGER and L.L.M. VAN DEENEN
London and Utrecht

1985
ELSEVIER

AMSTERDAMeNEW YORK-OXFORD


0 1985, Elsevier Science Publishers B.V. (Biomedical Division)
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Library of Congress Cataloging in Publication Data

Main entry under title:
Modern physical methods in biochemistry.
(New comprehensive biochemistry; v. 11)
Bibliography: p.
Includes index.
1. Spectrum analysis. 2. Biological chemistry Technique. I. Neuberger, Albert. 11. Deenen, Laurens L. M. van.
111. Series.
QD415.N48 VOI. 11 574.192 s [574.19'283] 85-4402
[QP5 19.9.S6]
ISBN 0-444-80649-0

Printed in The Netherlands


V

Preface
The great and, one might say without exaggerating, the amazing progress which has
been made in the biological sciences, particularly in biochemistry, over the last 20
years has been caused to a large extent by the development of sophisticated physical
methods and their application to biological problems. Our knowledge of the structure
and especially the conformation of protein and nucleic acids has been helped greatly
by the use of mass spectrometry and a variety of optical methods, such as circular
dichroism and the extension of optical rotary dispersion to low wavelengths. The use
of electron spin resonance has been of special use in our understanding of oxidation
and reduction processes, and also has been helpful in other problems affecting the
structure of important organic molecules.
The use of nuclear magnetic resonance has been another very important development in biological sciences. It is even being used to an increasing extent in
physiological investigations, and its application to clinical medicine is likely to be of

considerable benefit. The use of X-ray crystallography goes back to the 1930s, but in
recent years the techniques have been refined so that resolution has been increased to
a significant extent. Therefore, it seems reasonable to describe the techniques used in a
manner which is intelligible to the non-expert, and to describe at least some of the
applications of these techniques to important biological problems.
The present book will be followed by a second dealing with a variety of other
physical techniques. It would be quite impossible to deal with all physical methods
which will be used over the next 5 or 10 years, but we hope to cover most of the major
techniques which will be applied in solving important biological problems.
A, Neuberger
L.L.M. Van Deenen


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vii

Contents
Preface

V

Chapter I . Nuclear magnetic resonance spectroscopy in biochemistry, by J . K . M .
Roberts and 0. Jardetzky

1

1. Introduction
2. Theory

(a) Nuclear spin
(bj Nuclear precession
(c) Nuclear magnetic resonance
(i) In an isolated atomic nucleus
(ii) In an assembly of identical nuclei
(d) The free-induction decay and relaxation
(e) The chemical shift
(f) Spin-spin coupling
(g) Spin-decoupling
(h) Relaxation mechanisms
(ij Cross-relaxation and the nuclear Overhauser effect
(j)Chemical exchange
(k) The spectrometer
3. Biochemistry in vivo
(a) Introduction
(b) Experimental considerations
(c) Observation and quantitation of metabolites
(i) Assignment of resonances
(ii) Quantitation of metabolites
(d) Intracellular pH measurements
(e) Compartmentation of metabolites
( f j Measurement of unidirectional reaction rates by saturation transfer
(g) Tracing metabolic pathways by I3C- and "N-NMR
4. Macromolecules in vitro
(a) Introduction
(b) Analysis of macromolecular spectra
(i) Purely spectroscopic techniques
(ii) Techniques dependent o n the knowledge of the crystal structure
(iii) Combinations of chemical and spectroscopic methods independent of the knowledge of
the crystal structure

(c) The information content of macromolecular spectra
(i) Chemical shift
(ii) Coupling constants
(iii) Relaxation parameters
(iv) The problem of averaging

1
2
2
2
5
5
6
10

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33

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41

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51

52


...

