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Novel pulse methods for multidimensional NMR imaging and spectroscopy

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Novel pulse methods for multidimensional NMR imaging and spectroscopy
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Cockman, Michael D., 1961-
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xi, 189 leaves : ill. ; 28 cm.

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Cartesianism ( jstor )
Exchange rates ( jstor )
Imaging ( jstor )
Magnetic fields ( jstor )
Magnetic spectroscopy ( jstor )
Mathematical sequences ( jstor )
Signals ( jstor )
Solvents ( jstor )
Spectral methods ( jstor )
Spectral resolution ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Magnetic Resonance Spectroscopy ( mesh )
Magnetic resonance imaging ( lcsh )
Nuclear magnetic resonance ( lcsh )
Nuclear magnetic resonance spectroscopy ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
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Vita.
Statement of Responsibility:
by Michael D. Cockman.

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University of Florida
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NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY












By

MICHAEL D. COCKMAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1988




NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988


To my wife and best friend, Lisa
*U OF F LIBRARIES


ACKNOWLEDGMENTS
A number of people have contributed to my scientific growth during the past five
years. I would like to thank my chairman, Dr. Wallace S. Brey, Jr., for allowing me to
obtain my degree through the Chemistry Department while working with the Department
of Radiology at Shands Hospital. 1 also thank him for encouraging me to speak about
topics in NMR at informal and formal meetings. Those experiences were invaluable. I
also thank my co-advisor, Dr. Tom Mareci, who guided me virtually step-by-step through
my years as a member of his research group, letting me stand on my own when he thought
I would learn more that way. For providing generous financial support during my time at
the University of Florida, I thank Dr. Kate Scott. Finally, I thank these three and Drs. \V
Weltner and D. E. Richardson for the time which they have devoted to this dissertation.
Other members of the research group have helped me over the years. 1 especially
value one-on-one discussions with Dr. Richard Briggs and Dr. Ralph Brooker. Richard
often helped me define my personal goals, and Ralph, who rarely accepted a premise until
proven to his satisfaction, clarified much of the science of NMR for me. 1 also thank Pro
fessor E. Raymond Andrew and Dr. Jeff Fitzsimmons for their lectures and advice. Visit
ing professors have shown me applications of NMR other than those on which I focused
my research effort. These included Dr. Reszo Gaspar, Dr. Eugene Sczescniak, and Dr.
Attilio Rigamonti, who helped to inspire Chapter 2. The postdocs who have passed
through the group have combined science with fun and taught me a little about both dur
ing their time at the University of Florida. For this I thank Gareth Barker, Suite
D0nstrup, and Dikoma Shungu. 1 also acknowledge the students, Bill Brey, Randy Duens-
ing, Willie Kuan, Lori Lewis, Jintong Mao, Laura Pavesi, Dan Plant, and Bill Sattin. all of
whom have affected my way of thinking about things scientific and otherwise.
in


Without technical support, a grad students life becomes quite a bit more difficult.
For their contributions at the University of Florida, I thank Barbara Beck, Don Sanford,
and Ray Thomas. I also thank Dave Dalrymple, Chris Sotak, and Subramaniam Sukumar
of Nicolet Instruments and General Electric NMR for their enormous help in understand
ing and writing the software which drove our NMR instrumentation. Much of this
dissertation could not have been done without them. I also thank Katherine Nash and
Teresa Lyles, the wonderful secretaries of the Magnetic Resonance Imaging Department at
Shands, and Mike Ingeno, Jack Dionis, Jim Kassebaum, Tim Vinson, and Dr. Ray Ber
geron of the Health Center at the University of Florida.
Finally, I thank my parents for their steadfast support during those periods of doubt
which every graduate student feels from time to time, and my wifes parents, who have
taken such good care of
whose presence has made
me while I have lived in Florida. I also thank my wife, Lisa,
my life much happier during the production of this dissertation.
iv


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES viii
ABSTRACT x
CHAPTER
1 NMR THEORY AND MULTIDIMENSIONAL NMR I
1.0 Introduction 1
1.1 Formalism for the Description of NMR Experiments 2
1.1.1 The Density Operator 2
1.1.2 Equation of Motion of the Density Operator 3
1.1.3 The Density Operator at Thermal Equilibrium 5
1.1.4 Basis Operators 8
1.1.5 Hamiltonian Operators Describing
Spin System Perturbations 11
1.1.5.1 The Effect of a Static Magnetic Field 13
1.1.5.2 The Effect of a Linear Field Gradient 17
1.1.5.3 Application of Radiofrequency (RF) Pulses 18
1.1.5.4 Application of Phase-Shifted RF Pulses 19
1.1.6 Observable Magnetization 21
1.2 The Two-Dimensional NMR Experiment 21
1.2.1 Phase and Amplitude Modulation 22
1.2.2 General Description of the 2D Experiment 24
1.3 Multidimensional NMR 27
2 SPECTRAL IMAGING AND APPLICATIONS TO THE STUDY
OF DYNAMIC POLYMER-SOLVENT SYSTEMS 30
2.0 Introduction 30
2.1 Analysis of a Spectral Imaging Method 32
2.2 Experimental 38
2.3 Results 43
2.4 The Susceptibility Model 65
2.5 Conclusion 69
3 CONVOLUTION SPECTRAL IMAGING 75
3.0 Introduction 75
3.1 The Convolution Spectral Imaging Method 75
v


3.2 Experimental 86
3.3 Practical Aspects of the Method 87
3.3.1 The Effect of Sample Geometry 92
3.3.2 The Interaction of TE and and the Spatial Resolution 104
3.3.3 Extension to Three Spatial Dimensions 105
3.3.4 Signal-to-Noise Considerations 126
3.4 Convolution Spectral Imaging at High Field 128
3.5 Conclusion 140
4 QUANTIFICATION OF EXCHANGE RATES
WITH RED NOESY SPECTROSCOPY 142
4.0 Introduction 142
4.1 The NOESY Pulse Sequence 145
4.2 The RED NOESY Pulse Sequence 155
4.3 Problems Unique to the RED NOESY Sequence 155
4.4 Experimental 159
4.5 Results 164
4.6 Discussion 164
REFERENCES 184
BIOGRAPHICAL SKETCH 189
vi


LIST OF TABLES
TABLE page
1-1 Effects of the Single-Element Operators 10
1-2 Solutions to the Equation of Motion of the
Density Operator Expressed as Cartesian Space Rotations 14
1-3 Effect of a Product Operator Hamiltonian on Terms of
a Cartesian Product Operator Basis Set 15
1-4 Transformations of Cartesian Operators by Phase-Shifted
RF Pulses 20
2-1 2D Spectral Imaging: Relative Pulse and Receiver Phases 40
2-2 Calculated Volume Susceptibilities 67
2-3 Calculated Susceptibility Shifts at 2 Tesla 68
3-1 Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases 88
3-2 Three-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases 89
4-1 Relative Pulse and Receiver Phases for RED NOESY 160
4-2 Temperature Dependence of Relaxation Rates, Exchange Rates,
and Free Energies of Activation for DMF, DMA, and DMP 169
4-3 Activation Parameters of DMF Found Using RED NOESY Data 170
4-4 Activation Parameters of DMF: Literature Values 171
4-5 Activation Parameters of DMA: Literature Values 172
4-6 Activation Parameters of DMP: Literature Values 173
4-7 Relaxation Rates of DMF Methyl Protons 182
vii


LIST OF FIGURES
FIGURE pase
2-1 The pulse sequence corresponding to the evolution period of
a spectral imaging method 33
2-2 A pulse sequence for two-dimensional spectral imaging 36
2-3 Sample orientation for studies of PMMA solvation 42
2-4 Spectral images used to observe the quality of the
static field homogeneity 45
2-5 The spectrum of a piece of PMMA partially
dissolved in chloroform at 2 T 48
2-6 Spectral images of PMMA in deuterated chloroform
after 61 minutes of solvation 49
2-7 Spectral images of PMMA in deuterated chloroform
after 81 minutes of solvation 52
2-8 Spectral images of PMMA in deuterated chloroform
at late stages of solvation 54
2-9 Spectral images of PMMA in perdeuterated acetone
after 180 minutes of solvation 58
2-10 Spectral images of PMMA in perdeuterated acetone
after 200 minutes of solvation 61
2-11 Spectral images of PMMA in perdeuterated acetone
at late stages of solvation 63
2-12 PMMA solvation 71
3-1 The convolution of spectral and spatial information 80
3-2 Pulse sequences for convolution spectral imaging 83
3-3 The effect of sample geometry 93
3-4 The interaction of TE and the spatial resolution 106
3-5 Three-dimensional convolution spectral imaging 112
3-6 Convolution spectral imaging at high field 130
viii


146
4-1 The NOESY pulse sequence
4-2 Peak intensity behavior as a function of
the mixing time and exchange rate 152
4-3 The RED NOESY pulse sequence 156
4-4 A plot of ln(k) versus 1000/T for DMF 165
4-5 A plot of ln(k) versus 1000/T (corrected) for DMF 166
4-6 A plot of ln(k/T) versus 1000/T for DMF 167
4-7 A plot of ln(k/T) versus 1000/T (corrected) for DMF 168
4-8 Four NOESY spectra of DMF obtained using the RED NOESY sequence 174
4-9 Behavior of peak intensities of NOESY spectra of
DMF at 347 K 178
IX


Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
December, 1988
Chairman: Dr. Wallace S. Brey, Jr.
Major Department: Chemistry
This dissertation examines new methods and applications of existing methods in
multidimensional nuclear magnetic resonance. Chapter 1 contains the outline of a simpli
fied theory for the description of pulse NMR experiments. The theory is a mathematical
formalism based in quantum theory and describes the effects of the applications of a static
magnetic field, radiofrequency pulses, and linear field gradients to simple spin-1/2 sys
tems. In the context of the formalism, the basic principles of multidimensional NMR
experiments are described. The material of Chapter 1 is the basis for all subsequent
chapters. Chapter 2 outlines the application of a two-dimensional method to the study of
slices of solid polymethylmethacrylate dissolving in liquid deuterated chloroform and per-
deuterated acetone. The method allowed the correlation of NMR spectra with their spatial
positions. Time-dependent frequency shifts of the spectral frequencies of the polymer
were observed in the vicinity of the polymer-solvent interfaces. The direction of the fre
quency shifts were found to depend on the orientation of the slice relative to the static
magnetic field. The magnitudes of the shifts depended on the solvent. The effects could
be explained in part by a simple magnetic susceptibility model. Chapter 3 describes new
pulse sequences for two- and three-dimensional spectral imaging. The methods produce
convolutions of spatial and spectral information, which saves experimental time. It is
x


shown that the methods are best suited for small samples with slowly relaxing nuclei,
immersed in a strong static field. The methods appear to be ideal for NMR microimaging.
Finally, Chapter 4 describes a new multidimensional pulse sequence for the quantification
of exchange rates called RED NOESY. The sequence is applied to three N,N-dimethyl
amides to determine the exchange rates of the N,N-methyl groups. For one of these
molecules, N,N-dimethylformamide, the Arrhenius energy of activation and frequency
factor and the enthalpy and entropy of activation were found by obtaining RED NOESY
data at several temperatures. The values found were 20 kcal mol-1, 28, 19 kcal mol-1, and
-5 cal mol-1 K-1, respectively.
XI


CHAPTER 1
NMR THEORY AND MULTIDIMENSIONAL NMR
1.0 Introduction
Since the original concept of two-dimensional NMR spectroscopy was proposed
(Jee71), the basic principles of 2D NMR have been used to develop a large variety of mul
tipulse experiments which in turn have found almost limitless applications.
Two-dimensional spectroscopy has been used extensively to unravel networks of coupled
nuclear spins and to estimate distances between nuclei in molecules with molecular
weights up to several thousand daltons. The field has developed such that implementation
of certain 2D NMR techniques is becoming almost routine for organic and inorganic
chemists. Similarly, two-dimensional imaging techniques are now highly automated and
routinely used clinically. Clearly, multidimensional NMR is now a domain shared by
physicists, chemists, and physicians. Yet communication between the three groups has
been impeded in part by their different concerns and professional languages. However,
the multidimensional NMR experiments performed by these groups are united by theory.
One of the functions of this chapter is to describe a theory which can be used to analyze
many NMR experiments.
The chapter first outlines the origin of a formalism for the evaluation of pulsed
NMR experiments. The formalism is then used to describe the behavior of spin systems
when immersed in a static magnetic field and exposed to radiofrequency (RF) magnetic
field pulses and linear magnetic field gradients. In the context of the formalism, the con
cept of modulation, crucial to the understanding of multidimensional NMR, is explored in
the subsequent section. Next, a general discussion of two-dimensional methods is
presented. The final section of the chapter is devoted to the extension of the ideas of 2D
NMR to three and four dimensions and the problems associated with these experiments.
1


2
The material of this chapter forms the basis for the understanding of the multidimensional
experiments described in Chapters 2, 3, and 4.
1.1 Formalism for the Description of NMR Experiments
This dissertation is primarily concerned with the development and application of
pulse sequences to very simple spin-1/2 systems. Therefore, it is usually sufficient to
analyze an experiment applied to a single spin or to two weakly coupled spins. In this
section, a simplified formalism is developed from basic quantum theory to describe the
behavior of spin systems immersed in a static magnetic field and exposed to pulse
sequences made up of time delays, radiofrequency pulses, and linear field gradients. The
classical description of an NMR experiment which arises from the Bloch equations will
not be used because this approach ignores many of the effects created by NMR pulse
sequences. The quantum theory of NMR has been described by Howarth, et al. (How86),
Levitt (Lev88), Mareci (Mar80), Slichter (Sli80), and S0rensen, et al. (Sor83). The discus
sion of the following sections draws heavily from these references.
1.1.1 The Density Operator
The state of a spin system can be described by a wavefunction rp which in turn can
be represented as a linear combination of orthonormal functions un (the eigenstates)
weighted by coefficients cn,
V>=EcnUn- [1-1]
n
For a system of N spin-1/2 nuclei, there are 2N eigenstates. In the Dirac notation, the
expectation value of an observable, Q, is described by
= P-2a]
=
m n
[ 1-2b]


3
= E E CmCn [l-2c]
m n
The products of the coefficients, c^cn, may be arranged to form a matrix which is called
the density matrix. If 0 is time-dependent, then either the eigenstates or the coefficients
can carry the time dependence. In the "Schrodinger representation, the eigenstates are
allowed to be time-independent and the time-dependence is carried by the coefficients.
Thus, the time dependence of an expectation value is carried by the density matrix. The
density operator, p, is defined by
= cncm- t1'3]
With this definition, the expectation value for an observable can be written in terms of
the density operator. By Eqs. [1-2] and [1-3] and the orthonormality of the eigenstates un,
= E E [ 1-4a]
m n
= Tr [p Q] [l-4b]
where Tr is an abbreviation for the trace of the matrix. This important equation estab
lishes the relationship of the density operator to the operator, Q, which describes an
observable phenomenon. If the trace of the equation is nonzero, the phenomenon is
observed. Mathematical forms of Q and p relevant to NMR are shown in section 1.1.6.
1.1.2 Equation of Motion of the Density Operator
An alternative way of describing the state of a spin system at a particular time is to
use the density operator itself, rather than the density matrix. Thus, perturbations of the
spin system which occur during an NMR pulse sequence appear as changes in the density
operator. Each of the perturbations which will be discussed in this chapter can be
described by a time-independent Hamiltonian operator, H, whose mathematical form
depends on the nature of the perturbation. The behavior of a spin system under the
effect of a perturbation is described by the equation of motion of the density operator.


4
This is an expression for the rate of change of the density operator with time and may be
derived from the time-dependent Schrodinger equation using the method described by
Slichter (SH80). The time-dependent Schrodinger equation is
[1-5]
where H = h/2tr and h is Plancks constant. Substituting Eq. [1-1] into Eq. [1-5] produces
We
3cn
at u"
E cHun
[1-6]
since only the cns are time-dependent. Multiplying both sides of the equation from the
left by u leads to
We f uk*un = E cnuk*Hun [1-7]
n n
Integrating and using the orthonormality of the basis functions produces
W-^p- = E c [1-8]
The equation of motion of the density operator may be derived by taking the deriva
tive of Eq. [1-3] with respect to time and substituting Eq. [1-8] into the result:
3 3cm 3cn *
at "Cn at + at Cm
= C[(-i/H)E cm]* + [(-/H)e cn]c
m n
= (i/H)[E cnCm s cnc^]
[1-9a]
[1 -9b]
[1 -9c]
m n
= (i/tf)[E E ] [1_9d]
m n
= (i/H)[ ] [l-9e]
Thus the time derivative, or the equation of motion, of the density operator is


:>
p = (i/K) [p,H] .
[1-10]
When H is time-independent, a solution of the equation of motion is
pO) = eH/#)Htp(0) e(WHt,
[1-11]
which may be verified by taking the derivative with respect to time. This equation is of
prime importance because it describes the effect of applying a Hamiltonian operator to
the density operator. In terms of the NMR experiment, the equation describes the effect
of some perturbation, represented by the Hamiltonian, to the state of the spin system,
represented by the density operator. Analyzing the effect of an NMR pulse sequence on a
spin system requires knowledge of the mathematical forms of several operators. One of
these is the density operator which describes a spin system at thermal equilibrium. This is
the normal starting point for a density operator analysis since it describes a spin system at
rest in a static magnetic field. The other required operators are the Hamiltonian operators
describing the various perturbations which can occur. The perturbations commonly
found in NMR pulse sequences include the applications of a static magnetic field,
radiofrequency pulses and linear field gradients to a spin system.
1.1.3 The Density Operator at Thermal Equilibrium
To establish the starting point for a density operator analysis, the form of the density
operator which describes the spin system at thermal equilibrium, p(0), is needed. At ther
mal equilibrium, dp/dt = 0. This implies that p(0) has a form determined by the Hamil
tonian operator describing the interaction of the static magnetic field with the unper
turbed spin system. For a single spin, this Hamiltonian is
H*1H = -'yBol, = wl, ,
[1-12]
where the Larmor relationship,
w = -TfB0 ,
[1-13]
has been used; 7 is the nuclear gyromagnetic ratio, B0 is the strength of the applied static


6
field, It is the operator for the z component of angular momentum, and w is the preces-
sional frequency of the nuclear spin. For a nucleus of spin I in eigenstate um,
H|Um> = Emlum> = MJum> = U>m | Um> [1-14]
where m is one of 21 + 1 values in the range I, I-1, -I and Em is the energy of the
mth eigenstate.
The populations of the eigenstates are given by the diagonal terms of the density
matrix, cmc^. At thermal equilibrium these are described by the Boltzmann distribution
factors, pm:
Pm = cmcm = [1-15]
For a set of n states of energies, En, a Boltzmann factor expresses the probability that the
mth state is occupied:
e- Em/kT
Pn>= -E/kT [1-16]
D c
n
where k is the Boltzmann constant and T is the temperature of the spin system. If
kT En, Em, then the exponential terms can be approximated and the Boltzmann distri
bution factor becomes
l-(Em/kT)
Pm E l-(En/kT) I1'17]
n
Because there are 21 + 1 possible values of En, the sum in the denominator equals 21+1.
Collecting equations produces
= (2I+l)'1(l-(Em/kT)) [1-18]
Finally, because H|um> = Em|um>, it follows that
p = (2I+l)_1(l-(H/kT)) = (2I+1)'1 (2I+ir1(MI/kT) [1-19]
This is the form of the density operator at thermal equilibrium. The constant term,


7
(21+1 )_1, cannot be made observable and so it may be dropped. Making the definition,
^ = -(2I+l)-1(Hw/kT), [1-20]
the reduced density operator at thermal equilibrium may be written
o=p\t. [1-21]
Because the density operator, p, and the reduced density operator, a, are related by con
stant terms, the equation of motion of the reduced density operator may be written
directly:
= (i/H) [cr,H] [1-22]
A solution of this equation for a time-independent Hamiltonian is
o(t) = e(-¡MHt o(0) eO/#)1, [1-23]
and the expectation value of an observable is
= Tr [o Q] [1-24]
Equation [1-21] is the usual starting point for a pulse sequence analysis. For a sys
tem of N spins, the reduced density operator at thermal equilibrium is
~ + ^2^2i + + [1-25]
where the different spins are labeled by the subscripts. The spins may be treated
independently, applying a perturbation Hamiltonian to each separately. When all the
spins have the same gyromagnetic ratio, all of the /9s are nearly equivalent and the spin
system is referred to as homonuclear. This dissertation deals solely with such systems and
so the /9 terms will not be written explicitly for the pulse sequence analyses which follow.
The derivation of the expression for a at thermal equilibrium was made using the
assumption that kT Em = jtu;m. To test the validity of the assumption, let T = 298 K,
(j) = 2?r(300 x 106) rad sec'1 and m = 1/2. These are typical values for a proton precessing
in a magnetic field of 7.1 tesla at room temperature. A simple calculation with appropri-


8
ate values of the Boltzmann and Planck constants shows that kT is approximately 40000
times the value of tfwm. Thus the "high temperature" assumption is valid for this system.
The nuclei studied for this dissertation were 19F at 2 tesla (w = 27r(80.5 x 106) rad sec"1) at
room temperature, at 2 tesla (w = 27t(85.5 x 106) rad sec'1) at room temperature, and
at 7.1 tesla (w = 2tt(300 x 106) rad sec'1) at temperatures ranging from 293 to 363 K. For
all of these cases, the high temperature assumption holds and Eq. [1-21] is a valid starting
point for pulse sequence analysis.
1.1,4 Basis Operators
The density operator can be written as a linear combination of time-independent
basis operators, Bg, weighted by time-dependent coefficients, b8(t):
(0 = £ bs(t)Bg. [i-26]
S
This set of basis operators can also be used to describe the Hamiltonians which describe
possible perturbations to a spin system. The most popular set of basis operators has been
proposed by S0rensen, et al. (Sor83). These are the Cartesian "product operators" pro
duced by the multiplication of the single-spin, Cartesian angular momentum operators, Ix,
ly, and IB, and the unity operator, E. The basis set for N spin-1/2 nuclei consists of 4N
product operators. For a system of two spin-1/2 nuclei the basis consists of:
The unity operator:
(1/2)E
One-spin operators:
I lx ^ly ^li ^2x ^2y ^2r
Two-spin operators:
2Ilx^2x 2Ijxl2y 2Ilxl2t ,
^^ly^2x ^ly^2y ^ly^2z


9
^lt^2x ^liJy 21x^2*
The subscripts 1 and 2 are used to distinguish the two spins. For much of the pulse
sequence analysis of this dissertation, the density operator describing the state of a spin
system will be written in the Cartesian product operator basis. This basis is particularly
suitable for describing the effects of the applications of a static magnetic field, radiofre
quency pulses, and linear field gradients.
The Cartesian operator basis is not well-suited for describing observable terms of the
density operator, and so it is necessary to convert to another basis set consisting of pro
ducts of the single-element operators, I, 1^, I+, and I'. These operators are directly
related to the energy levels of the spin system. It has been shown that a nucleus of
spin-1/2 immersed in a static magnetic field can be in one of two eigenstates, um, with
energies Em = (l/2)jiia;. In this dissertation, the state of lower energy is called P and that
of higher energy, a. The application of a single-element operator to an eigenstate can
produce a change in the spin state. The effects of these operators are shown in Table 1-1.
Table 1-1 shows that the only operators which cause a change in the spin state and lead to
observable signals are the I+ and I" operators. Thus, the single-element basis is most use
ful for expressing the observable terms of the density operator (section 1.1.6).
The Cartesian operators are directly related to the single-element operators by the
following:
yEk= j(Ik+l,f),
11-27a]
lkx=yUk + I).
[1-27b)
Iky = -yUk
[1-27c]
Ik, = ydk Ilf).
[l-27d]
where k indicates a particular spin. The Cartesian product operator basis may be rewrit
ten as a single-element product operator basis by the use of these relationships.


Table 1-1
Effects of the Single-Element Operators
Operator
1+ j- ja
Initial Spin State Final Spin States
a
0
0 P a 0
Q 0 0 p


11
In summary, two basis sets of operators may be used to describe the density operator.
The effect of an NMR pulse sequence on a spin system is described most conveniently in
the Cartesian product operator basis. To determine which terms of the density operator
are observable, however, the single-element product operator basis is more convenient.
The pulse sequence analyses of this dissertation use whichever basis set is most convenient
for emphasizing a particular aspect of the sequence.
1.1.5 Hamiltonian Operators Describing Spin System Perturbations
The form of the density operator which describes a spin system at thermal equili
brium has been shown, as have two different basis sets of operators which can be used to
describe the density operator in general. Still needed for an NMR pulse sequence analysis
are the Hamiltonian operators which describe the various perturbations which can be
applied to a spin system. These will be shown later in this section, but first an alternative
way of expressing the solution to the equation of motion of the reduced density operator
is described.
The effect of a perturbation on a spin system is described by the solution to the
equation of motion of the reduced density operator. The form previously shown in
Eq. [1-23] is not very convenient for visualizing the behavior of the spin system under the
effect of a perturbation. However, the product operator basis is made up of Cartesian
angular momentum operators and so it is possible to describe the effect of a perturbation
on the density operator in Cartesian terms by showing that the solution to the equation of
motion of the density operator is equivalent to a rotation in a three-dimensional Cartesian
space. The following analysis follows the method described by Slichter (Sli80). Recall
that for a time-independent Hamiltonian a solution to the equation of motion of the
reduced density operator is a(t) = cr(0) e(*WHt. For this analysis, let the initial
state of the density operator be o(0) = Ix and the Hamiltonian be such that h_1Ht = lL. As
will be seen in section 1.1.5.1, this describes a spin, not at thermal equilibrium, whose


12
state is changing under the effect of chemical shift precession. Define
= e'*1* Ix eWt. [1-28]
By the relationship between operators A and B,
A elB = elB A when [A,B] = 0 [1-29]
and the commutators of the angular momentum operators, the first derivative of f($) is
f(^) = e^IMyei*I,1 [1-30]
and the second derivative of f($) is
n*) = e'^1* Ix e*1*. [1-31]
Thus the relationship of Eq. [1-28] is a solution of the second order differential equation
rW-m-0. [1-32]
Another solution to this equation is
f(<) = a cos() + b sin() [1-33]
which may be verified by substitution. Thus
e1^1 Ix e1*1* = a cos() + b sin() [1-34]
The coefficient, a, may be found by finding the solution of Eq. [1-33] when 4> equals 0
and using Eq. [1-28]. Finding the solution of the first derivative of Eq. [1-33] when <(>
equals 0 and using Eq. [1-30] gives the coefficient, b. The results are that
a = Ix [ 1 -35a]
b = ly [ 1 -35b]
Finally, the following is obtained:
e"1*1* lx e1*1* = Ixcos() + Iysin(<) .
[1-36]


13
Equation [1-36] shows that the quantum mechanical solution to the equation of
motion of the density operator has an analog in a Cartesian space described by axes
defined by the angular momentum operators. The angle of rotation, , originates from the
applied Hamiltonian. Similar expressions may be found for Hamiltonians containing Ix or
Iy terms. These are outlined in Table 1-2. To find the effect of a particular Hamiltonian
on one of the Cartesian operators, the function

tonian must be determined. A Hamiltonian consisting of a one-spin operator affects the
terms of a product of Cartesian operators separately. For example, let the initial state of
the density operator be the product operator IlxI2l and the Hamiltonian be such that
H_1Ht = Then the solution to the equation of motion of the reduced density operator
is
Iixl2. e*Ilz = dixcos<*) + Ilysin(*))I2, [1-37]
Only the terms of spin 1 have been affected.
It is also possible for the Hamiltonian to contain products of Cartesian operators.
The only such Hamiltonian relevant to this dissertation is such that h_1Ht = 2llt\2z and so
alternative expressions for the reduced density operator solution e ^21lzl2Et7(t) e1^"112'22
must be derived for the various product operators which make up a(t). Again this can be
done using the method described by Slichter under the assumption that operators belong
ing to different spins commute (Sli80). The results are shown in Table 1-3.
1.1.5.1 The Effect of a Static Magnetic Field
By using the expressions of the previous section, it is possible to examine the effects
of a static magnetic field on a weakly coupled spin system. This perturbation occurs after
a spin system is immersed in a static magnetic field and during an NMR pulse sequence
when all radiofrequency pulses and field gradients are removed. The Hamiltonian for the
interaction of the field with a system of N spins is


Table 1-2
Solutions to the Equation of Motion of the Reduced Density Operator
Expressed as Cartesian Space Rotations
<7(t)
e'^Ixa(t) e*Ix
e'^cKt) e*Iy
e'^otOe^
(1/2)E
(1/2)E
(1/2)E
(1/2)E
lx
lx
Ixcos(<) IEsin(<)
IXCOS(0) + lysin(^)
ly
Iycos(^) + IEsin ()
ly
lycos(0) Ixsin($)
h
lEcos(4>) Iysin(<)
IEC0S () + lxsin ()
K


Table 1-3
Effect of a Product Operator Hamiltonian on
Terms of a Cartesian Product Operator Basis Set
CT(t)
e~i^2 Ii*i2* £7(t) e*2 Ilzl2z
(1/2)E
(1/2)E
hx
Ilxcos(<) + 2Ilyl2zsin(0)
hy
Ilycos(^) 2IlxI2lsin(<0)
ll.
ht
^2x
I2xcos () + 2I1II2ysin()
hy
I2ycos(<) 2IlBI2xsin(0)
I2i
I2,
^^lx^2x
2Ilxl2x
^lxhy
21lxI2y
2I1xI2i
2I1xI2,cos(^) + Ilysin()
2IlyI2x
2llyl2x
2IlyI2y
21 ly12y
2IiyI2t
2IlyI2zcos(<) llxsin(0)
2Ii1I2xcos(<) + l2ysin(0)
2Iitl2y
2IlzI2ycos() l2xsin(tf>)
211,12.
2IuI2,


16
i=N j=(N 1) k=N
H'1 = E wiI¡E + E E ^JjkljA. (j i=l j=l k=l
where u>¡ is the angular precession frequency of the ith spin, Jjk is the coupling constant
between spins j and k, and the spins are labeled by the subscripts. The Hamiltonian may
be broken into two parts, which can be applied independently. The chemical shift pre
cession term is
i=N
X"1H=EwiIit. [1-39]
i=l
The spin-spin coupling term is
j=(N 1) k=N
H ^H = 2 £ 2trJ:kI¡Iki (j j=l k=l
which is rewritten in terms of the Cartesian product operator basis by moving the factor
of two (Sor83):
j=(N -1) k=N
H_1H = £ £ 5rJjk(2IjEIkz) (j j=l k=l
The evolution of the angular momentum operators under the chemical shift Hamil
tonian is described by the fourth column of Table 1-2 where = u>¡t :
Iix IixCOs(w¡t) + Iiysin(u>¡t) ,
[1-41a]
Iiy Iiycos(w¡t) Iixs¡n(w¡t) ,
[1-4 lb]
I*. Ii. -
[ 1-4lc]
Equation [1-41] shows that in the Cartesian frame transverse components of angular
momentum rotate through an angle x w,t under the effects of chemical shift precession.
The longitudinal component is not affected.
From the expressions of Table 1-3 where jk = 7rJjkt the evolutions of the one-spin
operators under the coupling Hamiltonian are given by
Ijx ^ IjxCOS^JjkO + 21jyIkEsin(7rJjkt) ,
[1-42a]


17
Ijy Ijycos(7rJjltt) 2Ijxlkesin(7rJjkt) [ 1 -42b]
ij.-V n-42c]
Two-spin operators also evolve under the Hamiltonian which describes spin-spin cou
pling. Examples are:
2IjxIkl 2IjxIklcos(irJjkt) + Ijysin(jrJjkt) [1 -43a]
21jyIkr 2IjyIklcos(7rJjkt) Ijxsin(jrJjkt) [ 1 -43b]
1.1.5.2 The Effect of a Linear Field Gradient
The application of a linear field gradient has effects similar to those of chemical shift
precession but the expression for the angle x is different. The Hamiltonian has the form
1ft.. I1'44]
where r = x, y, or z and T¡ is a vector describing the spatial position of the ith spin along
the r axis. The field gradient is defined by the partial derivative and may be written
H'1H = 'r
3Bo
dr
[1-45]
for the component of the gradient along the r axis. By this Hamiltonian, <£¡ = '/G^t ; thus
the angular momentum operators evolve as follows:
Iix Iixcos(']fGrrit) + IiySinTfGrTit), [ 1 -46a]
Iiy Iiycos(TfGrr¡t) I^sinTGrrjt) [1 -46b]
Ii. Ii. [l-46c]
The degree of rotation of the transverse components depends upon the strength of the
applied gradient, Gr, and the position of the nuclear spin along the r axis. Two-spin
operators also evolve under the effect of a linear field gradient. Examples are:
2Ijxlkl 2IjxIklcos(7Grrjt) + 21jyIklsin(ryGrrjt) [ 1 -47a]


18
2IjyIkl 2IjyIklcos('yGrrjt) 2IjxIktsin('yGrrjt) [ 1 -47b]
1.1.5.3 Application of Radiofrequency (RF) Pulses
NMR experiments are performed using an alternating magnetic field applied in the
plane transverse to the static field to create a torque on the magnetic moments. The Ham
iltonian for the total applied magnetic field (static plus alternating) is written
H_1H = -7B0I, 'yB1[Ixcos(n0t) + Iysin(fi0t)] [1-48]
where Bj is the strength of the applied alternating field, fi0 is its angular frequency and t
is the duration of application. This Hamiltonian is time-dependent, but on transformation
to a frame rotating with angular frequency fi0, the time-dependence of the alternating
field vanishes and the Hamiltonian becomes
*-1H--(iVw,)I, + n1Ix, [1-49]
where the Larmor relationship and the definition il1 = ')B1 have been used (Sli80). The
Hamiltonian can be simplified considerably under two conditions. At resonance, fl0 = w1
and the off-resonance It term vanishes. Also, the term becomes negligible when the
applied alternating field is strong enough that Hj (fio-c^) (a "hard" pulse). For these
cases, the Hamiltonian becomes
[1-50]
In terms of Table 1-2, = fijt. However, this product, called the "tip angle", is more com
monly labeled 6. In this notation, the application of an RF pulse to each of the Cartesian
operators produces:
1X -* lx
Iy lycos(0) + IjSin(fl),
I, l,cos(0) lysin(0) .
[1 51 a]
[1-51b]
[l-51c]


19
1.1.5.4 Application of Phase-Shifted RF Pulses
If the alternating field is applied at an angle ( with respect to the x axis in the rotat
ing frame, the Hamiltonian is rotated in the transverse plane by this angle. The applied
field is said to be "phase-shifted" relative to the x axis. In one-dimensional NMR experi
ments, phase-shifting of the RF pulses of an NMR pulse sequence is done commonly in
conjunction with signal averaging to isolate certain terms of the density operator. This
process is called phase-cycling and has been used for all of the experiments described in
this dissertation. Phase cycles can vary from one pulse sequence to another and so a
description of the effect of a particular cycle will accompany the description of each of
the NMR pulse sequences used.
The density operator under the effect of a phase-shifted RF pulse may be written
o(t) = e' i1 e'iWx eifI a(0) e"iiIe eiiIx ei{1* [ 1 -52]
By using the method outlined in section 1.1.5, the following relationships can be derived:
Ix -Ix (cos2(0 + sin2(£)cos(0)) + iy cos(£)sin(O(l-cos(0)) It sin(£)sin(0) [ 1 -53a]
Iy - Ix cos(Osin(£)(l-cos(0)) + ly (cos2(£)cos(0) + sin2(£)) + I* cos(£)sin(0) [ 1 -53b]
It -* Ix sin(£)sin(0) iy cos(£)sin(0) + It cos(0) [ 1 -53c]
Most older NMR spectrometers are capable of executing RF phase shifts in 90 degree
increments only, that is £ can equal 0, 90, 180, or 270 degrees. The work in this disserta
tion has been done within this limitation. Table 1-4 summarizes the effects of RF pulses
with these shifts on the three angular momentum operators. The terms of this table are
valid only when the phase shifts are perfect increments of 90 degrees and when
off-resonance effects are negligible. These assumptions are made throughout this work.


Table 1-4
Transformations of Cartesian Operators by Phase-Shifted RF Pulses
Initial Operator
0
£(degrees)
90 180
270
Final Operators
lx
lx
IxC*-IeS0
lx
1^0+1^
ly
IyC0+IES0
ly
Iyc0-Its0
ly
I.
itce-iyse
Itc0+Ixs0
ltC0+IyS0
Itc0-lxs0
Notation: c =
cos and s =
sin.


