Table Of ContentFourier Transforms in
NMR, Optical, and
Mass Spectrometry
A User's Handbook
Alan G. Marshall and Francis R. Verdun
Departments of Chemistry and Biochemistry, The Ohio State
University, Columbus, OH 43210-1173, U.S.A.
Amsterdam — Oxford — New York — Tokyo 1990
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress Catalog1ng-1n-Publ1cat1on Data
Marshal 1, Alan G., 1944-
Fourier transforms 1n NMR, optical, and mass spectrometry : a
user's handbook / by Alan G. Marshall and Francis R. Verdun.
p. cm.
Includes Index.
ISBN 0-444-87360-0
1. Fourier transform spectroscopy. 2. Nuclear magnetic resonance
spectroscopy. 3. Spectrum analysis. 4. Mass spectrometry.
I. Verdun, Francis R. II. Title.
QD96.F68M37 1989
543'.0877~dc20 89-1518
CIP
ISBN 0-444-87360-0 (hard bound)
ISBN 0-444-87412-7 (paperback)
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Dedicated to Devon W. Meek
for helping to bring one of
the authors (A.G.M.) to The Ohio
State University in 1980, and
more generally for his
unstinting and selfless efforts on
behalf of faculty, staff, and
students during his terms as
Chairman of the OSU Chemistry
Department.
XV
PREFACE
This book is offered as a teaching and reference text for Fourier transform
methods as they are applied in spectroscopy. Whereas several other books treat
either the mathematics or one type of spectroscopy, or offer a collection of
chapters by several authors, the present monograph offers a unified treatment of
the three most popular types of FT/spectroscopy, with uniform notation and
complete indexing of specialized terms. All mathematics is self-contained, and
requires only a knowledge of simple calculus. The main emphasis is on pictures
(-200 original illustrations) and physical analogs, with sufficient supporting
algebra to enable the reader to enter the literature. Instructive problems offer
extensions from the basic treatment. Solutions are given or outlined for all
problems.
Because the book aims to inform practicing spectroscopists, non-ideal effects
are treated in detail: noise (source- and detector-limited); non-linear response;
limits to spectrometer performance based on finite detection period, finite data
size, mis-phasing, etc. Common puzzles and paradoxes are explained: e.g., use of
mathematically complex variables to represent physically real quantities;
interpretation of negative-frequency signals; on-resonance vs. off-resonance
response; interpolation (when it helps and when it doesn't); ultimate accuracy of
discrete representation of an analog signal; differences between linearly- and
circularly-polarized radiation; multiplex advantage or disadvantage, etc.
Chapter 1 introduces the fundamental line shapes encountered in
spectroscopy, from a simple classical mass-on-a-spring model. The Fourier
transform relationship between the time-domain response to a sudden impulse
and the steady-state frequency-domain response (absorption and dispersion
spectra) to a continuous oscillation are established and illustrated. Chapters 2 and
3 summarize the basic mathematics (definitions, formulas, theorems, and
examples) for continuous (analog) and discrete (digital) Fourier transforms, and
their practical implications. Experimental aspects which are common to the
signal (Chapter 4) and noise (Chapter 5) in all forms of Fourier transform
spectrometry are followed by separate treatments of those features which are
unique to FT/MS (Chapter 7), FT/NMR (Chapter 8), FT/optical (Chapter 9), other
types (Chapter 10) of FT/spectrometry. In Chapter 6, non-FT methods (e.g.,
autoregression, maximum entropy) are presented and critically compared to FT
methods.
The list of references includes both historical and comprehensive reviews
and monographs, along with articles describing several key developments. The
Appendices include Fast Fourier and Fast Hartley Transform algorithms in Fortran
and Basic, a look-up table of useful integrals (definite and indefinite), and a
pictorial atlas of the Fourier transform time/frequency functions most commonly
encountered in FT spectroscopy. The comprehensive Index is designed to enable
the reader to locate particular key words, including those with more than one
name—e.g., foldover (aliasing), throughput (étendue or Jacquinot) advantage;
apodization (windowing); Hubert transform (Kramers-Kronig transform. Bode
relation), etc.
xvi
ACKNOWLEDGMENTS
The authors acknowledge, with deep appreciation, the inspiration and
assistance provided by past and present collaborators, particularly M. B.