Vlll

(d) Solution structure of proteins and nucleic acids
(e) Dynamics of protein and nucleic acids
(i) Hydrogen exchange between solvent and biopolymers
(ii) Motion of aromatic side chains in proteins
(iii) Information from relaxation data
References

53
57

51
59
60
64

Chapter 2. Electron spin resonance, b y R.C. Sealy, J . S . Hyde and W.E. Antholine

69

I . Introduction
(a) Classification with respect to technique
(b) Classification with respect to order, motion and stability
2. Nitroxide radical spin labels and spin probes
(a) Labels and probes
(b) Physical properties of spin labels
(i) Intramolecular magnetic interactions
(ii) Relaxation times
(iii) Intramolecular motional modes
(c) Spin-label information content
(i)
Intensity
(ii)
Lineshapes and rotational motions
(iii) Spectral diffusion of saturation and rotational motions
(iv) Translational diffusion (homospecies) and line broadening
(v) Translational diffusion (heterospecies), line broadening, and saturation
(vi) pH detection
(vii) Polarity probes
(viii) Distance determinations (fixed interaction distance)
(ix) Distance determination (distribution of fixed interaction distances)

(x) Concluding remarks
3. Biological free radicals
(a) Physical and chemical properties
(b) Radicals from chemical oxidation/reduction
(c) Radicals from enzymes, their substrates, and other macromolecular radicals
(i) One-electron oxidation
(ii) Rearrangement and related reactions
(iii) One-electron reductions
(iv) Mixed reaction mechanisms, redox equilibria
(d) Radicals in drug metabolism
(i) Oxidation reactions
(ii) Reduction reactions
4. Metal ions
(a) General remarks
(b) ESR of metalloproteins and metalloenzymes
(c) Complementary probes
(i) Isolated metal centers
(ii) Coupled metal centers
(d) Extensions of the standard ESR methods
(i) S-band
(ii) Spin echo spectroscopy
(iii) ENDOR
5. Instrumentation and methodology
(a) The reference arm microwave bridge
(b) Sensitivity

69
69
71
72

73
14
75
79
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81
81
82
82
83
83
84
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92
96
97
102
106
106
107
109
109
114

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121
122
122
125
127
129
129
132


ix
(c) Resonators
(d) Field modulation
(e) Accessories
(f) ENDOR, ELDOR, time domain ESR and multifrequency ESR
(g) ESR and computers
References

Chapter 3. Mass spectroscopy, by J.C. Tabet and M . Fetizon
I . General
(a) Peripheral techniques in mass spectrometry
(b) Chemical ionization (CI)
(i) Positive CI
(i-a) Protonation reactions (and the formation of adducts)
(i-b) Adduct ion formation reactions and their decompositions
(i-c) Charge-exchange reactions
(ii) Negative chemical ionization
(c) Chemical ionization at atmospheric pressure (API)

(d) Thermal desorption
(i) Flash desorption
(ii) Desorption by ‘electron (or ion) beam’ technique
(iii) Formation and ionization of aerosols
(e) Field ionization and desorption
(i) Field ionization (FI)
(ii) Field desorption (FD)
(iii) Desorption by chemical ionization (DCI)
(f) Other types of desorption
(i) 25ZCfplasma desorption (PDMS)
(ii) Laser-induced desorption (LDMS)
(iii) Desorption by ionic bombardment (SIMS)
2. Ion metastable studies and MS/MS methodology
(a) Detections of metastable ions
(i) Methods involving the variation of one field
(i-a) Variation of accelerating voltage (HV scan or defocused metastable scanning)
(i-b) Variation of the electric field (IKE technique)
(i-c) MIKE (or DADI) technique
(ii) Linked scan methods
(ii-a) E Z I V linked scan (simulated MIKE)
(ii-b) B/E linked scan method (daughter ml,, ions of ml)
(ii-c) B 2 / E linked scan method (precursors of ml: ions decomposing in the first FFR)
(ii-d) B/E
linked scan spectra
(b) Collisionally activated fragmentations
(c) Special case of negative ions
(i) I K E spectra
(ii) MIKE spectra and charge inversion reactions induced by collisions
(d) Use of computers for processing unimolecular and collisional-induced decomposition
spectra