21
1.1.6 Observable Magnetization
The forms of the operators necessary to analyze an NMR pulse sequence have been
shown. However, the form of an observable term of the density operator has not yet been
derived. In section 1.1.3, it was shown that in terms of the reduced density operator the
expectation value of an observable equals Tr [a Q], In most modern NMR experiments,
the signal is detected in quadrature. The operator corresponding to the detectable signal
may be written as the sum of the two orthogonal transverse Cartesian angular momentum
operators:
I+=(Ix + iIy). [1-54]
Thus the expectation value for the detected signal is
= Tr [a(lx + ily)] [1-55]
or transforming to the single-element operator basis,
= Tr [al*] [1-56]
This implies that o(t) must consist of I' operators to give a nonzero trace and thus observ
able magnetization. In terms of the two-spin Cartesian product operator basis, only the
terms Ilx, Ily, I2x, and I2y are directly observable. However, the terms 2IlxI2z, 2IlyI2z,
2IlzI2x, and 2IlzI2y can evolve under the effects of the chemical shift and coupling Hamil
tonians to produce observable terms. The unity operator is never observable. Each of the
remaining seven operators of the basis can be made observable only in an indirect way by
subjecting it to an RF pulse Hamiltonian.
1.2 The Two-Dimensional NMR Experiment
The theory necessary for the analysis of most NMR experiments has been intro
duced. This section makes use of the theory to describe the basic principles of
two-dimensional NMR methods. First, the required concepts of phase and amplitude
modulation are introduced using the density operator formalism. The results of the


22
application of the Fourier transform to phase- and amplitude-modulated signals are
shown. Next, a general description of the 2D experiment is presented, and a mathematical
expression is derived in terms of the density operator. Finally, a short discussion of some
of the restrictions of 2D NMR follows.
1.2.1 Phase and Amplitude Modulation
The concepts of phase and amplitude modulation are necessary for the understanding
of any NMR experiment and in particular are the keys to understanding two-dimensional
experiments. Both types of modulation can be created by a pulse sequence and appear in
a density operator expression. From the expressions for the evolution of the density
operator introduced in section 1.1, precession due to chemical shifts, spin-spin coupling,
and applied gradients corresponds to the multiplication of product operators by cosine
and sine terms. For example, applying an RF pulse of tip angle 6 = tt/2 and phase £ = 0 to
a single spin initially at thermal equilibrium produces a transverse term which evolves
under chemical shift precession for a time, t, to produce
C7(t) = Iy cos(wt) + Ix sin(wt) [ 1 -57]
The amplitude of each of the transverse components oscillates during t according to the
angular frequency, w, of the spin. Thus the components are "amplitude-modulated". If
the components are combined using the single-element operator basis, then
(7(t)=y(lViu*-reiut). [1-58]
The phases of the 1+ and I' terms oscillate according to w and the terms are said to be
"phase-modulated". This example shows that signal phase modulation may be produced
by properly combining amplitude-modulated signals.
The type of modulation is important because the Fourier transforms of the two types
of signal are very different. By using the notation of Keeler and Neuhaus (Kee85), the
Fourier transform (FT) of a damped phase-modulated signal may be written as


23
FT [eiwte't/T2] = A" + iD ,
where A and D are absorption and dispersion
functions have the forms:
A(w) =
D-(w) =
(l+((u; n0)2T22))
(<> n0)T22
(l+((w fyj)2T22))
[1-59]
Lorentzian functions, respectively. These
[1-60a]
[1-60b]
where T2 is the damping constant, w is the angular frequency of the spin, and n0 is the
angular frequency of the rotating frame. By these functions, the sign of w relative to fl0 is
unambiguous. For a damped amplitude modulation term,
FT [c(wt)e't/T2] = FT [y(eilJt + e-,)e't/T2] [ 1 -61a]
= y[A+ + iD+ + A* + iD" ] [1 -61 b]
where A* and D' have been defined and
A =
D =
t2
(i+((iv + o0)2t22))
(a; + n0)T22
(i+((w + n0)2T22)) '
[1-62a]
[1 -62b]
These results show that amplitude modulation does not allow discrimination of the sign of
a signal. The Fourier transform is a mixture of the two types of absorption and dispersion
functions and so the sign of w relative to the ft0 is ambiguous. This has important conse
quences for many 2D NMR methods where the density operator describing the detected
signal contains amplitude-modulated terms; these include 2D spectroscopy methods such
as COSY (Jee71) and NOESY (Jee79), and 2D imaging methods such as the "rotating
frame" method of Hoult (Hou79). Because the sign of a signal is desirable information,
methods have been developed to convert the amplitude-modulated signals into


24
phase-modulated ones. This idea will be used for the experiments of Chapter 4.
1.2,2 General Description of the 2D Experiment
The goal of any 2D NMR experiment is the correlation of two processes which occur
during the experiment. This requires that the signal be doubly-modulated as a function
of the two processes. Such a signal can be produced by a pulse sequence made up of at
least three periods, commonly called preparation, evolution, and detection (Aue76). In
accord with the accepted notation, the evolution and detection periods will be labeled tj
and t2, respectively. Some 2D pulse sequences contain an additional period, often called
mixing, which is sandwiched between the evolution and detection periods (Bax82). The
mixing period will be labeled rm. Each of these periods can be made up of a number of
spin perturbations, including time delays, RF pulses, and linear field gradients.
Each of the four periods of a 2D pulse sequence has a unique function. Occurring at
the outset of the pulse sequence, the preparation period perturbs the spin system from
thermal equilibrium. Assuming the use of the Cartesian product operators, the events of
the preparation period produce transverse terms in the density operator expression which
describes the effect of the pulse sequence. After the preparation period, the terms of the
density operator can change according to the effects of the perturbations of the evolution
period. The terms of the evolution period do not have to be observable. It is the function
of the perturbations of the optional mixing period to convert such terms, if desired, into
observable terms. These are detected when the spectrometer receiver is switched on. The
terms of the density operator then evolve under the effects of the perturbations of the
detection period.
For any two-dimensional NMR method, the density operator expression which
describes the detected signal of a 2D pulse sequence contains modulation terms which
reflect the effects of the perturbations which preceded detection. This can be shown by
using the density operator formalism to derive a general mathematical description of a 2D


25
NMR experiment. The density operator just before detection begins, oftj), is a function
of the perturbations of the evolution period, tj. By the equation of motion of the density
operator, c^tj) evolves during the detection period, t2, under the effect of some Hamil
tonian to produce
^,t2) = e(-i/)H%(.1)e(i/|,)Ht. [1-63]
The detected signal is found by using the trace relation
Tr(o(t1,t2)I+] [1-64]
This equation is simply the 2D case of the expression derived earlier for any a, Eq. [1-56].
Substituting for a(tj,t2),
= Tr[e('i/,<)Ht2<7(ti) e(i/*)Ht2I+] [1-65]
Because the trace is invariant to cyclic permutation of the operators, the expression for
becomes
TrWt1)e/>)H,Ve(-/'""!]. [1-66]
This relationship shows that the initial phases and amplitudes of the terms which evolve
during t2 are determined by the events preceding detection. Pulse sequences can be
designed to control what arrives at the spectrometer receiver, and the detected signal will
have a "memory" of the perturbations which occurred before the receiver was switched
on.
Although 0(tltt2) contains information about the events which preceded detection,
the application of a Fourier transform with respect to the detection period would produce
a frequency spectrum which describes the behavior of the spin system only during detec
tion. This occurs because the modulation functions corresponding to the events preceding
detection are constants for each application of a pulse sequence. To map out the behavior
of the spin system under the effects of the perturbations which precede detection, the
pulse sequence must be repeated with a change in one or more of the perturbations of the


26
evolution period, while holding the perturbations of the detection period constant. Build
ing up a matrix of such signals is equivalent to sampling the signal during a changing tj
interval. Application of a Fourier transform with respect to the change made during the
evolution period produces a frequency spectrum of the behavior of the spin system during
that period. This is the principle behind every multidimensional NMR experiment. Since
the terms of the density operator describing the detected signal of a 2D experiment can be
written as a function of the evolution and detection periods, ta and t2, the double Fourier
transform of the signal matrix is a function of two variables, Fj and F2, which have units
of frequency.
Because the Fourier transform applied with respect to the evolution period produces
a frequency spectrum corresponding to only the perturbation which has changed during
the interval, the effects of different spin system perturbations can be completely
separated during evolution. The key to these experiments and others is that although
many perturbations may act simultaneously during evolution, it may be possible to
arrange the pulse sequence such that only one perturbation is changed when the pulse
sequence is repeated while other perturbations remain constant. Thus, the effect of the
changing perturbation can be mapped out. For example, it is possible to produce a fre
quency spectrum of only the spin-spin coupling information (Aue76). Another possibil
ity, found in many imaging experiments, is the production of a frequency spectrum
related to the spatial positions of the nuclei (Ede80).
The effects of different perturbations acting during the detection period cannot be
separated, unlike those of the evolution period. Thus, the Fourier transform with respect
to the detection period is a convolution of the various frequency spectra corresponding to
the effects which occur during that period. It may be possible to adjust experimental
parameters such that the effect of a desired perturbation dominates the effects of
unwanted ones. The application of broadband decoupling is a good example of this;
enough decoupling power must be applied to remove the coupling information. Other


27
examples appear in some NMR imaging experiments where a field gradient term must
dominate the static field term. This can be ensured in many cases by using a strong gra
dient; just how strong the gradient must be is discussed in Chapter 3. It is possible to take
advantage of convolutions for certain applications, also shown in Chapter 3.
1.3 Multidimensional NMR
The basic 2D NMR method may be extended to include more dimensions by
appending more evolution periods to a 2D pulse sequence. In principle, an N-dimensional
space could be described by a density operator of the form o(t11,t12, . t1(N.1),t2) where
N 1 processes occur during N 1 evolution periods and are correlated with the events of
the detection period. Data processing requires an N-dimensional (ND) Fourier transform.
At present, the largest value of N described in the literature is four (Hal85).
A number of limitations have impeded the implementation of ND NMR methods.
One of the most troublesome is the amount of spectrometer time required to acquire the
data matrix. Typically, several seconds are required to acquire a single time-domain sig
nal and allow the spin system to return to equilibrium. If signal averaging is required,
several minutes to several hours may be needed to collect a 2D data set. This time
increases dramatically as the number of dimensions increases, possibly encompassing
several days if good resolution is required in each dimension. The data provided by a
multidimensional data set must be worth the time required to obtain it. This limitation is
not surmountable by technological improvements.
Currently available technology has also imposed some limitations on the implementa
tion of ND NMR methods. Normally, multidimensional data sets are stored in digital
form on magnetic media. As the number of dimensions increases, so do the media storage
requirements. For example, a typical 2D data set might consist of 256 time-domain sig
nals, each of which is digitized into IK, 16-bit words. Thus this matrix requires one-half
megabyte for storage. The data required to describe a third dimension would then require


28
several megabytes. Already this approaches the current limits of modern NMR spectrom
eters. Thus even if the time is available to acquire large data sets, there is restricted space
to store them. Speedy processing and the display of multidimensional data are also prob
lems. A solution which is gaining acceptance is processing and display using large
off-line computers with dedicated mathematics hardware. These difficulties are identical
to those encountered in the the early years of 2D NMR, when data handling was done
using small-memory minicomputers.
Because of the difficulties, examples of ND NMR methods are rare but do exist for
both imaging and spectroscopy. Multidimensional NMR methods were first proposed for
imaging applications, possibly because these were concerned with the correlation of spatial
information from the three dimensions of Cartesian space, which is easily visualized. The
first ND NMR experiment was proposed by Kumar and coworkers (Kum75). This was a
method for obtaining a three-dimensional spatial image, but technological limits at the
time precluded its implementation. True three-dimensional results were obtained later by
Maudsley, et al. (Mau83). However, instead of correlating information from the three
spatial dimensions, the method of Maudsley, et al. was used to correlate NMR spectra with
their two-dimensional spatial spin distributions to produce some of the earliest "chemical
shift images". In contrast, the development of ND NMR spectroscopic methods has been
pursued only recently, possibly because these are not concerned with the physical space
defined by imaging methods. Some pseudo-3D methods have been implemented (Bod81,
Bod82, Bol82). These are actually 3D methods in which the effects of two different Ham
iltonians acting during two different evolution periods are multiplied by stepping two
time intervals in concert. Thus three-dimensional data is compressed into two dimen
sions. True 3D spectroscopy was first implemented by Plant, et al. by combining the 2D
COSY and J-resolved spectroscopy pulse sequences into one (Pla86). A very similar
experiment has been described by Vuister and Boelens (Vui87). In a similar manner,
Griesinger and coworkers have described a number of 3D pulse sequences created by the


29
combination of 2D sequences (Gri87a, Gri87b, Osc88). One of the justifications for these
experiments is that the use of more than two dimensions may help to separate peaks which
overlap even in two dimensions.


CHAPTER 2
SPECTRAL IMAGING AND APPLICATIONS
TO THE STUDY OF DYNAMIC POLYMER-SOLVENT SYSTEMS
2,0 Introduction
One of the challenges of NMR imaging is the correlation of spatial and spectral
information. This chapter is not concerned with methods in which a single spatial point is
chosen as the source of a spectrum ("localized spectroscopy"); a review of these methods
may be found elsewhere (Aue86). The focus here is on methods in which image informa
tion is correlated with spectral information. Many of these methods have been reviewed
by Aue (Aue86), Brateman (Bra86b), and Brady, et al. (Bra86a). The methods fall into
two broad categories. One category includes those techniques in which a single resonance
is chosen as the source of the spectral information contained in an image (Bot84, Dix84,
Haa85a, Hal84, Jos85, Ord85). These methods will be referred to as selective spectral
imaging techniques. Most require a separate experiment for each resonance of interest, an
obvious disadvantage if several resonances are to be examined and time is at a premium
Other methods circumvent this problem by obtaining spatial information simultaneously
for every spectral resonance (Bro82, Cox80, Man85, Sep84). These will be referred to as
nonselective spectral imaging techniques. The majority of these are Fourier imaging
methods. Most encode the entire spectrum, which may include spin-spin coupling infor
mation, by allowing free precession in the absence of applied gradients. Fourier transfor
mation with respect to the precession period produces the spectrum which can be corre
lated with the spectral information encoded during other time periods in the pulse
sequence. A potential disadvantage of nonselective spectral imaging methods is that in
acquiring data from the entire spectrum, regions which do not contain resonances must be
sampled, resulting in some amount of unused data matrix. Also the sampling of the spec
tral information in addition to the spatial information necessitates a longer total
30


31
acquisition time. Both selective and nonselective spectral imaging methods suffer from
sensitivity to B0 inhomogeneity. Variations in the static field can cause separate reso
nances to broaden to the extent that overlap of the spectral lines occurs. Clean separation
of spatial images as a function of the resonance frequency then becomes difficult or
impossible.
The focus of this chapter is the use of spectral imaging to study dynamic systems
consisting of a solid polymer dissolving in liquid solvents. The use of NMR imaging to
study solid or near solid materials has not been widely applied. The resonance linewidths
of solids are usually very broad and the application of pulsed field gradients cannot dom
inate the chemical shift dispersion without severe penalties in signal-to-noise. In addi
tion, the T2 relaxation times of solids are often very short, precluding the use of
spin-echo imaging methods. Thus most studies of solids using NMR imaging methods
have been observations of an NMR-detectable liquid, usually water, which has become
distributed within the solid either by diffusion or force. Studies of woods (Hal86b), oil
cores (Rot85), glass-reinforced epoxy resin composites (Rot84), nylon (Bla86), rock
(Vin86), various building materials (Gum79), and ceramics (Ack88) have been done in this
way. Virtually no work has been done in which spectra are correlated with spatial posi
tions in solids, although Hall and coworkers have examined a piece of sandstone soaked
with n-dodecane and water using a spectral imaging method (Hal86c).
This chapter describes the application of a nonselective spectral imaging method to
the observation of polymethylmethacrylate (PMMA) dissolving in deuterated chloroform
and in perdeuterated acetone. The work was based in part on the experiments of Mareci,
et al. in which two-dimensional proton NMR images of PMMA dissolving in chloroform
and deuterated chloroform were obtained at staggered time intervals during the solvation
process (Mar88). These authors used an imaging pulse sequence which produced images
whose contrast depended on the nuclear relaxation times, Tj and T2. It was found that
the spin-lattice relaxation times of the protons of the dissolved polymer were essentially


32
constant during solvation. Because changes in relaxation times can be related to changes
in nuclear mobility, Mareci, et al. were able to deduce that polymer solvation in the
PMMA-chloroform system is a first-order phase transition. During the course of their
work, Mareci, et al. were able to obtain a well-resolved proton spectrum of the dissolved
PMMA at a static field strength of 2 tesla. This indicated that it might be possible to
observe chemical changes occurring at the polymer-solvent interface by using a spectral
imaging method to observe the spectra corresponding to various spatial positions in the
polymer-solvent system. This chapter describes such studies. These differed from most
other NMR imaging studies of solid-liquid systems in two ways:
(1) Sample spectra were correlated with their spatial position.
(2) The spectra observed were from the dissolved polymer, not from the solvent.
The results of these experiments showed large frequency shifts of the spectral resonances
of PMMA in the vicinity of the polymer-solvent interface. The magnitudes and direc
tions of the frequency shifts could be explained by a simple theoretical model based on
changes in magnetic susceptibility at the polymer-solvent interface. Thus the shifts in the
resonance frequencies were probably not the result of chemical changes.
2.1 Analysis of a Spectral Imaging Method
The spectral imaging method chosen for these studies was a variation of that of
Maudsley, et al. (Mau83). This section describes a density operator analysis of the
preparation and evolution periods of the pulse sequence. The sequence has no mixing
period. The preparation and evolution periods of the method are described by the pulse
sequence shown in Fig. 2-1. Consider the application of this sequence to a system of two
coupled spins, labeled 1 and 2, with coupling constant J and located at two spatial posi
tions, rj and r2. The spins are assumed to be at thermal equilibrium initially so the den
sity operator is
o(0) = p(\u + 12e) .
[2-1]


33
FIG. 2-1. The pulse sequence corresponding to the evolution period of a spectral imaging
method. The timing of the sequence and the labels given to various time intervals are
shown at the bottom. RF radiofrequency transmitter, g gradient.


34
For the rest of the analysis, the /? term will be implied, as described in Chapter 1. section
1.1.3. The preparation period begins with the application of the 7t/2 pulse with phase
£ = 0, which produces transverse terms in the density operator expression. The evolution
period starts immediately after the first pulse. The tranverse terms evolve under the com
bined effects of the static field and linear field gradient Hamiltonians during the time
period tj = r + (tj/2). For spin 1, the result is
cr(t,spin 1) = cos^tj) cos(TrJti) [Ily cos^n-k^!) Ilx sin(2trkrrj)] [2-2]
+ cos^ti) sin(7rJti) [2I1xI2i cos(27rkrr1) + 2IlyI2l sin(27rkr r:)]
+ siniwjtj) cos(trJti) [llx cos(27rkrr!) + Ily sin(27tkrr1)]
+ sin(o>iti) sin(7rJti) [2IlyI2e cos(27rkrrj) 2IlxI2z sin(27rkr rj)] ,
where Wj is the angular precessional frequency of spin 1 and kr = T(Grr/(27r), where Gr is
the gradient amplitude. After application of a tr pulse with phase £ = 0, the density
operator evolves in the absence of gradients during t = T (t1/2) to produce
cr(ti,spin 1) = Ilx sin^x-k^ + Wj(t t)) cos(7rJ(t + t'))
[2-3a]
+ Ily cos(2trkrr1 + Wj(t t')) cos(7rJ(ti + tj))
- 2hxht cos(2jrkrr1 + Wj(ti t)) sin(trJ(t + t'))
+ 2IlyI2l sin(2jrkrrj + w^t t)) sin(7rj(t + t')) .
Finally, this expression can be rewritten in the single-element basis by applying Eq. [1-27]
to produce
cr(t',spinl) = -yla+ cos(7rJ(t + t')) e
i(2krri + ui(t t))
[2-3b]
+ ylf cos(7rJ(ti + t')) e~l(
-ylIf sin(trj(t + t')) e1
+yli+lf sin(7rj(t + t')) e1
-i(2jrkrri + wjti t))
i(25riirri + (Ji(ti tj))
i(2irkrr1 + W1(ti tj))


35
-|ll2 sin(7rJ(t + t¡')) e'i(2rkrri + u'i(ti t)}
+|l\§ sin(jrJ(t + t')) ei(2^rri + Wl(ti't)) .
This analysis shows that modulation produced by the pulse sequence of Fig. 2-1 is a
function of three properties of a spin: spatial position, spectral frequency, and coupling
constant. All of the terms of Eq. [2-3] are phase-modulated as a function of the spatial
position and the spectral frequency of spin 1. The terms are amplitude-modulated as a
function of the coupling constant, J. The modulation functions can be mapped out by
changing the functions in a stepwise fashion, as described in Chapter 1, section 1.2.2.
This process is often referred to as "phase-encoding". The spatially-dependent modula
tion functions can be mapped out by changing the value of kr, which can be done by
altering the gradient amplitude and holding the time delays, t and tj, constant. The value
of kr could be altered by changing the duration of the gradient, but because t and t'i are
constants, r must be constant, and so to allow a change in the spatially-dependent modula
tion functions, the gradient amplitude must be varied. The modulation functions which
depend on spectral frequency are not eliminated except in the special case where t = t^,
as shown by Eq. [2-3]. For the spectral imaging method of this chapter, experimental
conditions were chosen such that t = t'. In an alternative method, the terms of Eq. [2-3]
which depend on spectral frequency can be mapped out by altering the time delays, t and
t', in a stepwise fashion and holding the value of kr constant. This is the basis of a spec
tral imaging method which is described in Chapter 3. This type of phase-encoding does
not eliminate the modulation terms which depend on spatial position. For either
phase-encoding method, the modulation terms which are dependent on the coupling con
stant cannot be removed by adjusting the time delays. Thus, coupling always affects the
signal amplitudes.
The pulse sequence used for the studies described in this chapter is shown in
Fig. 2-2. It is a 2D spectral imaging method whose evolution period is identical to the


selective ni2
RF O
g
pe
\+\A/2+\\l T t,/2
TE
TR
FIG. 2-2. A pulse sequence for two-dimensional spectral imaging. The timing of the
sequence and the labels given to the various timing intervals are given at the bottom. Spa
tial phase-encoding is accomplished by altering the gradient amplitude, gpe, in a series of
M steps. RF = radiofrequency transmitter, gp, = phase-encode gradient, g,, = slice gra
dient, acq = data acquisition.


evolution period of the pulse sequence of Fig. 2-1. The time delays r, t1/2, and T tj/2
are held constant. For the experiments of this chapter, the values of these delays were
chosen such that, in the notation of Eq. [2-3], the sum of tj and one-half the duration of
the selective 7r/2 pulse equalled tj'. Under the assumption that the duration of the selec
tive pulse is negligible relative to the durations of the delays, r and tj/2, the terms of
Eq. [2-3] which are dependent on the spectral frequencies of the spins vanish. The cou
pling terms remain but can vanish when
where n = 0, 1/2, 1, 3/2, .... Thus for studies of coupled spin systems using the pulse
sequence of Fig. 2-2, the delays must be chosen with care. Since r is fixed,
phase-encoding of the spatial information is accomplished by altering the gradient ampli
tude. This is indicated in Fig. 2-2 by the multiple bars describing the different ampli
tudes of the gradient, gpe. The signal is sampled in the absence of gradients during the
time period, t2. Modulation with respect to the detection period is a function of the pre
cession frequencies of the spins and not of their spatial position. Thus the pulse sequence
is a method for correlating spectra with spatial position.
The preparation periods of the pulse sequences of Figs. 2-1 and 2-2 are different.
The former consists of a nonselective 7t/2 RF pulse which excites the portion of the sam
ple which lies within the RF transmitter coil. The preparation period of the pulse
sequence of Fig. 2-2 is a "slice-selective" tt/2 pulse. The combination of a
frequency-selective, "soft" RF pulse (indicated by the diamond) and a field gradient (the
"slice" gradient, gsl) allows the excitation of a plane of sample spins. Only spins in the
plane experience a 7t/2 tip angle. The slice thickness of the plane is controlled by the
duration of the RF pulse, the pulse shape, and the amplitude of the slice gradient.
Because the Hamiltonians describing the effects of a static field and a linear field gradient
both act during the slice-selection process, different but adjacent slices are selected for


38
each chemically-shifted species in the sample. The result is slice misregistration. This
effect can be overcome by applying a strong gradient, but this restricts the choice of slice
widths. This effect is an example of the convolution of spatial and spectral information
discussed in Chapter 3.
A final aspect of the pulse sequence of Fig. 2-2 is the effect of the time periods, TE
and TR, on the signal amplitude. The echo time, TE, is the period from the center of the
soft RF pulse to the center of the spin echo. The period between successive initiations of
the pulse sequence is the repetition time, TR, and includes a delay to allow the perturbed
spin system to relax toward thermal equilibrium. The amplitude of the echo is weighted
by the product e TE/,T2(i e TR/,Tl), where Tj and T2 are the nuclear spin-lattice and
spin-spin relaxation times, respectively. Thus two conditions must exist for the echo to
have appreciable amplitude. The TE must be short relative to T2 and the TR must be on
the order of or greater than Tx. These conditions are not so easily met when obtaining
images of solid or semisolid materials, because the T2 values can be short.
2.2 Experimental
All experiments were carried out using a General Electric CS1-2 NMR imaging spec
trometer equipped with a 2 T Oxford Instruments superconducting magnet with a 31 cm
clear bore diameter. With shim and gradient coils installed, the working clear bore was
reduced to a 23 cm diameter. The maximum attainable gradient strength was 0.03
mT mm'1. The RF coil was a slotted tube resonator built in house by the author; it was
tunable to both and 19F frequencies. For these studies, only protons were detected.
The RF coil had a length of 80 mm and a diameter of 56 mm with an effective RF mag
netic field over a length of 40 mm.
The pulse sequence used is shown in Fig. 2-2. Quadrature detection was used for all
experiments. Table 2-1 shows the phase cycle which was used; it performed several func
tions. Signals which had not felt the effects of phase-encoding were moved to the edges


39
of the spatial axis by phase alternation of the n pulse with every other phase-encode step
(Gra86). These signals arose because of imperfect n/2 and n pulses and spin relaxation
during t¡. The phase alternation of the n/2 pulse and receiver with signal averaging can
celled out the effects of imperfect n/2 pulses and imbalance in gain between the two qua
drature detection channels. Slice misregistration due to the chemical shift effect was
assumed to be negligible and a frequency-selective sine-shaped n/2 RF pulse and a gra
dient normal to the imaging plane were used to select slices for all images. All n pulses
were nonselective. Only the second half of the echo was acquired to circumvent problems
of centering the echo in the acquisition window and to allow digitization of the signal
until it decayed fully.
For all images, the soft n/2 pulse duration was 1 millisecond and the n pulse duration
was 86 microseconds. The spectral width was 2000 Hz and 256 complex points were
acquired for each phase-encode step, producing a spectral resolution of 7.8 Hz per point.
Thirty-two phase-encode steps of two signal averages each were performed. The
field-of-view along the spatial axis was 64 mm and so the resolution was 2 mm per data
point. The slice width was 5 mm. The phase-encode gradient duration was 4 mil
liseconds. The repetition time, TR, was 15 seconds and the echo time, TE, was 20 mil
liseconds.
An experiment was initiated by pouring approximately 4 ml of deuterated chloro
form or perdeuterated acetone into a circular Pyrex dish (5 cm diameter, 1.5 cm deep),
then centering a polymethylmethacrylate (PMMA) block in the solvent. The polymer and
solvents were at ambient temperature. This point was time zero for the solvation process
which ensued. The blocks were made of commercial grade PMMA, each approximately 1
cm on each side and 0.5 cm thick. The solvent covered approximately the lower half of
the block. The amount of solvent was based on the amount of chloroform empirically
found sufficient to dissolve the block slowly. Immediately after positioning the block, the
dish was then covered with a tight-fitting nylon cap to retard solvent evaporation. This


Table 2-1
2D Spectral Imaging:
Relative Pulse and Receiver Phases
Phase-Encode
Step
Signal
Average
Phases
Pulses
Receiver
7r/2
7T
1
1
0
90
0
1
2
180
90
180
2
1
0
270
0
2
2
180
270
180
M
1
0
270
0
M
2
180
270
180
Refer to Fig. 2-2 and its legend for the notation corresponding to this table.


41
assembly was placed on a plexigls support and inserted into the RF coil. The coil was
then placed in the magnet bore. The sample and coil could be positioned reproducibly
and imaging could be started within a few minutes after placing the block in the solvent.
The sample orientation relative to the gradient axes is shown in Fig. 2-3. The thinness of
the sample along the y axis meant that slice selection along this axis was not necessary.
Imaging was initiated at approximately 20-minute intervals, each image requiring 16
minutes to acquire. The slow acquisition was necessary to reduce ridges parallel to the
phase-encode axis. These appeared if TR was so short that the transverse magnetization
could not relax to near equilibrium before the pulse sequence was applied again. Thus
some time-averaging of the solvation process was unavoidable. Initially, the x gradient
was used for choosing the slice and phase-encoding was done with the z gradient. For the
next image, the x and z gradients were interchanged, the z becoming the slice axis and the
x the phase-encode axis. This interleaving of images was continued until no more shifts
in the spectral resonances were observed or until the sample lines broadened considerably
due to solvent evaporation and subsequent sample solidification.
Because static field inhomogeneity could possibly cause frequency shifts of reso
nance peaks, care was taken to ensure that this did not occur. The static field inhomo
geneity was reduced by adjusting the electronic shim coils, using the same Pyrex dish,
nylon cap, dish support, and RF coil as those used for the PMMA-solvent imaging experi
ment, but replacing the sample with a few milliliters of CuS04-doped water. Crude
adjustments were made using a one-pulse sequence, attempting to increase the time con
stant of the signal decay. The spectral imaging sequence of Fig. 2-2 was then used to
assess the field homogeneity. Because water has a single XH resonance line, the homo
geneity was considered good when the frequency of the line did not change with position.
Several iterations of this procedure were adequate to set the shim currents. These current
settings were then used without modification during spectral imaging of PMMA solvation.
Examples of the spectral images obtained after shimming are shown in the contour plots


PMMA Sample
FIG. 2-3. Sample orientation for studies of PMMA solvation. The Cartesian coordinates
were defined by the axes of the gradient coils.


43
of Fig. 2-4. There was some shifting of the water resonance frequency with position
along the x axis. Shifts of the resonance frequency with z axis position were less notice-
able. The falloff of signal intensity along the z axis was the result of the sample extending
slightly outside of the active region of the RF coil.
2.3 Results
The proton spectrum of a piece of PMMA partially dissolved in deuterated chloro
form is shown in Fig. 2-5 along with a diagram of the methylmethacrylate monomer unit
and the assignments of the three peaks. Because the amplitudes of the peaks in the 2D
spectral images were weighted by the Tj and T2 values of their corresponding protons, it
was important to know these before imaging commenced. Based on the measurements of
Mareci, et al., estimates of the Tx values were 215 milliseconds for the ester methyl pro
tons, 83 milliseconds for the methylene protons, and 52 milliseconds for the methyl pro
tons (Mar88). Because the TR of the spectral imaging sequence was set to 15 seconds, TR
had virtually no effect on the amplitudes of the spectral peaks. However, since the T,
values of the protons could not be greater than their Ta values and since the pulse
sequence TE was set to 20 milliseconds, all of the peaks were attenuated by spin-spin
relaxation. The degree of attenuation relative to the case where TE was infinitely short
was estimated using the formula 100 x (1 eTE^T2). Assuming that each protons T2
equalled its Tx, the ester methyl, methylene, and methyl peaks should have been reduced
by at least 9%, 21%, and 32%, respectively. This was a best case estimate; in the actual
system the T2s could have been much shorter than the Tjs, resulting in a greater percen
tage of attenuation.
A spectral image of a PMMA block dissolving in deuterated chloroform is shown in
Fig. 2-6a as a stacked plot. Data acquisition was begun after 61 minutes of solvation. The
spatial dimension was defined by the x axis. The three spectral peaks of PMMA were
visible and changed in amplitude, linewidth, and resonance frequency along the spatial


FIG. 2-4. Spectral images used to observe the quality of the static field inhomogeneity.
Parameters and processing are given in the text, (a) A spectral image produced by
phase-encoding along the x axis. The frequency of the water resonance shifted slightly
downfield at positions away from the x axis origin.


|~ n i i i i i i i i | i I
1000
500
Water (doped)
7Tr
32 mm
o X
32 mm
0
-500
-1000 Hz


FIG. 2-4continued, (b) A spectral image produced by phase-encoding along the z axis.
The frequency of the water resonance did not change appreciably with spatial position.
Because the sample did not fit completely inside the RF coil, the image intensity was
reduced at the ends of the spatial axis.


Water (doped)
1000
500
0
-500
-1000 Hz


FIG. 2-5. The *H spectrum of a piece of PMMA partially dissolved in deuterated chloro
form at 2 T. The methylmethacrylate monomer unit is shown at the left and the three
proton resonance peak assignments are given by the arrows. The small spike at zero fre
quency was probably the result of quadrature channel imbalance. The shoulder in the
most downfield position belonged to residual chloroform.
-U
OO


PMMA in CDCI3
time of solvation: 61 min.
32 mm
X
FIG. 2-6. Spectral images of PMMA in deuterated chloroform after 61 minutes of solva
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
produced by phase-encoding along the x axis. The three spectral peaks of PMMA were
visible at regions distant from the origin where the polymer was more mobile.
v£>


50
axis. The changes in linewidth and amplitude arose in part from the different degrees of
sample solidness. In the vicinity of the undissolved PMMA, near x = 0 mm, linewidths
were very broad and spread into the baseline. At the polymer-solvent interfaces where
solvation began, the mixture was gel-like and the linewidths began to narrow. Further
away from the polymer block a higher concentration of solvent was present and so the
three spectral lines were resolved. Because the polymer and solvent were clear, the degree
of solvation could not be followed visually. However, the spectral image showed that
some polymer had diffused to the edges of the sample dish. The data of the stacked plot
of Fig. 2-6a is shown as a contour plot in Fig. 2-6b to emphasize the changes in the reso
nance frequencies with spatial position. An overall curvature was seen which was due to
static field inhomogeneity as shown by comparison of Fig. 2-6b with Fig. 2-4a. In addi
tion, the resonance frequencies of the spectral peaks all shifted strongly downfield
(toward positive frequency) near the polymer-solvent interface. The maximum shift was
roughly 70 Hz. This number could not be measured exactly because of the contributions
of the static field inhomogeneity and the varying linewidths.
A second spectral image of the same sample was obtained by initiating data acquisi
tion after 81 minutes of solvation. The z axis defined the spatial dimension, and the
stacked plot result is shown in Fig. 2-7a. As in Fig. 2-6a, the three spectral lines
broadened as the concentration of solvent decreased near the polymer-solvent interface.
However, the contour plot corresponding to Fig. 2-7a, shown in Fig. 2-7b, shows that the
resonance frequencies shifted upfield near the interface then sharply downfield at the
interface. The shifts were about equal to those found for Fig. 2-6.
Spectral images of much later stages of solvation are shown in Fig. 2-8. At this
point, the polymer-solvent interface no longer existed and the mixture was distributed
fairly evenly throughout the sample dish. This is seen most clearly in Fig. 2-8a where the
spatial axis is the x. The rolloff at the ends of the z axis seen in Fig. 2-8b was due to the
sample not being completely inside the RF coil. Continued spectral imaging showed little


PMMA in CDCI/
time of solvation: 61 min.
32 mm
L o X
l i i i i i i ii|iiiiiiiii|iiiiiiiii| *"32 mm
500 0 -500 -1000 Hz
FIG. 2-6--continued. (b) A contour plot of the data of (a). Downfield shifts of the three
spectral peaks were visible near the polymer-solvent interfaces.


PMMA ¡n CDCI3
time of solvation: 81 min.
32 mm
z
1000 0 -1000 Hz
FIG. 2-7. Spectral images of PMMA in deuterated chloroform after 81 minutes of solva
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
produced by phase-encoding along the z axis. As in Fig. 2-6a, the three spectral peaks of
PMMA were visible at regions away from the polymer-solvent interfaces.
v-n
K>


PMMA in CDCI3
time of solvation: 81 min.
I
|iiii1iii1|iiiiiiiii|iiiir
32 mm
- 0 Z
-32 mm
500 0 -500 -1000 Hz
FIG. 2-7continued, (b) A contour plot of the data of (a). Upfield shifts of the three
spectral peaks were observed near the polymer-solvent interfaces and downfield shifts in
the vicinity of the solid polymer.


PMMA in CDCI3
1000 0 -1000 Hz
FIG. 2-8. Spectral images of PMMA in deuterated chloroform at late stages of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x axis after 482 minutes of solvation. The
polymer-solvent interfaces had vanished and the resulting solution had become homo
geneous.