Comisarow, T. L. Ricca, D. C. Roe, T.-C. L. Wang, L. Chen, A. T. Hsu, C. E. Cottrell, J.
E. Meier, C. P. Williams, M. Wang, P. B. Grosshans, Z. Liang, G. M. Alber, E. C. Craig,
J. Skilling, and S. Goodman. We also thank several individuals for their critical
and helpful comments about the manuscript: M. B. Comisarow, P. K. Dutta, T.
Gäumann, P. R. Griffiths, T. L. Gustafson, J. A. de Haseth, M. Wang, G. M. Alber, P.
B. Grosshans, N. M. M. Nibbering, and E. Williams. We thank G. M. Alber, E. J.
Behrman, R. B. Cody, C. E. Cottrell, R. Doskotch, W. G. Fateley, B. S. Freiser, T.
Gäumann, G. Horlick, D. Horton, D. Hunt, F. W. McLafferty, M. D. Morris, P.
Schmalbrock, M. Wang, and J. Wu for providing illustrations. F.R.V. especially
thanks T. Gäumann for his help and indulgence during the preparation of the
manuscript. Finally, A.G.M. wishes to thank his family for their patience and
support.
1
CHAPTER 1
Spectral line shape derived from the motion of a
damped mass on a spring
1.1 Terminology
Virtually everything we need to know about line shapes in Fourier transform
spectra can be understood from the simple physical model of a mass on a spring.
It is easy to see why. In general, if the (time-independent) force, F, acting on a
particle of mass, m, varies with distance, x, we can express Newton's second law
as a Taylor series in x, in which x is measured from some arbitrary origin (e.g., the
center of an atom).
d2x
Net force = m -jr^ =ao + aix + a2 x2 + ··· (1.1)
However, the representation of even a relatively simple force, such as the
Coulomb force, m d2x/dt2 = q2/x2, between two particles each of charge, q, may
require an infinite number of terms in Eq. 1.1. Fortunately, two simplifications
render the problem tractable and useful.
First, we can dispense with the constant term, ao, simply by choosing a
suitable "reference" frame. For example, we can neglect the (constant) force of
gravity in most problems. Second, if the force is so weak that the observed
displacement, x, is very small (i.e., x » x2 » x3 ...), then we may conveniently
neglect all of the higher order terms, leaving
d2x
Net force = m ,, = a l x (1.2)
2
If we then rename the constant, a i, as a new constant, -Jc, then Eq. 1.2 becomes
identical to the equation for the (restoring) force for a weight (of mass, m) on a
spring of force constant, k.
d2 x
Net force = m ^+2 = - k x (1.3)
For example, even though we know that the force binding an electron to an
atom or molecule is a Coulomb force, we may still treat the electron as if it were
bound by springs, provided that whatever forces we apply do not displace the
electron very far from its equilibrium position. Thus, because the electric and
magnetic forces from electromagnetic waves (light, x-rays, radiofrequencies,
microwaves, infrared, etc.) on atoms and molecules are indeed weak, we can
successfully describe many effects of electromagnetic radiation on ions, atoms,
and molecules using the same mathematics that describe the response of a simple
mechanical weight on a spring to some externally applied "jiggling".
Before we discuss the motion of a "jiggled" spring, it is useful to review the
vocabulary of wave motion, since electromagnetic radiation can be described by
the same language as that for ordinary water waves.
2
A transverse wave is a disturbance whose displacement oscillates in a
direction perpendicular to the direction of propagation of the wave. For example,
a Cork floating on water moves up and down vertically as the water wave moves
along horizontally, as shown in Figure 1.1.
Amplitude
E
Figure 1.1 Transverse monochromatic waves. Top: water wave, viewed according
either to its displacement as a function of distance at a given time instant (left), or
to its displacement as a function of time at a particular point in space. Bottom:
electromagnetic wave (plane-polarized for simplest display). Both waves are
generated continuously at the left of the figure, so that propagation is from left to
right. See text for definitions of terms.