(e) New generation of mass spectrometers for MS/MS techniques
(i) Magnet and electric analyzer instrument as tandems
(ii) Triple quadrupole instruments
(iii) Hybrid instruments
(f) A new methodology for the study of mixtures: MS/MS

Jw

i35
136
137
138
139
140

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149
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154
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157
157
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159
160

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190
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192
193
193
194
195
196

X

3. Applications
(a) Analysis of steroid compounds
(b) Analysis of peptide compounds
(c) Analysis of polysaccharide and antibiotic compounds
(d) Analysis of heterocycles and alkaloids
4. Conclusion
References

20 1
20 1
218
236
246
262
263

Chapter 4. Absorption, circular dichroism and optical rotatory dispersion of
polypeptides, proteins, prosthetic groups and biomembranes, by
D.W. Urry
2 75
1. Introduction
2. Fundamental aspects of absorption and optical rotation
(a) Absorption of ultraviolet and visible light
(i) Electric transition dipole moment and experimental determination of dipole strength
(ii) Magnetic transition dipole moment
(iii) Effects of polymeric arrays of interacting chromophores
(iii-a) The shifting and splitting of absorption bands and excitation resonance interactions
(iii-b)Hypochromism and hyperchromism and dispersion force interactions

(iii-c) The heme chromophore and heme-heme association
(b) Refractive index (ordinary dispersion)
(c) Optical rotation
(i) Plane polarization and the physical optics of rotatory polarization
(ii) Circular dichroism
(ii-a) Ellipticity and experimental determination of rotational strength
(iii) Optical rotatory dispersion
(iii-a) Molar rotation
(iii-b)Rotational strengths from O R D data
(iv) Analysis of optical rotation data in terms of rotational strengths
(iv-a) Strong absorption bands: Large electric transition dipole moments
(iv-b) Weak absorption bands with large magnetic transition dipole moments
(iv-c) The inherently dissymmetric chromophore
3. Circular dichroism and absorption spectra of polypeptide conformations and prosthetic groups
(a) Polypeptide conformations
(i) The a-helix
(ii) The /&pleated sheet conformations
(iii) The collagen triple-stranded helix
(iv) !-turns and /]-spirals
(iv-a) The type I1 /)-turn
(iv-b)The 8-spiral of the polypentapeptide of elastin
(v) /j-helices
(vi) Estimations of conformational fractions in a protein
(b) Prosthetic groups
(i) Heme moieties
(i-a) Aggregation of heme peptides (heme-heme interactions)
(i-b) Applications to multiheme proteins
(ii) Dinucleotides
4. Circular dichroism, absorption and optical rotatory dispersion of biomembranes
(a) Poly-L-glutamic acid as a model particulate system

(b) Obtaining an equivalent solution absorbance from a suspension absorbance
(c) Circular dichroism of suspensions

215
276
216
211
219
280
28 1
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285
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29 1
29 1
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309
31 1

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312
314
318
319
319
320
322
323
325
326
328
331


xi
(i) Differential absorption flattening and differential absorption obscuring
(ii) Differential light scattering
(iii) Calculation of [O]susp for poly-L-glutamic acid
(d) Application to the purple membrane of Halobacterium halohiurn: The pseudoreference state
approach
(i) The pseudoreference state approach
5. Acknowledgements
References