PMMA in CDCI3
FIG. 2-8--continued. (b) A stacked plot spectral image obtained by phase-encoding
along the z axis after 502 minutes of solvation. The change in signal amplitude along the
spatial axis was caused by the RF inhomogeneity over the sample.
kyi
KJt


56
change in the spatial distribution of the spectra. As the solvent evaporated and the sample
mixture hardened, the linewidths broadened to the extent that the peaks could no longer
be seen.
To test whether the appearance of the spatially-dependent frequency shifts was a
function of the solvent used, the previous experiments were repeated using perdeuterated
acetone in place of deuterated chloroform. The time required for solvation was much
longer since perdeuterated acetone was a poorer solvent. In addition, the solvent had
some water contamination which contributed a fourth peak to the three-peak PMMA
spectrum. A stacked plot spectral image and its corresponding contour plot are shown in
Fig. 2-9. Data acquisition was begun 180 minutes after placing the PMMA block in the
solvent. An anomalous water peak is seen in the third most downfield position. Despite
the differences in solvent and solvation duration, Fig. 2-9 agrees qualitatively with
Fig. 2-6. The slight curvature of the resonance lines due to static field inhomogeneity
along the x axis was present as well as the downfield shifts of the peaks near the
polymer-solvent interface. However, the magnitudes of the shifts were roughly half that
found when using deuterated chloroform as the solvent. The spectral image obtained 20
minutes later using the z gradient for phase-encoding is shown in Fig. 2-10. It agrees
qualitatively with Fig. 2-7 but again the magnitudes of the resonance shifts are smaller.
Spectral images obtained later in the solvation process are shown in Fig. 2-11. The
linewidths narrowed somewhat, indicating a more liquid-like sample mixture, but other
wise the qualitative features of Fig. 2-10 remained. The amount of perdeuterated acetone
used was not sufficient to dissolve the polymer block and so spectral images of
near-homogeneous mixtures like that of Fig. 2-8 were never seen.
In summary, for a given solvent, the appearance of spatially-dependent resonance
shifts depended on the time of solvation. The signs of the shifts depended on the spatial
axis observed. Finally, the magnitudes of the shifts depended on the solvent used. Origi
nally, it was suspected that the resonance shifts were the result of a chemical interaction at


FIG. 2-9. Spectral images of PMMA in deuterated acetone after 180 minutes of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x axis. The gross features of this image were simi
lar to those of Fig. 2-6a; however, a fourth peak due to water in the solvent was seen in
the third most downfield position.


PMMA in Acetone-dg
time of solvation: 180 min.
T
0
-1000 Hz
oo


PMMA ¡n Acetone-dg
time of solvation: 180 min.
500 0 -500 -1000 Hz
FIG. 2-9continued, (b) A contour plot of (a).
N-0


FIG. 2-10. Spectral images of PMMA in deuterated acetone after 200 minutes of solva
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the z-axis. The features of this image were similar to
those of Fig. 2-7a.


PMMA in Acetone-dg
time of solvation: 200 min.


I'r
500
PMMA in Acetone-dg
time of solvation: 200 min.
V
I
i i I i ii|iiiliiiii|i|iir
0 -500
32 mm
z
y32 mm
FIG. 2-10--continued. (b) A contour plot of (a).
-1000 Hz
O'
NJ


PMMA in Acetone-dg
time of solvation: 460 min
32 mm
X
1000 0 -1000 Hz
FIG. 2-11. Spectral images of PMMA in deuterated acetone at late stages of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x-axis after 460 minutes of solvation.
O'


PMMA in Acetone-dg
time of solvation: 480 min.
32 mm
z
1000 0 -1000 Hz
FIG. 2-11continued, (b) A stacked plot spectral image obtained by phase-encoding
along the z axis after 480 minutes of solvation. Only small differences existed between
these images and those of Figs. 2-9 and 2-10 because the solvent concentration was low.


65
the polymer-solvent interfaces. However, this possibility was eliminated when the signs
of the shifts were found to be dependent on the spatial axis. This behavior indicated
instead that the shift effect might be the manifestation of the differences in magnetic
field susceptibility between the polymer and solvent. Thus estimates of the shifts caused
by susceptibility differences were calculated using a crude model applied to each of the
samples studied.
2.4 The Susceptibility Model
The change in the static field which results when a sample of a particular geometry
and susceptibility is immersed in the field is given by
B0'=B0[ 1+(^-/c)Xv], [2-5]
where B0 is the static field in the absence of the sample, /c is a factor which depends on
the bulk sample geometry, and xv *s the volume susceptibility, a dimensionless quantity
dependent on the sample molecule (Pop59). Using the Larmor equation, the frequency
difference between two chemical species possessing identical bulk sample geometries due
solely to differences in volume susceptibility is
Ai/ = i/2 j/j = -^:B0(4p /t)(xv2 Xvi) [2-6]
This shift is not identical to the chemical shift phenomenon but behaves the same way in
terms of the evolution of product operators. Thus for the spectral imaging method used
here, a susceptibility effect appears as the addition of a constant frequency shift to each
of the chemical shift frequencies of a sample.
To analyze a particular problem using Eq. [2-6], the values of k are needed. These
have been described for several sample geometries (And69):
(1) For a cylinder whose length is infinitely greater than its diameter and oriented
transverse to the static field,


66
k = 2tt [2-7a]
(2) For an identical cylinder oriented parallel to the static field,
/c = 0 [2-7b]
The greatest susceptibility shifts are seen in cylinders oriented parallel to the static field.
It has been shown that for a cylinder oriented parallel to the static field and whose
length is about 10 times its diameter, k is not zero, but about 0.2 (Boz51). Thus the infin
ite cylinder approximation is a good one for such a sample. In the spectral imaging
experiments of the dissolving polymer, the excited region was a bar about 5 mm wide (the
slice width), 2 mm deep (the solvent depth), and 50 mm long (the dish diameter). Thus
the length was at least 10 times the width. By considering the polymer-solvent bar to be
an infinite cylinder and using the analytical values of k in Eqs. [2-7], approximate suscep
tibility shifts at the PMMA-solvent interface could be calculated if the volume suscepti
bilities of the molecules of the system were known. These were estimated using Pascal
constants and the densities and molecular weights of the solute and solvents, according to
the procedure of Pople, Schneider, and Bernstein (Pop59). The results are shown in Table
2-2. Using the data of Table 2-2 and Eq. [2-6], the susceptibility shifts for coaxial
cylinders of various pairs of substances were calculated. The results are shown in Table
2-3. The polymer-solvent interfaces were considered parallel to the static field if the x
gradient was used to define the slice axis and transverse if the z gradient was used. Using
these criteria, the calculated susceptibility shifts of Table 2-3 were compared with the
frequency shifts seen in Figs. 2-6, 2-7, 2-9, and 2-10. The magnitudes and signs of the
frequency shifts were measured with reference to the peaks belonging to the ester methyl
protons, because these were the least-attenuated and their spatially-dependent frequencies
could be measured fairly accurately. At some spatial positions, the ester methyl peak was
so broad that its spectral frequency could not be measured. The spatially-dependent fre
quencies of the methylene and methyl peaks could not be measured accurately because
they were quite broad and more attenuated than the ester methyl peaks. The frequencies


Table 2-2
Calculated Volume Susceptibilities
Molecule
Xv x io6
Acetone
-0.461
Chloroform
-0.853
Methylmethacrylate (MMA)
-0.527


Table 2-3
Calculated Susceptibility Shifts at 2 Tesla
System
Orientation
(relative to B0)
Shift (Hz)
MMA, Acetone
transverse
-12
MMA, Acetone
parallel
24
MMA, Chloroform
transverse
58
MMA, Chloroform
parallel
-116


69
of the ester methyl peaks were corrected to remove the static field inhomogeneity contri
bution. Using the data of Fig. 2-4, the difference between the frequencies of the water
peak at 0 mm and at some other position were calculated for each spatial position. These
differences were assumed to be due to static field inhomogeneity and were subtracted
from the spatially-dependent frequencies of the ester methyl peaks found from the poly
mer solvation experiments. The data of Fig. 2-4a were used to correct Figs. 2-6 and 2-9,
and Figs. 2-7 and 2-10 were corrected using the data of Fig. 2-4b. The corrected fre
quency shifts which could be measured were plotted versus spatial position relative to one
side of the polymer block and are shown in Fig. 2-12.
2,5 Conclusion
The susceptibility model successfully predicted a number of the features of the spec
tral images of the polymer-solvent systems studied. In general, the observed frequency
shifts were in the range calculated. More specifically, for a particular slice axis, the mag
nitude of the frequency shift observed when deuterated chloroform was used as the sol
vent was greater than that observed when perdeuterated acetone was used. This observa
tion was consistent for both slice axes, as seen by comparison of Fig. 2-6b with Fig. 2-9b
and Fig. 2-7b with Fig. 2-10b. For a particular solvent, the shift observed when the z
gradient defined the slice axis was of opposite sign from that observed when the slice axis
was defined by the x gradient. These observations are summarized for the ester methyl
peak of PMMA in Fig. 2-12.
Although the model was partially successful at explaining the observed frequency
shifts, several discrepancies existed between the model and the observed spectral images.
For a particular slice axis, the spectral images showed no difference in the signs of the
frequency shifts observed for the two solvents. The susceptibility calculations shown in
Table 2-3 predicted otherwise. The disagreement possibly was due to the close
equivalence of the calculated volume susceptibilities of acetone and methylmethacrylate.


FIG. 2-12. PMMA solvation. The data for this plot were obtained by measuring the
spatially-dependent spectral frequency of the ester methyl peak of PMMA at several spa
tial positions then subtracting the inhomogeneity contribution as described in the text.
Key: , Data obtained from Fig. 2-6 corrected using the data of Fig. 2-4a; the
polymer-solvent interface was perpendicular to the static field and the solvent was deu-
terated chloroform. +, Data obtained from Fig. 2-9 corrected using the data of Fig. 2-4a;
the polymer-solvent interface was perpendicular to the static field and the solvent was
perdeuterated acetone. A, Data obtained from Fig. 2-10 corrected using the data of
Fig. 2-4b; the polymer-solvent interface was parallel to the static field and the solvent was
perdeuterated acetone, x. Data obtained from Fig. 2-7 corrected using the data of
Fig. 2-4b; the polymer-solvent interface was parallel to the static field and the solvent was
deuterated chloroform.


Frequency Shift (Hz)
-50
-100
-150
-200
-30 -26 -22 -18 -14 -10 -6 -2
t 1 t 1 1 1 1 r 1 1 1 r
Spatial Position (mm)


72
These calculations were too poor to say truly whether the susceptibility difference was
positive or negative. A second discrepancy existed between the magnitudes of the calcu
lated shifts and the observed values. For a particular solvent, the magnitudes of the
observed shifts should have changed substantially with the slice axis according to the sus
ceptibility model. Some support for this could be seen by comparing Fig. 2-6b with
Fig. 2-7b and Fig. 2-9b with Fig. 2-1 Ob; the observed shift magnitudes appeared to be
greater when the x gradient was used to define the slice. This is consistent with the model
but is not conclusive because the observed shift magnitudes could not be accurately meas
ured. The chief reason for this obstacle was that the observed shift magnitudes were on
the order of the linewidths in the vicinity of the polymer-solvent interfaces. This coupled
with the effect of static field inhomogeneity and the variations of peak amplitudes made
the exact measurement of frequency shifts impossible.
Thus far, the behavior of the frequency shifts which were observed in Fig. 2-7 has
not been explained. The shifts first moved upfield in the vicinity of the polymer-solvent
interface as predicted by the model, but then went unexpectedly downfield in the vicinity
of the as-yet-undissolved PMMA. The appearance of spectral peaks in the vicinity of
solid indicated that some solvent had entered the slice. Far more likely, though, was the
possibility that the polymer block had become loose and moved out of the slice region
along the phase-encoded spatial axis. This would explain two observations. First, the
slice would have encompassed both solid polymer and liquid solvent, which would explain
the appearance of peaks at spatial positions where none should have been seen. Secondly,
with the polymer slightly out of the slice, two interfaces would have been observable, one
parallel to the static field and the other transverse to it. This would explain the upfield
shift observed near the polymer-solvent interface and the downfield shift seen where the
polymer should have not yet dissolved. Figure 2-6 provides some evidence that the poly
mer block was not centered at the x axis origin; the polymer-solvent interfaces are not
distributed symetrically with respect to the x = 0 mm position.


73
Although this work failed to observe any chemical changes at the polymer-solvent
interface, it has some implications for similar studies. A number of papers have described
the application of the basic spectral imaging technique described in this chapter to the
study of living systems (Bai87, Has83, Pyk83). One of the proposed uses is the measure
ment of in vivo tissue pH at particular spatial locations by the measurement of the fre
quency difference between the inorganic phosphate and phosphocreatine peaks detected
with 31P spectral imaging. However, very little has been said about the contribution of
susceptibility to the results of such studies. If an interface, such as one between a muscle
and an organ, exists along the phase-encoded spatial axis, then susceptibility changes
could cause anomalous frequency shifts of the corresponding spectral peaks. If both tis
sues have the same orientation relative to the static field and differ only in their suscepti
bilities, then all spectral peaks would be affected equally and the relative frequency shifts
would be unaffected. However, if the tissues do not have the same susceptibilities and
orientations relative to the static field, errors in the estimate of tissue pH could result. It
can be shown that this error is probably very small using the susceptibility model of sec
tion 2.4. Equation [2-6] can be rewritten in units of parts per million (ppm) as
Ai/ = i/2-t'1=lx 106(^- /c)(xv2 Xvi) [2-8]
In the worst case, two cylindrical samples with two different susceptibilities would be
oriented parallel and perpendicular to the static magnetic field. When /c = 0 as for a
cylinder oriented parallel to the static magnetic field, then
"parallel = 4-19 106(XV2 Xvl) [2-9]
Also, when /c = 27r as for a cylinder oriented perpendicular to the static magnetic field,
then
"perpendicular = 209 ,()6^v2 Xvl) [2-10]
Subtracting Ai/perpendicular from Ai/parallel produces


74
Ai/ = 6.28 x 106(xv2 Xvi) [2-H]
The value of Ai/ is the frequency difference between two samples due to differences in
their volume susceptibility and sample orientation and is a source of error in the measure
ment of the true frequency separation. As an example, the true frequency separation
between inorganic phosphate and phosphocreatine varies over about a 2.5 ppm range
between pH 6 and pH 7 (Gad82). If the maximum allowable error in the frequency
difference measurement is 0.25 ppm, and this is set equal to Ai/, then (xv2 Xvi) must be
less than 4 x 10'8. This is very likely in biological tissues, since the inorganic phosphate
and phosphocreatine are in very dilute solution and their volume susceptibilities are prob
ably very similar.
Finally, the spectral imaging experiments of this chapter have important implications
for materials science. In particular, spectral imaging could be used as a rheological tool to
observe deformation and flow during the solvation process, possibly uncovering some
chemical process occurring at an interface. The experiments discussed in this chapter
have shown that it may be possible to observe semisolid materials directly, complementing
the observation of solvents as they penetrate solids. It may also be possible to observe the
curing process, observing spatially-localized chemical changes as a polymer hardens.


CHAPTER 3
CONVOLUTION SPECTRAL IMAGING
3.0 Introduction
In the previous chapter spectral imaging techniques were categorized as being either
selective or nonselective. In this chapter, a new set of nonselective spectral imaging tech
niques is introduced. Like other methods in that class, they are sensitive to static field
inhomogeneity. However, they differ from the other techniques in that the number of
dimensions required for image acquisition and display equals the number required to
define the spatial image. For example, a three-dimensional experiment in which the spa
tial information from each of two spatial dimensions and the spectral information are col
lected separately may be compressed into a two-dimensional experiment. This is accom
plished by including the spectral information with the spatial dimensions rather than let
ting the spectrum comprise a separate dimension. Because the convolution theorem for
Fourier transformation is central to the applicability of these techniques, they have been
grouped under the term "convolution spectral imaging". It is shown that under certain
conditions these methods drastically reduce the time required to obtain spectral and spa
tial information without a loss in spectral or spatial resolution.
3.1 The Convolution Spectral Imaging Method
In Chapter 2, section 2.1, an expression for a density operator was given which
described the evolution period of a 2D spectral imaging method. This was derived by
using the density operator formalism to describe the effect of the pulse sequence of
Fig. 2-1 applied to a system of two weakly-coupled spins. The result appeared in
Eq. [2-3J. The method of Chapter 2 mapped out the spatial modulation functions of the
evolution period by changing the amplitude of a gradient with each pass of the pulse
75


76
sequence. The durations of the time intervals were fixed during an experiment and so the
modulation functions related to the spectral frequencies and coupling constants of the
spins were also constant. The spectral dimension, which was correlated with the spatial
dimension, was defined by allowing free precession during detection.
Using the same pulse sequence shown in Fig. 2-1, a different type of modulation
function can be mapped out by varying the time interval, tx; in effect the n pulse is
moved through the time window, T. The gradient amplitude, Gr, and its duration, r, are
fixed. Under these conditions, the density operator expression for two weakly-coupled
spins, given by Eq. [2-3], shows that the phase modulation function which is mapped out
is a function of the spectral frequencies of the spins. The spatial modulation functions are
constant. Also, the modulation functions related to spin-spin coupling are constant
because the start of detection occurs at a fixed time after the initial n/2 excitation pulse
(Bax79). The phase-encoded spectral width is given by the inverse of the amount, Atj, by
which the tj interval is changed with each pass of the pulse sequence. The desired spec
tral width and the number of phase-encode steps place strict limits on the TE value. This
can have serious consequences since the acquired signal amplitude depends on the TE and
the T2s of the sample, as shown in Chapter 2.
Phase-encoding of spectral information is the basis of the three-dimensional method
of Sepponen and coworkers (Sep84), from which convolution spectral imaging methods
are derived. With the Sepponen method, two spatial dimensions are correlated with a
spectral dimension. The spectral modulation functions produced during evolution are
mapped out by the process just described. In addition, the spatial modulation functions
produced during evolution are also mapped out by using the stepping of a phase-encode
gradient as described in Chapter 2. To maintain the independence of the phase-encoded
spectral and spatial information, for each step in the variation of the phase-encode gra
dient, the time interval, tl5 is stepped a number of times equal to the desired number of
points in the spectral dimension. The time interval is then reset to its initial value and the


77
phase-encode gradient is stepped to its next value. A complete cycle of time interval
stepping is repeated for each phase-encode gradient step.
The second spatial dimension of the three dimensions defined by the Sepponen tech
nique is produced by the process of frequency-encoding, where a gradient is turned on
during detection to create spatial modulation. However, as mentioned in Chapter 1,
modulation due to free precession also occurs during detection. The Hamiltonian describ
ing the effect of the applied gradient must dominate the unwanted Hamiltonian which
describes the effect of the static field or spatial and spectral information will be mixed.
In contrast to the Sepponen method, for convolution spectral imaging techniques the
phase-encode gradient amplitude and the time interval are stepped simultaneously during
the evolution period. The result of this simultaneous stepping is a phase modulation of
the acquired signal which is a function of the spatial and spectral details of the object
being imaged. This phase modulation is the product of two phase modulations, one of
which is a function of the gradient stepping and the other a function of the time interval
stepping. From Eq. [2-3], which was derived for a single spin, the phase modulation
function corresponding to the gradient stepping has the form
h(kr,ra)
e-i2xkrri
[3-1]
where kr is a function of the applied gradient magnitude and r2 is the position of the spin
along the r axis. The function kr has the form
W
TfGr(t)dt
2rr
[3-2]
where q is the gyromagnetic ratio, Gr(t) is a function describing the time-dependent
amplitude of the applied gradient, and r is the time during which the gradient is applied.
In general, for a distribution of spins along the r axis, S(r¡), the modulated signal has the
form


78
s(kr) = ^S(ri)h(kr,ri)dr¡ p_3]
where R expresses the limits of integration imposed by the extent of the spin distribution
in r space. The phase modulation function corresponding to the stepping of the time
interval has the following form for a single spin not J-coupled to another:
h,(t1,'1) = e'12*1"1, [3-4]
where vx = uJI-k is the resonance frequency of the spin and t2 is the time during which
free precession occurs in the absence of gradients. In general, for a spectral distribution of
spins, S'(^¡), the modulated signal can be expressed by
s'(tx) = £S'[*/i)h'[t1,i/i)di/i [3_5]
where N expresses the limits of integration imposed by the extent of the spectral distribu
tion. If one takes FT to mean "the Fourier transform of, then ideally
FT[s(kr)] = S(r¡) ,
[3-6]
FTIs'itx)] = S>i) .
[3-7]
In the convolution spectral imaging technique, the amplitude of the phase-encode
gradient and the time interval are stepped simultaneously. Thus kr and t2 are related by a
constant, rj:
h = t?kr [3-8]
The resulting doubly-modulated signal may be expressed by the product of Eqs. [3-3] and
[3-5]. By the convolution theorem (Bri74), the Fourier transform of this product is the
convolution of the spatial and spectral spin distributions. This transformed result may be
mapped into frequency space in which case the spatial spin distribution appears as a func
tion of frequency scaled by r¡. Denoting the convolution by an asterisk and using
Eq. [3-8], the Fourier transform can be written


79
FT[s(kr)s'(t1)] = r?[S(r?^)*S'(^i)] [3-9]
Alternatively, the Fourier transform of the product may be mapped into r space. The
spectral spin distribution then appears as a function of spatial position scaled by the
inverse of rj:
FT[s(kr)s'(t1)] = (l/?)lS(rj)*S'(r¡/7)] [3-10]
Each of the convolution functions shown in Eqs. [3-9] and [3-10] is the spatial distri
bution of the sample spins offset by their spectral resonance frequencies or vice versa. A
plot of one of these functions for a hypothetical two-compartment sample containing two
different chemical species is shown in Fig. 3-la. The function is the convolution of the r
space information shown in Fig. 3-lb with the spectrum shown in Fig. 3-lc. The two
resonance lines are assumed to have widths much smaller than their chemical shift differ
ence and B0 inhomogeneity is assumed to be negligible. Another way of viewing Fig. 3-1 a
is as a series of spectral frequencies, each of which has an identical spatial field-of-view
spread around it. The origin in spatial coordinates for a particular chemical species coin
cides with its spectral frequency. Any spatial offset of a nucleus from the origin appears
as an offset from its spectral frequency. The idea of convolving two types of information
has been exploited in spectroscopy by the accordion experiment of Bodenhausen and
Ernst (Bod81) and the three-frequency experiment of Bolton (Bol82). The convolution of
a spatial axis and the zero-quantum spectrum has been accomplished recently in a similar
manner by Hall and Norwood (Hal86a).
Although Eqs. [3-9] and [3-10] were derived for phase-encoding, the mathematics
are similar for frequency-encoding. The two processes differ in that free precession,
which is the source of spectral information, occurs in the presence of a gradient during
frequency-encoding. The result is still a doubly-modulated signal, and Eqs. [3-9] and
[3-10] still hold. These equations represent the origin of the "chemical shift artifact"
(Bab85, Dwy85, Soi84) which is observed when a gradient applied during signal acquisi
tion is too weak to obliterate the separation between spectral resonances.


b)
A/2
+
B
B/2
-ti
c)
1
1
1
1
>
'a 0 >
1
!b
FIG. 3-1. The convolution of spectral and spatial information, (a) The convolution of the
spatial and spectral information shown in (b) and (c), respectively, (b) The spatial profile
of a pair of vessels containing differing amounts of two molecular species, A and B. (c)
The spectrum corresponding to (b).
OO
O


81
The pulse sequence for the acquisition of a two-dimensional convolution spectral
image is shown in Fig. 3-2a. The interval between the initiation of the pulse sequence at
successive phase-encode steps, TR, is kept constant. The time to the echo formation, TE,
is also kept constant; thus, the T2 relaxation weightings of the phase-encoding steps are
equivalent. Phase-encoding of spectral information is accomplished by moving the 7r
pulse through the window labelled T by incrementing t2 by an amount Atj in a stepwise
fashion. The spectral width of the phase-encoded axis is 1/Atj.
The implementation of the pulse sequence of Fig. 3-2a with the frequency-encode
gradient, gfe, defined as the z gradient and the phase-encode gradient, gpe, defined as the
x gradient, would result in an z,vt versus \,ux image where z and x define the spatial axes
and vt and vx define the impressed spectral information. Each planar zx image
corresponding to a spectral resonance will lie on a diagonal passing through the origin if
the sample is centered on the crossing point of the z and x axes as defined by the z and x
gradients. The planar images will be located on a diagonal parallel to that passing through
the origin if the sample is spatially offset from this crossing point. The center of each
image will be separated from that of its chemically shifted neighbor along the diagonal by
the chemical shift difference of the two species multiplied by a scaling factor. If the spec
tral widths convolved with each spatial dimension were identical, this scaling factor would
equal VI.
Like other nonselective spectral imaging methods, convolution spectral imaging tech
niques collect spectral data from all points of the spectrum including those which do not
contain resonances. The advantage lies in the ability to compress three dimensions of data
into two dimensions for planar spectral imaging or four dimensions into three for volume
spectral imaging with a corresponding saving in total acquisition and processing time. It
will be shown that the techniques are best suited for small samples whose frequency spec
tra are composed of well-separated resonances and that these techniques optimize the
available signal-to-noise ratio. Thus convolution spectral imaging may be most applicable


FIG. 3-2. Pulse sequences for convolution spectral imaging, (a) The pulse sequence of
two-dimensional convolution spectral imaging. The timing of the sequence and the labels
given to the various timing intervals are indicated at the bottom. The non-selective n
pulse is moved through the window labelled T by an amount, Atj, in a series of M steps at
the same time as the amplitude of the phase-encode gradient, g^, is varied stepwise. RF =
radiofrequency transmitter, gfe = frequency-encode gradient, gpe = phase-encode gra
dient, gs) = slice gradient, acq = data acquisition.


selective 7t 12
RF O
K
g
fe
g,
pe
gsl
kvHh
-T
TE
T -
t-j/2
TR


FIG. 3-2continued, (b) The pulse sequence for three-dimensional convolution spectral
imaging. The non-selective t pulse is moved through the window labelled T by an
amount, Atlt for each of a series of M steps at the same time that the amplitude of the
phase-encode gradient, gp^, is varied stepwise. The ir pulse is then moved through the
window by an amount, Ats, for each L step as the amplitude of the phase-encode gra
dient, gp^j, is changed simultaneously. The cycle of M phase-encode steps is repeated for
each L step, gp^ phase-encode gradient (M steps), gp.2 = phase-encode gradient (L
steps), all other abbreviations as in (a).


HI *|
j bOB \j 1
\*Z} z/l-t\+-z/l\ 2/l+||Z/l*-^Z/eJ-*|
l.adB
9*6
dd


86
to the field of NMR microimaging (Agu86).
3.2 Experimental
Experiments were carried out using the spectrometer and RF coil described in
Chapter 2, section 2.2. The pulse sequences used are shown in Fig. 3-2. Quadrature
detection was employed in all experiments. Slice selectivity was used if the spectral reso
nances of the sample under investigation were sufficiently close that the application of a
strong gradient during slice selection could eliminate slice misregistation due to the chem
ical shift effect. Otherwise, the entire sample volume was excited. Thus a
frequency-selective sine-shaped tt/2 RF pulse and a gradient normal to the imaging plane
were used to select slices for all JH images. The 7r/2 RF pulse was nonselective in all 19F
imaging experiments. All tt pulses were nonselective.
The effects of imperfect n pulses and DC imbalance between the quadrature chan
nels were corrected by using a two-step phase cycle during signal averaging impressed on
the w/2 pulse and receiver. This phase cycle had the effect of cancelling out artifacts due
to magnetization which was not phase-encoded. However, as pulse repetition times were
decreased for more rapid image data acquisition, the ability of the cycle to suppress
artifacts was diminished. The origins of these residual artifacts were from
non-steady-state magnetization which was not phase-encoded and instrumental errors
causing DC offset. Therefore gradient proportional phase incrementation (GPPI) of the n
pulse was also added to the two-step phase cycle (Gra86). This had the effect of moving
artifacts which appeared at the midpoint of the phase-encoded axis to the edges of that
axis without increasing the image acquisition time by requiring signal averaging. The ini
tial desire was to impress GPPI on the rr/2 pulse and the receiver to shift both types of
artifact. However, this was not allowed by the instrument design. Thus GPPI had to be
impressed on the n pulse. This could only shift artifacts resulting from magnetization
which had not been phase-encoded. Artifacts due to instrumental DC effects remained at


87
the midpoint of the phase-encoded axis. Fortunately, these were quite minimal. The phase
cycle for the pulse sequence of Fig. 3-2a is shown in Table 3-1, and that for the sequence
of Fig. 3-2b is shown in Table 3-2.
To reduce truncation artifacts, it was ensured that the echoes corresponding to the
middle steps in the phase-encoding process were centered in the acquisition window. This
was accomplished in a set-up mode before acquisition of the image data by turning all
phase-encode gradients off and placing the it pulse in the center of the time delay, T (see
Fig. 3-2). The amplitude of the frequency-encoding gradient on during the time, r, was
then carefully adjusted. This set-up mode was also used for pulse calibration and for
choosing transmitter and receiver gain settings. In most cases the values of TR and TE
were chosen such that very little relaxation attenuation was allowed.
3.3 Practical Aspects of the Method
Convolution spectral imaging is most effective for small objects containing molecular
species with a spectrum of well-separated single peak resonances, which implies that the
technique may find its greatest application in the field of NMR microimaging. The effec
tiveness of the method is enhanced if short phase-encode gradient times are employed.
These conclusions become apparent if one considers the sources of signal modulation and
if one makes the restriction that the frequency separation between resonances is greater
than the frequency spread caused by the applied gradients. This restriction is necessary to
prevent the overlap of the spatial images comprising the convolution spectral image. How
ever, it will be shown that this restriction may be relaxed for certain sample geometries.
Consider a single spin, not spin-spin coupled to another, which is subjected to the
two-dimensional technique of Fig. 3-2a, again letting z be the frequency-encode dimen
sion and x be the phase-encode dimension. During the acquisition time, t2, the signal is
modulated in part by the function
h"(mt2)
-¡2^(1/! + mi)t2
[3-11]


Table 3-1
Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases
Phase-Encode
Step
Signal
Average
Phases
Pulses
Receiver
tt/2
7T
1
1
0
0
0
1
2
180
0
180
2
1
0
180
0
2
2
180
180
180
M
2
180
180
180
Refer to Fig. 3-2a and its legend for the notation corresponding to this table.


Full Text

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UNIVERSITY OF FLORIDA
3 1262 08554 2917


NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988

To my wife and best friend, Lisa
*U OF F LIBRARIES

ACKNOWLEDGMENTS
A number of people have contributed to my scientific growth during the past five
years. I would like to thank my chairman, Dr. Wallace S. Brey, Jr., for allowing me to
obtain my degree through the Chemistry Department while working with the Department
of Radiology at Shands Hospital. 1 also thank him for encouraging me to speak about
topics in NMR at informal and formal meetings. Those experiences were invaluable. I
also thank my co-advisor, Dr. Tom Mareci, who guided me virtually step-by-step through
my years as a member of his research group, letting me stand on my own when he thought
I would learn more that way. For providing generous financial support during my time at
the University of Florida, I thank Dr. Kate Scott. Finally, I thank these three and Drs. \V
Weltner and D. E. Richardson for the time which they have devoted to this dissertation.
Other members of the research group have helped me over the years. 1 especially
value one-on-one discussions with Dr. Richard Briggs and Dr. Ralph Brooker. Richard
often helped me define my personal goals, and Ralph, who rarely accepted a premise until
proven to his satisfaction, clarified much of the science of NMR for me. 1 also thank Pro¬
fessor E. Raymond Andrew and Dr. Jeff Fitzsimmons for their lectures and advice. Visit¬
ing professors have shown me applications of NMR other than those on which I focused
my research effort. These included Dr. Reszo Gaspar, Dr. Eugene Sczescniak, and Dr.
Attilio Rigamonti, who helped to inspire Chapter 2. The postdocs who have passed
through the group have combined science with fun and taught me a little about both dur¬
ing their time at the University of Florida. For this I thank Gareth Barker, Suite
D0nstrup, and Dikoma Shungu. I also acknowledge the students, Bill Brey, Randy Duens-
ing, Willie Kuan, Lori Lewis, Jintong Mao, Laura Pavesi, Dan Plant, and Bill Sattin. all of
whom have affected my way of thinking about things scientific and otherwise.
in

Without technical support, a grad student’s life becomes quite a bit more difficult.
For their contributions at the University of Florida, I thank Barbara Beck, Don Sanford,
and Ray Thomas. I also thank Dave Dalrymple, Chris Sotak, and Subramaniam Sukumar
of Nicolet Instruments and General Electric NMR for their enormous help in understand¬
ing and writing the software which drove our NMR instrumentation. Much of this
dissertation could not have been done without them. I also thank Katherine Nash and
Teresa Lyles, the wonderful secretaries of the Magnetic Resonance Imaging Department at
Shands, and Mike Ingeno, Jack Dionis, Jim Kassebaum, Tim Vinson, and Dr. Ray Ber¬
geron of the Health Center at the University of Florida.
Finally, I thank my parents for their steadfast support during those periods of doubt
which every graduate student feels from time to time, and my wife’s parents, who have
taken such good care of
whose presence has made
me while I have lived in Florida. I also thank my wife, Lisa,
my life much happier during the production of this dissertation.
iv

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES viii
ABSTRACT x
CHAPTER
1 NMR THEORY AND MULTIDIMENSIONAL NMR I
1.0 Introduction 1
1.1 Formalism for the Description of NMR Experiments 2
1.1.1 The Density Operator 2
1.1.2 Equation of Motion of the Density Operator 3
1.1.3 The Density Operator at Thermal Equilibrium 5
1.1.4 Basis Operators 8
1.1.5 Hamiltonian Operators Describing
Spin System Perturbations 11
1.1.5.1 The Effect of a Static Magnetic Field 13
1.1.5.2 The Effect of a Linear Field Gradient 17
1.1.5.3 Application of Radiofrequency (RF) Pulses 18
1.1.5.4 Application of Phase-Shifted RF Pulses 19
1.1.6 Observable Magnetization 21
1.2 The Two-Dimensional NMR Experiment 21
1.2.1 Phase and Amplitude Modulation 22
1.2.2 General Description of the 2D Experiment 24
1.3 Multidimensional NMR 27
2 SPECTRAL IMAGING AND APPLICATIONS TO THE STUDY
OF DYNAMIC POLYMER-SOLVENT SYSTEMS 30
2.0 Introduction 30
2.1 Analysis of a Spectral Imaging Method 32
2.2 Experimental 38
2.3 Results 43
2.4 The Susceptibility Model 65
2.5 Conclusion 69
3 CONVOLUTION SPECTRAL IMAGING 75
3.0 Introduction 75
3.1 The Convolution Spectral Imaging Method 75
v

3.2 Experimental 86
3.3 Practical Aspects of the Method 87
3.3.1 The Effect of Sample Geometry 92
3.3.2 The Interaction of TE and and the Spatial Resolution 104
3.3.3 Extension to Three Spatial Dimensions 105
3.3.4 Signal-to-Noise Considerations 126
3.4 Convolution Spectral Imaging at High Field 128
3.5 Conclusion 140
4 QUANTIFICATION OF EXCHANGE RATES
WITH RED NOESY SPECTROSCOPY 142
4.0 Introduction 142
4.1 The NOESY Pulse Sequence 145
4.2 The RED NOESY Pulse Sequence 155
4.3 Problems Unique to the RED NOESY Sequence 155
4.4 Experimental 159
4.5 Results 164
4.6 Discussion 164
REFERENCES 184
BIOGRAPHICAL SKETCH 189
vi

LIST OF TABLES
TABLE page
1-1 Effects of the Single-Element Operators 10
1-2 Solutions to the Equation of Motion of the
Density Operator Expressed as Cartesian Space Rotations 14
1-3 Effect of a Product Operator Hamiltonian on Terms of
a Cartesian Product Operator Basis Set 15
1-4 Transformations of Cartesian Operators by Phase-Shifted
RF Pulses 20
2-1 2D Spectral Imaging: Relative Pulse and Receiver Phases 40
2-2 Calculated Volume Susceptibilities 67
2-3 Calculated Susceptibility Shifts at 2 Tesla 68
3-1 Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases 88
3-2 Three-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases 89
4-1 Relative Pulse and Receiver Phases for RED NOESY 160
4-2 Temperature Dependence of Relaxation Rates, Exchange Rates,
and Free Energies of Activation for DMF, DMA, and DMP 169
4-3 Activation Parameters of DMF Found Using RED NOESY Data 170
4-4 Activation Parameters of DMF: Literature Values 171
4-5 Activation Parameters of DMA: Literature Values 172
4-6 Activation Parameters of DMP: Literature Values 173
4-7 Relaxation Rates of DMF Methyl Protons 182
vii

LIST OF FIGURES
FIGURE pase
2-1 The pulse sequence corresponding to the evolution period of
a spectral imaging method 33
2-2 A pulse sequence for two-dimensional spectral imaging 36
2-3 Sample orientation for studies of PMMA solvation 42
2-4 Spectral images used to observe the quality of the
static field homogeneity 45
2-5 The spectrum of a piece of PMMA partially
dissolved in chloroform at 2 T 48
2-6 Spectral images of PMMA in deuterated chloroform
after 61 minutes of solvation 49
2-7 Spectral images of PMMA in deuterated chloroform
after 81 minutes of solvation 52
2-8 Spectral images of PMMA in deuterated chloroform
at late stages of solvation 54
2-9 Spectral images of PMMA in perdeuterated acetone
after 180 minutes of solvation 58
2-10 Spectral images of PMMA in perdeuterated acetone
after 200 minutes of solvation 61
2-11 Spectral images of PMMA in perdeuterated acetone
at late stages of solvation 63
2-12 PMMA solvation 71
3-1 The convolution of spectral and spatial information 80
3-2 Pulse sequences for convolution spectral imaging 83
3-3 The effect of sample geometry 93
3-4 The interaction of TE and the spatial resolution 106
3-5 Three-dimensional convolution spectral imaging 112
3-6 Convolution spectral imaging at high field 130
viii

146
4-1 The NOESY pulse sequence
4-2 Peak intensity behavior as a function of
the mixing time and exchange rate 152
4-3 The RED NOESY pulse sequence 156
4-4 A plot of ln(k’) versus 1000/T for DMF 165
4-5 A plot of ln(k’) versus 1000/T (corrected) for DMF 166
4-6 A plot of ln(k’/T) versus 1000/T for DMF 167
4-7 A plot of ln(k’/T) versus 1000/T (corrected) for DMF 168
4-8 Four NOESY spectra of DMF obtained using the RED NOESY sequence 174
4-9 Behavior of peak intensities of NOESY spectra of
DMF at 347 °K 178
IX

Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
December, 1988
Chairman: Dr. Wallace S. Brey, Jr.
Major Department: Chemistry
This dissertation examines new methods and applications of existing methods in
multidimensional nuclear magnetic resonance. Chapter 1 contains the outline of a simpli¬
fied theory for the description of pulse NMR experiments. The theory is a mathematical
formalism based in quantum theory and describes the effects of the applications of a static
magnetic field, radiofrequency pulses, and linear field gradients to simple spin-1/2 sys¬
tems. In the context of the formalism, the basic principles of multidimensional NMR
experiments are described. The material of Chapter 1 is the basis for all subsequent
chapters. Chapter 2 outlines the application of a two-dimensional method to the study of
slices of solid polymethylmethacrylate dissolving in liquid deuterated chloroform and per-
deuterated acetone. The method allowed the correlation of NMR spectra with their spatial
positions. Time-dependent frequency shifts of the spectral frequencies of the polymer
were observed in the vicinity of the polymer-solvent interfaces. The direction of the fre¬
quency shifts were found to depend on the orientation of the slice relative to the static
magnetic field. The magnitudes of the shifts depended on the solvent. The effects could
be explained in part by a simple magnetic susceptibility model. Chapter 3 describes new
pulse sequences for two- and three-dimensional spectral imaging. The methods produce
convolutions of spatial and spectral information, which saves experimental time. It is
x

shown that the methods are best suited for small samples with slowly relaxing nuclei,
immersed in a strong static field. The methods appear to be ideal for NMR microimaging.
Finally, Chapter 4 describes a new multidimensional pulse sequence for the quantification
of exchange rates called RED NOESY. The sequence is applied to three N,N-dimethyl
amides to determine the exchange rates of the N,N-methyl groups. For one of these
molecules, N,N-dimethylformamide, the Arrhenius energy of activation and frequency
factor and the enthalpy and entropy of activation were found by obtaining RED NOESY
data at several temperatures. The values found were 20 kcal mol-1, 28, 19 kcal mol-1, and
-5 cal mol-1 K-1, respectively.
xi

CHAPTER 1
NMR THEORY AND MULTIDIMENSIONAL NMR
1.0 Introduction
Since the original concept of two-dimensional NMR spectroscopy was proposed
(Jee71), the basic principles of 2D NMR have been used to develop a large variety of mul¬
tipulse experiments which in turn have found almost limitless applications.
Two-dimensional spectroscopy has been used extensively to unravel networks of coupled
nuclear spins and to estimate distances between nuclei in molecules with molecular
weights up to several thousand daltons. The field has developed such that implementation
of certain 2D NMR techniques is becoming almost routine for organic and inorganic
chemists. Similarly, two-dimensional imaging techniques are now highly automated and
routinely used clinically. Clearly, multidimensional NMR is now a domain shared by
physicists, chemists, and physicians. Yet communication between the three groups has
been impeded in part by their different concerns and professional languages. However,
the multidimensional NMR experiments performed by these groups are united by theory.
One of the functions of this chapter is to describe a theory which can be used to analyze
many NMR experiments.
The chapter first outlines the origin of a formalism for the evaluation of pulsed
NMR experiments. The formalism is then used to describe the behavior of spin systems
when immersed in a static magnetic field and exposed to radiofrequency (RF) magnetic
field pulses and linear magnetic field gradients. In the context of the formalism, the con¬
cept of modulation, crucial to the understanding of multidimensional NMR, is explored in
the subsequent section. Next, a general discussion of two-dimensional methods is
presented. The final section of the chapter is devoted to the extension of the ideas of 2D
NMR to three and four dimensions and the problems associated with these experiments.
1

2
The material of this chapter forms the basis for the understanding of the multidimensional
experiments described in Chapters 2, 3, and 4.
1.1 Formalism for the Description of NMR Experiments
This dissertation is primarily concerned with the development and application of
pulse sequences to very simple spin-1/2 systems. Therefore, it is usually sufficient to
analyze an experiment applied to a single spin or to two weakly coupled spins. In this
section, a simplified formalism is developed from basic quantum theory to describe the
behavior of spin systems immersed in a static magnetic field and exposed to pulse
sequences made up of time delays, radiofrequency pulses, and linear field gradients. The
classical description of an NMR experiment which arises from the Bloch equations will
not be used because this approach ignores many of the effects created by NMR pulse
sequences. The quantum theory of NMR has been described by Howarth, et al. (How86),
Levitt (Lev88), Mareci (Mar80), Slichter (Sli80), and S0rensen, et al. (Sor83). The discus¬
sion of the following sections draws heavily from these references.
1.1.1 The Density Operator
The state of a spin system can be described by a wavefunction rp which in turn can
be represented as a linear combination of orthonormal functions un (the eigenstates)
weighted by coefficients cn,
V>=EcnUn- [1-1]
n
For a system of N spin-1/2 nuclei, there are 2N eigenstates. In the Dirac notation, the
expectation value of an observable, Q, is described by
= [ 1 -2a]
=
m n
[ 1-2b]

3
= E E CmCn • [l-2c]
m n
The products of the coefficients, c^cn, may be arranged to form a matrix which is called
the density matrix. If 0 is time-dependent, then either the eigenstates or the coefficients
can carry the time dependence. In the "Schrodinger representation”, the eigenstates are
allowed to be time-independent and the time-dependence is carried by the coefficients.
Thus, the time dependence of an expectation value is carried by the density matrix. The
density operator, p, is defined by
= cnc¿. [1-3]
With this definition, the expectation value for an observable can be written in terms of
the density operator. By Eqs. [1-2] and [1-3] and the orthonormality of the eigenstates un,
= E E ’ [ 1-4a]
m n
= Tr [p Q], [l-4b]
where Tr is an abbreviation for the trace of the matrix. This important equation estab¬
lishes the relationship of the density operator to the operator, Q, which describes an
observable phenomenon. If the trace of the equation is nonzero, the phenomenon is
observed. Mathematical forms of Q and p relevant to NMR are shown in section 1.1.6.
1.1.2 Equation of Motion of the Density Operator
An alternative way of describing the state of a spin system at a particular time is to
use the density operator itself, rather than the density matrix. Thus, perturbations of the
spin system which occur during an NMR pulse sequence appear as changes in the density
operator. Each of the perturbations which will be discussed in this chapter can be
described by a time-independent Hamiltonian operator, H, whose mathematical form
depends on the nature of the perturbation. The behavior of a spin system under the
effect of a perturbation is described by the equation of motion of the density operator.

4
This is an expression for the rate of change of the density operator with time and may be
derived from the time-dependent Schrodinger equation using the method described by
Slichter (SH80). The time-dependent Schrodinger equation is
[1-5]
where H = h/2tr and h is Planck’s constant. Substituting Eq. [1-1] into Eq. [1-5] produces
*£ %-un = scnHun [1-6]
n 01 n
since only the cn’s are time-dependent. Multiplying both sides of the equation from the
left by u¿ leads to
WE k*un = E cnuk*Hun . [1-7]
n n
Integrating and using the orthonormality of the basis functions produces
â–  E cn . [1-8]
The equation of motion of the density operator may be derived by taking the deriva¬
tive of Eq. [1-3] with respect to time and substituting Eq. [1-8] into the result:
3 dcm 3cn .
at ~Cn at + at Cm
= C„[(-i/H)E cm]* + [(-Í/H)e cn]c¿
m n
= (i/l*)[E cnCm - s cnc^]
[1-9a]
[1 -9b]
[1 -9c]
= (i/tf)[E - E ]
m n
= (i/^)[ - ] .
[1-9d]
[1-9e]
Thus the time derivative, or the equation of motion, of the density operator is

:>
f> = (i/H) [P,H].
[1-10]
When H is time-independent, a solution of the equation of motion is
p(t) = eH/tf)Ht p(0) e^)»1,
[1-11]
which may be verified by taking the derivative with respect to time. This equation is of
prime importance because it describes the effect of applying a Hamiltonian operator to
the density operator. In terms of the NMR experiment, the equation describes the effect
of some perturbation, represented by the Hamiltonian, to the state of the spin system,
represented by the density operator. Analyzing the effect of an NMR pulse sequence on a
spin system requires knowledge of the mathematical forms of several operators. One of
these is the density operator which describes a spin system at thermal equilibrium. This is
the normal starting point for a density operator analysis since it describes a spin system at
rest in a static magnetic field. The other required operators are the Hamiltonian operators
describing the various perturbations which can occur. The perturbations commonly
found in NMR pulse sequences include the applications of a static magnetic field,
radiofrequency pulses and linear field gradients to a spin system.
1.1.3 The Density Operator at Thermal Equilibrium
To establish the starting point for a density operator analysis, the form of the density
operator which describes the spin system at thermal equilibrium, p(0), is needed. At ther¬
mal equilibrium, dp/dt = 0. This implies that p(0) has a form determined by the Hamil¬
tonian operator describing the interaction of the static magnetic field with the unper¬
turbed spin system. For a single spin, this Hamiltonian is
X^H = -'yBol, = wl, ,
[1-12]
where the Larmor relationship,
u = -TfB0 ,
[1-13]
has been used; 7 is the nuclear gyromagnetic ratio, B0 is the strength of the applied static

6
field, It is the operator for the z component of angular momentum, and w is the preces-
sional frequency of the nuclear spin. For a nucleus of spin I in eigenstate um,
H|Um> = Emlum> = MJum> = U>m | Um> , [1-14]
where m is one of 21 + 1 values in the range I, I-1, • • • , -I and Em is the energy of the
mth eigenstate.
The populations of the eigenstates are given by the diagonal terms of the density
matrix, cmc^. At thermal equilibrium these are described by the Boltzmann distribution
factors, pm:
Pm = cmcm = . [1-15]
For a set of n states of energies, En, a Boltzmann factor expresses the probability that the
mth state is occupied:
e- Em/kT
Pn>= -E„/kT ’ [1-16]
D c
n
where k is the Boltzmann constant and T is the temperature of the spin system. If
kT » En, Em, then the exponential terms can be approximated and the Boltzmann distri¬
bution factor becomes
l-(Em/kT)
Pm E l-(En/kT) ' I1'17]
n
Because there are 21 + 1 possible values of En, the sum in the denominator equals 21+1.
Collecting equations produces
= (2I+l)'1(l-(Em/kT)) . [1-18]
Finally, because H|um> = Em|um>, it follows that
p = (2I+l)_1(l-(H/kT)) = (2I+1)'1 - (2I+ir1(MI/kT) . [1-19]
This is the form of the density operator at thermal equilibrium. The constant term,

7
(21+1 )_1, cannot be made observable and so it may be dropped. Making the definition,
^ = -(21+ir1(Hw/kT), [1-20]
the reduced density operator at thermal equilibrium may be written
o = 0lt. [1-21]
Because the density operator, p, and the reduced density operator, a, are related by con¬
stant terms, the equation of motion of the reduced density operator may be written
directly:
ó = (i/H) [a,H]. [1-22]
A solution of this equation for a time-independent Hamiltonian is
o(t) = e(-¡MHt o(0) eO/#)»1, [1-23]
and the expectation value of an observable is
= Tr [o Q] . [1-24]
Equation [1-21] is the usual starting point for a pulse sequence analysis. For a sys¬
tem of N spins, the reduced density operator at thermal equilibrium is
° ~ + ^2^2i + • • • + » [1-25]
where the different spins are labeled by the subscripts. The spins may be treated
independently, applying a perturbation Hamiltonian to each separately. When all the
spins have the same gyromagnetic ratio, all of the /9’s are nearly equivalent and the spin
system is referred to as homonuclear. This dissertation deals solely with such systems and
so the /9 terms will not be written explicitly for the pulse sequence analyses which follow.
The derivation of the expression for a at thermal equilibrium was made using the
assumption that kT » Em = jtu;m. To test the validity of the assumption, let T = 298 K,
(j) = 2?r(300 x 106) rad sec'1 and m = 1/2. These are typical values for a proton precessing
in a magnetic field of 7.1 tesla at room temperature. A simple calculation with appropri-

8
ate values of the Boltzmann and Planck constants shows that kT is approximately 40000
times the value of tfwm. Thus the "high temperature" assumption is valid for this system.
The nuclei studied for this dissertation were 19F at 2 tesla (w = 27r(80.5 x 106) rad sec"1) at
room temperature, at 2 tesla (w = 27t(85.5 x 106) rad sec'1) at room temperature, and
at 7.1 tesla (w = 2tt(300 x 106) rad sec'1) at temperatures ranging from 293 to 363 K. For
all of these cases, the high temperature assumption holds and Eq. [1-21] is a valid starting
point for pulse sequence analysis.
1.1,4 Basis Operators
The density operator can be written as a linear combination of time-independent
basis operators, Bg, weighted by time-dependent coefficients, b8(t):
^0 = £ bs(t)Bg . [ i -26]
8
This set of basis operators can also be used to describe the Hamiltonians which describe
possible perturbations to a spin system. The most popular set of basis operators has been
proposed by S0rensen, et al. (Sor83). These are the Cartesian "product operators" pro¬
duced by the multiplication of the single-spin, Cartesian angular momentum operators, Ix,
ly, and IB, and the unity operator, E. The basis set for N spin-1/2 nuclei consists of 4N
product operators. For a system of two spin-1/2 nuclei the basis consists of:
The unity operator:
(1/2)E
One-spin operators:
I lx ' ^ly ’ ^li ’ ^2x ’ ^2y » ^2r
Two-spin operators:
2Ilx^2x » 2Ijxl2y , 2Ilxl2t ,
^^ly^2x * ^ly^2y * ^ly^2z ’

9
^lt^2x » ^lt^2y ’ •
The subscripts 1 and 2 are used to distinguish the two spins. For much of the pulse
sequence analysis of this dissertation, the density operator describing the state of a spin
system will be written in the Cartesian product operator basis. This basis is particularly
suitable for describing the effects of the applications of a static magnetic field, radiofre¬
quency pulses, and linear field gradients.
The Cartesian operator basis is not well-suited for describing observable terms of the
density operator, and so it is necessary to convert to another basis set consisting of pro¬
ducts of the single-element operators, I“, 1^, I+, and I\ These operators are directly
related to the energy levels of the spin system. It has been shown that a nucleus of
spin-1/2 immersed in a static magnetic field can be in one of two eigenstates, um, with
energies Em = ±( 1 /2)|4cl). In this dissertation, the state of lower energy is called P and that
of higher energy, a. The application of a single-element operator to an eigenstate can
produce a change in the spin state. The effects of these operators are shown in Table 1-1.
Table 1-1 shows that the only operators which cause a change in the spin state and lead to
observable signals are the I+ and I" operators. Thus, the single-element basis is most use¬
ful for expressing the observable terms of the density operator (section 1.1.6).
The Cartesian operators are directly related to the single-element operators by the
following:
yEk = y(I£ + 1(f) ,
[1-27a]
lkx=yUk + I¿).
[1-27b]
Iky = -yUk - Jk) >
[1-27c]
Ik, = ydk“ - Ilf).
[l-27d]
where k indicates a particular spin. The Cartesian product operator basis may be rewrit¬
ten as a single-element product operator basis by the use of these relationships.

Table 1-1
Effects of the Single-Element Operators
Operator
1+ j- ja
Initial Spin State Final Spin States
a
0
0 P a 0
Q 0 0 p

11
In summary, two basis sets of operators may be used to describe the density operator.
The effect of an NMR pulse sequence on a spin system is described most conveniently in
the Cartesian product operator basis. To determine which terms of the density operator
are observable, however, the single-element product operator basis is more convenient.
The pulse sequence analyses of this dissertation use whichever basis set is most convenient
for emphasizing a particular aspect of the sequence.
1.1.5 Hamiltonian Operators Describing Spin System Perturbations
The form of the density operator which describes a spin system at thermal equili¬
brium has been shown, as have two different basis sets of operators which can be used to
describe the density operator in general. Still needed for an NMR pulse sequence analysis
are the Hamiltonian operators which describe the various perturbations which can be
applied to a spin system. These will be shown later in this section, but first an alternative
way of expressing the solution to the equation of motion of the reduced density operator
is described.
The effect of a perturbation on a spin system is described by the solution to the
equation of motion of the reduced density operator. The form previously shown in
Eq. [1-23] is not very convenient for visualizing the behavior of the spin system under the
effect of a perturbation. However, the product operator basis is made up of Cartesian
angular momentum operators and so it is possible to describe the effect of a perturbation
on the density operator in Cartesian terms by showing that the solution to the equation of
motion of the density operator is equivalent to a rotation in a three-dimensional Cartesian
space. The following analysis follows the method described by Slichter (Sli80). Recall
that for a time-independent Hamiltonian a solution to the equation of motion of the
reduced density operator is a(t) = cr(0) e(*WHt. For this analysis, let the initial
state of the density operator be o(0) = Ix and the Hamiltonian be such that h_1Ht = lL. As
will be seen in section 1.1.5.1, this describes a spin, not at thermal equilibrium, whose

12
state is changing under the effect of chemical shift precession. Define
f(¿) = e'*1' Ix eWt. [1-28]
By the relationship between operators A and B,
A elB = elB A when [A,B] = 0 , [1-29]
and the commutators of the angular momentum operators, the first derivative of f($) is
f(^) = e^IMyei*I,1 [1-30]
and the second derivative of f($) is
n*) = - e'^1* Ix e^1*. [1-31]
Thus the relationship of Eq. [1-28] is a solution of the second order differential equation
f"(¿) - m = 0 . [1-32]
Another solution to this equation is
f(<¿) = a cos(^) + b sin() , [1-33]
which may be verified by substitution. Thus
e’1^1 Ix e1*1* = a cos() + b sin() . [1-34]
The coefficient, a, may be found by finding the solution of Eq. [1-33] when 4> equals 0
and using Eq. [1-28]. Finding the solution of the first derivative of Eq. [1-33] when <(>
equals 0 and using Eq. [1-30] gives the coefficient, b. The results are that
a = Ix , [ 1 -35a]
b = ly . [ 1 -35b]
Finally, the following is obtained:
e"1*1* lx e1*1* = Ixcos(<¿) + Iysin(<¿) .
[1-36]

13
Equation [1-36] shows that the quantum mechanical solution to the equation of
motion of the density operator has an analog in a Cartesian space described by axes
defined by the angular momentum operators. The angle of rotation, , originates from the
applied Hamiltonian. Similar expressions may be found for Hamiltonians containing Ix or
Iy terms. These are outlined in Table 1-2. To find the effect of a particular Hamiltonian
on one of the Cartesian operators, the function

tonian must be determined. A Hamiltonian consisting of a one-spin operator affects the
terms of a product of Cartesian operators separately. For example, let the initial state of
the density operator be the product operator IlxI2l and the Hamiltonian be such that
H_1Ht = Then the solution to the equation of motion of the reduced density operator
is
e'^Ilz Iixl2. e*Ilz = dixcos(*) + Ilysin(*))I2, . [1-37]
Only the terms of spin 1 have been affected.
It is also possible for the Hamiltonian to contain products of Cartesian operators.
The only such Hamiltonian relevant to this dissertation is such that h^Ht = 2llt\2z and so
alternative expressions for the reduced density operator solution e~‘^21lzl2Ec7(t) e“*’"llzl2z
must be derived for the various product operators which make up a(t). Again this can be
done using the method described by Slichter under the assumption that operators belong¬
ing to different spins commute (Sli80). The results are shown in Table 1-3.
1.1.5.1 The Effect of a Static Magnetic Field
By using the expressions of the previous section, it is possible to examine the effects
of a static magnetic field on a weakly coupled spin system. This perturbation occurs after
a spin system is immersed in a static magnetic field and during an NMR pulse sequence
when all radiofrequency pulses and field gradients are removed. The Hamiltonian for the
interaction of the field with a system of N spins is

Table 1-2
Solutions to the Equation of Motion of the Reduced Density Operator
Expressed as Cartesian Space Rotations
<7(t)
e'^Ixa(t) e‘*Ix
e'^cKt) e‘*Iy
e'^otOe^
(1/2)E
(1/2)E
(1/2)E
(1/2)E
lx
lx
Ixcos(<¿) - IEsin(<¿)
lxcos(0) + lySin(^)
ly
IyCos(^) + IEsin ()
ly
lycos(0) - lxsin(0)
h
lEcos(4>) - Iysin(i^)
I*cos(^) + Ixsin(<¿)
I,

Table 1-3
Effect of a Product Operator Hamiltonian on
Terms of a Cartesian Product Operator Basis Set
CT(t)
g-i^2 Iltl2i gi^2 Iljl2z
(1/2)E
(1/2)E
Ilx
Ilxcos(<¿) + 2Ilyl2zsin(0)
!iy
Ilycos(^) - 2IlxI2lsin(<0)
>1.
Iu
^2x
I2xcos () + 2I1II2ysin(¿)
^2y
I2ycos(<¿) - 2IlBI2xsin(0)
I2,
^ixhx
2Ilx12x
^lxhy
21lxI2y
2I1xI2i
2i1xI2zcos(^) + llysin(tf>)
2IlyI2x
2llyl2x
2IlyI2y
21 ly12y
2IiyI2r
2IlyI2zcos(<¿) - llxsin(0)
21al2x
2IilI2xcos(<0) + l2ysin(0)
2Iltl2y
2IlzI2ycos(¿) - l2xsin(tf>)
2IuI2.
2Iu12i

16
i=N j=(N - 1) k=N
P'1H=E^1¡,+ E E ^JjkVk* (j i=l j=l k=l
where u>¡ is the angular precession frequency of the ith spin, Jjk is the coupling constant
between spins j and k, and the spins are labeled by the subscripts. The Hamiltonian may
be broken into two parts, which can be applied independently. The chemical shift pre¬
cession term is
i=N
X-1H=EWiIit. [1-39]
i=l
The spin-spin coupling term is
j=(N - 1) k=N
H ^H = 2 e 2trJ I Ikt (j j=l k=l
which is rewritten in terms of the Cartesian product operator basis by moving the factor
of two (Sor83):
j=(N -1) k=N
H_1H = j] s trJjk(2IjEIkz) (j j=i k=l
The evolution of the angular momentum operators under the chemical shift Hamil¬
tonian is described by the fourth column of Table 1-2 where 4>x = u>¡t :
Iix —■ IixCOs(w¡t) + Iiysin(ü)¡t) ,
[1-41a]
Iiy — Iiycos(w¡t) - Iixs¡n(w¡t) ,
[1-4 Ib]
Ita-Iu-
[ 1-4 le]
Equation [1-41] shows that in the Cartesian frame transverse components of angular
momentum rotate through an angle x - w,t under the effects of chemical shift precession.
The longitudinal component is not affected.
From the expressions of Table 1-3 where jk = 7rJjkt , the evolutions of the one-spin
operators under the coupling Hamiltonian are given by
Ijx ^ IjxCOS^Jjkt) + 21jyIkEsin(7rJjkt) ,
[1-42a]

17
1 jy — Ijycos(7rJjltt) - 2Ijxlkesin(7rJjkt) , [ 1 -42b]
Ij.-V n-42c]
Two-spin operators also evolve under the Hamiltonian which describes spin-spin cou¬
pling. Examples are:
2IjxIk. —• 2IjxIklcos(irJjkt) + Ijysin(jrJjkt) , [1 -43a]
21jyIkr — 2IjyIklcos(7Tjjkt) - Ijxsin(jrJjkt) . [ 1 -43b]
1.1.5.2 The Effect of a Linear Field Gradient
The application of a linear field gradient has effects similar to those of chemical shift
precession but the expression for the angle x is different. The Hamiltonian has the form
1ft.. I1'44]
where r = x, y, or z and T¡ is a vector describing the spatial position of the ith spin along
the r axis. The field gradient is defined by the partial derivative and may be written
H_1H-i
3Bo
dr
[1-45]
for the component of the gradient along the r axis. By this Hamiltonian, <¿¡ = ; thus
the angular momentum operators evolve as follows:
Iix — Iixcos(']fGrrit) + IiySinÍTfGrTit), [ 1 -46a]
Iiy — Iiycos(TfGrr¡t) - I^sinÍTGrrjt) , [1 -46b]
— Ii. - [l-46c]
The degree of rotation of the transverse components depends upon the strength of the
applied gradient, Gr, and the position of the nuclear spin along the r axis. Two-spin
operators also evolve under the effect of a linear field gradient. Examples are:
2IjxIkl — 2IjxIklcos(nrGrrjt) + 21jyIklsin('yGrrjt) , [ 1 -47a]

18
2IjyIkl — 2IjyIklcos('yGrrjt) - 2IjxIktsin('|Grrjt) . [ 1 -47b]
1.1.5.3 Application of Radiofrequency (RF) Pulses
NMR experiments are performed using an alternating magnetic field applied in the
plane transverse to the static field to create a torque on the magnetic moments. The Ham¬
iltonian for the total applied magnetic field (static plus alternating) is written
H_1H = -7B0It - 'yB1[Ixcos(n0t) + Iysin(fi0t)] , [1-48]
where Bj is the strength of the applied alternating field, fi0 is its angular frequency and t
is the duration of application. This Hamiltonian is time-dependent, but on transformation
to a frame rotating with angular frequency fi0, the time-dependence of the alternating
field vanishes and the Hamiltonian becomes
*-1H--(iVw,)I, + n1Ix, [1-49]
where the Larmor relationship and the definition flj = ')B1 have been used (Sli80). The
Hamiltonian can be simplified considerably under two conditions. At resonance, fl0 =
and the off-resonance It term vanishes. Also, the term becomes negligible when the
applied alternating field is strong enough that Hj » (fio-c^) (a "hard" pulse). For these
cases, the Hamiltonian becomes
[1-50]
In terms of Table 1-2, = fijt. However, this product, called the "tip angle", is more com¬
monly labeled 6. In this notation, the application of an RF pulse to each of the Cartesian
operators produces:
lX -* lx »
ly — lycos(0) + l£sin(0) ,
I. - l,cos(0) - lysin(0) .
[1 - 51 a]
[1-51b]
[ 1-51c]

19
1.1.5.4 Application of Phase-Shifted RF Pulses
If the alternating field is applied at an angle ( with respect to the x axis in the rotat¬
ing frame, the Hamiltonian is rotated in the transverse plane by this angle. The applied
field is said to be "phase-shifted" relative to the x axis. In one-dimensional NMR experi¬
ments, phase-shifting of the RF pulses of an NMR pulse sequence is done commonly in
conjunction with signal averaging to isolate certain terms of the density operator. This
process is called phase-cycling and has been used for all of the experiments described in
this dissertation. Phase cycles can vary from one pulse sequence to another and so a
description of the effect of a particular cycle will accompany the description of each of
the NMR pulse sequences used.
The density operator under the effect of a phase-shifted RF pulse may be written
o(t) = e' i€1‘ e'iWx eifI‘ a(0) e"iiIe eiiIx ei{1*. [ 1 -52]
By using the method outlined in section 1.1.5, the following relationships can be derived:
Ix -*■ Ix (cos2(0 + sin2(£)cos(0)) + ¡y cos(£)sin(£)(l-cos(0)) - ir sin(£)sin(0) , [ 1 -53a]
Iy -► Ix cos(£)sin(£)(l-cos(0)) + ly (cos2(£)cos(0) + sin2(£)) + I* cos(£)sin(0) , [ 1 -53b]
It -* Ix sin(£)sin(0) - iy cos(£)sin(0) + It cos(0) . [ 1 -53c]
Most older NMR spectrometers are capable of executing RF phase shifts in 90 degree
increments only, that is £ can equal 0, 90, 180, or 270 degrees. The work in this disserta¬
tion has been done within this limitation. Table 1-4 summarizes the effects of RF pulses
with these shifts on the three angular momentum operators. The terms of this table are
valid only when the phase shifts are perfect increments of 90 degrees and when
off-resonance effects are negligible. These assumptions are made throughout this work.

Table 1-4
Transformations of Cartesian Operators by Phase-Shifted RF Pulses
Initial Operator
0
£(degrees)
90 180
270
Final Operators
I*
lx
IxC*-IeS0
lx
Ixc0+Its0
ly
IyC0+IES0
ly
IyC*-I,Sfl
ly
h
I,C0-IyS0
Itc0+Ixs0
Itc0+Iys0
Iec0-lxs0
Notation: c =
cos and s =
sin.

21
1.1.6 Observable Magnetization
The forms of the operators necessary to analyze an NMR pulse sequence have been
shown. However, the form of an observable term of the density operator has not yet been
derived. In section 1.1.3, it was shown that in terms of the reduced density operator the
expectation value of an observable equals Tr [a Q], In most modern NMR experiments,
the signal is detected in quadrature. The operator corresponding to the detectable signal
may be written as the sum of the two orthogonal transverse Cartesian angular momentum
operators:
I+=(Ix + iIy). [1-54]
Thus the expectation value for the detected signal is
= Tr [a(lx + ily)] , [1-55]
or transforming to the single-element operator basis,
= Tr [al*] . [1-56]
This implies that o(t) must consist of I' operators to give a nonzero trace and thus observ¬
able magnetization. In terms of the two-spin Cartesian product operator basis, only the
terms Ilx, Ily, I2x, and I2y are directly observable. However, the terms 2IlxI2z, 2IlyI2z,
2IlzI2x, and 2IlzI2y can evolve under the effects of the chemical shift and coupling Hamil¬
tonians to produce observable terms. The unity operator is never observable. Each of the
remaining seven operators of the basis can be made observable only in an indirect way by
subjecting it to an RF pulse Hamiltonian.
1.2 The Two-Dimensional NMR Experiment
The theory necessary for the analysis of most NMR experiments has been intro¬
duced. This section makes use of the theory to describe the basic principles of
two-dimensional NMR methods. First, the required concepts of phase and amplitude
modulation are introduced using the density operator formalism. The results of the

22
application of the Fourier transform to phase- and amplitude-modulated signals are
shown. Next, a general description of the 2D experiment is presented, and a mathematical
expression is derived in terms of the density operator. Finally, a short discussion of some
of the restrictions of 2D NMR follows.
1.2.1 Phase and Amplitude Modulation
The concepts of phase and amplitude modulation are necessary for the understanding
of any NMR experiment and in particular are the keys to understanding two-dimensional
experiments. Both types of modulation can be created by a pulse sequence and appear in
a density operator expression. From the expressions for the evolution of the density
operator introduced in section 1.1, precession due to chemical shifts, spin-spin coupling,
and applied gradients corresponds to the multiplication of product operators by cosine
and sine terms. For example, applying an RF pulse of tip angle 6 = n/2 and phase £ = 0 to
a single spin initially at thermal equilibrium produces a transverse term which evolves
under chemical shift precession for a time, t, to produce
C7(t) = - Iy cos(wt) + Ix sin(wt) . [ 1 -57]
The amplitude of each of the transverse components oscillates during t according to the
angular frequency, w, of the spin. Thus the components are "amplitude-modulated". If
the components are combined using the single-element operator basis, then
(7(t)=y(lViu*-reiut). [1-58]
The phases of the 1+ and I' terms oscillate according to w and the terms are said to be
"phase-modulated". This example shows that signal phase modulation may be produced
by properly combining amplitude-modulated signals.
The type of modulation is important because the Fourier transforms of the two types
of signal are very different. By using the notation of Keeler and Neuhaus (Kee85), the
Fourier transform (FT) of a damped phase-modulated signal may be written as

23
FT [eiwte't/T2] = A" + iD‘ ,
where A and D are absorption and dispersion
functions have the forms:
A‘(w) =
D-(w) =
(l+((u; - n0)2T22)) ’
(<â– > - n0)T22
(l+((a;-n0)2T22)) ’
[1-59]
Lorentzian functions, respectively. These
[1-60a]
[1-60b]
where T2 is the damping constant, u> is the angular frequency of the spin, and n0 is the
angular frequency of the rotating frame. By these functions, the sign of u relative to fl0 is
unambiguous. For a damped amplitude modulation term,
FT [c(wt)e't/T2] = FT [y(eilJt + e‘ihlt)e't/T2] [ 1 -61a]
= y[A+ + iD+ + A* + iD" ] , [1 -61 b]
where A* and D' have been defined and
A» =
D» =
t2
(i+((iv + n0)2T22» ’
(a; + n0)T22
(l+((w + O0)2T22)) '
[1-62a]
[1 -62b]
These results show that amplitude modulation does not allow discrimination of the sign of
a signal. The Fourier transform is a mixture of the two types of absorption and dispersion
functions and so the sign of w relative to the ft0 is ambiguous. This has important conse¬
quences for many 2D NMR methods where the density operator describing the detected
signal contains amplitude-modulated terms; these include 2D spectroscopy methods such
as COSY (Jee71) and NOESY (Jee79), and 2D imaging methods such as the "rotating
frame" method of Hoult (Hou79). Because the sign of a signal is desirable information,
methods have been developed to convert the amplitude-modulated signals into

24
phase-modulated ones. This idea will be used for the experiments of Chapter 4.
1.2,2 General Description of the 2D Experiment
The goal of any 2D NMR experiment is the correlation of two processes which occur
during the experiment. This requires that the signal be doubly-modulated as a function
of the two processes. Such a signal can be produced by a pulse sequence made up of at
least three periods, commonly called preparation, evolution, and detection (Aue76). In
accord with the accepted notation, the evolution and detection periods will be labeled tj
and t2, respectively. Some 2D pulse sequences contain an additional period, often called
mixing, which is sandwiched between the evolution and detection periods (Bax82). The
mixing period will be labeled rm. Each of these periods can be made up of a number of
spin perturbations, including time delays, RF pulses, and linear field gradients.
Each of the four periods of a 2D pulse sequence has a unique function. Occurring at
the outset of the pulse sequence, the preparation period perturbs the spin system from
thermal equilibrium. Assuming the use of the Cartesian product operators, the events of
the preparation period produce transverse terms in the density operator expression which
describes the effect of the pulse sequence. After the preparation period, the terms of the
density operator can change according to the effects of the perturbations of the evolution
period. The terms of the evolution period do not have to be observable. It is the function
of the perturbations of the optional mixing period to convert such terms, if desired, into
observable terms. These are detected when the spectrometer receiver is switched on. The
terms of the density operator then evolve under the effects of the perturbations of the
detection period.
For any two-dimensional NMR method, the density operator expression which
describes the detected signal of a 2D pulse sequence contains modulation terms which
reflect the effects of the perturbations which preceded detection. This can be shown by
using the density operator formalism to derive a general mathematical description of a 2D

25
NMR experiment. The density operator just before detection begins, tTftj), is a function
of the perturbations of the evolution period, tj. By the equation of motion of the density
operator, oitj) evolves during the detection period, t2, under the effect of some Hamil¬
tonian to produce
^t1,t2) = e(-i/>»H%(.1)e(i/|,)Hti. [1-63]
The detected signal is found by using the trace relation
= Trloftj.ta)!4] . [1-64]
This equation is simply the 2D case of the expression derived earlier for any a, Eq. [1-56].
Substituting for a(tj,t2),
= Tr[e( i/^)Ht2o(t1) e(i/*)Ht2r] . [1-65]
Because the trace is invariant to cyclic permutation of the operators, the expression for
becomes
- |l-66]
This relationship shows that the initial phases and amplitudes of the terms which evolve
during t2 are determined by the events preceding detection. Pulse sequences can be
designed to control what arrives at the spectrometer receiver, and the detected signal will
have a "memory" of the perturbations which occurred before the receiver was switched
on.
Although o(tltt2) contains information about the events which preceded detection,
the application of a Fourier transform with respect to the detection period would produce
a frequency spectrum which describes the behavior of the spin system only during detec¬
tion. This occurs because the modulation functions corresponding to the events preceding
detection are constants for each application of a pulse sequence. To map out the behavior
of the spin system under the effects of the perturbations which precede detection, the
pulse sequence must be repeated with a change in one or more of the perturbations of the