333
335
337
339
339

343
343

Chapter 5. Protein crystallography, by L. Johnson

34 7

I . lntroduction
2. Protein crystallographic methods
(a) Basic X-ray diffraction equations
(b) Crystallisation
(i) Supersaturation: Factors affecting the solubility of proteins
(ii) Nucleation and seeding
(iii) Crystal growth and cessation of growth
(iv) Practical techniques for crystallisation
(v) Crystallisation of membrane proteins
(c) Data collection
(d) Preparation of heavy atom derivatives
(e) Calculation of phases
(i) Use of heavy atom isomorphous derivatives
(ii) Use of anomalous scattering
(iii) Molecular replacement
(iv) Treatment of errors
(f) Interpretation of electron density maps
(9) Refinement
(i) Restrained least-squares
(ii) Constrained-restrained refinement
(iii) Fast-Fourier least-squares
(iv) Simultaneous energy and least-squares refinement
(h) Difference Fourier syntheses

(if Use in refinement
(ii) Use in ligand binding studies
(i) The solvent structure
3. Recent developments
(a) The relationship between the crystal structure and the solution structure
(i) Evidence that the gross structure of the protein is not altered by crystallisation
(ii) Cases where differences have been observed
(iii) Activity in the crystal
(iv) NMR evidence
(v) Summary
(b) Dynamics and flexibility
(c) Low temperature studies
(d) Synchrotron radiation
(e) Neutron diffraction
(f) Maximum entropy and direct methods in protein crystallography
4. Acknowledgements
References

347
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356
357
358
359
359
360
363
364

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371
373
374
376
376
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377
377
379
380
382
382
383
385
386
387
389
390
395
40 1
404
406
408
40R

Subject Index

417


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NeubergerlVun Deenen feds.) Modern Physical Methods in Biochemistry, Purt A
0 Elsevier Science Publishers B.V., 1985

CHAPTER 1

Nuclear magnetic resonance spectroscopy in
biochemistry
JUSTIN K.M. ROBERTS and OLEG JARDETZKY
Stanford Magnetic Resonance Laboratory, Stanford University, Stanford, C A 94305.
U.S.A.

1. Introduction
The absorption and re-emission of radiofrequency radiation by atomic nuclei of
substances placed in a strong magnetic field is referred to as nuclear magnetic
resonance (NMR). This phenomenon was first detected in bulk matter independently
by the groups of Bloch and Purcell in 1946. The discovery by Knight in 1949 that the
resonance frequency of a given nucleus is dependent on the chemical group in which it
is located - a phenomenon known as chemical shift - led the way for NMR
spectroscopy to become a powerful technique for molecular structure elucidation.
Other parameters sensitive to chemical environment and molecular motions measured from NMR spectral lines (such as line splitting due to coupling of magnetic nuclei,
the line width, and the related relaxation parameters, TI, T,, and the Nuclear
Overhauser Enhancement) have also become useful probes of molecular structure and
dynamics. Furthermore, kinetics of chemical reactions and exchange can be studied

by a variety of NMR techniques. Because of these attributes, this form of spectroscopy
occupies an important place among methods to study molecules.
The field of biological application of NMR consists of such a large body of work
that it is not feasible to summarize the working knowledge of the subject in a single
introductory chapter. This chapter, intended for the beginner, accordingly aims to
provide no more than an orienting overview of the main directions in which the field
has developed, the kinds of biochemical or biological questions which can be studied
by NMR, and the major specific NMR techniques useful for this purpose. This
discussion is preceded by a brief exposition of the elementary concepts of NMR and
supplemented by references to the literature that treats each topic in greater depth.
Applications of NMR of interest in biochemistry can be grouped into three major
categories: (1) determination of the structure of biologically active compounds especially new natural products; (2) studies of biochemical reactions, or processes,
especially in vivo; and (3) studies of macromolecular structure and dynamics. In the


2

first two categories of applications, NMR is used largely as an analytical tool to
identify compounds, assay their concentrations and measure reaction rates. An
elementary understanding of the relationship between line intensity and concentration
and empirical information on chemical shifts characteristic of different molecular
species suffices for most studies of this type. In the third category, NMR is used as a
structural tool, and a more elaborate theoretical analysis of the experimentally
measured NMR parameters is required to obtain the desired information on the
details of molecular events.

2. Theory
( a ) Nuclear spin

Observation of nuclear magnetic resonance relies on two properties of nuclei: charge

and spin. The movement of charge in a spinning nucleus produces a magnetic field
whose vector is parallel to the spin axis. In other words, the nucleus possesses a
magnetic moment, p. The fundamental property of spin is described by the nuclear
spin quantum number, I (in units of h/2, where h is Planck's constant), its value being
determined by the atomic mass number and the atomic number according to Table 1.
Thus, nuclear magnetic resonance cannot be observed in such important nuclei as
"C, l6O and 32S. The vast majority of NMR studies in biochemistry have utilized
nuclei of spin number 1/2: 'H, I3C, lSN, I9F and 31P. Hence, we will consider such
nuclei almost exclusively. Nuclei with 12 1 possess an electric quadrupole moment
(from non-spherical nuclear charge distribution) leading, in general, to broad lines
compared to nuclei with I = 1/2, due to rapid relaxation. Where the quadrupole
moment is small, for example with 'H and "B, broadening is not-excessive, and, for
certain purposes, the nuclei can be treated as if I = 1/2.
( b ) Nuclear precession

+

In a stationary external magnetic field, H,, a nucleus of spin I has 21 1 quantitized
energy levels. This means that there is only one possible energy transition for a
nucleus I = 1/2, a vastly simpler situation compared to energy transition of electrons in

TABLE 1
The relationship between atomic number, atomic mass and nuclear spin number
Mass number

Atomic number

Spin number, 1

Odd

Even
Even

odd or even
even
odd

half integral: 1/2, 3/2, 5/2
0
integral: 1, 2, 3


3

\

............

.-

fP

4

Ground
state ,a

..- Excited

..............

state,

p

E

Figure 1. Quantization of the magnetic moment, p, and the energy of interaction, E, in a magnetic field, H ,
for a nucleus of spin I = 1/2.

molecules. In the classical mechanical description of NMR, these two energy levels are
considered as the alignment of p with or against H , (Fig. 1).
The nucleus in Figure 1 will experience a torque, T, due to interaction of p and Ho,
expressed in vector notation as:
-+

+

T=,iixHo

Since the nucleus is spinning, the nucleus also possesses angular momentum, L, whose
vector is co-linear with and linearly proportional to p (the spinning motion being
common to both nuclear charge and mass), i.e.
--t

ji=yL

where y is an empirically derived constant for each nucleus, the magnetogyric ratio.
Newton’s law of conservation of angular momentum requires that:
dL

-=?.
dt

(3)

where c = time. So, from equations 1 and 2:

or

These equations indicate that at any instant, changes in p are perpendicular to both 71


4
and Go, i.e., they describe the precession* of
velocity, oo,defined by:
dz
dt

t

and ji about I?, with an angular

+

- =Lo,
or
dji
dt

-=jiw,

hence,
o,=yH,

(units of rad-sec-')

the Larmor equation. Larmor precession of a nuC.Gus at a frequency oo,
where:
Yfio
o,= __
2n

(7)

is shown in Figure 2.

Z
\

Figure 2. Nuclear precession about the magnetic field axis. The nucleus is in the ground state.

*Precession is defined as the rotation of an axis of rotation about another axis.


5

( c ) Nuclear magnetic resonance
( i ) In an isolated atomic nucleus
To each of the discrete orientations assumed by the nuclear magnetic moment vector
in the external magnetic field corresponds an energy of interaction E (Fig. 1):

+

+

+

E = - ji * H= - jiHo cos 0 = -pzH

(8)

0

where p, is the projection of the true nuclear magnetic moment on the z axis, the
direction of the applied magnetic field, H,. (In fact, p is not measurable since the
magnetic properties of particles can only be detected by their interaction with a
magnetic field, hence magnetic moments given in tables are the maximum observable
values, pz.) The energy AE associated with a transition between energy levels E, and
E , (Fig. 1) is defined by:

( H , = HO).
If the transition is to result from the absorption of electromagnetic radiation, the
frequency, v, of this radiation must be such that the transition energy for one nucleus
can be expressed as the energy of one absorbed quantum, i.e.