26
evolution period, while holding the perturbations of the detection period constant. Build¬
ing up a matrix of such signals is equivalent to sampling the signal during a changing t!
interval. Application of a Fourier transform with respect to the change made during the
evolution period produces a frequency spectrum of the behavior of the spin system during
that period. This is the principle behind every multidimensional NMR experiment. Since
the terms of the density operator describing the detected signal of a 2D experiment can be
written as a function of the evolution and detection periods, ta and t2, the double Fourier
transform of the signal matrix is a function of two variables, Fj and F2, which have units
of frequency.
Because the Fourier transform applied with respect to the evolution period produces
a frequency spectrum corresponding to only the perturbation which has changed during
the interval, the effects of different spin system perturbations can be completely
separated during evolution. The key to these experiments and others is that although
many perturbations may act simultaneously during evolution, it may be possible to
arrange the pulse sequence such that only one perturbation is changed when the pulse
sequence is repeated while other perturbations remain constant. Thus, the effect of the
changing perturbation can be mapped out. For example, it is possible to produce a fre¬
quency spectrum of only the spin-spin coupling information (Aue76). Another possibil¬
ity, found in many imaging experiments, is the production of a frequency spectrum
related to the spatial positions of the nuclei (Ede80).
The effects of different perturbations acting during the detection period cannot be
separated, unlike those of the evolution period. Thus, the Fourier transform with respect
to the detection period is a convolution of the various frequency spectra corresponding to
the effects which occur during that period. It may be possible to adjust experimental
parameters such that the effect of a desired perturbation dominates the effects of
unwanted ones. The application of broadband decoupling is a good example of this;
enough decoupling power must be applied to remove the coupling information. Other

27
examples appear in some NMR imaging experiments where a field gradient term must
dominate the static field term. This can be ensured in many cases by using a strong gra¬
dient; just how strong the gradient must be is discussed in Chapter 3. It is possible to take
advantage of convolutions for certain applications, also shown in Chapter 3.
1.3 Multidimensional NMR
The basic 2D NMR method may be extended to include more dimensions by
appending more evolution periods to a 2D pulse sequence. In principle, an N-dimensional
space could be described by a density operator of the form <7(tn,t12, • ■ • t^N-i)’^) where
N - 1 processes occur during N - 1 evolution periods and are correlated with the events of
the detection period. Data processing requires an N-dimensional (ND) Fourier transform.
At present, the largest value of N described in the literature is four (Hal85).
A number of limitations have impeded the implementation of ND NMR methods.
One of the most troublesome is the amount of spectrometer time required to acquire the
data matrix. Typically, several seconds are required to acquire a single time-domain sig¬
nal and allow the spin system to return to equilibrium. If signal averaging is required,
several minutes to several hours may be needed to collect a 2D data set. This time
increases dramatically as the number of dimensions increases, possibly encompassing
several days if good resolution is required in each dimension. The data provided by a
multidimensional data set must be worth the time required to obtain it. This limitation is
not surmountable by technological improvements.
Currently available technology has also imposed some limitations on the implementa¬
tion of ND NMR methods. Normally, multidimensional data sets are stored in digital
form on magnetic media. As the number of dimensions increases, so do the media storage
requirements. For example, a typical 2D data set might consist of 256 time-domain sig¬
nals, each of which is digitized into IK, 16-bit words. Thus this matrix requires one-half
megabyte for storage. The data required to describe a third dimension would then require

28
several megabytes. Already this approaches the current limits of modern NMR spectrom¬
eters. Thus even if the time is available to acquire large data sets, there is restricted space
to store them. Speedy processing and the display of multidimensional data are also prob¬
lems. A solution which is gaining acceptance is processing and display using large
off-line computers with dedicated mathematics hardware. These difficulties are identical
to those encountered in the the early years of 2D NMR, when data handling was done
using small-memory minicomputers.
Because of the difficulties, examples of ND NMR methods are rare but do exist for
both imaging and spectroscopy. Multidimensional NMR methods were first proposed for
imaging applications, possibly because these were concerned with the correlation of spatial
information from the three dimensions of Cartesian space, which is easily visualized. The
first ND NMR experiment was proposed by Kumar and coworkers (Kum75). This was a
method for obtaining a three-dimensional spatial image, but technological limits at the
time precluded its implementation. True three-dimensional results were obtained later by
Maudsley, et al. (Mau83). However, instead of correlating information from the three
spatial dimensions, the method of Maudsley, et al. was used to correlate NMR spectra with
their two-dimensional spatial spin distributions to produce some of the earliest "chemical
shift images". In contrast, the development of ND NMR spectroscopic methods has been
pursued only recently, possibly because these are not concerned with the physical space
defined by imaging methods. Some pseudo-3D methods have been implemented (Bod81,
Bod82, Bol82). These are actually 3D methods in which the effects of two different Ham¬
iltonians acting during two different evolution periods are multiplied by stepping two
time intervals in concert. Thus three-dimensional data is compressed into two dimen¬
sions. True 3D spectroscopy was first implemented by Plant, et al. by combining the 2D
COSY and J-resolved spectroscopy pulse sequences into one (Pla86). A very similar
experiment has been described by Vuister and Boelens (Vui87). In a similar manner,
Griesinger and coworkers have described a number of 3D pulse sequences created by the

29
combination of 2D sequences (Gri87a, Gri87b, Osc88). One of the justifications for these
experiments is that the use of more than two dimensions may help to separate peaks which
overlap even in two dimensions.

CHAPTER 2
SPECTRAL IMAGING AND APPLICATIONS
TO THE STUDY OF DYNAMIC POLYMER-SOLVENT SYSTEMS
2,0 Introduction
One of the challenges of NMR imaging is the correlation of spatial and spectral
information. This chapter is not concerned with methods in which a single spatial point is
chosen as the source of a spectrum ("localized spectroscopy"); a review of these methods
may be found elsewhere (Aue86). The focus here is on methods in which image informa¬
tion is correlated with spectral information. Many of these methods have been reviewed
by Aue (Aue86), Brateman (Bra86b), and Brady, et al. (Bra86a). The methods fall into
two broad categories. One category includes those techniques in which a single resonance
is chosen as the source of the spectral information contained in an image (Bot84, Dix84,
Haa85a, Hal84, Jos85, Ord85). These methods will be referred to as selective spectral
imaging techniques. Most require a separate experiment for each resonance of interest, an
obvious disadvantage if several resonances are to be examined and time is at a premium
Other methods circumvent this problem by obtaining spatial information simultaneously
for every spectral resonance (Bro82, Cox80, Man85, Sep84). These will be referred to as
nonselective spectral imaging techniques. The majority of these are Fourier imaging
methods. Most encode the entire spectrum, which may include spin-spin coupling infor¬
mation, by allowing free precession in the absence of applied gradients. Fourier transfor¬
mation with respect to the precession period produces the spectrum which can be corre¬
lated with the spectral information encoded during other time periods in the pulse
sequence. A potential disadvantage of nonselective spectral imaging methods is that in
acquiring data from the entire spectrum, regions which do not contain resonances must be
sampled, resulting in some amount of unused data matrix. Also the sampling of the spec¬
tral information in addition to the spatial information necessitates a longer total
30

31
acquisition time. Both selective and nonselective spectral imaging methods suffer from
sensitivity to B0 inhomogeneity. Variations in the static field can cause separate reso¬
nances to broaden to the extent that overlap of the spectral lines occurs. Clean separation
of spatial images as a function of the resonance frequency then becomes difficult or
impossible.
The focus of this chapter is the use of spectral imaging to study dynamic systems
consisting of a solid polymer dissolving in liquid solvents. The use of NMR imaging to
study solid or near solid materials has not been widely applied. The resonance linewidths
of solids are usually very broad and the application of pulsed field gradients cannot dom¬
inate the chemical shift dispersion without severe penalties in signal-to-noise. In addi¬
tion, the T2 relaxation times of solids are often very short, precluding the use of
spin-echo imaging methods. Thus most studies of solids using NMR imaging methods
have been observations of an NMR-detectable liquid, usually water, which has become
distributed within the solid either by diffusion or force. Studies of woods (Hal86b), oil
cores (Rot85), glass-reinforced epoxy resin composites (Rot84), nylon (Bla86), rock
(Vin86), various building materials (Gum79), and ceramics (Ack88) have been done in this
way. Virtually no work has been done in which spectra are correlated with spatial posi¬
tions in solids, although Hall and coworkers have examined a piece of sandstone soaked
with n-dodecane and water using a spectral imaging method (Hal86c).
This chapter describes the application of a nonselective spectral imaging method to
the observation of polymethylmethacrylate (PMMA) dissolving in deuterated chloroform
and in perdeuterated acetone. The work was based in part on the experiments of Mareci,
et al. in which two-dimensional proton NMR images of PMMA dissolving in chloroform
and deuterated chloroform were obtained at staggered time intervals during the solvation
process (Mar88). These authors used an imaging pulse sequence which produced images
whose contrast depended on the nuclear relaxation times, Tj and T2. It was found that
the spin-lattice relaxation times of the protons of the dissolved polymer were essentially

32
constant during solvation. Because changes in relaxation times can be related to changes
in nuclear mobility, Mareci, et al. were able to deduce that polymer solvation in the
PMMA-chloroform system is a first-order phase transition. During the course of their
work, Mareci, et al. were able to obtain a well-resolved proton spectrum of the dissolved
PMMA at a static field strength of 2 tesla. This indicated that it might be possible to
observe chemical changes occurring at the polymer-solvent interface by using a spectral
imaging method to observe the spectra corresponding to various spatial positions in the
polymer-solvent system. This chapter describes such studies. These differed from most
other NMR imaging studies of solid-liquid systems in two ways:
(1) Sample spectra were correlated with their spatial position.
(2) The spectra observed were from the dissolved polymer, not from the solvent.
The results of these experiments showed large frequency shifts of the spectral resonances
of PMMA in the vicinity of the polymer-solvent interface. The magnitudes and direc¬
tions of the frequency shifts could be explained by a simple theoretical model based on
changes in magnetic susceptibility at the polymer-solvent interface. Thus the shifts in the
resonance frequencies were probably not the result of chemical changes.
2.1 Analysis of a Spectral Imaging Method
The spectral imaging method chosen for these studies was a variation of that of
Maudsley, et al. (Mau83). This section describes a density operator analysis of the
preparation and evolution periods of the pulse sequence. The sequence has no mixing
period. The preparation and evolution periods of the method are described by the pulse
sequence shown in Fig. 2-1. Consider the application of this sequence to a system of two
coupled spins, labeled 1 and 2, with coupling constant J and located at two spatial posi¬
tions, rj and r2. The spins are assumed to be at thermal equilibrium initially so the den¬
sity operator is
o(0) = p(\u + 12e) .
[2-1]

33
FIG. 2-1. The pulse sequence corresponding to the evolution period of a spectral imaging
method. The timing of the sequence and the labels given to various time intervals are
shown at the bottom. RF â–  radiofrequency transmitter, g - gradient.

34
For the rest of the analysis, the /? term will be implied, as described in Chapter 1. section
1.1.3. The preparation period begins with the application of the 7t/2 pulse with phase
£ = 0, which produces transverse terms in the density operator expression. The evolution
period starts immediately after the first pulse. The tranverse terms evolve under the com¬
bined effects of the static field and linear field gradient Hamiltonians during the time
period tj = r + (tj/2). For spin 1, the result is
cr(tí,spin 1) = - cosO^tj) cos(TrJti) [Ily cos^n-k^!) - Ilx sin(2trkrrj)] [2-2]
+ cos^ti) sin(7rJti) [2I1xI2i cos(27rkrr1) + 2IlyI2l sin(27rkr r:)]
+ siniwjtj) cos(trJti) [llx cos(27rkrr1) + Ily sin(27tkrr1)]
+ sin(o>ití) sin(7rJt¡) [2IlyI2e cos(27rkrrj) - 2IlxI2z sin(27rkr rx)] ,
where is the angular precessional frequency of spin 1 and kr = T(Grr/(27r), where Gr is
the gradient amplitude. After application of a tr pulse with phase £ = 0, the density
operator evolves in the absence of gradients during tí = T - (t1/2) to produce
cr(ti,spin 1) = Ilx sin^x-k^ + Wj(tí - tí)) cos(7rJ(tí + tí'))
[2-3a]
+ Ily cos(2trkrr1 + Wj(tí - tí')) cos(7rJ(ti + tj))
- 2I1xI2i cos(27rkrr1 + Wj(tí - tí)) sin(trJ(tí + tí'))
+ 2IlyI2l sm^k^ + w^tí - tí)) sin(7rj(tí + tí')) .
Finally, this expression can be rewritten in the single-element basis by applying Eq. [1-27]
to produce
cr(tí',spinl) = -yla+ cos(ttJ(tí + tí')) e
¡(2»krri + wi(tí - tí))
[2-3b]
+ ylf COS(7Tj(tí + tí')) e~‘(
-ylí’If sin(trj(tí + tí')) e1
+yli+lf sin(7Tj(ti + tí')) e1
-i(2jrkrri + wjíti - tí))
i(25riirri + (Ji(ti - tj))
i(2irkrr1 + W1(ti - tj))

35
- jlflf sin(7rJ(tí + t¡')) e'i(2,rkrri + u'i(tí ‘ *3)
+jlflf sin(7rJ(tí + tí')) e’i(2^rri + Wl(ti -t{)] .
This analysis shows that modulation produced by the pulse sequence of Fig. 2-1 is a
function of three properties of a spin: spatial position, spectral frequency, and coupling
constant. All of the terms of Eq. [2-3] are phase-modulated as a function of the spatial
position and the spectral frequency of spin 1. The terms are amplitude-modulated as a
function of the coupling constant, J. The modulation functions can be mapped out by
changing the functions in a stepwise fashion, as described in Chapter 1, section 1.2.2.
This process is often referred to as "phase-encoding". The spatially-dependent modula¬
tion functions can be mapped out by changing the value of kr, which can be done by
altering the gradient amplitude and holding the time delays, tí and tí', constant. The value
of kr could be altered by changing the duration of the gradient, but because tí and tj are
constants, r must be constant, and so to allow a change in the spatially-dependent modula¬
tion functions, the gradient amplitude must be varied. The modulation functions which
depend on spectral frequency are not eliminated except in the special case where tí = tj,
as shown by Eq. [2-3]. For the spectral imaging method of this chapter, experimental
conditions were chosen such that tí = tí'. In an alternative method, the terms of Eq. [2-3]
which depend on spectral frequency can be mapped out by altering the time delays, tí and
tí', in a stepwise fashion and holding the value of kr constant. This is the basis of a spec¬
tral imaging method which is described in Chapter 3. This type of phase-encoding does
not eliminate the modulation terms which depend on spatial position. For either
phase-encoding method, the modulation terms which are dependent on the coupling con¬
stant cannot be removed by adjusting the time delays. Thus, coupling always affects the
signal amplitudes.
The pulse sequence used for the studies described in this chapter is shown in
Fig. 2-2. It is a 2D spectral imaging method whose evolution period is identical to the

selective ni2
RF O
g
pe
\+\A/2+\\l T - t,/2
TE
TR
FIG. 2-2. A pulse sequence for two-dimensional spectral imaging. The timing of the
sequence and the labels given to the various timing intervals are given at the bottom. Spa¬
tial phase-encoding is accomplished by altering the gradient amplitude, gpe, in a series of
M steps. RF = radiofrequency transmitter, gp, = phase-encode gradient, g,, = slice gra¬
dient, acq = data acquisition.

evolution period of the pulse sequence of Fig. 2-1. The time delays r, t1/2, and T - tj/2
are held constant. For the experiments of this chapter, the values of these delays were
chosen such that, in the notation of Eq. [2-3], the sum of tj and one-half the duration of
the selective 7t/2 pulse equalled tj'. Under the assumption that the duration of the selec¬
tive pulse is negligible relative to the durations of the delays, r and tj/2, the terms of
Eq. [2-3] which are dependent on the spectral frequencies of the spins vanish. The cou¬
pling terms remain but can vanish when
where n = 0, 1/2, 1, 3/2, .... Thus for studies of coupled spin systems using the pulse
sequence of Fig. 2-2, the delays must be chosen with care. Since r is fixed,
phase-encoding of the spatial information is accomplished by altering the gradient ampli¬
tude. This is indicated in Fig. 2-2 by the multiple bars describing the different ampli¬
tudes of the gradient, gpe. The signal is sampled in the absence of gradients during the
time period, t2. Modulation with respect to the detection period is a function of the pre¬
cession frequencies of the spins and not of their spatial position. Thus the pulse sequence
is a method for correlating spectra with spatial position.
The preparation periods of the pulse sequences of Figs. 2-1 and 2-2 are different.
The former consists of a nonselective 7t/2 RF pulse which excites the portion of the sam¬
ple which lies within the RF transmitter coil. The preparation period of the pulse
sequence of Fig. 2-2 is a "slice-selective" tt/2 pulse. The combination of a
frequency-selective, "soft" RF pulse (indicated by the diamond) and a field gradient (the
"slice" gradient, gsl) allows the excitation of a plane of sample spins. Only spins in the
plane experience a 7t/2 tip angle. The slice thickness of the plane is controlled by the
duration of the RF pulse, the pulse shape, and the amplitude of the slice gradient.
Because the Hamiltonians describing the effects of a static field and a linear field gradient
both act during the slice-selection process, different but adjacent slices are selected for

38
each chemically-shifted species in the sample. The result is slice misregistration. This
effect can be overcome by applying a strong gradient, but this restricts the choice of slice
widths. This effect is an example of the convolution of spatial and spectral information
discussed in Chapter 3.
A final aspect of the pulse sequence of Fig. 2-2 is the effect of the time periods, TE
and TR, on the signal amplitude. The echo time, TE, is the period from the center of the
soft RF pulse to the center of the spin echo. The period between successive initiations of
the pulse sequence is the repetition time, TR, and includes a delay to allow the perturbed
spin system to relax toward thermal equilibrium. The amplitude of the echo is weighted
by the product e TE/,T2(i - e TR/,Tl), where Tj and T2 are the nuclear spin-lattice and
spin-spin relaxation times, respectively. Thus two conditions must exist for the echo to
have appreciable amplitude. The TE must be short relative to T2 and the TR must be on
the order of or greater than Tj. These conditions are not so easily met when obtaining
images of solid or semisolid materials, because the T2 values can be short.
2.2 Experimental
All experiments were carried out using a General Electric CS1-2 NMR imaging spec¬
trometer equipped with a 2 T Oxford Instruments superconducting magnet with a 31 cm
clear bore diameter. With shim and gradient coils installed, the working clear bore was
reduced to a 23 cm diameter. The maximum attainable gradient strength was 0.03
mT mm'1. The RF coil was a slotted tube resonator built in house by the author; it was
tunable to both and 19F frequencies. For these studies, only protons were detected.
The RF coil had a length of 80 mm and a diameter of 56 mm with an effective RF mag¬
netic field over a length of 40 mm.
The pulse sequence used is shown in Fig. 2-2. Quadrature detection was used for all
experiments. Table 2-1 shows the phase cycle which was used; it performed several func¬
tions. Signals which had not felt the effects of phase-encoding were moved to the edges

39
of the spatial axis by phase alternation of the n pulse with every other phase-encode step
(Gra86). These signals arose because of imperfect n/2 and n pulses and spin relaxation
during t¡. The phase alternation of the n/2 pulse and receiver with signal averaging can¬
celled out the effects of imperfect n/2 pulses and imbalance in gain between the two qua¬
drature detection channels. Slice misregistration due to the chemical shift effect was
assumed to be negligible and a frequency-selective sine-shaped n/2 RF pulse and a gra¬
dient normal to the imaging plane were used to select slices for all images. All n pulses
were nonselective. Only the second half of the echo was acquired to circumvent problems
of centering the echo in the acquisition window and to allow digitization of the signal
until it decayed fully.
For all images, the soft n/2 pulse duration was 1 millisecond and the n pulse duration
was 86 microseconds. The spectral width was 2000 Hz and 256 complex points were
acquired for each phase-encode step, producing a spectral resolution of 7.8 Hz per point.
Thirty-two phase-encode steps of two signal averages each were performed. The
field-of-view along the spatial axis was 64 mm and so the resolution was 2 mm per data
point. The slice width was 5 mm. The phase-encode gradient duration was 4 mil¬
liseconds. The repetition time, TR, was 15 seconds and the echo time, TE, was 20 mil¬
liseconds.
An experiment was initiated by pouring approximately 4 ml of deuterated chloro¬
form or perdeuterated acetone into a circular Pyrex dish (5 cm diameter, 1.5 cm deep),
then centering a polymethylmethacrylate (PMMA) block in the solvent. The polymer and
solvents were at ambient temperature. This point was time zero for the solvation process
which ensued. The blocks were made of commercial grade PMMA, each approximately 1
cm on each side and 0.5 cm thick. The solvent covered approximately the lower half of
the block. The amount of solvent was based on the amount of chloroform empirically
found sufficient to dissolve the block slowly. Immediately after positioning the block, the
dish was then covered with a tight-fitting nylon cap to retard solvent evaporation. This

Table 2-1
2D Spectral Imaging:
Relative Pulse and Receiver Phases
Phase-Encode
Step
Signal
Average
Phases
Pulses
Receiver
7r/2
7T
1
1
0
90
0
1
2
180
90
180
2
1
0
270
0
2
2
180
270
180
M
1
0
270
0
M
2
180
270
180
Refer to Fig. 2-2 and its legend for the notation corresponding to this table.

41
assembly was placed on a plexiglás support and inserted into the RF coil. The coil was
then placed in the magnet bore. The sample and coil could be positioned reproducibly
and imaging could be started within a few minutes after placing the block in the solvent.
The sample orientation relative to the gradient axes is shown in Fig. 2-3. The thinness of
the sample along the y axis meant that slice selection along this axis was not necessary.
Imaging was initiated at approximately 20-minute intervals, each image requiring 16
minutes to acquire. The slow acquisition was necessary to reduce ridges parallel to the
phase-encode axis. These appeared if TR was so short that the transverse magnetization
could not relax to near equilibrium before the pulse sequence was applied again. Thus
some time-averaging of the solvation process was unavoidable. Initially, the x gradient
was used for choosing the slice and phase-encoding was done with the z gradient. For the
next image, the x and z gradients were interchanged, the z becoming the slice axis and the
x the phase-encode axis. This interleaving of images was continued until no more shifts
in the spectral resonances were observed or until the sample lines broadened considerably
due to solvent evaporation and subsequent sample solidification.
Because static field inhomogeneity could possibly cause frequency shifts of reso¬
nance peaks, care was taken to ensure that this did not occur. The static field inhomo¬
geneity was reduced by adjusting the electronic shim coils, using the same Pyrex dish,
nylon cap, dish support, and RF coil as those used for the PMMA-solvent imaging experi¬
ment, but replacing the sample with a few milliliters of CuS04-doped water. Crude
adjustments were made using a one-pulse sequence, attempting to increase the time con¬
stant of the signal decay. The spectral imaging sequence of Fig. 2-2 was then used to
assess the field homogeneity. Because water has a single XH resonance line, the homo¬
geneity was considered good when the frequency of the line did not change with position.
Several iterations of this procedure were adequate to set the shim currents. These current
settings were then used without modification during spectral imaging of PMMA solvation.
Examples of the spectral images obtained after shimming are shown in the contour plots

PMMA Sample
FIG. 2-3. Sample orientation for studies of PMMA solvation. The Cartesian coordinates
were defined by the axes of the gradient coils.

43
of Fig. 2-4. There was some shifting of the water resonance frequency with position
along the x axis. Shifts of the resonance frequency with z axis position were less notice-
able. The falloff of signal intensity along the z axis was the result of the sample extending
slightly outside of the active region of the RF coil.
2.3 Results
The proton spectrum of a piece of PMMA partially dissolved in deuterated chloro¬
form is shown in Fig. 2-5 along with a diagram of the methylmethacrylate monomer unit
and the assignments of the three peaks. Because the amplitudes of the peaks in the 2D
spectral images were weighted by the Tj and T2 values of their corresponding protons, it
was important to know these before imaging commenced. Based on the measurements of
Mareci, et al., estimates of the Tx values were 215 milliseconds for the ester methyl pro¬
tons, 83 milliseconds for the methylene protons, and 52 milliseconds for the methyl pro¬
tons (Mar88). Because the TR of the spectral imaging sequence was set to 15 seconds, TR
had virtually no effect on the amplitudes of the spectral peaks. However, since the T,
values of the protons could not be greater than their Ta values and since the pulse
sequence TE was set to 20 milliseconds, all of the peaks were attenuated by spin-spin
relaxation. The degree of attenuation relative to the case where TE was infinitely short
was estimated using the formula 100 x (1 - e’TE^T2). Assuming that each proton’s T2
equalled its Tx, the ester methyl, methylene, and methyl peaks should have been reduced
by at least 9%, 21%, and 32%, respectively. This was a best case estimate; in the actual
system the T2’s could have been much shorter than the Tj’s, resulting in a greater percen¬
tage of attenuation.
A spectral image of a PMMA block dissolving in deuterated chloroform is shown in
Fig. 2-6a as a stacked plot. Data acquisition was begun after 61 minutes of solvation. The
spatial dimension was defined by the x axis. The three spectral peaks of PMMA were
visible and changed in amplitude, linewidth, and resonance frequency along the spatial

FIG. 2-4. Spectral images used to observe the quality of the static field inhomogeneity.
Parameters and processing are given in the text, (a) A spectral image produced by
phase-encoding along the x axis. The frequency of the water resonance shifted slightly
downfield at positions away from the x axis origin.

1000
500
Water (doped)
T~rr
32 mm
o X
”32 mm
0
-500
-1000 Hz

FIG. 2-4—continued, (b) A spectral image produced by phase-encoding along the z axis.
The frequency of the water resonance did not change appreciably with spatial position.
Because the sample did not fit completely inside the RF coil, the image intensity was
reduced at the ends of the spatial axis.

Water (doped)
1000
500
0
-500
-1000 Hz

FIG. 2-5. The *H spectrum of a piece of PMMA partially dissolved in deuterated chloro¬
form at 2 T. The methylmethacrylate monomer unit is shown at the left and the three
proton resonance peak assignments are given by the arrows. The small spike at zero fre¬
quency was probably the result of quadrature channel imbalance. The shoulder in the
most downfield position belonged to residual chloroform.
-U
OO

PMMA in CDCI3
time of solvation: 61 min.
32 mm
X
FIG. 2-6. Spectral images of PMMA in deuterated chloroform after 61 minutes of solva¬
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
produced by phase-encoding along the x axis. The three spectral peaks of PMMA were
visible at regions distant from the origin where the polymer was more mobile.
VO

50
axis. The changes in linewidth and amplitude arose in part from the different degrees of
sample solidness. In the vicinity of the undissolved PMMA, near x = 0 mm, linewidths
were very broad and spread into the baseline. At the polymer-solvent interfaces where
solvation began, the mixture was gel-like and the linewidths began to narrow. Further
away from the polymer block a higher concentration of solvent was present and so the
three spectral lines were resolved. Because the polymer and solvent were clear, the degree
of solvation could not be followed visually. However, the spectral image showed that
some polymer had diffused to the edges of the sample dish. The data of the stacked plot
of Fig. 2-6a is shown as a contour plot in Fig. 2-6b to emphasize the changes in the reso¬
nance frequencies with spatial position. An overall curvature was seen which was due to
static field inhomogeneity as shown by comparison of Fig. 2-6b with Fig. 2-4a. In addi¬
tion, the resonance frequencies of the spectral peaks all shifted strongly downfield
(toward positive frequency) near the polymer-solvent interface. The maximum shift was
roughly 70 Hz. This number could not be measured exactly because of the contributions
of the static field inhomogeneity and the varying linewidths.
A second spectral image of the same sample was obtained by initiating data acquisi¬
tion after 81 minutes of solvation. The z axis defined the spatial dimension, and the
stacked plot result is shown in Fig. 2-7a. As in Fig. 2-6a, the three spectral lines
broadened as the concentration of solvent decreased near the polymer-solvent interface.
However, the contour plot corresponding to Fig. 2-7a, shown in Fig. 2-7b, shows that the
resonance frequencies shifted upfield near the interface then sharply downfield at the
interface. The shifts were about equal to those found for Fig. 2-6.
Spectral images of much later stages of solvation are shown in Fig. 2-8. At this
point, the polymer-solvent interface no longer existed and the mixture was distributed
fairly evenly throughout the sample dish. This is seen most clearly in Fig. 2-8a where the
spatial axis is the x. The rolloff at the ends of the z axis seen in Fig. 2-8b was due to the
sample not being completely inside the RF coil. Continued spectral imaging showed little

PMMA in CDCI3
time of solvation: 61 min.
32 mm
500 0 -500 -1000 Hz
FIG. 2-6--continued. (b) A contour plot of the data of (a). Downfield shifts of the three
spectral peaks were visible near the polymer-solvent interfaces.

PMMA ¡n CDCI3
time of solvation: 81 min.
1000 0 -1000 Hz
FIG. 2-7. Spectral images of PMMA in deuterated chloroform after 81 minutes of solva¬
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
produced by phase-encoding along the z axis. As in Fig. 2-6a, the three spectral peaks of
PMMA were visible at regions away from the polymer-solvent interfaces.
v-n
K>

PMMA in CDCI3
time of solvation: 81 min.
I
|—i—i—i—i—1—i—i—i—1—|—i—i—i—i—i—i—i—i—i—|—i—i—i—i—r
500 0 -500
32 mm
- 0 Z
1—1—1 -32 mm
-1000 Hz
FIG. 2-7—continued, (b) A contour plot of the data of (a). Upfield shifts of the three
spectral peaks were observed near the polymer-solvent interfaces and downfield shifts in
the vicinity of the solid polymer.

PMMA in CDCI3
1000 0 -1000 Hz
FIG. 2-8. Spectral images of PMMA in deuterated chloroform at late stages of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x axis after 482 minutes of solvation. The
polymer-solvent interfaces had vanished and the resulting solution had become homo¬
geneous.

PMMA in CDCI3
FIG. 2-8--continued. (b) A stacked plot spectral image obtained by phase-encoding
along the z axis after 502 minutes of solvation. The change in signal amplitude along the
spatial axis was caused by the RF inhomogeneity over the sample.
kyi
KJt

56
change in the spatial distribution of the spectra. As the solvent evaporated and the sample
mixture hardened, the linewidths broadened to the extent that the peaks could no longer
be seen.
To test whether the appearance of the spatially-dependent frequency shifts was a
function of the solvent used, the previous experiments were repeated using perdeuterated
acetone in place of deuterated chloroform. The time required for solvation was much
longer since perdeuterated acetone was a poorer solvent. In addition, the solvent had
some water contamination which contributed a fourth peak to the three-peak PMMA
spectrum. A stacked plot spectral image and its corresponding contour plot are shown in
Fig. 2-9. Data acquisition was begun 180 minutes after placing the PMMA block in the
solvent. An anomalous water peak is seen in the third most downfield position. Despite
the differences in solvent and solvation duration, Fig. 2-9 agrees qualitatively with
Fig. 2-6. The slight curvature of the resonance lines due to static field inhomogeneity
along the x axis was present as well as the downfield shifts of the peaks near the
polymer-solvent interface. However, the magnitudes of the shifts were roughly half that
found when using deuterated chloroform as the solvent. The spectral image obtained 20
minutes later using the z gradient for phase-encoding is shown in Fig. 2-10. It agrees
qualitatively with Fig. 2-7 but again the magnitudes of the resonance shifts are smaller.
Spectral images obtained later in the solvation process are shown in Fig. 2-11. The
linewidths narrowed somewhat, indicating a more liquid-like sample mixture, but other¬
wise the qualitative features of Fig. 2-10 remained. The amount of perdeuterated acetone
used was not sufficient to dissolve the polymer block and so spectral images of
near-homogeneous mixtures like that of Fig. 2-8 were never seen.
In summary, for a given solvent, the appearance of spatially-dependent resonance
shifts depended on the time of solvation. The signs of the shifts depended on the spatial
axis observed. Finally, the magnitudes of the shifts depended on the solvent used. Origi¬
nally, it was suspected that the resonance shifts were the result of a chemical interaction at

FIG. 2-9. Spectral images of PMMA in deuterated acetone after 180 minutes of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x axis. The gross features of this image were simi¬
lar to those of Fig. 2-6a; however, a fourth peak due to water in the solvent was seen in
the third most downfield position.

PMMA in Acetone-dg
time of solvation: 180 min.

PMMA ¡n Acetone-dg
time of solvation: 180 min.
500 0 -500 -1000 Hz
FIG. 2-9—continued, (b) A contour plot of (a).
N-0

FIG. 2-10. Spectral images of PMMA in deuterated acetone after 200 minutes of solva¬
tion. Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the z-axis. The features of this image were similar to
those of Fig. 2-7a.

PMMA in Acetone-dg
time of solvation: 200 min.

I—'—r
500
PMMA in Acetone-dg
time of solvation: 200 min.
V
I
i i I i i—i—|—i—i—i—l—i—i—i—|—i—|—i—|—i—i—r
0 -500
32 mm
z
y—32 mm
FIG. 2-10--continued. (b) A contour plot of (a).
-1000 Hz
O'
NJ

PMMA in Acetone-dg
time of solvation: 460 min
32 mm
X
1000 0 -1000 Hz
FIG. 2-11. Spectral images of PMMA in deuterated acetone at late stages of solvation.
Parameters and processing are given in the text, (a) A stacked plot spectral image
obtained by phase-encoding along the x-axis after 460 minutes of solvation.
O'

PMMA in Acetone-dg
time of solvation: 480 min.
32 mm
z
1000 0 -1000 Hz
FIG. 2-11—continued, (b) A stacked plot spectral image obtained by phase-encoding
along the z axis after 480 minutes of solvation. Only small differences existed between
these images and those of Figs. 2-9 and 2-10 because the solvent concentration was low.