Hence, equation 9 may be rearranged as:

We now want to show that the frequency of radiation necessary for a transition
between nuclear energy levels is equal to the Larmor frequency, wo (defined in
equation 7).
The reorientation of a nuclear dipole with respect to the external field fiz is

accomplished by the magnetic field component H , of electromagnetic radiation
applied to the sample, oriented in the x-y plane (Fig. 2). This field will exert a torque
on the dipole according to equation 1 (H, substituting for Ho). In an NMR
experiment, H,is much smaller than H, (by a factor of > lo3),so if H is stationary,
there will be no net torque forcing ji into the x-y plane, because the direction of
torque is reversed every 180", as p precesses about the external field H (in a nonquantized system, such as a gyroscope, a force equivalent to H , would lead to
nutation: precession, together with an up and down oscillation). HI can only
continually force toward the x-y plane if H I rotates about H , (Fig. 2) with the same
angular frequency and the same sense as the precessing dipole, wo. This criterion is
met by circularly polarized radiofrequency radiation of frequency w0/2n (although

,


6

linearly polarized radiation can interact with the nuclear dipole, as it can be
considered to be a superimposition of two circular polarized fields, of equal amplitude,
wavelength and phase but opposite handedness - only one of these components
interacting with the dipole). Thus, we may conclude that transition of a nucleus from
the ground to the excited state (Fig. 1) occurs when the frequency of radiation, v,
equals the Larmor frequency w,, for the nucleus in a given applied magnetic field H,.
So, we can extend equation 11 as:

Including a representation of precession, one may illustrate the resonance condition
for a nucleus of spin 1/2, as in Figure 3.
( i i ) In an assembly of identical nuclei
In practice, nuclear magnetic resonance is observed in large populations of identical
nuclei ( 10l6- 10' per sample).The distribution ofidentical nuclei of spin 1/2 between the
two possible energy rates shown in Figure 1 is defined, under conditions of thermal

equilibrium, by the Boltzmann equation:

where N , and N , are the number of nuclei with their magnetic moments aligned
parallel (ground state) and anti-parallel (excited state) to the external magnetic field,
respectively. It should be noted that since AE < kT, only a very small excess of nuclei

Figure 3. The resonance phenomenon.


7

will be in the lowest energy state at thermal equilibrium, the excess being of the order
of 1 in 7 x lo5 for protons in an external field of 100 kG.This excess of nuclei in the
ground state gives rise to a net nuclear magnetization vector
in the direction of the
external magnetic field ( z axis). The absorption of radiofrequency radiation and the net
excitation of a certain fraction of the population of spins results in a decrease in the z
component of fi. According to Einstein's law of transition probabilities under the
influence of a radiation field, the probabilities of excitation and emission are equal.
Therefore, absorption can occur only to the extent to which there is an excess of nuclei
in the lower energy state. Hence, the small excess given by the Boltzmann distribution
accounts for the low sensitivity of the NMR method compared to spectroscopic
methods using higher frequencies (infrared, visible) where AE is much larger; in a
population of 1 O I 6 nuclei, only 10'O are actually 'seen' by NMR. The properties of an
assembly of identical nuclei just described may be represented as in Figure 4.
The explanation of the effect that absorption of RF radiation has on this system is
greatly simplified if one considers the assembly depicted in Figure 4 using a rotating
coordinate system. If x and y axes of Figure 4 are rotated about the z axis with an
angular velocity R, when R equals coo, the angular velocity of the nuclear magnetic
moments in the assembly, precession of nuclear moments about z will apparently

Figure 4. Precession of an ensemble of identical nuclei ( I = 1/2) at thermal equilibrium. The net macroscopic
magnetization, M , is oriented along the z axis (the direction of H ) , components of magnetization along x
and y being zero (the dipoles are randomly oriented in the x, y plane).


8
cease. The external magnetic field, H o , has therefore been effectively reduced to zero;
or, in other words, the operation of rotating the x, y plane introduces a ‘fictitious’
magnetic field that cancels H , which, by analogy to equation 6, is equal to R/g. We
save space by omitting a rigorous derivation of this conclusion because it is intuitively
valid (see Refs. 1 and 2). Thus, the motion of p in the rotating frame obeys
equations 4-6 (for the laboratory system) provided H , is replaced by the effective
magnetic field H e , where:

R
H = H --

=

(14)

Y

Absorption of radio waves by this assembly, as discussed in the previous section and
illustrated in Figure 2, occurs when the magnetic field component of the radiation, H I ,
rotates in the x, y plane at the Larmor frequency w0/2x. In the rotating frame just
described (Q = 0,) H , will appear to be stationary; it is convenient here to arbitrarily
assign H , along the rotating x axis, designated x’. Because, in this rotating frame, H , is
effectively reduced to zero, individual magnetic moments p, and the net macroscopic

magnetization M , can only interact with H , (i.e., H e = H , ) . Substituting M for p, and
H , for H,, equation 4a becomes:
dM
=yMxX,
dt

~

indicating that at resonance, the net macroscopic magnetic moment precesses about
Hl.
The vast majority of NMR experiments (viz., all Fourier transform NMR
techniques) are performed using short pulses of radiation. It is clear that by varying
the duration of the pulse, t,, and the field intensity H , contained in the pulse of
radiation, one can rotate M in the zy’ plane by any desired angle to the z axis
according to:

+

Typical values of t , range from 1 to 50 pseconds. Figure 5 illustrates the degree of
precession for two pulses of different length ( H , constant).
Many NMR experiments are described using this model. For example, the Hahn
spin-echo experiment involves measurement of the signal (or ‘echo’)following a 90”, z,
180”, z sequence, 7 being the interval between two pulses. The behavior of the spin
system in the spin echo experiment is shown in Figure 6.
One might now ask: how can precession of individual nuclear moments in the upper
and lower quantum energy levels shown in Figure 4 permit continuous precession of
the net macroscopic magnetization in the zy’ plane? It is possible to obtain such


9

(0)

(b)

Figure 5. Precession of M about H , in the rotating frame following: a, 90" pulse; b, 180" pulse.

continuous precession by a combination of the excess of nuclei in the ground or excited
state (Fig. 3), and the introduction of phase coherence in the precession of nuclear
moments about the external magnetic field. This is illustrated in Figure 7 for different
pulse angles.
Thus, the quantum mechanical and classical mechanical treatments of nuclear
magnetic resonance closely correspond, as has been demonstrated mathematically

PI.

Figure 6. The Hahn spin echo experiment in the rotating frame. (a) Tipping of M into the x'y' plane by 90"
pulse. (b) Decrease in My.as spins dephase. (c) Application of a second (180") pulse. (d) Increase in M y . as
spins 'refocus'. (e) Complete refocusing. (f) Decay in M y ,as spins dephase. From 121.


10

Figure 7. Positioning of individual nuclear magnetic moments to give apparent continuous precession of
the net magnetic moment about x’.