65
the polymer-solvent interfaces. However, this possibility was eliminated when the signs
of the shifts were found to be dependent on the spatial axis. This behavior indicated
instead that the shift effect might be the manifestation of the differences in magnetic
field susceptibility between the polymer and solvent. Thus estimates of the shifts caused
by susceptibility differences were calculated using a crude model applied to each of the
samples studied.
2.4 The Susceptibility Model
The change in the static field which results when a sample of a particular geometry
and susceptibility is immersed in the field is given by
B0'=B0[ 1+(^-/c)Xv], [2-5]
where B0 is the static field in the absence of the sample, /c is a factor which depends on
the bulk sample geometry, and xv *s the volume susceptibility, a dimensionless quantity
dependent on the sample molecule (Pop59). Using the Larmor equation, the frequency
difference between two chemical species possessing identical bulk sample geometries due
solely to differences in volume susceptibility is
At/ = v2 - vl = ^-B0(^p - /t)(xv2 - Xvi) • [2-6]
This shift is not identical to the chemical shift phenomenon but behaves the same way in
terms of the evolution of product operators. Thus for the spectral imaging method used
here, a susceptibility effect appears as the addition of a constant frequency shift to each
of the chemical shift frequencies of a sample.
To analyze a particular problem using Eq. [2-6], the values of k are needed. These
have been described for several sample geometries (And69):
(1) For a cylinder whose length is infinitely greater than its diameter and oriented
transverse to the static field,

66
k = 2tt . [2-7a]
(2) For an identical cylinder oriented parallel to the static field,
/c = 0 . [2-7b]
The greatest susceptibility shifts are seen in cylinders oriented parallel to the static field.
It has been shown that for a cylinder oriented parallel to the static field and whose
length is about 10 times its diameter, k is not zero, but about 0.2 (Boz51). Thus the infin¬
ite cylinder approximation is a good one for such a sample. In the spectral imaging
experiments of the dissolving polymer, the excited region was a bar about 5 mm wide (the
slice width), 2 mm deep (the solvent depth), and 50 mm long (the dish diameter). Thus
the length was at least 10 times the width. By considering the polymer-solvent bar to be
an infinite cylinder and using the analytical values of k in Eqs. [2-7], approximate suscep¬
tibility shifts at the PMMA-solvent interface could be calculated if the volume suscepti¬
bilities of the molecules of the system were known. These were estimated using Pascal
constants and the densities and molecular weights of the solute and solvents, according to
the procedure of Pople, Schneider, and Bernstein (Pop59). The results are shown in Table
2-2. Using the data of Table 2-2 and Eq. [2-6], the susceptibility shifts for coaxial
cylinders of various pairs of substances were calculated. The results are shown in Table
2-3. The polymer-solvent interfaces were considered parallel to the static field if the x
gradient was used to define the slice axis and transverse if the z gradient was used. Using
these criteria, the calculated susceptibility shifts of Table 2-3 were compared with the
frequency shifts seen in Figs. 2-6, 2-7, 2-9, and 2-10. The magnitudes and signs of the
frequency shifts were measured with reference to the peaks belonging to the ester methyl
protons, because these were the least-attenuated and their spatially-dependent frequencies
could be measured fairly accurately. At some spatial positions, the ester methyl peak was
so broad that its spectral frequency could not be measured. The spatially-dependent fre¬
quencies of the methylene and methyl peaks could not be measured accurately because
they were quite broad and more attenuated than the ester methyl peaks. The frequencies

Table 2-2
Calculated Volume Susceptibilities
Molecule
Xv x io6
Acetone
-0.461
Chloroform
-0.853
Methylmethacrylate (MMA)
-0.527

Table 2-3
Calculated Susceptibility Shifts at 2 Tesla
System
Orientation
(relative to B0)
Shift (Hz)
MMA, Acetone
transverse
-12
MMA, Acetone
parallel
24
MMA, Chloroform
transverse
58
MMA, Chloroform
parallel
-116

69
of the ester methyl peaks were corrected to remove the static field inhomogeneity contri¬
bution. Using the data of Fig. 2-4, the difference between the frequencies of the water
peak at 0 mm and at some other position were calculated for each spatial position. These
differences were assumed to be due to static field inhomogeneity and were subtracted
from the spatially-dependent frequencies of the ester methyl peaks found from the poly¬
mer solvation experiments. The data of Fig. 2-4a were used to correct Figs. 2-6 and 2-9,
and Figs. 2-7 and 2-10 were corrected using the data of Fig. 2-4b. The corrected fre¬
quency shifts which could be measured were plotted versus spatial position relative to one
side of the polymer block and are shown in Fig. 2-12.
2,5 Conclusion
The susceptibility model successfully predicted a number of the features of the spec¬
tral images of the polymer-solvent systems studied. In general, the observed frequency
shifts were in the range calculated. More specifically, for a particular slice axis, the mag¬
nitude of the frequency shift observed when deuterated chloroform was used as the sol¬
vent was greater than that observed when perdeuterated acetone was used. This observa¬
tion was consistent for both slice axes, as seen by comparison of Fig. 2-6b with Fig. 2-9b
and Fig. 2-7b with Fig. 2-10b. For a particular solvent, the shift observed when the z
gradient defined the slice axis was of opposite sign from that observed when the slice axis
was defined by the x gradient. These observations are summarized for the ester methyl
peak of PMMA in Fig. 2-12.
Although the model was partially successful at explaining the observed frequency
shifts, several discrepancies existed between the model and the observed spectral images.
For a particular slice axis, the spectral images showed no difference in the signs of the
frequency shifts observed for the two solvents. The susceptibility calculations shown in
Table 2-3 predicted otherwise. The disagreement possibly was due to the close
equivalence of the calculated volume susceptibilities of acetone and methylmethacrylate.

FIG. 2-12. PMMA solvation. The data for this plot were obtained by measuring the
spatially-dependent spectral frequency of the ester methyl peak of PMMA at several spa¬
tial positions then subtracting the inhomogeneity contribution as described in the text.
Key: â–¡, Data obtained from Fig. 2-6 corrected using the data of Fig. 2-4a; the
polymer-solvent interface was perpendicular to the static field and the solvent was deu-
terated chloroform. +, Data obtained from Fig. 2-9 corrected using the data of Fig. 2-4a;
the polymer-solvent interface was perpendicular to the static field and the solvent was
perdeuterated acetone. A, Data obtained from Fig. 2-10 corrected using the data of
Fig. 2-4b; the polymer-solvent interface was parallel to the static field and the solvent was
perdeuterated acetone, x, Data obtained from Fig. 2-7 corrected using the data of
Fig. 2-4b; the polymer-solvent interface was parallel to the static field and the solvent was
deuterated chloroform.

Frequency Shift (Hz)
-50
-100
-150
-200
-30 -26 -22 -18 -14 -10 -6 -2
t 1 —t 1 1 1 1 1 1 1 1 r
Spatial Position (mm)

72
These calculations were too poor to say truly whether the susceptibility difference was
positive or negative. A second discrepancy existed between the magnitudes of the calcu¬
lated shifts and the observed values. For a particular solvent, the magnitudes of the
observed shifts should have changed substantially with the slice axis according to the sus¬
ceptibility model. Some support for this could be seen by comparing Fig. 2-6b with
Fig. 2-7b and Fig. 2-9b with Fig. 2-1 Ob; the observed shift magnitudes appeared to be
greater when the x gradient was used to define the slice. This is consistent with the model
but is not conclusive because the observed shift magnitudes could not be accurately meas¬
ured. The chief reason for this obstacle was that the observed shift magnitudes were on
the order of the linewidths in the vicinity of the polymer-solvent interfaces. This coupled
with the effect of static field inhomogeneity and the variations of peak amplitudes made
the exact measurement of frequency shifts impossible.
Thus far, the behavior of the frequency shifts which were observed in Fig. 2-7 has
not been explained. The shifts first moved upfield in the vicinity of the polymer-solvent
interface as predicted by the model, but then went unexpectedly downfield in the vicinity
of the as-yet-undissolved PMMA. The appearance of spectral peaks in the vicinity of
solid indicated that some solvent had entered the slice. Far more likely, though, was the
possibility that the polymer block had become loose and moved out of the slice region
along the phase-encoded spatial axis. This would explain two observations. First, the
slice would have encompassed both solid polymer and liquid solvent, which would explain
the appearance of peaks at spatial positions where none should have been seen. Secondly,
with the polymer slightly out of the slice, two interfaces would have been observable, one
parallel to the static field and the other transverse to it. This would explain the upfield
shift observed near the polymer-solvent interface and the downfield shift seen where the
polymer should have not yet dissolved. Figure 2-6 provides some evidence that the poly¬
mer block was not centered at the x axis origin; the polymer-solvent interfaces are not
distributed symetrically with respect to the x = 0 mm position.

73
Although this work failed to observe any chemical changes at the polymer-solvent
interface, it has some implications for similar studies. A number of papers have described
the application of the basic spectral imaging technique described in this chapter to the
study of living systems (Bai87, Has83, Pyk83). One of the proposed uses is the measure¬
ment of in vivo tissue pH at particular spatial locations by the measurement of the fre¬
quency difference between the inorganic phosphate and phosphocreatine peaks detected
with 31P spectral imaging. However, very little has been said about the contribution of
susceptibility to the results of such studies. If an interface, such as one between a muscle
and an organ, exists along the phase-encoded spatial axis, then susceptibility changes
could cause anomalous frequency shifts of the corresponding spectral peaks. If both tis¬
sues have the same orientation relative to the static field and differ only in their suscepti¬
bilities, then all spectral peaks would be affected equally and the relative frequency shifts
would be unaffected. However, if the tissues do not have the same susceptibilities and
orientations relative to the static field, errors in the estimate of tissue pH could result. It
can be shown that this error is probably very small using the susceptibility model of sec¬
tion 2.4. Equation [2-6] can be rewritten in units of parts per million (ppm) as
Ai/ = i/2-t'1=)x 106(^- - k)(xv2 " Xvi) • [2-8]
In the worst case, two cylindrical samples with two different susceptibilities would be
oriented parallel and perpendicular to the static magnetic field. When /c = 0 as for a
cylinder oriented parallel to the static magnetic field, then
^parallel = 4-19 * 106(XV2 * Xvl) • [2-9]
Also, when k. = 2n as for a cylinder oriented perpendicular to the static magnetic field,
then
¿"perpendicular = ’ 209 * ,()6(Xv2 ' Xvl) • [2~ 10]
Subtracting Ai/perpendicular from Ai/parallel produces

74
A¿/ = 6.28 x 106(xv2 - Xvi) • [2-H]
The value of Ai/ is the frequency difference between two samples due to differences in
their volume susceptibility and sample orientation and is a source of error in the measure¬
ment of the true frequency separation. As an example, the true frequency separation
between inorganic phosphate and phosphocreatine varies over about a 2.5 ppm range
between pH 6 and pH 7 (Gad82). If the maximum allowable error in the frequency
difference measurement is 0.25 ppm, and this is set equal to Ai/, then (xv2 - xvi) must be
less than 4 x 10'8. This is very likely in biological tissues, since the inorganic phosphate
and phosphocreatine are in very dilute solution and their volume susceptibilities are prob¬
ably very similar.
Finally, the spectral imaging experiments of this chapter have important implications
for materials science. In particular, spectral imaging could be used as a rheological tool to
observe deformation and flow during the solvation process, possibly uncovering some
chemical process occurring at an interface. The experiments discussed in this chapter
have shown that it may be possible to observe semisolid materials directly, complementing
the observation of solvents as they penetrate solids. It may also be possible to observe the
curing process, observing spatially-localized chemical changes as a polymer hardens.

CHAPTER 3
CONVOLUTION SPECTRAL IMAGING
3.0 Introduction
In the previous chapter spectral imaging techniques were categorized as being either
selective or nonselective. In this chapter, a new set of nonselective spectral imaging tech¬
niques is introduced. Like other methods in that class, they are sensitive to static field
inhomogeneity. However, they differ from the other techniques in that the number of
dimensions required for image acquisition and display equals the number required to
define the spatial image. For example, a three-dimensional experiment in which the spa¬
tial information from each of two spatial dimensions and the spectral information are col¬
lected separately may be compressed into a two-dimensional experiment. This is accom¬
plished by including the spectral information with the spatial dimensions rather than let¬
ting the spectrum comprise a separate dimension. Because the convolution theorem for
Fourier transformation is central to the applicability of these techniques, they have been
grouped under the term "convolution spectral imaging". It is shown that under certain
conditions these methods drastically reduce the time required to obtain spectral and spa¬
tial information without a loss in spectral or spatial resolution.
3.1 The Convolution Spectral Imaging Method
In Chapter 2, section 2.1, an expression for a density operator was given which
described the evolution period of a 2D spectral imaging method. This was derived by
using the density operator formalism to describe the effect of the pulse sequence of
Fig. 2-1 applied to a system of two weakly-coupled spins. The result appeared in
Eq. [2-3]. The method of Chapter 2 mapped out the spatial modulation functions of the
evolution period by changing the amplitude of a gradient with each pass of the pulse
75

76
sequence. The durations of the time intervals were fixed during an experiment and so the
modulation functions related to the spectral frequencies and coupling constants of the
spins were also constant. The spectral dimension, which was correlated with the spatial
dimension, was defined by allowing free precession during detection.
Using the same pulse sequence shown in Fig. 2-1, a different type of modulation
function can be mapped out by varying the time interval, tx; in effect the n pulse is
moved through the time window, T. The gradient amplitude, Gr, and its duration, r, are
fixed. Under these conditions, the density operator expression for two weakly-coupled
spins, given by Eq. [2-3], shows that the phase modulation function which is mapped out
is a function of the spectral frequencies of the spins. The spatial modulation functions are
constant. Also, the modulation functions related to spin-spin coupling are constant
because the start of detection occurs at a fixed time after the initial n/2 excitation pulse
(Bax79). The phase-encoded spectral width is given by the inverse of the amount, Atj, by
which the tj interval is changed with each pass of the pulse sequence. The desired spec¬
tral width and the number of phase-encode steps place strict limits on the TE value. This
can have serious consequences since the acquired signal amplitude depends on the TE and
the T2’s of the sample, as shown in Chapter 2.
Phase-encoding of spectral information is the basis of the three-dimensional method
of Sepponen and coworkers (Sep84), from which convolution spectral imaging methods
are derived. With the Sepponen method, two spatial dimensions are correlated with a
spectral dimension. The spectral modulation functions produced during evolution are
mapped out by the process just described. In addition, the spatial modulation functions
produced during evolution are also mapped out by using the stepping of a phase-encode
gradient as described in Chapter 2. To maintain the independence of the phase-encoded
spectral and spatial information, for each step in the variation of the phase-encode gra¬
dient, the time interval, tl5 is stepped a number of times equal to the desired number of
points in the spectral dimension. The time interval is then reset to its initial value and the

77
phase-encode gradient is stepped to its next value. A complete cycle of time interval
stepping is repeated for each phase-encode gradient step.
The second spatial dimension of the three dimensions defined by the Sepponen tech¬
nique is produced by the process of frequency-encoding, where a gradient is turned on
during detection to create spatial modulation. However, as mentioned in Chapter 1,
modulation due to free precession also occurs during detection. The Hamiltonian describ¬
ing the effect of the applied gradient must dominate the unwanted Hamiltonian which
describes the effect of the static field or spatial and spectral information will be mixed.
In contrast to the Sepponen method, for convolution spectral imaging techniques the
phase-encode gradient amplitude and the time interval are stepped simultaneously during
the evolution period. The result of this simultaneous stepping is a phase modulation of
the acquired signal which is a function of the spatial and spectral details of the object
being imaged. This phase modulation is the product of two phase modulations, one of
which is a function of the gradient stepping and the other a function of the time interval
stepping. From Eq. [2-3], which was derived for a single spin, the phase modulation
function corresponding to the gradient stepping has the form
h(kr,r!)
e-i2xkrri
[3-1]
where kr is a function of the applied gradient magnitude and r2 is the position of the spin
along the r axis. The function kr has the form
«W
TfGr(t)dt
2rr
[3-2]
where q is the gyromagnetic ratio, Gr(t) is a function describing the time-dependent
amplitude of the applied gradient, and r is the time during which the gradient is applied.
In general, for a distribution of spins along the r axis, S(r¡), the modulated signal has the
form

78
s(kr) = ^S(ri)h(kr,ri)dr¡ , [3.3]
where R expresses the limits of integration imposed by the extent of the spin distribution
in r space. The phase modulation function corresponding to the stepping of the time
interval has the following form for a single spin not J-coupled to another:
h'Oj,^) = e'12*1"1, [3-4]
where vx = uJI-k is the resonance frequency of the spin and t2 is the time during which
free precession occurs in the absence of gradients. In general, for a spectral distribution of
spins, S'(i/¡), the modulated signal can be expressed by
s'(tx) = £S'[*/i)h'[t1,i/i)di/i , [3_5]
where N expresses the limits of integration imposed by the extent of the spectral distribu¬
tion. If one takes FT to mean "the Fourier transform of, then ideally
FT[s(kr)] = S(r¡) ,
[3-6]
FTIs'itx)] = S'(i/¡) .
[3-7]
In the convolution spectral imaging technique, the amplitude of the phase-encode
gradient and the time interval are stepped simultaneously. Thus kr and t2 are related by a
constant, rj:
h = t?kr . [3-8]
The resulting doubly-modulated signal may be expressed by the product of Eqs. [3-3] and
[3-5]. By the convolution theorem (Bri74), the Fourier transform of this product is the
convolution of the spatial and spectral spin distributions. This transformed result may be
mapped into frequency space in which case the spatial spin distribution appears as a func¬
tion of frequency scaled by r¡. Denoting the convolution by an asterisk and using
Eq. [3-8], the Fourier transform can be written

79
FT[s(kr)s'(t1)] = r?[S(r?^)*S'(^i)] . [3-9]
Alternatively, the Fourier transform of the product may be mapped into r space. The
spectral spin distribution then appears as a function of spatial position scaled by the
inverse of tj:
FT[s(kr)s'(ta)] = (IMSirjmr,/*)] . [3-10]
Each of the convolution functions shown in Eqs. [3-9] and [3-10] is the spatial distri¬
bution of the sample spins offset by their spectral resonance frequencies or vice versa. A
plot of one of these functions for a hypothetical two-compartment sample containing two
different chemical species is shown in Fig. 3-la. The function is the convolution of the r
space information shown in Fig. 3-lb with the spectrum shown in Fig. 3-lc. The two
resonance lines are assumed to have widths much smaller than their chemical shift differ¬
ence and B0 inhomogeneity is assumed to be negligible. Another way of viewing Fig. 3-1 a
is as a series of spectral frequencies, each of which has an identical spatial field-of-view
spread around it. The origin in spatial coordinates for a particular chemical species coin¬
cides with its spectral frequency. Any spatial offset of a nucleus from the origin appears
as an offset from its spectral frequency. The idea of convolving two types of information
has been exploited in spectroscopy by the accordion experiment of Bodenhausen and
Ernst (Bod81) and the three-frequency experiment of Bolton (Bol82). The convolution of
a spatial axis and the zero-quantum spectrum has been accomplished recently in a similar
manner by Hall and Norwood (Hal86a).
Although Eqs. [3-9] and [3-10] were derived for phase-encoding, the mathematics
are similar for frequency-encoding. The two processes differ in that free precession,
which is the source of spectral information, occurs in the presence of a gradient during
frequency-encoding. The result is still a doubly-modulated signal, and Eqs. [3-9] and
[3-10] still hold. These equations represent the origin of the "chemical shift artifact"
(Bab85, Dwy85, Soi84) which is observed when a gradient applied during signal acquisi¬
tion is too weak to obliterate the separation between spectral resonances.

-r O r -r 0 r
b)
A/2
+
B
B/2
-t t l
c)
1
1
1
1
>
'a 0 >
1
!b
FIG. 3-1. The convolution of spectral and spatial information, (a) The convolution of the
spatial and spectral information shown in (b) and (c), respectively, (b) The spatial profile
of a pair of vessels containing differing amounts of two molecular species, A and B. (c)
The spectrum corresponding to (b).
OO
O

81
The pulse sequence for the acquisition of a two-dimensional convolution spectral
image is shown in Fig. 3-2a. The interval between the initiation of the pulse sequence at
successive phase-encode steps, TR, is kept constant. The time to the echo formation, TE,
is also kept constant; thus, the T2 relaxation weightings of the phase-encoding steps are
equivalent. Phase-encoding of spectral information is accomplished by moving the 7r
pulse through the window labelled T by incrementing t2 by an amount Atj in a stepwise
fashion. The spectral width of the phase-encoded axis is 1/Atj.
The implementation of the pulse sequence of Fig. 3-2a with the frequency-encode
gradient, gfe, defined as the z gradient and the phase-encode gradient, gpe, defined as the
x gradient, would result in an z,vt versus x,ux image where z and x define the spatial axes
and vt and vx define the impressed spectral information. Each planar zx image
corresponding to a spectral resonance will lie on a diagonal passing through the origin if
the sample is centered on the crossing point of the z and x axes as defined by the z and x
gradients. The planar images will be located on a diagonal parallel to that passing through
the origin if the sample is spatially offset from this crossing point. The center of each
image will be separated from that of its chemically shifted neighbor along the diagonal by
the chemical shift difference of the two species multiplied by a scaling factor. If the spec¬
tral widths convolved with each spatial dimension were identical, this scaling factor would
equal VI.
Like other nonselective spectral imaging methods, convolution spectral imaging tech¬
niques collect spectral data from all points of the spectrum including those which do not
contain resonances. The advantage lies in the ability to compress three dimensions of data
into two dimensions for planar spectral imaging or four dimensions into three for volume
spectral imaging with a corresponding saving in total acquisition and processing time. It
will be shown that the techniques are best suited for small samples whose frequency spec¬
tra are composed of well-separated resonances and that these techniques optimize the
available signal-to-noise ratio. Thus convolution spectral imaging may be most applicable

FIG. 3-2. Pulse sequences for convolution spectral imaging, (a) The pulse sequence of
two-dimensional convolution spectral imaging. The timing of the sequence and the labels
given to the various timing intervals are indicated at the bottom. The non-selective n
pulse is moved through the window labelled T by an amount, Atj, in a series of M steps at
the same time as the amplitude of the phase-encode gradient, g^, is varied stepwise. RF =
radiofrequency transmitter, gfe = frequency-encode gradient, gpe = phase-encode gra¬
dient, gs) = slice gradient, acq = data acquisition.

selective zt/2
RF O
K
g
fe
g,
pe
gsl
KvHh
-T
TE
T -
ti/2
TR

FIG. 3-2—continued, (b) The pulse sequence for three-dimensional convolution spectral
imaging. The non-selective t pulse is moved through the window labelled T by an
amount, Atlt for each of a series of M steps at the same time that the amplitude of the
phase-encode gradient, g^j, is varied stepwise. The ir pulse is then moved through the
window by an amount, Ats, for each L step as the amplitude of the phase-encode gra¬
dient, gp^j, is changed simultaneously. The cycle of M phase-encode steps is repeated for
each L step, g^ * phase-encode gradient (M steps), gp.2 = phase-encode gradient (L
steps), all other abbreviations as in (a).

RF
9fe
9pe1
9pe2
(*-t3/2+|<-t1/2*|l<-T/2 - t,/2-^*-T/2 - ^2-*^ t2—H
K«K
4
acq
TE
TR

86
to the field of NMR microimaging (Agu86).
3.2 Experimental
Experiments were carried out using the spectrometer and RF coil described in
Chapter 2, section 2.2. The pulse sequences used are shown in Fig. 3-2. Quadrature
detection was employed in all experiments. Slice selectivity was used if the spectral reso¬
nances of the sample under investigation were sufficiently close that the application of a
strong gradient during slice selection could eliminate slice misregistation due to the chem¬
ical shift effect. Otherwise, the entire sample volume was excited. Thus a
frequency-selective sine-shaped tt/2 RF pulse and a gradient normal to the imaging plane
were used to select slices for all JH images. The 7r/2 RF pulse was nonselective in all 19F
imaging experiments. All tt pulses were nonselective.
The effects of imperfect n pulses and DC imbalance between the quadrature chan¬
nels were corrected by using a two-step phase cycle during signal averaging impressed on
the w/2 pulse and receiver. This phase cycle had the effect of cancelling out artifacts due
to magnetization which was not phase-encoded. However, as pulse repetition times were
decreased for more rapid image data acquisition, the ability of the cycle to suppress
artifacts was diminished. The origins of these residual artifacts were from
non-steady-state magnetization which was not phase-encoded and instrumental errors
causing DC offset. Therefore gradient proportional phase incrementation (GPPI) of the n
pulse was also added to the two-step phase cycle (Gra86). This had the effect of moving
artifacts which appeared at the midpoint of the phase-encoded axis to the edges of that
axis without increasing the image acquisition time by requiring signal averaging. The ini¬
tial desire was to impress GPPI on the rr/2 pulse and the receiver to shift both types of
artifact. However, this was not allowed by the instrument design. Thus GPPI had to be
impressed on the n pulse. This could only shift artifacts resulting from magnetization
which had not been phase-encoded. Artifacts due to instrumental DC effects remained at

87
the midpoint of the phase-encoded axis. Fortunately, these were quite minimal. The phase
cycle for the pulse sequence of Fig. 3-2a is shown in Table 3-1, and that for the sequence
of Fig. 3-2b is shown in Table 3-2.
To reduce truncation artifacts, it was ensured that the echoes corresponding to the
middle steps in the phase-encoding process were centered in the acquisition window. This
was accomplished in a set-up mode before acquisition of the image data by turning all
phase-encode gradients off and placing the 7r pulse in the center of the time delay, T (see
Fig. 3-2). The amplitude of the frequency-encoding gradient on during the time, r, was
then carefully adjusted. This set-up mode was also used for pulse calibration and for
choosing transmitter and receiver gain settings. In most cases the values of TR and TE
were chosen such that very little relaxation attenuation was allowed.
3.3 Practical Aspects of the Method
Convolution spectral imaging is most effective for small objects containing molecular
species with a spectrum of well-separated single peak resonances, which implies that the
technique may find its greatest application in the field of NMR microimaging. The effec¬
tiveness of the method is enhanced if short phase-encode gradient times are employed.
These conclusions become apparent if one considers the sources of signal modulation and
if one makes the restriction that the frequency separation between resonances is greater
than the frequency spread caused by the applied gradients. This restriction is necessary to
prevent the overlap of the spatial images comprising the convolution spectral image. How¬
ever, it will be shown that this restriction may be relaxed for certain sample geometries.
Consider a single spin, not spin-spin coupled to another, which is subjected to the
two-dimensional technique of Fig. 3-2a, again letting z be the frequency-encode dimen¬
sion and x be the phase-encode dimension. During the acquisition time, t2, the signal is
modulated in part by the function
h"(m„t2)
-¡2^(1/! + mi)t2
[3-11]

Table 3-1
Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases
Phase-Encode
Step
Signal
Average
Phases
Pulses
Receiver
tt/2
7T
1
1
0
0
0
1
2
180
0
180
2
1
0
180
0
2
2
180
180
180
M
2
180
180
180
Refer to Fig. 3-2a and its legend for the notation corresponding to this table.

Table 3-2
Three-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases
Phase-Encode
Step
(Spei)
Phase-Encode
Step
(Spel)
Signal
Average
Phases
Pulses
Receiver
rr/2
â– n
1
1
1
0
0
0
1
1
2
180
0
180
1
2
1
0
180
0
1
2
2
180
180
180
1
M
2
180
180
180
2
1
1
0
180
0
2
1
2
180
180
180
2
2
1
0
0
0
2
2
2
180
0
180
L
M
2
180
0
180
Refer to Fig. 3-2b and its legend for the notation corresponding to this table.

90
where vl is the resonance frequency of the spin in Hz and = (TfGIzI)/27r where t is the
gyromagnetic ratio in rad sec'1 mT'1, Gt is the strength of the applied z gradient in
mT mm'1, and za is the position of the spin in mm along the z direction. The contribution
of spin-spin coupling to this modulation function has been ignored because it is usually
obscured in the frequency-encode dimension by the effect of the applied gradients.
The signal is modulated differently during the combined time interval r + T, the
relevant function having the form
h"'(mx,t1,r,T) = e12^1*1 + m*V2’ r) , [3-12]
where has been previously defined and, assuming that the x gradient amplitude is con¬
stant during r, mx = ('yGxx1)/2tr. The parameters Gx and Xj are defined in analogy to the
definitions of Gt and zv Free precession during (T - r) appears in Eq. [3-12] as a phase
shift term. This term is constant because T and r are fixed and will be ignored. Also,
because the time T is fixed for a given measurement, modulation by spin-spin coupling
could appear as a constant term in Eq. [3-12], As shown by the analysis of Chapter 2,
section 2.1, for two coupled spins with coupling constant J, the modulation represented by
Eq. [3-12] would have an amplitude dependent on cos(trJ(r + T)) when the
frequency-encode gradient is sufficient to cancel the two-spin operator terms. This means
that the choice of these time delays requires care for coupled spin systems. The contribu¬
tion of spin-spin coupling to Eq. [3-12] will be ignored for the remainder of this discus¬
sion.
The convolution spectral imaging technique requires that the frequency separation
between spectral resonances must be greater than the frequency spread caused by the
applied gradients. Applying this restriction to Eq. [3-11] and Eq. [3-12] and including the
effect of B0 inhomogeneity, it is found that the following inequalities must hold to obtain
a convolution spectral image.

91
During t2 :
I Ai/| > |M,| =
9GeR,
9(AB)
2tt
2x
[3-13]
Here Av is the frequency separation between the spectral resonances of two chemically
shifted species, Mt is the frequency spread across the object of size, RE, and AB is the
greatest linewidth in millitesla units in the spectrum. For a fixed t and Rt, this inequality
states that the applied z gradient must be weak enough to prevent obliteration of the spec¬
tral information but strong enough to overcome main field inhomogeneity.
During r + T:
lAKAtj)! > |(AMx)r| >
^(ABKAtj)
27T
[3-14a]
Since 7 and the size, Rx, are constants for a particular object, then AMX = Tf((AGx)Rx)/27r.
Substituting this into inequality [3-14a] and simplifying produces
I AH >
7(AGx)Rxr
Tf(AB)
2jr(At1)
2n
[3-14b]
Rearranging inequalities [3-13] and [3-14b] shows the limits of the applied gradient mag¬
nitudes in the frequency- and phase-encode dimensions.
For frequency-encoding:
AB !
R.
I< IG.I <
I
2tt(AH
nR,
For phase-encoding:
ABtAtj)
Rxr
< I AGX| <
27r(Ai^)(At1)
[3-15]
[3-16]
Inequalities [3-15] and [3-16] express the conditions under which convolution spec¬
tral imaging is most effective. Inequality [3-16] is satisfied as t, the duration of the
phase-encode gradient, is reduced. This reduction allows a decrease in TE, shown in

92
Fig. 3-2, thus reducing the attenuation of the acquired echo caused by transverse relaxa¬
tion processes. Another way to satisfy inequalities [3-15] and [3-16] is to use a sample
whose spins are distributed over a small range; this allows a reduction in Rz and Rx.
Unfortunately, reductions in r and the sample size also increase the magnitude of the
inhomogeneity terms, the leftmost terms of inequalities [3-15] and [3-16], Thus for con¬
volution spectral imaging experiments it becomes important to minimize the AB term,
which can be done by ensuring that the static magnetic field is homogeneous. Inequalities
[3-15] and [3-16] also indicate that a large spectral width is desirable, a condition which is
determined by the nuclear spin system which is being imaged and the strength of the
static magnetic field. Spin systems possessing large frequency separations between peaks
require larger spectral bandwidths and are better able to satisfy [3-15] and [3-16]. In sum¬
mary, convolution spectral imaging experiments will be most effective at high field using
small samples, indicating that the technique is most applicable to NMR microimaging.
3.3.1 The Effect of Sample Geometry
The result viewed along the diagonal through the centers of the spatial images
comprising a convolution spectral image may be thought of as representing an exaggerated
spectrum where the frequency differences between resonances are multiplied by a factor
greater than unity. This factor may be found using trigonometry and depends on both the
number of spatial dimensions and the spectral widths convolved with them. For certain
sample geometries, this exaggerated spectral resonance separation may be exploited to
increase the spatial resolution of the convolution spectral image without destroying the
spectral information. Fig. 3-3a shows the spectrum of a mixture of hexafluorobenzene and
hexafluoroacetone sesquihydrate contained in a section of a 9 mm inside-diameter NMR
tube (Norell 1005) about 3 cm in length. The tube and its orientation relative to the gra¬
dient coil set axes are shown in Fig. 3-3b. The transmitter frequency was set to the 19F
frequency and placed between and equidistant from each of the two spectral frequencies

-10000 -5000 0 5000 10000 Hz
FIG. 3-3. The effect of sample geometry, (a) The 19F spectrum of a mixture of hex-
afluorobenzene and hexafluoroacetone sesquihydrate contained in a single tube; Av = 6S78
Hz.

<—10 mm^
FIG. 3-3--continued. (b) The sample tube and its orientation relative to the gradient
axes.

95
of the sample liquid mixture. Thus, in the convolution spectral imaging experiments
which followed, any offset of the spatial information from a spectral frequency for a par¬
ticular spatial axis could be interpreted as a physical offset of the sample along that spatial
axis. The associated convolution spectral image is shown in Fig. 3-3c. A projection of the
image onto the x,vx axis is shown in Fig. 3-3d. Had the sample been centered on the ori¬
gin of the x axis, the center of the projection would be located at 0 Hz. Thus this projec¬
tion shows that the sample was slightly offset from the x axis origin. A projection of the
image onto the z,vt axis, shown in Fig. 3-3e, had a nearly identical appearance but the
spatial offset was not as great.
Clearly, the spatial resolution of the image of Fig. 3-3c could be increased without
spatial image overlap. In the case of a circular sample this increase could be up to a factor
of Vl. This maximum was used to obtain the convolution spectral image of Fig. 3-3f. The
projections of this image onto the x,vx and z,vt axes, depicted in Figs. 3-3g and 3-3h,
respectively, did not exhibit good spectral separation but the two-dimensional image
clearly shows two separated subimages. This simple example illustrates how a careful con¬
sideration of the sample geometry may be used to optimize the spatial resolution of a con¬
volution spectral image.
Convolving spectral and spatial information along the phase-encode direction
becomes more important as the ratio of sample length to width increases. Compared with
a square sample of equal area, a long, thin sample requires fewer phase-encoding steps
and more frequency-encoded points to maintain spatial resolution if the short axis lies
along the phase-encoded direction. In this case it is best to allow complete separation of
spatial and spectral information along the phase-encode axis using convolution spectral
imaging while allowing spectral and spatial information along the frequency-encode axis
to overlap. This would produce an image similar to that produced by conventional
spin-echo imaging (Ede80) if the frequency-encode gradient were weak and the object
were rotated such that the short axis lay along the frequency-encode direction. However.

FIG 3-3—continued, (c) A 19F two-dimensional convolution spectral image. Total time
for acquisition = 8.5 min, N - 256, M = 128, TR = 2 sec, TE = 30 msec. Resolutions. z(fe)
119 microns/point, i/.(fe) 78 Hz/point, x(pe) 119 microns/point, i>x(pe) 78 Hz/point.

10 mm —r— 10 mm

FIG. 3-3—continued, (d) The projection of (c) onto the x,i/x axis.
o
OO

FIG. 3-3--continued. (e) The projection of (c) onto the z,ut axis.

FIG 3-3—continued (f) A 19F two-dimensional convolution spectral image. Total time
for acquisition - 8.5 min. N - 256, M - 128. TR - 2 sec. TE - 30 msec Resolutions: site)
84 microns/point, i/,(fe) 78 Hz/point, s(pe) 84 microns/point. ox(pe) 78 Hz/point.

101
UIUJ l.'/ —JL UiUJ I.*/

FIG. 3-3—continued, (g) The projection of (f) onto the x,i/x axis.

| 1 1 1 T 1 1 T 1 1 1 1 1 1 1 1 1 1 1 1 1
-10000 -5000 0 5000 10000 Hz
FIG. 3-3—continued, (h) The projection of (f) onto the z,vt axis. The images of (c) and
(f) were obtained using the pulse sequence of Fig. 3-2a with the phase cycle of Table 3-1.
The slice gradient was turned off and the selective jr/2 pulse was replaced with a non-
selective x/2 pulse. The images were processed using a two-dimensional Fourier transfor¬
mation with no apodization and one zerofill in the phase-encode dimension and are
displayed in magnitude mode. The phase-encoded resolutions reflect the effect of zero-
filling used in processing.

104
this would not be optimum since more phase-encode steps would be required to maintain
spatial resolution, which would significantly increase measurement time.
3.3.2 The Interaction of TE and the Spatial Resolution
Since the amplitude of the acquired echo decreases due to T2 relaxation processes as
the echo time, TE, increases, the desire to conserve signal demands that TE be as short as
possible. The minimization of TE is limited in part by the length of the time delay, T,
shown in Fig. 3-2a, which is controlled by the number of phase-encode steps, M, and the
length of the time step, Atj. Reducing the TE may be accomplished by reducing M, r. or
At^ The latter option is not used in practice because a reduction in Atj increases the
phase-encoded spectral width. This increases the possibility of acquiring spectral regions
containing no spectral data and can cause the subimages of a convolution spectral image to
overlap.
The signal-to-noise ratio and image contrast of a convolution spectral image are
highly dependent on the number of phase-encode steps and the magnitude of the spectral
encoding time step. An image with high resolution of a small spectral width will neces¬
sarily require a longer TE than an image with lower resolution over the same spectral
width. This is true of the technique of Sepponen and coworkers as well (Sep84). Thus the
relaxation properties of the nuclear spin system determine the possible spectral and spatial
resolution of the phase-encoded information for a given spectral width. Molecular species
with short relaxation times will require short TE values so that they may be observed, and
consequently, lower spectral and spatial resolution of the phase-encoded information must
be used. Increasing TE weakens this restriction and greater resolution of the
phase-encoded spectral and spatial information is possible. However, a relaxation-caused
drop in the signal can occur as well. Figure 3-4 illustrates this effect for a mixture of
acetone and CuS04-doped water contained in a 4 mm inside-diameter NMR tube (Wilmnd
507-PP) about 3 cm long. The sample orientation was identical to that shown in Fig. 3-3b.