( d ) The free-induction decay and relaxalion

In Fourier transform (FT)-NMR experiments, the signal from excited nuclei is
observed following the pulse via voltage changes, induced by the net macroscopic

magnetization in the x’y’ plane (‘nuclear induction’), in a coil around the sample
tuned to the resonance frequency. This signal decreases in intensity to zero with time
as the nuclei return, or relax, to their original state of thermal equilibrium. Hence, the
signal is termed the free-induction decay (FID). Fourier transform of the FID, or a
summation of FIDs, yields a conventional absorption-type spectrum (Fig. 8). The
intensity of the signal from a population of identical nuclei (‘peak area’) is linearly
proportional to the population size, i.e., concentration (not chemical activity). In other
words, Beer’s law is valid over all concentrations above the detection limit of the
spectrometer. Moreover, the extinction coefficient of a nuclear species is independent
of its chemical environment, in contrast to the absorption of visible and ultraviolet
light - hence, relative peak areas in a spectrum can be directly converted to relative
concentrations (provided saturation is avoided, see Section 3(c)).
It is useful to identify two components of nuclear relaxation. One is termed spinspin, or transverse, relaxation, by which energy is transferred from one nucleus to
another (mutual spin flips or spin-spin exchange). This process leads to a decrease in
the phase coherence induced by the pulse, and so to a decrease in the x‘y’ component
of the sample magnetization (i.e., the signal). Spin-spin exchange cannot affect the
magnitude of the z component of the sample magnetization, for no change in the
distribution of spins between the upper and lower energy levels occurs via this
mechanism (i.e., no loss of energy from the sample). In homogeneous liquids, but not
solids or in complex systems where there are strong interactions between different
types of nuclei, this relaxation process can be described by a simple exponential decay,
characterized by a time constant, T,. Since, in the NMR experiment, the signal
measured is the net magnetization in the x’y’ plane, M y , , T2 characterizes the decay of


11

>
L
Figure 8. (A) Free induction decay. (B) Its Fourier transform, a Lorentzian line (from [61]).

the FID from a population of identical nuclei in a pulse experiment. Loss of phase
coherence in the x’y’ plane also arises because of inhomogeneity of the stationary
applied magnetic field. Such inhomogeneity results in nuclei in different portions of
the sample precessing at different frequencies, since they experience different field
strengths, so that the phase of one nucleus relative to others necessarily changes.
Hence, if inhomogeneity effects are significant, the time constant for the decay of the
FID from an assembly of identical nuclei is T2*,where T2*< T2.It can readily be seen
that as T,, increases, the line-width of a resonance at half-height, vf, gets narrower, in
fact:
v3=

1
nT,,

~

This direct effect of T,, on line-widths is also evident on considering the Heisenberg
uncertainty principle; when applied to the simultaneous measurement of energy and
time we may write:

where A indicates the uncertainty in the measurement of parameters E, v and t .
Concerning spectroscopic lines, this relation states that the uncertainty in
measurement of the frequency corresponding to a transition between two energy
levels is greater than or equal to the uncertainty in the frequency of transitions


12

between the two energy levels, characterised by 1/T2*.Hence, we can define v ~ , ~ ,

according to equation 17.
Line-widths can also be influenced by chemical exchange processes (see
Section 2(j)).
The second relaxation process is termed spin-lattice, thermal or longitudinal
relaxation, in which energy contained in the nuclear spin system is lost to surrounding
molecules (or 'lattice') in the form of heat (i.e., rotational and translational motion).
Such energy loss leads to a decrease in the number of nuclei in the excited state, and a
corresponding increase in the z component of the net magnetization, M,. Spin-lattice
relaxation, like spin-spin relaxation, is also an exponential phenomenon in homogeneous liquids, characterized by a time constant, Tl. Unlike T2,TI is not influenced by
magnetic field inhomogeneity. One can note that Tl 2 T', for M , cannot be at its
equilibrium value before M y , equals zero.
Figure 9 illustrates these relaxation processes in the rotating frame.
Z

Y'

Figure 9. Excitation and relaxation in a population ofspins. (a) Before pulse. (b)Induction ofphase coherence
along y' by H , , and consequent tipping of macroscopic magnetization, M . (c) Dephasing of nuclear
magnetic moments by spin-spin relaxation, i.e., My.= 0. (d) Re-establishment of the Boltzmann distribution
( M , is at its equilibrium value)(a = d).