105
The JH spectrum of the mixture is shown in Fig. 3-4a, illustrating the quality of the main
field homogeneity. Figure 3-4b is a convolution spectral image with TE = 90 msec
obtained using 64 phase-encode steps, and Fig. 3-4c is a convolution spectral image
obtained using 128 phase-encode steps with TE = 180 msec. Since At! was fixed for both
Figs. 3-4b and 3-4c and since more phase-encode steps were used to obtain Fig. 3-4c, the
TE of the latter was forced to be longer. The faster-relaxing water began to vanish from
the image as the phase-encoded resolution and TE increased.
The loss in signal due to relaxation is ameliorated somewhat in convolution spectral
imaging techniques since small gradient magnitudes are used in the frequency-encoding
process. The use of a small gradient magnitude allows a decrease in the magnitude of the
spectral width per data point with an increase in the signal-to-noise ratio in the resulting
image. This aspect is discussed further in section 3.3.4.
The sample volume represented by the images of Figs. 3-4b and 3-4c occupied less
than 1% of the active volume of the RF coil. Despite this disadvantage, the signal strength
was sufficient to obtain good signal-to-noise with only two signal averages per
phase-encode step and spatial resolutions on the order of 150-300 microns per data point
while retaining spectral information. This suggests that convolution spectral microimaging
should be possible at main field strengths of 2 tesla if the B0 homogeneity is quite good,
especially since the technique is most suitable for small samples, as has been previously
shown.
3.3.3 Extension to Three Spatial Dimensions
The two-dimensional convolution spectral imaging technique is easily extended to
three dimensions by inserting a second phase-encoding gradient, time interval pair into
the pulse sequence of Fig. 3-2a. The result is shown in Fig. 3-2b. The second pair is
stepped after each complete cycle of stepping the first pair. The result is a volume image
in which spectral information is mixed into all spatial dimensions. Again, TR, TE, T, and

Acetone
■*— Av
Water
T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-400 -200 0 200 400 Hz
FIG. 3-4. The interaction of TE and the spatial resolution, (a) The *H spectrum of a
mixture of CuS04-doped water and acetone in a 4 mm inside-diameter tube; Av = 203 Hz.

FIG. 3-4—continued, (b) A *H two-dimensional convolution spectral image. Total time
for acquisition — 8.5 min, N * 64, M = 32, TR = 8 sec, TE = 90 msec, slice thickness = 2
mm. Resolutions: z(fe) 280 microns/point, i/t(fe) 16 Hz/point, x(pe) 280 microns/point,
i/x(pe) 16 Hz/point.

X
O
oo

FIG. 3-4--continued. (c) A *H two-dimensional convolution spectral image. Total time
for acquisition = 17 min N = 128, M = 64, TR = 8 sec, TE = 180 msec, slice thickness = 2
mm. Resolutions: z(fe) 140 microns/point, i/,(fe) 8 Hz/point, x(pe) 140 microns/point,
i/x(pe) 8 Hz/point. The convolution spectral images of (b) and (c) were obtained using the
pulse sequence of Fig. 3-2b and the phase cycle of Table 3-1. The images were processed
using a two-dimensional Fourier transformation with no apodization and one zerofill in
the phase-encode dimension and are displayed in magnitude mode. The phase-encoded
resolutions reflect the effect of zerofilling used in processing.

T
1 r
o
1 i
200
T
-400
-200
N
I
o
o
o
- o
CM
: r
- O 3
:l
o
_ o
r cm
i
o
o
I
T 1 1 1 1
400 Hz
o

t are fixed during an experiment. There are two more delays than in the two-dimensional
experiment and the magnitude of each of these is controlled by the size of the time steps,
Ata and At3, and the number of phase-encode steps, M and L. In the three-dimensional
technique the restrictions of spatial resolution and relaxation make the minimization of
TE a more difficult task.
As in the two-dimensional case, the sample geometry should be considered to optim¬
ize the spatial resolution. Trigonometry shows that if the convolved spectral widths are
the same in all three spatial dimensions, then the center of each spatial volume image
corresponding to a spectral resonance will be separated from its neighbor by times the
frequency difference between the two neighbors. Again, sample geometry may allow
increases in the spatial resolution without concurrent image overlap. A spherical sample
would be most favored.
An example of the use of three-dimensional convolution spectral imaging is shown in
Fig. 3-5. It was obtained with the three-dimensional convolution spectral imaging method
of Fig. 3-2b using the phase cycle of Table 3-2. The tube was identical to that of Fig. 3-3
but a glass spike was inserted into it along the long axis, which was aligned with the y axis
of the gradient coil set. The sample tube orientation relative to the gradient coil set axes is
shown in Fig. 3-5a. The transmitter frequency was placed as for Fig. 3-3. The resulting
images are displayed as a series of two-dimensional slices in the z,vt-\,vx plane. Each slice
corresponds to a small band of positions along the y axis; these bands are also functions of
the spectral frequencies of the two types of nuclei comprising the sample. The central fre¬
quency of each of these bands is indicated in the upper left corner of each of the images
of Fig. 3-5. By comparing the resonance frequencies of the bands indicated with these
images with the spectrum in Fig. 3-3a it may be seen that the sample was spatially offset
along the y axis. In Figs. 3-5b-e, cross sections normal to the y,uy axis are shown for the
lower frequency resonance (hexafluorobenzene); the size of the hole due to the presence
of the glass spike increased. Irregularities in the hole shape reflected irregularities in the


FIG. 3-5—continued, (b) A slice of the 19F volume image of showing the spatial distribu¬
tion of the lower frequency spectral resonance.

-6250 Hz
-10000
N
X
o
o
o
o
o
o
o
in
° 5
o
o
o
in
o
o
o
o
10000 Hz

FIG. 3-5—continued, (c) A second slice of the 19F volume image of showing the spatial
distribution of the lower frequency spectral resonance.

7.1 mm
7.1 mm
-10000 -5000 0 5000 10000

FIG. 3-5—continued, (d) A slice of the 19F volume image of showing the spatial distribu¬
tion of both spectral resonances.


FIG. 3-5—continued, (e) A second slice of the 19F volume image of showing the spatial
distribution of both spectral resonances.

7.1 mm
X
1250 Hz
T
t—r
i
5000
10000
T
T
0
Av
T
' I ' 1 1 1 I
5000 10000 Hz
-10000 -5000

121
shape of the glass spike. Figures 3-5d-g show cross sections normal to the y,vy axis for the
higher frequency resonance (hexafluoroacetone sesquihydrate). The hole in the images
started out small and increased again. The experimental conditions were chosen intention¬
ally such that two subimages appeared in Figs. 3-5d-e, demonstrating the increase in spa¬
tial resolution along the y axis allowed by sample geometry. Thus with a single
three-dimensional data matrix and three-dimensional Fourier transform, a complete
volume image for each spectral resonance was defined. The short TE and the high resolu¬
tion of this data matrix were made possible by the large separation between the two spec¬
tral resonances. To have maintained the same resolution for two resonances which were
closer together in frequency would have required a longer TE. The resulting signal
attenuation due to relaxation might be unacceptable, since as described in Chapter 2, sec¬
tion 2.1, the signal amplitude is weighted by e-TE^T2. For example, the parameters used to
obtain Figs. 3-5b-g could have been used to obtain a 3D convolution spectral image of the
sample tube containing the mixture of acetone and water of Fig. 3-4. However, the dura¬
tions of the time steps, Atx and Ats would have had to be increased to 1 msec each such
that a spectral width of 1000 Hz would have been convolved with each of the
phase-encoded spatial dimensions. In addition, the acquisition time, t2, would also have
had to be increased to about 128 msec for the same reason. Thus, the TE would have had
to be about 200 msec to allow for the 128 steps of t2 and the 8 steps of t3. As can be
deduced from the image of Fig. 3-4c, which was obtained using TE = 180 msec, a
200 msec TE would have resulted in a great attenuation of the water signal and some
attenuation of the more slowly decaying acetone signal.
Three-dimensional convolution spectral imaging could also be used with
slice-selective excitation to obtain very thin slices without slice misregistration due to
chemical shifts. This could be accomplished by replacing the hard jt/2 pulse of Fig. 3-2b
with a frequency-selective one in the presence of a slice gradient, as in Fig. 3-2a. The
slice gradient axis would coincide with one of the phase-encode gradient axes. In the

FIG. 3-5—continued, (f) A slice of the 19F volume image of showing the spatial distribu¬
tion of the higher frequency spectral resonance.

X
N
x

FIG. 3-5—continued, (g) A second slice of the 19F volume image of showing the spatial
distribution of the higher frequency spectral resonance; M = 128, L = 8, TR = 5 sec, TE =
30 msec. Resolutions: z(fe) 84 microns/point, yt(fe) 78 Hz/point, x(pe) 84 microns/point,
i/x(pe) 78 Hz/point, y(pe) 4559 microns/point, i/y(pe) 2500 Hz/point. The data of (b)-(g)
were obtained using the pulse sequence of Fig. 3-2b with the phase cycle of Table 3-2.
The data were processed with a three-dimensional Fourier transformation with no apodi-
zation and one zerofill in the x,vx dimension. The results are displayed in magnitude
mode. The resolution of the x,vx phase-encoded dimension reflects the effect of zerofil¬
ling during processing.

X
N
I

126
presence of spectral separation and a relatively weak slice gradient, a different slice posi¬
tion would be excited for each spectral resonance. If the slice thickness and the
phase-encode field-of-view along the slice axis were identical, spatial phase-encoding
alone along that axis would result in aliasing. The problem could be avoided by convolu¬
tion spectral imaging phase-encoding. Folding along the slice axis would not occur when
the slice thickness and phase-encode field-of-view were equal, and the number of
phase-encode steps could be maintained without a loss in spatial resolution. The convolu¬
tion process would not remove slice overlap; this would have to be done in the other two
dimensions.
3.3.4 Signal-to-Noise Considerations
The effect of changes in controllable variables on the signal-to-noise ratio (S/N) in
the convolution spectral imaging technique may be seen by considering the frequencv-
and phase-encoding processes separately. The S/N in an image or a projection is defined
as the ratio of a peak amplitude to the amplitude of the root-mean-square noise of the
Fourier-transformed signal. Throughout this discussion it will be assumed that the
linewidths in a spectrum are dominated by the effect of the applied gradients. Now con¬
sider the generation of a spin-echo from an ensemble of spins with a tt/2 - t - ir pulse
sequence. The experiment is modified to include the frequency-encoding process by
applying a gradient of constant amplitude Glx during the time r and applying a second
gradient of constant amplitude G2x along the same direction as Glx for a time t2 immedi¬
ately following the 7r pulse. A gradient echo will also be formed. Under ideal conditions,
this gradient echo and the spin-echo will reach their maxima simultaneously when
Glx(r) = G2x(t2/2) . [3-17]
Since data is acquired during t2 , it is assumed that t2 is long enough to permit complete
decay of the signal. Thus truncation artifacts in the Fourier-transformed signal, which
would alter the S/N value, may be ignored. If the Nyquist sampling rate for quadrature

127
detection is chosen, then t2 = N/F = AF"1, where F is the total spectral bandwidth placed
around the transmitter carrier frequency and N is the number of sampled complex data
points. After Fourier transformation with respect to t2, the noise in the spectrum is pro¬
portional to the square root of the bandwidth per frequency-encode data point, AF
(Ern66). Since the signal is proportional to the spatial width per data point, Arfe , the
conclusion is that
S/N oc Arfe(AF)_1/2 [3-18]
under these conditions. For frequency-encoding, when the linewidths are dominated by
the effect of applied field gradients,
AF=7G^ArIl [3_19]
27T
By proportionality [3-18], S/N is maximized when AF is minimized if the spatial resolu¬
tion, Arfe, is fixed. To maintain the spatial resolution, however, Eq. [3-19] shows that
G2x must be minimized with AF. As G2x is reduced (while maintaining the relationship of
Eq. [3-17]), spectral information can appear, as shown by inequality [3-15], Inequality
[3-15] must be obeyed to produce a convolution spectral image. This contrasts with other
techniques such as spin-warp imaging (Ede80) and the chemical shift imaging technique
of Sepponen and coworkers (Sep84) in which inequality [3-15] is violated. The S/N will be
maximized for a technique where the spectral information is not obliterated by the effect
of applied frequency-encode gradients.
For the phase-encode process, a separate signal consisting of N data points is
obtained for each of M phase-encoding steps. Since the M signals are acquired separately,
the noise between them is uncorrelated. Thus this noise sums as the square root of the
number of phase-encode steps regardless of what transpires during the pulse sequence
(Bro87). For an imaging pulse sequence in which the linewidths in the phase-encode
dimension are dominated by the effect of applied gradients, the signal is proportional to
the spatial width per phase-encode data point, Arpe. Also the phase-encode process

128
averages the signal from each step. Therefore the S/N obeys the following proportionality:
S/N oc Ar^M)1/2 . [3-20]
This expression has important implications for convolution spectral imaging. Since pro¬
portionality [3-20] has no dependence on the phase-encode spectral width, the convolu¬
tion of phase-encoded spectral information with phase-encoded spatial information will
not alter the S/N in the phase-encode dimension for a fixed Ar^.
3.4 Convolution Spectral Imaging at High Field
Inequalities [3-15] and [3-16] expressed the constraints necessary to perform convo¬
lution spectral imaging. It was noted that these inequalities are more easily satisfied when
the separation between spectral resonances, Av, is large and when the sample dimensions,
Rt and Rx, are small. The reduction in sample size also can have the benefit of helping to
reduce the inhomogeneity term, AB. Under these conditions the applied gradient magni¬
tudes can span a wider range, allowing more control over the field-of-view of the image.
To demonstrate these ideas, experiments similar to those corresponding to Fig. 3-4
were performed at a much higher static field, 7.1 T, using a slightly smaller sample. The
spectrometer was a modified Nicolet NT-300 high resolution unit operating at a proton
resonance frequency of 300 MHz and interfaced to a Nalorac superconducting magnet
with a 49 mm diameter vertical bore. Modifications and additions to the existing
hardware were performed by W.W. Brey, B. Beck, J.R. Fitzsimmons, T.H. Mareci, and
R.G. Thomas. The additions included: a probe completely manufactured in-house, a set
of 3 Techron model 7540 power amplifiers, a logic box, and appropriate cabling. The
probe contained an RF coil made of copper foil and a Teflon film dielectric mounted on a
thin-walled glass tube with a 5 mm diameter; it was single-tuned to 300 MHz and had an
active volume of 196 mm3. The coil was a copy of a standard coil supplied with the spec¬
trometer. A set of three orthogonal gradient coils consisting of two Golay pairs, one for x
and one for y, and a Helmholtz pair for z was mounted on a plexiglás tube of inner

129
diameter 12 mm which surrounded the RF coil. The gradient coils were driven by con¬
stant voltage outputs from the Techron amplifiers. Input to the amplifiers came from
digital-to-analog converters (DACs) located in the spectrometer’s 293B pulse programmer.
The output of these DACs was normally routed to the shim coils for computer-controlled
adjustment of the static field homogeneity. The logic box re-routed the DAC output to
the gradient amplifiers for imaging experiments. Maximum gradient output was found to
be 0.041 mT mm'1 for the x, 0.036 mT mm'1 for the y, and 0.054 mT mm"1 for the z.
Control of the imaging facilities was accomplished by modifying the assembly language
software package provided with the spectrometer, which was done by the author, based
on work done earlier by R.G. Thomas and T.H. Mareci. Software was written to control
the logic box output routing and load the DACs with appropriate output voltages. In
addition, timing software was written to enable the user to control the relative steppings
of gradient amplitudes and time delay lengths. Finally, the modified software package
allowed the processing of multidimensional data files.
Images of a 3 mm diameter glass microsphere were obtained using the pulse sequence
of Fig. 3-2a with the slice-select gradient removed and the frequency-selective 7r/2 pulse
replaced by a non-selective one. The phase cycle is shown in Table 3-1. The sphere was
filled with a CuS04-doped mixture of water and acetone. The measured Ta values of the
two liquids were approximately 130 and 1600 msec, respectively. The sphere was centered
along the z axis by using a set-up sequence similar to that described in section 3.2. The
transmitter was centered between the water and acetone resonance frequencies; the spec¬
trum is shown in Fig. 3-6a. Comparison of Figs. 3-4a and 3-6a clearly shows that, as
expected, the separation between the water and acetone resonances increased by a factor
of 3.5 on going to the higher static field. The effect on the convolution spectral imaging
experiment was to allow more spectral room to put spatial information. Much greater
resolution could be obtained while keeping the TE fairly short. For Figs. 3-6b-d, the
fields-of-view in the x and y dimensions were slightly less than twice the diameter of the

I I I I I I I
1000
Water
Av-
J V
i i i I i i i i i
500
~i—r
Acetone
t i i i i—i i i | -i r i i i i i i i |
-500 -1000 Hz
FIG. 3-6. Convolution spectral imaging at high field, (a) The spectrum of a CuS04-
doped mixture of water and acetone contained in a 3 mm diameter glass microsphere;
Ai/ = 714 Hz.

131
microsphere. The tt pulse was not stepped for the image of Fig. 3-6b. Thus, no spectral
information was mixed into the phase-encode dimension, seen along the y axis, and the
two subimages overlapped. The chemical shift effect was so large and the field-of-view
was so small that the overlap was substantial. For Fig. 3-6c, the convolution effect along
the y axis was used to separate the overlapping subimages while maintaining spatial and
spectral resolution. Relative to the previous image, the price paid for removing the over¬
lap was roughly a doubling of the TE. Finally, Fig. 3-6d is a convolution spectral image
obtained by doubling the number of phase-encode steps used to obtain Fig. 3-6c, which
increased the TE by roughly 50% to a value of 96 msec, which is marginally acceptable for
most proton imaging of liquids. The effect of the increase in TE was seen by comparison
of Figs. 3-6e and 3-6f, which are the projections of Figs. 3-6c and 3-6d, respectively,
onto the x,vx axis. Some slight attenuation at the greater TE was evident but not nearly so
dramatic as that seen in Fig. 3-4. These profiles should have been convex, symmetric
curves since the sample was spherical. However, since a small amount of liquid was con¬
tained within a glass stem used to support the microsphere, the change in the sample
geometry at the junction of the stem and the sphere was enough to distort the static field
and thus the profiles were skewed.
The larger chemical shift difference between the water and acetone peaks afforded
by the high static field also allowed high spatial resolution along the phase-encoded axis
without a loss in signal. In Fig. 3-6d, the spatial resolution along the y axis was 23
microns per point. This resolution is better by a factor of six than that obtained for the
image of Fig. 3-4c, and the relaxation attenuation is not nearly as great. However, this
comparison is not perfect. For equal numbers of phase-encode steps, the spatial resolu¬
tion of the image of Fig. 3-4c would have been worse than that of Fig. 3-6d even if the
images had been obtained using the same equipment because of the larger field-of-view
required for the former. Better spatial resolution could have been achieved for the images
of Fig. 3-4, but the TE values could not have been kept reasonably short.

FIG. 3-6—continued, (b) Conventional spin-echo image of the microsphere. Due to the
shorter TR, this image is more highly Tj-weighted than (c) and (d). Total time for
acquisition = 4.47 min, N â–  64, M = 128, TR = 1.05 sec, TE = 32.4 msec. Resolutions:
x(fe) 93 microns/point, t/x(fe) 31.25 Hz/point, y(pe) 46 microns/point.

5.89 mm
X
5.95 mm
r
1000
500
1-1 I
-500
~T 1
-1000 Hz
Ai/
UJ

FIG. 3-6—continued, (c) Convolution spectral image of the microsphere. The resolution
is identical to (b), but the subimages are separated. Total time for acquisition = 32.34
min, N = 64, M = 128, TR * 7.58 sec, TE « 64.1 msec. Resolutions: x(fe) 93
microns/point, i/x(fe) 31.25 Hz/point, y(pe) 46 microns/point, i/y(pe) 15.625 Hz/point.

>
5.95 mm

FIG. 3-6—continued, (d) Convolution spectral image of the microsphere with greater
spatial and spectral resolution in y and uy. Attenuation of the water subimage due to T2
processes is evident. Total time for acquisition = 64.96 min, N = 64, M = 256, TR = 7.61
sec, TE - 96.1 msec. Resolutions: x(fe) 93 microns/point, i/x(fe) 31.25 Hz/point, y(pe) 23
microns/point, vy(pe) 7.8125 Hz/point.

E
E
>“o>
00
in
1—r
1000
I
-500
o
o
-1000 Hz

T
t—i—i—r
i—i—i—r
r~r
500
i—i—r
T
0
Au
i—i—r
I
-500
1000
5.95 mm
X
-1000
-H
FIG. 3-6—continued, (e) The projection of (c) onto the x,vx axis.

,—r
-500 -1000 Hz
5.95 mm
X
FIG. 3-6—continued, (f) The projection of (d) onto the x,i/x axis using the same scaling
as for (e). The data of (b)-(f) were processed using a two-dimensional Fourier transform
with sine-bell multiplication in both dimensions and no zerofilling and are displayed in
magnitude mode.

140
3.5 Conclusion
Convolution spectral imaging substantially reduces the total time required for data
acquisition and processing without a loss of information. The use of weak gradients makes
the technique inherently more sensitive than techniques in which large gradients are used
to destroy spectral information. However, because the minimum value of the echo time,
TE, is restricted in part by the number of data points in the phase-encoded dimension(s),
fast-relaxing nuclei whose resonances are not well-resolved could possibly not be imaged
at high resolution. Convolution spectral imaging appears to be best suited for slowly
relaxing nuclei whose spectral resonances are well-separated and whose spatial distribu¬
tions are small. These criteria suggest that the technique might find its greatest applica¬
tions in the relatively new field of NMR microimaging (Agu86). The spatial resolutions
obtained in the studies presented here are very encouraging. The speed of convolution
spectral imaging could be increased by extending the convolution spectral imaging con¬
cept to the fast spectral imaging method of Stroebel and Raetzel (Str86), reducing the total
time required for data set acquisition by a factor of n where n is the number of data
points in the spectral dimension.
The separation of spatial and spectral phase-encode processes, as is done in the
three-dimensional technique of Sepponen, et al. (Sep84), has a S/N advantage over the
convolution of the two phase-encode processes but only because more phase-encode data
points are being sampled to create the three-dimensional data set. This requires a longer
total acquisition time. The spectral dimension is separated from the spatial dimensions,
but the technique is usually applied to cases where the spectral resolution is sacrificed so
that greater spatial resolution may be obtained in a given total acquisition time. The
number of data points sampled during spectral phase-encoding is usually kept small to
reduce the total acquisition time. In this case, the peaks of the Fourier transformed spec¬
tra, each of which corresponds to a two-dimensional spatial location, may not coincide
with the true frequency positions of the spectral resonances due to the finite digital reso-

141
lution. The observation of well-resolved, spatially localized spectra is not the goal of the
Sepponen technique. Therefore, if two-dimensional spatial images corresponding to indi¬
vidual resonances are desired, and if inequalities [3-15] and [3-16] can be satisfied, then
convolution spectral imaging is the more efficient technique.

CHAPTER 4
QUANTIFICATION OF EXCHANGE RATES WITH RED NOESY SPECTROSCOPY
4,0 Introduction
Chapters 2 and 3 used the density operator formalism and the basic ideas of multidi¬
mensional NMR to describe methods for the correlation of spectral and spatial informa¬
tion. This chapter uses the same formalism and ideas but applies them to the correlation
of different types of spectral information. Specifically, two-dimensional methods for the
observation of exchange processes are described. Exchange processes which may be stu¬
died by NMR rely on the magnetic inequivalence of two or more nuclear sites, each of
which may consist of a number of magnetically equivalent nuclei. Each site has a charac¬
teristic resonance frequency. Thus the NMR spectrum is sensitive to processes where the
nuclei exchange sites. Because the NMR timescale is slow relative to the timescales of
other forms of spectroscopy, the rates which can be studied are also slow, on the order of
1 O'x-10s sec'1.
NMR methods have been used to study a variety of exchange processes including
hindered internal rotation, nuclear transfer, and molecular rearrangement. Some of these
are intramolecular. Examples are the rotation of N-methyl groups about the
nitrogen-carbon bond in N,N-dimethyl amides, the exchange of axial and equatorial pro¬
tons during ring inversion in cyclohexane, and proton transfer in the keto-enol tautomer-
ism of acetylacetone (Joh65). Other exchange processes studied by NMR are intermolecu¬
lar. These include studies of proton transfer in alcohol solutions (Loe63). Exchange has
also been observed in physiological systems using NMR (Gar83).
Attempts to quantify exchange rates using NMR spectroscopy have been made
almost since the discovery of NMR itself. Gutowsky, et al. are generally credited with the
first efforts (Gut53a). These workers realized that exchange processes which are rapid
142

143
relative to the difference in spectral frequencies of the exchanging nuclei would produce
broadened spectral lines. The degree of broadening could be predicted by solving a set of
Bloch equations which had been modified to include exchange terms. These solutions, the
lineshape equations, could be fitted to experimental spectra and the exchange rate deter¬
mined. The earliest lineshape equations were derived for the case of two
equally-populated sites where the exchanging nuclei spent equal amounts of time in each
site. It was assumed that the nuclei had the same relaxation behavior and spin-spin cou¬
pling was ignored. This model predicted three types of lineshapes. In the slow-exchange
limit, where the exchange rate was much less than the chemical shift difference of the two
sites, two distinct lines appeared. At coalescence, when the exchange rate was near the
chemical shift difference, a single broad line resulted. Finally, in the fast-exchange limit,
where the exchange rate was much greater than the chemical shift difference, a sharp line
was predicted.
Analytical solutions to modified Bloch equations were derived later for other simple
systems, such as three exchanging sites (Gut53b). However, the complexity of the
lineshape equations and the lack of computing power at the time brought about the
development of a number of approximate methods for determining exchange rates.
Loewenstein and Connor (Loe63), Johnson (Joh65), and Sandstrom (San82) have discussed
a number of these. The idea was to relate the exchange-dependent change of some spec¬
tral parameter to the exchange rate and so avoid the need to fit a lineshape equation to an
experimental spectrum. For the case of nuclei in two sites exchanging below the coales¬
cence exchange rate, some of the parameters which were tried included the ratio of the
peak separation near coalescence to that in the slow-exchange limit and the ratio of the
maximum peak intensity to the intensity at the center of the two peaks. For the same case
above coalescence, parameters included the ratio of the half-height linewidth in the
fast-exchange limit to that at coalescence and the ratio of the peak height in the
fast-exchange limit to that at coalescence (San82). Most one-parameter methods have

144
been applied only to two-site systems and ignore the effects of spin-spin coupling. Few
can be used over a wide range of temperatures. Somewhat surprisingly, in many studies
temperature-dependent exchange rates found using one-parameter methods have been
fitted very well to Arrhenius theory. However, for particular molecules the calculated
energies of activation have varied enormously from study to study. Rabinovitz and Pines
have concluded that the goodness-of-fit of exchange rates determined by one-parameter
methods to the curve of an Arrhenius equation is no indicator of the accuracy of the data
(Rab69).
With the advent of more powerful computers, approximate methods could be aban¬
doned and the idea of complete lineshape analysis could be extended to more complex
systems of sites. Exchange rates could be calculated by finding numerical solutions to
exchange-perturbed Bloch equations. Multiple spectral parameters could be varied to
produce calculated lineshapes which could be compared with an experimental result. The
parameters included coupling constants, chemical shifts, relaxation and exchange rates,
and site populations. These could be adjusted for each site. In this chapter, data obtained
using complete lineshape analysis are regarded as the standards against which the results
of other methods are measured.
The methods alluded to so far have been one-dimensional NMR techniques for
determining exchange rates. This chapter introduces a new two-dimensional method. It
is based on the well-known two-dimensional pulse sequence called NOESY (Jee79).
Measuring an exchange rate using this sequence requires the time-consuming process of
acquiring several NOESY spectra. The new pulse sequence, called RED NOESY, allows
the near-simultaneous acquisition of these NOESY spectra and, under certain conditions,
a large reduction in the required experimental time (Mar86a). Because the RED NOESY
sequence is derived from the NOESY sequence, many of the aspects of NOESY are dis¬
cussed before introducing RED NOESY. The chapter then describes the limitations of
RED NOESY and possible solutions. Finally, applications of the method to the measure-

145
ment of exchange rates of methyl groups in three N,N-dimethyl amides are shown. For
one of these, dimethylformamide, the exchange rates were measured as a function of tem¬
perature, enabling calculations of activation properties from RED NOESY data. The data
are compared with literature values found using other techniques.
4.1 The NOESY Pulse Sequence
With the advent of multidimensional NMR came the development of a
two-dimensional method for measuring exchange rates, the NOESY (Nuclear Overhauser
Enhancement SpectroscopY) technique of Jeener and coworkers (Jee79). This method has
the potential of simultaneously allowing the observation of all of the exchange processes
occurring within a spin system. The name of the method comes from the fact that
NOESY can also be used to observe the nuclear Overhauser effect, which can arise from
cross-relaxation between nuclei and can be used to determine internuclear distances. This
application of the method will not be discussed in this chapter.
The NOESY pulse sequence is shown in Fig. 4-1. It consists of three 7r/2 pulses
separated by the time delays tj and rm and a linear field gradient (a ’homospoil’ pulse)
applied for a time, rh, during the delay, rm. Data is acquired during t2. The mixing time,
rm, and the data acquisition time are fixed for a particular NOESY experiment. As
described in Chapter 1, t! is incremented to map out the signal modulation related to
events occurring during that time interval. Thus a matrix of signals, each having been
acquired with a different value of tlt is generated. Two-dimensional Fourier transforma¬
tion of this data matrix produces a 2D spectrum which correlates the precessional fre¬
quencies of the spins during t2 with their frequencies during t2. Qualitatively, it may be
said that the spectrum correlates the pre-exchange and post-exchange spectral frequencies
of the sites; commonly, the axes of the spectrum are labeled Fa and F2, respectively. The
mixing time, rm, is the amount of time allowed for exchange to take place.

146
*72 *72 */2
FIG. 4-1. The NOESY puke sequence. This is a method for correlating the pre- and
post-exchange spectral frequencies of exchanging spins. The timing of the sequence and
the labels given to the various timing intervals are indicated at the bottom. The prepara¬
tion period consists of the first jt/2 pulse and a delay between successive repetitions of the
puke sequence. The modulation which is produced during the evolution period, tlt is a
function of the pre-exchange spectral frequencies of the exchanging spins. Immediately
following the evolution period, the mixing period consists of a pair of x/2 pulses
separated by the mixing time, rm; spin exchange occurs during this interval. A field gra¬
dient of fixed amplitude and duration is applied during the mixing period to diminish
unwanted magnetization which could disturb peak intensities. Modulation with respect to
the detection period, t2, is a function of the post-exchange spectral frequencies of the
exchanging spins. RF - radiofrequency transmitter, g * field gradient.

147
The origins of the peaks of the 2D NOESY spectrum may be seen by performing an
analysis of the pulse sequence when applied to a system of two nuclei using the density
operator formalism introduced in Chapter 1. Relaxation and spin-spin coupling are
ignored in this treatment. The initial state of the density operator is
a(0) = 0(llt + I2e) , [4-1]
where the subscripts 1 and 2 label the exchanging nuclei. After the application of a tt/2
pulse with phase £ = 0 and chemical shift precession during t2, the density operator
becomes
cr(if) = - IjyCOSÍWjtj) + I^sinCwjtj) - Ijycosiwjtj) + I2xsin(w2t1) , [4-2]
where o(tf) is the density operator before the second t/2 pulse. The f) term has been
dropped for convenience but will be implied for the rest of the analysis. Applying a
second 7r/2 pulse with the same phase as the first rotates the y components to the z axis of
the rotating frame:
= - IuCosO^tj) + IixSiniwjtj) - Ij.cosiwjtj) + I^sin^tj) , [4-3]
where oit^) is the density operator after the second t/2 pulse. The magnetization has
been split such that the x components continue to evolve in the transverse plane.
If the NOESY pulse sequence is applied to a spin system which is in the slow-
exchange limit, where the exchange rate between two sites is much less than their chemi¬
cal shift difference, the effect of exchange on the transverse components of the density
operator can be ignored (Jee79). However, it is necessary to introduce the kinetic matrix
for a two-site system to see the effect which exchange has on the z components. During
the mixing time, z magnetization of spin 1 becomes z magnetization of spin 2 and vice
versa:

148
*<12
Ii. Ü *2. . [4-4]
k21
where k12 and k21 are the forward and reverse exchange rates, respectively. The change
with time of each of the z components is expressed by the kinetic matrix equation:
dlu
dt
C4
H
1
1
1
pl.l
dl2.
CVJ
H
1
1
*—1
cs
1
hi.
dt
[4-5]
This system of linear differential equations can be solved as an eigenvalue problem to see
how IlE and I2e evolve under the effects of exchange. The analysis can be simplified by
defining a new exchange rate, k, such that k/2 = k12 = k21. This is a valid substitution
when the mole fractions of exchanging nuclei in each site are identical and when the for¬
ward and reverse exchange rates are identical. Solving the differential equations shows
that exchange has the following effects:
IlE(l + e - kt) + I2e( 1 -e-kt),
[4-6a]
IlEd - e - kt) + I2e( 1 + e-kt).
[4-6b]
Since the effects of exchange have been described, the density operator analysis of
the NOESY pulse sequence can be continued. Under the effects of chemical shift preces¬
sion, exchange, and the field gradient shown in Fig. 4-1, c^t^) evolves during rm to
become
°(Tm) = - lu(l + e ' krm)cos(w1t1) - I2t(l - ekTm)cos(w1t1) [4-7]
+ Ilxsin(w1t1)cos(w1rm)cos(2jrkhr1) + Ilysin(w1t1)cos(w1rm)sin(2jrkhr1)
+ llysin(w1t1)sin(u;Irm)cos(27rkhr1) - Ilxsin(w1tI)sin(o;irm)sin(27rkhr1)
- IlE(l - ekTm)cos(w2t1) - 12e(1 + e ' kTm)cos(w2t1)

149
+ I2xsin(w2t1)cos(u;2rrn)cos(27rkhr2) + I2ysin(w2t1)cos(w2Tm)sin(27rkhr2)
+ I2ysin(w2t1)sin(w2rm)cos(27rkhr2) - I2xsin(w2t1)sin(w2Tm)sin(25rkhr2) ,
where rj and r2 are the spatial positions of the exchanging nuclei along the gradient axis
and kh = TGrrh/2tr, where Gr and rh are the gradient amplitude and duration, respectively.
The symbol kh has been chosen to be consistent with the k-space notation commonly used
in the NMR imaging literature and should not be confused with the exchange rates k, k12,
and k2j. A final tr/2 pulse, again with phase ( = 0, brings the z components which have
undergone exchange into the transverse plane for data acquisition:
^m) = + e kTm)cos(w1ti) + I2y(l - ekTm)cos(wit1) [4-8]
+ Ilxsin(wit1)cos(w1rm)cos(2irkhr1) + lltsin(w1ti)cos(c;1rin)sin(2irkhr1)
+ Iltsin(a;1t1)sin(a;1rm)cos(27rkhr1) - llxsin(w1t1)sin(w1rm)sin(2Tkhr1)
+ Ily(l - ekTm)cos(w2t1) + l2y(l + e ' kTm)cos(w2t1)
+ l2xsin(a;2t1)cos(a;2rm)cos(27rkhr2) + I2lsin(w2t1)cos(w2rm)sin(27rkhr2)
+ l2lsin(w2t1)sin(w2rm)cos(27rkhr2) - I2xsin(w2t1)sin(w2rm)sin(27rkhr2) .
Terms 1, 2, 7, and 8 of this expression represent the desired peaks of the NOESY spec¬
trum, which are directly related to the kinetic matrix. Terms 1 and 8 represent magneti¬
zation components which precessed at the angular frequencies of spins 1 and 2, respec¬
tively, during tj and will precess at the same angular frequencies during detection. Thus
the peaks which these terms represent will appear on the diagonal of the NOESY spectrum
and are called diagonal peaks. The diagonal peaks decrease in intensity as the exchange
rate and the mixing time increase. Terms 2 and 7 represent magnetization components
which precessed at the angular frequencies of spins 1 and 2, respectively, during tj but
will precess at the angular frequencies of spins 2 and 1, respectively, during detection.
The peaks which these terms represent lie off the diagonal of the NOESY spectrum and
are called cross peaks. In the absence of spin relaxation, the intensities of the cross peaks

150
increase as the exchange rate and the mixing time increase. Terms 3, 6, 9, and 12 are
terms which carry no exchange information and can perturb the diagonal peak intensities.
Terms 4, 5, 10, and 11 do not evolve into observable magnetization during detection.
The analysis which produced Eq. [4-8] was performed using the assumption that
exchange does not perturb the transverse terms of the density operator. This assumption
is true only when the exchanging nuclei are in the slow-exchange limit (Jee79). As the
exchange rate increases, to the point of peak coalescence and on to the fast-exchange
limit, the lineshapes of the NOESY spectrum would behave just as in a series of one¬
dimensional lineshape analysis experiments. The diagonal and cross peaks for each pair of
exchanging sites would broaden in both dimensions, coalesce, and finally collapse into a
single diagonal peak. Thus the NOESY experiment can be run successfully only when the
spin system is in the slow-exchange limit.
Equation [4-8] shows a number of other features of the NOESY experiment. First,
the detected signal is amplitude-modulated as a function of tj. By the equations of
Chapter 1, section 1.2.1, the Fourier transform of the signal with respect to tx produces
sign ambiguities of the resonance frequencies along the Fj axis. The NOESY spectrum is
then a superposition of two NOESY spectra; one has diagonal peaks where oij = u2 and the
other where - = u2. This is an undesirable situation because it could create confusion
in the interpretation of the spectrum, especially if the spin system is complex. A number
of phase cycling procedures have been developed to remove the sign ambiguity in Fx
(Der87). All of them work by using phase shifts of the RF pulses and receiver to produce
a second signal whose phase is shifted 90 degrees relative to that of Eq. [4-8]. Often, this
signal is added to the first during signal averaging to simulate quadrature detection.
A second feature of the NOESY sequence is that the field gradient applied for a time
rh during the mixing time, rm, affects only the magnetization components which carry no
exchange information. This is shown by Eq. [4-8]. This provides a way of eliminating the
effects of these components without perturbing the desired exchange-weighted com-

151
ponents. For a fixed gradient amplitude, the gradient duration can be adjusted such that
the unwanted terms destructively interfere when integrated over the range of nuclear
positions. The field gradient does not destroy these terms but does remove them from the
detected signal.
Equation [4-8] was not derived with an accounting for relaxation effects, and so it
describes diagonal peaks which decay and cross peaks which grow according to the
exchange rates and the mixing time. However, Jeener, et al. have shown that spin-lattice
relaxation also perturbs these peak intensities (Jee79). If the relaxation rates of the two
exchanging nuclei are the same and the exchange rates are such that k/2 = k21 = k12, then
In = I22 = Ce'RlTm( 1+e'kTm), [4-9a]
I12 = I21 = Ce'RlTm( 1 -e'kTm), [4-9b]
where Rj is the relaxation rate, k is the exchange rate, C is a constant, and In and I22 are
the diagonal peak intensities and I12 and l21 are the cross peak intensities. These equa¬
tions are valid for a two-site system where the forward and reverse exchange rates
between the sites are equal, the mole fractions of nuclei which occupy each site are equal,
and the nuclei of the sites experience the same local field fluctuations and thus have the
same relaxation rates. These intensities are not peak areas as in one-dimensional NMR
spectroscopy; instead they are peak volumes. Plots of these intensities versus the product
of rm and k for different Rj values are shown in Fig. 4-2; C has been chosen to be 1/4 so
that the sum of the four peak intensities equals unity when rm = 0. The intensities behave
in the manner described by Eq. [4-9]. As Rj increases relative to k, the intensity curves
are damped more by relaxation. By Eq. [4-9], the peak intensities of several NOESY
plots, each obtained with a different mixing time, can be used to find the values of R2
and k.
Several obstacles can impede successful implementation of the NOESY method.
First, for appreciable cross peak intensity to appear, k must be greater than or equal to Rj

Peak Intensities
FIG. 4-2. Peak intensity behavior as a function of the mixing time and exchange rate.
These are plots of Eq. [4-9] of the text for two different conditions. The upper pair of
curves describe the behaviors of the diagonal and cross peaks of a NOESY spectrum when
I*! = k/4. The behaviors of the diagonal and cross peaks when Rj = k are described by
the lower pair of curves.

153
so that exchange takes place before the signal decays due to relaxation. This restriction
places a lower limit on the temperature range at which the method can be used since tem¬
perature adjustment is the usual way of controlling the exchange rate. For many liquids,
this is not a serious obstacle because in general Rj decreases and k increases with increas¬
ing temperature. However, physical properties of the sample may limit the usable tem¬
perature range. It is possible that the boiling point of a sample may be reached before k
becomes on the order of Rx. The upper limit of k is also limited by peak coalescence;
above temperatures where k is approximately equal to the frequency separation between
peaks belonging to the exchanging sites, the peaks merge and no intensity information can
be obtained.
A second difficulty with the NOESY sequence is the residual transverse magnetiza¬
tion produced by the second tt/2 pulse. This corresponds to terms 3, 6, 9, and 12 of
Eq. [4-8]. From the equation it can be seen that these components, if not removed, can
interfere with the diagonal peak intensities. It has also been shown that the application of
a field gradient during the mixing time can be used to suppress these transverse terms.
However, the application of the field gradient temporarily destroys the static field homo¬
geneity necessary for the maintenance of the field/frequency lock, precluding use of the
lock system. Thus, peak frequency measurements are subject to error due to the field
drift which occurs during acquisition of the 2D signal matrix.
Another problem with the NOESY sequence is the existence of "axial peaks". The
origin of these was not shown by the analysis which led to Eq. [4-8] because relaxation
was not taken into account. However, the peaks can be explained qualitatively. Longitu¬
dinal magnetization which returns to thermal equilibrium during the mixing period is
placed in the transverse plane by the third n/2 pulse, where it is detected. This magneti¬
zation is not modulated as a function of tj and so is represented by peaks which appear at
zero frequency of the Fa axis and the resonance frequencies of the sample nuclei on the F,
axis. Axial peaks become more intense as rm increases since more relaxation occurs.

154
Phase cycles have been developed which remove axial peaks because they can interfere
with the desired peaks (Der87). These cycles can be combined with those used to simulate
quadrature detection in F1.
A final obstacle is the correct choice of the mixing time. If rm is too short, cross
peaks will not appear. If rm is too long, the cross and diagonal peaks will have decayed
substantially due to spin relaxation and possibly spin diffusion. Jeener, et al. derived an
equation relating the maximum cross peak intensity to rm, which for the system of two
exchanging spins discussed earlier has the form (Jee79)
r
m,max
Rx + k
[4-10]
Unfortunately, calculation of rmmax requires prior knowledge of k and R:. Good esti¬
mates can be made for these parameters, but normally several NOESY experiments are
run, each with a different rm but at a fixed temperature to ensure that cross peaks are
seen. This can be very time-consuming, especially if the temperature-dependent behavior
of the sample is to be investigated.
While all the problems of the NOESY technique may not be solved by pulse sequence
modifications, a number of them can be minimized. Phase cycling procedures for the
removal of axial peaks and for the simulation of quadrature detection in Fj have already
been indicated. The rest of this chapter is concerned with a modification of the NOESY
pulse sequence which allows the near-simultaneous acquisition of several NOESY spectra,
each having a different mixing time. Under certain experimental conditions, the method
can be used to reduce the time needed to acquire several different NOESY spectra. This
can ease the search for a mixing time which gives maximum cross peak intensity and the
acquisition of temperature-dependent data for the calculation of activation parameters.

155
4.2 The RED NOESY Pulse Sequence
The RED NOESY pulse sequence is shown in Fig. 4-3. It is based on the same prin¬
ciples as the TART (Mar86b) and STEAM (Haa85b) NMR imaging sequences for produc¬
ing a series of ^-weighted images. RED NOESY is very much like the NOESY sequence;
however, the third pulse is replaced by a series of N ’read’ pulses with different tip
angles, an. Each of these pulses samples only a portion of the exchanged magnetization.
The portion not sampled continues to undergo exchange. Thus, by acquiring data after
each read pulse, a series of N NOESY exchange spectra may be produced, each with a
different rm. The read pulses are all of tip angle less than 90 degrees, hence the origin of
the name RED (REDuced tip angle) NOESY. Ignoring relaxation and exchange, the
amounts of exchanged magnetization sampled by the read pulses will be equal when the
pulses obey the recursive relation
an . j = arctan(sin(an)) , [4-11]
where n is the read pulse index and a is the read pulse tip angle (Mar86b). If the nth read
pulse is set to 90 degrees, then previous pulses have the values 45, 35.3, 30 ... . degrees.
The sequence can have an arbitrary number of read pulses, allowing the acquisition of an
arbitrary number of NOESY spectra.
4.3 Problems Unique to the RED NOESY Sequence
Although potentially very powerful, the RED NOESY sequence possesses all of the
experimental restraints of the NOESY sequence plus a few additional. The first of these is
directly related to the multipulse nature of the RED NOESY experiment. Each read pulse
separates the magnetization into a number of parts. Only some of these components con¬
tain information about exchange. The other components remain, however, and, after
application of additional read pulses, can form unwanted echoes. The field gradient pulse
used in the NOESY pulse sequence minimizes this effect. Yet, for a RED NOESY experi¬
ment, merely using a series of field gradient pulses of equal amplitude and duration after

rmN
acqN
FIG. 4-3. The RED NOESY pulse sequence. The timing of the sequence and the labels
given to the various timing intervals are indicated at the bottom. This is an extension of
the NOESY sequence in which only a portion of the magnetization which exchanges dur¬
ing the mixing time is sampled by each of a series of "read" pulses, labeled an. The por¬
tion not sampled continues to exchange. The durations of the applications of the field
gradient are doubled with every mixing period to reduce the effects of spurious echoes.
RF = radiofrequency transmitter, g = field gradient.
Os

157
each acquisition will not reduce spurious echoes. This is because succeeding field gradient
pulses tend to refocus the dephasing effect of previous ones, causing the formation of
"gradient" echoes. Barker and Mareci have studied the problem in detail using coherence
transfer pathway theory and have suggested that doubling the duration of each succeeding
field gradient pulse while maintaining the amplitude will suppress unwanted echoes
(Bar88). This lengthening of the gradient durations restricts the minimum values of the
mixing times. If the field gradient is not powerful enough to diminish unwanted
transverse magnetization quickly, then the mixing times may be too long to observe the
growth and decay of cross peaks. As the number of read pulses increases, the field gra¬
dient must be even more efficient.
A second and possibly more serious limitation of the RED NOESY sequence is that
the acquisition times, t2, are part of the mixing times, rm. Thus the mixing times impose a
limit on the maximum acquisition times. To ensure adequate peak digitization, acquisition
times are often chosen to be on the order of the inverse of the smallest peak width at half
maximum. For high resolution spectra of liquids where linewidths may be less than 1 Hz,
acquisition times should be several seconds. This forces the mixing times to be several
seconds. As shown by Fig. 4-2, this implies that under these conditions only slow
exchange processes where k is on the order of a few tenths of a second can be studied
with RED NOESY. As exchange occurs more quickly and k increases, it becomes neces¬
sary to use shorter mixing times and the acquisition times must be reduced. This forced
truncation of data acquisition can lead to poor peak digitization and "sine-wiggles" in the
baseplane of a NOESY spectrum. This makes accurate measurement of peak intensities
more difficult. One possible solution is the use of strong apodization of the time-domain
data before Fourier transformation. This broadens the peaks, effectively allowing each
data point to define more of the spectrum. The disadvantage of this approach is that high
resolution is lost and peak overlap can occur.

158
Tip angle calibration is especially crucial to the RED NOESY method. Deviations of
the read pulse tip angles from the values calculated using Eq. [4-11] will perturb peak
intensities. Normally the duration of a ?r/2 or tr pulse is determined and the values of the
read pulse durations are calculated as fractions of one of these values. For a well-tuned
RF coil on many NMR spectrometers, the duration of the tr/2 pulse is a few microseconds
which is accurate to the tenths place. If a large number of read pulses is desired for a
RED NOESY experiment, the tip angles of the first few read pulses will differ by such a
small amount that accurately defining them may be impossible. For these cases, it may be
necessary to detune the coil or to insert an attenuator into the RF transmission line to
increase the duration of the tt/2 and the read pulses. These measures will reduce the
effective spectral width excited by the RF pulses, which can also perturb peak intensities.
A fourth concern of the RED NOESY method is its sensitivity, defined as as the
signal-to-noise per unit time, relative to the NOESY method. The signal-to-noise (S/N) is
reduced in a RED NOESY experiment relative to the NOESY experiment because the read
pulses of the former place only part of the exchanged magnetization into the transverse
plane. When the number of signal averages is fixed, each of the N NOESY spectra
obtained using a RED NOESY sequence containing N read pulses has a S/N equal to l/v'N
times the S/N of the equivalent NOESY experiment. However, to provide the same infor¬
mation as the RED NOESY experiment, N NOESY experiments would need to be run. If
one RED NOESY experiment requires a time T to complete, then the N NOESY experi¬
ments would require a time approximately equal to NT’. If the RED NOESY experiment
were modified such that N signal averages were performed at each increment of tlt the
time required to complete the experiment would also be NT’. Since the S/N increases as
the square root of the number of signal averages, the sensitivities of the NOESY and
RED NOESY methods are about the same.

159
4,4 Experimental
RED NOESY experiments were performed using three N,N-dimethyl amides: DMF
(N,N-dimethylformamide, Aldrich), DMA (N,N-dimethylacetamide, Sigma), and DMP
(N,N-dimethylpropanamide, Aldrich). All three were used without purification. All
spectra were acquired and processed using an NT-300 NMR spectrometer operating at a
proton resonance frequency of 300 MHz. Assembly language software was written by the
author to allow acquisition of up to 16 signals corresponding to 16 rm values per tj incre¬
ment for a single RED NOESY experiment. Additional post-acquisition processing com¬
mands were programmed in assembly language for data shuffling, baseline correction,
apodization, Fourier transformation, and scaling of the multiple NOESY data files which
made up a RED NOESY data file. All curve fitting was done using the Lotus 1-2-3
(Release 2) software package. Best fits were calculated using the method of least squares
with no weighting of the data points.
Proton data were acquired using the pulse sequence of Fig. 4-3 and the phase cycle
of Table 4-1. The phase cycle used four signal averages to cancel out axial peaks and to
simulate quadrature detection in Fj. This was done for the experiments using DMF and
DMA, since the spectral widths were such that the tails of the axial peaks could overlap
with the diagonal and cross peaks, making it difficult to quantify their intensities.
Without cancelling out axial peaks, quadrature detection in Fj could be simulated using
only the first two steps of the phase cycle. This was done for the experiments using DMP,
since the spectral widths were large and the axial peaks caused no interference. True qua¬
drature detection was used during data acquisition. Four read pulses of tip angle 30, 35.3,
45, and 90 degrees were used. The tip angles were found by first determining the 7r pulse
duration using a one pulse experiment in which the tip angle was adjusted to give a null.
The durations of the read pulses were then found by calculating fractions of the tr pulse
duration.

160
Table 4-1
Relative Pulse and Receiver Phases for RED NOESY
tj Step
Signal
Average
Phases
Pulses
Receiver
First
Second
Qn
*/2
»/2
1
1
0
0
0
0
1
2
270
0
0
90
1
3
180
0
0
180
1
4
90
0
0
270
2
1
0
0
0
0
2
2
90
0
0
270
2
3
180
0
0
180
2
4
270
0
0
90
M
4
270
0
0
90
Refer to Fig. 4-3 and its legend for the notation corresponding to this table.

161
The nominal duration of the field gradient pulse was found by applying a single RF
pulse of rr/2 radians or less followed by data acquisition in the presence of the gradient.
The duration required to eliminate the transverse magnetization was estimated by visual
inspection of the decay of the signal. Assembly language software based on work done by
R.G. Thomas and T.H. Mareci was written by the author to allow control of the amplitude
of the field gradient pulse through the voltage output of a digital-to-analog converter.
For these experiments, the maximum available voltage of 5 V was applied to the Zx shim
coil. A TTL logic line of the spectrometer’s 293B pulse programmer which could be
placed under software control was dedicated to gating of the field gradient pulse. Gating
was controlled by a logic box constructed by R.G. Thomas and modified by W.W. Brey.
The temperature at the sample was established in the following manner. A sealed
NMR tube containing pure ethylene glycol was allowed to equilibrate at a temperature
maintained by the variable temperature (VT) unit of the spectrometer. A proton spec¬
trum was obtained and the frequency separation between the hydroxyl and methylene
resonances was measured. The value of the frequency separation was known to be linear
over the temperature range, 310-410 CK (Van68). Seven readings of the frequency
separation at seven different temperatures were made at approximately 5 °K intervals over
the range 333-363 °K. A plot of the frequency separation versus temperature was made
and a line fitted to the data. The equation for the fitted line was
T(°K) = 473.436 - 0.3515(Ai/) , [4-12]
where T is the temperature and Ai/ is the frequency separation. The temperature calcu¬
lated from a measured value of Ai/ using this equation was about 3 °K greater than that
found using the equation of Van Geet (Van68):
T(°K) = 466 - 0.339(Ay) . [4-13]
Because Van Geet’s equation was found using a thermocouple immersed in ethylene
glycol, it was assumed that this equation was more valid. Therefore the temperature read-

162
ings of the VT unit were converted to temperatures at the sample. First, a VT unit read¬
ing was converted to a value of Av using Eq. [4-12]. This value was inserted into Van
Geet’s equation to obtain the corrected temperature. This procedure could not be used at
temperatures below 310 °K (Van68) and so these temperatures were not corrected.
All RED NOESY data sets consisted of 256 complex data points for each of the four
t2 intervals and 128 values of tj for a total matrix size of 128 by 1024 points. The data
were processed first by applying the same exponential apodization function to the tx and
t2 time domains. The time constant of the exponential was chosen to match the
half-height linewidth of the broadest line in the NMR spectrum. This procedure did not
always force the signals to decay into the noise and so some truncation artifacts were una¬
voidable. After apodization, a single zerofill was applied in the tx domain. The data
matrix was then subjected to four two-dimensional Fourier transformations to produce
four NOESY spectra. The magnitude spectra were then calculated and all four were
scaled identically and plotted using the same number of contours.
To extract the information necessary for measuring exchange rates, the peak intensi¬
ties were measured by using the digital integration software of the spectrometer to find
the relative areas of peaks in slices parallel to the F2 axis. These were the nearest approx¬
imations to the peak volumes which could be made with the spectrometer software. The
F2 axis was chosen because it had better resolution than the Fj axis. The slices were
chosen by visually searching through the RED NOESY spectra for the slices with the
greatest peak amplitudes. Early attempts at measuring the peak intensities were done by
fitting a Lorentzian function to the peaks of a slice taken parallel to the F2 axis and
recording the amplitudes of the fitted Lorentzians. However, this method did not give
reproducible results because the magnitude spectra did not have Lorentzian lineshapes
and because the amplitudes probably poorly approximated peak volumes.
The values of the relaxation rates, Ra, and the exchange rates, k, were found for
each RED NOESY experiment using Eq. [4-9]. The sum of the four peak intensities is

163
sum = e RlTm. [4-14]
Subtraction of the cross peak intensities from the diagonal peak intensities produces
diff = e'(Rl + k)Tm. [4-15]
To determine Rj and k, plots of ln(sum) versus rm and In(diff) versus rm were prepared
for each RED NOESY experiment. By Eq. [4-14] the slope of the former was - R1? and
by Eq. [4-15] the slope of the latter was - Rx + k; k could be obtained from the difference
of the slopes. By the theory of Jeener, et al., discussed in section 4.1, this value of k
equalled twice the forward or reverse exchange rates, k12 and k21, respectively (Jee79).
Thus for the calculation of activation parameters, k was divided by two and the result was
called k\ For DMF, the relaxation rates of the methyl protons were measured at several
different temperatures using the inversion recovery method (Vol68) for comparison with
the Rj values found using RED NOESY.
Using the values of k’ and the nominal and corrected values of T, several activation
parameters were calculated. These included the free energy of activation, AG*, the
Arrhenius activation energy, Ea, and its frequency factor, In (A), and the activation
enthalpy, AH*, and activation entropy, AS*. Values of AG* were found using the Eyring
equation as described by Sandstrom (San82):
AG* = 4.575 x 10 s T (10.319 + log(T/k')) , [4-16]
where AG* is in units of kcal mole'1, T is the temperature in Kelvin and k’ is the
exchange rate in sec'1. For DMF several values of k’ were found at different tempera¬
tures. The values of Ea and A were found from the slope and intercept, respectively, of a
line fitted to a plot of ln(k’) versus 1000/T. The slope was multiplied by the constant
-R = -1.98726 kcal mol'1 K'1 to obtain Ea in kcal mol'1; In (A) was equal to the intercept.
Similarly, AH* and AS* were found from the slope and intercept, respectively, of line fit¬
ted to a plot of ln(k’/T) versus 1000/T. The slope was multiplied by R to obtain AH* in
kcal mol'1. The value of AS* in cal mol'1 K'1 was found by the formula

164
AS* = R (intercept - ln(/ckB/h)) , [4-17]
where the transmission coefficient, /c, was assumed to be unity, kB, the Boltzmann con¬
stant, was 1.38045 x 10'16 erg deg"1, and h, the Planck constant, was 6.6252 x 10'27 erg
sec. The plots of ln(k’) versus 1000/T and ln(k’/T) versus 1000/T were made using the
nominal and corrected temperatures and are shown in Figs. 4-4, 4-5, 4-6, and 4-7.
4.5 Results
Table 4-2 summarizes the experimental results obtained using the RED NOESY
sequence. Activation parameters calculated from RED NOESY data for DMF are shown
in Table 4-3. Each error shown in Tables 4-2 and 4-3 is one standard deviation. For
comparison, some literature values of activation parameters are shown in Tables 4-4, 4-5,
and 4-6 for DMF, DMA, and DMP, respectively. The gaps in these tables indicate
parameters which were not given in the corresponding journal articles. Most of these data
were obtained using approximate lineshape analysis methods. Only the works of Ingle-
field, et al. (Ing68), Rabinovitz and Pines (Rab69), Drakenberg, et al. (Dra72), and
Isbrandt, et al. (Isb73) are complete lineshape analysis studies. Therefore, these are prob¬
ably the most accurate data.
4.6 Discussion
The four NOESY spectra obtained from a single RED NOESY experiment are shown
in Fig. 4-8. The sample was DMF maintained at a corrected temperature of 344 °K. The
mixing times were 1, 3, 5, and 7 seconds. The diagonal and cross peaks behaved accord¬
ing to Eq. [4-9]. The diagonal peak intensities decayed from a maximum at rm = 1 sec as
the mixing time increased. The cross peak intensities grew initially as rm increased and
then decayed as relaxation began to contribute more to the intensities. The plots of
ln(sum) versus rm and ln(diff) versus rm for these data are shown in Fig. 4-9. From these,
the relaxation rate of the methyl group protons was found to be 0.18 sec"1 and their

In (k')
FIG. 4-4. A plot of ln(k’) versus 1000/T for DMF. Each of the data points of the
Arrhenius plot were obtained by analyzing the diagonal and cross peak intensities of a
single RED NOESY experiment. The slope produced an energy of activation
Ea = 20 kcal mole-1 and the intercept produced a frequency factor ln(A) = 27.3.
o

In (k')
FIG. 4-5. A plot of ln(k’) versus 1000/T (corrected) for DMF. Each of the data points of
the Arrhenius plot were obtained by analyzing the diagonal and cross peak intensities of a
single RED NOESY experiment. The temperatures were corrected using the method
described in the text. The slope produced an energy of activation Ea = 20 kcal mole-1 and
the intercept produced a frequency factor ln(A) = 28.0.
o
O'

In (k'/T)
FIG. 4-6. A plot of ln(k’/T) versus 1000/T for DMF. Each of the data points of this plot
were obtained by analyzing the diagonal and cross peak intensities of a single
RED NOESY experiment. The slope produced an enthalpy of activation
AH* = 19 kcal mole-1 and the intercept produced an entropy of activation
AS* = -6.6 cal mol-1 K_1.
O'
--J

In (k'/T)
-5.00
D
-6.00-
-7.00-
-8.00+—
2.74
â–¡
i 1 1 1 1 1 1 1 1 1 1—
2.78 2.82 2.86 2.90 2.94 2.98
1000/T
FIG. 4-7. A plot of ln(k’/T) versus 1000/T (corrected) for DMF. Each of the data points
of this plot were obtained by analyzing the diagonal and cross peak intensities of a single
RED NOESY experiment. The temperatures were corrected as described in the text. The
slope produced an enthalpy of activation AH* = 19 kcal mole-1 and the intercept produced
an entropy of activation AS* = -5.0 cal mol-1 K-1.

169
Table 4-2
Temperature Dependence of Relaxation Rates, Exchange Rates, and
Free Energies of Activation for DMF, DMA, and DMP
Molecule Ta Ta Rj>
corrected
kb AG*C AG*C
corrected
DMF
DMF
DMF
DMF
DMF
DMF
DMF
DMF
DMF
DMF
DMA
DMA
DMP
338
336
343
340
347
344
352
349
357
354
357
354
358
355
362
359
362
359
363
360
293
293
293
0.14 ± 0.01
0.17 ± 0.01
0.18 ± 0.01
0.11 ± 0.02
0.07 ± 0.07
0.02 ± 0.05
0.08 ± 0.02
0.10 ± 0.01
-0.02 ± 0.04
0.03 ± 0.01
0.18 ± 0.02
0.18 ± 0.03
0.26 ± 0.07
0.31 ± 0.01
0.44 ± 0.01
0.60 ± 0.01
0.6 ± 0.2
1.4 ± 0.1
1.7 ± 0.2
0.7 ± 0.2
0.1 ± 0.6
2.90 ± 0.05
2.18 ± 0.07
0.45 ± 0.03
0.46 ± 0.03
1.75 ± 0.12
20.7 ± 0.3
20.8 ± 0.2
20.8 ± 0.2
21 ± 1
20.8 ± 0.4
20.7 ± 0.5
21 ± 1
23 ± 17
20.6 ± 0.3
20.9 ± 0.3
17.6 ± 0.3
17.6 ± 0.3
16.8 ± 0.3
20.5 ± 0.3
20.6 ± 0.2
21 ± 2
21 ± 1
20.6 ± 0.4
20.5 ± 0.5
21 ± 1
23 ± 17
20.4 ± 0.3
20.6 ± 0.3
Units for table entries:
a: °K
b: sec'1
c: kcal mol"1

170
Table 4-3
Activation Parameters of DMF Found Using RED NOESY Data
Eaa
In (A)
AH*a
AS*b
T (nominal)
20 ± 4
27.3 ± 0.4
19 ± 4
-6.6 ± 0.7
T (corrected)
20 ± 4
28.0 ± 0.4
19 ± 4
-5.0 ± 0.7
Units for table entries:
a: kcal mol'1
b: cal mol'1 K'1

171
Table 4-4
Activation Parameters of DMF:
Literature Values
Eaa
In (A)
AH*a
AS*b
Ref.
9.6
15.0
Fra60
18.3
24.9
Rog62
26
36.8
Fry65
18.7
27.2
Neu65
15.9
Whi65
28.2
39.6
Geh66
20.8
39.6
0
Ing68
20.5
29.2
20.2
-1.7
Rab69
26
35
11
Cal70
24.5
Ram71
21.3
20.5
-1.4
Dra72
20.88
0
Gut75
20
28.0
19
-5
this work
Units for table entries:
a: kcal mol'1
b: cal mol*1 K'1

172
Table 4-5
Activation Parameters of DMA:
Literature Values
Eaa
In (A)
AH*a
AS*b
Ref.
10.6
18.0
Rog62
23.2
37
Fry65
20.2
37.1
Geh66
23
37
11
Cal70
19.6
18.9
2.2
Ram71
19.0
18.3
0.7
Dra72
19.7
32.0
19.0
2.9
Isb73
17.98
0
Gut75
Units for table entries:
a: kcal mol'1
b: cal mol'1 K"1

Table 4-6
Activation Parameters of DMP:
Literature Values
Eaa
In (A)
AH*a
AS*b
Ref.
9.2
16.8
Woo62
17
28
Fry65
16.6
16.0
-4.1
Dra72
18.9
32.0
18.2
3.1
Isb73
17.12
0
Gut75
Units for table entries:
a: kcal mol'1
b: cal mol"1 K.'1

174
2
Q.
Q.
d
i
o
0.4
T r r
0.2
1—I—'—'—'—T—'—*—'—1~
0.0 “0.2 “0.4 PPM
FIG. 4-8. Four NOESY spectra of DMF obtained using the RED NOESY sequence.
These data were obtained at 347 *K and required 3.96 hours to obtain. The parameters
and processing are described in the text, (a) The NOESY spectrum of the exchanging
methyl groups; the mixing time was 1 second.

175
1 1 1 I r
0.4
l~~\ 1 1 1 r1 â–  1 i 1
0.2 0.0 -0.2
-l—r
-0.4 PPM
FIG. 4-8—continued, (b) The NOESY spectrum of the exchanging methyl groups of
DMF; the mixing time was 3 seconds.
0.4 0.2 0.0 -0.2 -0.4 PPM

176
* +
T
0.4
o
i | i r i | i—i—i—|—i—i—i—|—
0.2 0.0 -0.2 -0.4 PPM
FIG. 4-8--continued. (c) The NOESY spectrum of the exchanging methyl groups of
DMF; the mixing time was 5 seconds.

177
t—r
T—
0.4
1 I 1 ' 1 I r
0.2 0.0
1 I 1 1 1 I
-0.2 -0.4 PPM
FIG. 4-8—continued, (d) The NOESY spectrum of the exchanging methyl groups of
DMF; the mixing time was 7 seconds.
0.4 0.2 0.0 -0.2 -0.4 PPM

In (sum)
4.50-
fig. 4-9. Behavior of the peak intensities of NOESY spectra of DMF at 347 °K. The
data points of these plots were obtained from the single RED NOESY experiment of
Fig. 4-8 as described in the text, (a) A plot of In(sum) versus rm. The slope of the fitted
line produced the relaxation rate Rj = 0.18 sec-1.

In (diff)
5.00
0.00
2.00
4.00
6.00
8.00
'm
FIG. 4-9--continued. (b) A plot of In(diff) versus rm. The slope of the fitted line pro¬
duced the sum of the relaxation and exchange rates Rj + k = 0.78 sec-1. By subtracting
the contribution of the relaxation rate found from the plot of (a), the exchange rate, k,
was found to be k = 0.60 sec-1.

180
exchange rate was found to be 0.60 sec'1. Using these values in Eq. [4-10] showed that a
mixing time of 2.44 seconds would correspond to maximum cross peak intensity. Thus the
choice of mixing times for this RED NOESY experiment were such that the curves of
Eq. [4-9] were well-described. This was not always possible because of experimental res¬
trictions. In obtaining temperature-dependent data for DMF, it was found that holding
the mixing times of the RED NOESY experiments fixed while changing the temperature
for each experiment produced a series of exchange rates which could not be fitted to
Arrhenius theory. This condition became worse as the temperature was increased but
could be explained in part on the basis of the curves of Fig. 4-2, which shows that as the
exchange rate increased due to an increase in temperature, the mixing times should have
been reduced to better define the curves of peak intensity versus mixing time. Experi¬
mentally, it was found that at a fixed temperature a change in the mixing times chosen for
a RED NOESY experiment could change the values of k and Rj. For example, values of k
obtained at 354 °K for DMF were 1.4 and 1.7 sec"1 for mixing time ranges of 0.125 - 1
and 0.5 - 1.5 seconds, respectively. The respective values of Rj were 0.07 and 0.02 sec"1.
These results indicate that for good quantification of k and Rj using RED NOESY (or
NOESY), the mixing times must be chosen to cover as wide a range as possible. Possibly
the data of Table 4-2 could have been improved if more than four mixing times had been
used at each temperature.
As an aside to the quantification of exchange rates using RED NOESY, the relaxa¬
tion rates found using RED NOESY were compared to those found using the inversion
recovery method for DMF. The results are shown in Table 4-7. Curiously, the differ¬
ences between the inversion recovery and RED NOESY measurements became larger as
the temperature increased. This might have been the result of the choice of mixing times
for the RED NOESY experiments, as discussed above. A contribution from spin diffusion
to the relaxation rate would not explain the anomaly. The rate of spin diffusion, which
would have increased as the temperature increased, would have produced greater apparent

181
relaxation rates, not the smaller rates seen in the second column of Table 4-7.
The energy barrier to the exchange of the methyl groups in N,N-dimethylamides has
been thought to be affected by inductive, steric, and associative effects. One popular
assumption has been that substituents attached to the carbonyl carbon which destabilize
resonance forms of the amide tend to lower the energy barrier. Relative to the proton,
better electron-donating substituents such as alkyl groups should reduce the barrier. The
values of AG* found using the RED NOESY method and seen in Table 4-2 seemed to
support this idea. For DMF, where the substituent is a proton, the energy barrier was
about 3-4 kcal mole'1 greater than that of DMA, where the substituent is the methyl
group which is a better electron donor. The energy barriers of DMA and DMP are simi¬
lar, corresponding to the similar electron-donating abilities of their methyl and ethyl sub¬
stituents, respectively. Qualitatively, these features are also seen in the literature values of
Tables 4-4, 4-5, and 4-6 although these data are quite scattered. If the complete lineshape
analyses of Inglefield, et al. (Ing68), Rabinovitz and Pines (Rab69), Drakenberg, et al.
(Dra72), and Isbrandt, et al. (Isb73) are considered to be benchmark calculations, then the
RED NOESY data appear to be extremely good quantitatively as well. Thus, it appears
that the RED NOESY experiment and the two-site equations of Jeener, et al. can be used
together to calculate accurate exchange parameters.
Based on the experiments of this chapter, the RED NOESY method is a worthwhile
alternative or supplement to NOESY and lineshape analysis, but several possible barriers
to a successful RED NOESY experiment should be considered before attempting it. First,
because RED NOESY uses tip angles less than jt/2 radians, the concentration of nuclei
must be sufficient to make tolerable the resulting signal-to-noise loss relative to a NOESY
experiment which uses the same number of signal averages and t: steps. Secondly, the
exchange rate should be slow enough such that long mixing times, and thus long acquisi¬
tion times, can be used to achieve adequate spectral resolution. Finally, good quantifica¬
tion of exchange rates can be assured if the number of mixing times and their values are

182
Table 4-7
Relaxation Rates of DMF Methyl Protons
Ta (°K)
corrected
RED NOESY
Inversion Recovery
Rb (sec'1)
Rb (sec'1)
downfield methyl
Rb (sec'1)
upfield methyl
331
0.19
0.17
335
0.18
0.17
336
0.14
339
0.16
0.16
340
0.17
344
0.16
0.16
344
0.18
349
0.11
350
0.15
0.15
354
0.07
354
0.02
355
0.15
0.15
355
0.08
359
0.10
359
-0.02
360
0.14
0.14
360
0.03
363
0.14
0.14
369
0.13
0.13
373
0.13
0.13

183
chosen such that plots of peak intensity versus mixing time adequately define the curve.
It is not necessary that the mixing times be equally spaced to execute the RED NOESY
pulse sequence. The spacing could be adjusted such that a particular range of mixing
times is observed and better curve fits obtained.
In summary, a number of features make the RED NOESY method desirable for the
calculation of exchange rates. First, the RED NOESY pulse sequence must be run at tem¬
peratures below coalescence of the lines corresponding to exchanging sites. This makes
possible the measurement of exchange rates which are slower than those measurable by
lineshape analysis. Certainly RED NOESY measurements could be used to complement
those of lineshape analysis. Secondly, RED NOESY is superior when multiple exchange
rates must be investigated. This can be done with lineshape analysis techniques, but if a
large range of exchange rates is possible, the sample temperature (or some other exchange
rate-altering parameter) might have to be changed to perturb the spectrum enough to use
lineshape analysis. By itself, the NOESY technique is more flexible than this because the
the observation of cross peaks, which are directly related to the kinetic matrix, depends
on the exchange rates and the mixing time. However, RED NOESY has one more degree
of freedom since it has the ability to obtain NOESY spectra with different mixing times.
Finally, relative to NOESY, RED NOESY has several advantages. First, it could be used
as a survey technique to check more rapidly a series of mixing times to find one at which
cross peak intensity is sufficiently observable. Secondly, assuming that the proper range
of mixing times can be used, RED NOESY could accelerate the acquisition of
temperature-dependent data for a kinetic study. Finally, as pointed out by Meyerhoff, et
al., the NOESY spectra obtained from a RED NOESY experiment are less susceptible to
spectrometer instabilities since the data for all mixing times at each tj step are acquired
nearly simultaneously (Mey87).

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BIOGRAPHICAL SKETCH
Michael D. Cockman was born September 19, 1961, in Tallahassee, Florida, to Boyce
R. and Vivian M. Cockman. He grew up in the Carolinas, graduating from Wade Hamp¬
ton High School in Greenville, South Carolina, in 1979. He then attended Furman
University, also in Greenville, from 1979 to 1983, graduating with a B.S. in Chemistry. A
desire to enter the field of magnetic resonance imaging led Mr. Cockman to the Univer¬
sity of Florida, where he began graduate school in the Department of Chemistry in
August, 1983. In June, 1988, he was married to Lisa Deju in Altamonte Springs, Florida.
189

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a disser¬
tation for the degree of Doctor of Philosophy.
W.S. Brey, Jr., Chair
Professor of Chemistr>
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a disser¬
tation for the degree of Doctor of Philosophy. ! ft , Í)
W. Weltner, Jr.
Professor of Chemistry
I certify that 1 have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a disser¬
tation for the degree of Doctor of Philosophy. . _
X. IM. Stiff—
K.N. Scott
Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a disser¬
tation for the degree of Doctor of Philosophy.
T.H. Mareci
Associate Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a disser¬
tation for the degree of Doctor of Philosophy. —v % >r_
7 ¿¿/j
D.E. Richardson
Assistant Professor
of Chemistry

This dissertation was submitted to the Graduate Faculty of the Department of
Chemistry in the College of Liberal Arts and Sciences and to the Graduate School and was
accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.
December, 1988
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08554 2917