3
The amplitude of the wave is the maximum displacement from the
equilibrium position. The wavelength, X, for a monochromatic (see below) wave is
the distance (along the direction of propagation) between two successive points of
maximal displacement at a given instant in time. The velocity, c, of a
monochromatic wave is defined as the distance a given wave crest moves per unit
time. The frequency, v (in cycles per second, or Hz), of a monochromatic wave is
the number of times per second that the wave displacement at a given point in
space passes through its maximum value. The period, T, is the time (in seconds)
required to complete one cycle of the oscillation of a monochromatic wave.
Finally, the phase (or "phase angle"), <p, of a wave at a given time instant, t, can be
defined as the number of radians of oscillation accumulated since time zero (there
are 2π radians per oscillation cycle). Beginning at an (arbitrary) zero time, phase
angle accumulates at an "angular" velocity (or "angular" frequency), ω = 2πν radians
per second, for a wave of frequency, v oscillations/second. The relations between
the previous definitions are given by the following equations:
c = Xv meter/second = (meter/cycle)«(cycles/second) (1.4)
T = — seconds/cycle (1.5)
v
φ = φο + ω t radians accumulated since time zero (1.6)
ω = 2 π v radians/second = (radians/cycle)-(cycles/second) (1.7)
The principle of superposition states that when two waves travel through the
same region of space, their displacements add, as, for example, when two light
waves of different frequency (color) travel together. The ways (coherent,
incoherent) in which two or more waves combine (interference, diffraction) will
be discussed in Chapter 9.2.1. A wave is said to be monochromatic if all of its
components have the same frequency (and thus the same wavelength). The
intensity (joule m~2 s_1) of a wave is the energy flow per unit time across unit area
perpendicular to the direction of propagation. Radiation power (Watts) is the
product of intensity and cross-sectional area across which the wave passes.
Intensity is proportional to the square of the wave amplitude, as may be seen by
analogy to the water wave. If the amplitude of a water wave doubles, then the water
molecules must travel twice as far in a given length of time, and therefore have on
the average twice as much velocity. Since kinetic energy is proportional to the
square of velocity, intensity must be proportional to the square of wave amplitude.
Intensity ~ (amplitude)2 (1.8)
Returning to Eq. 1.3, we recall that the equation of motion of a mass, m,
attached to a spring of stiffness (spring constant), k, in the absence of any
damping or driving force is
d2x
m^T2-+fcx=0 (1.3)
The reader can quickly verify by substitution that Eq. 1.9 is a solution of Eq.
1.3 for the undamped, undriven mass on a spring.
x= xo cos ω t (1.9)
0
4
in which
ω» °° == VVm rn == natura^ angular frequency for mass on a spring (1.10)
for a maximum initial displacement, x = x o at time, t = 0. Since there is no
frictional "damping", the displacement (position) of the mass continues to
oscillate sinusoidally indefinitely at the natural frequency, νο=ωο/2π Hz. Eq. 1.9
also shows why angular frequency is almost always used in the mathematical
description of spring motion, since the wave frequency, v o, appears in the equa
tions in the form, ω o = 2πν o · Armed with the necessary vocabulary, we are now
prepared to discuss the motion of a mass on a spring in the presence of additional
driving and damping forces.
There are two ways to discover the natural frequency of a weight on a spring.
One could simply strike the weight to displace it suddenly from its equilibrium
position, and then count the oscillations per unit time as the mass moves up and
down. Alternatively, one could jiggle the spring steadily at each of various
"driving" frequencies and wait until the system settles into a steady-state motion;
when the jiggling frequency matches the natural frequency, the system will
oscillate with maximal amplitude (Just as a tuning fork is set into motion by sound
applied at its natural frequency). These two approaches are shown schematically
in Figure 1.2.
If the driven spring were completely free to move, then when the driving
frequency is set equal to (in "resonance" with) the natural spring frequency, vo, we
would expect the spring to absorb energy continuously and execute larger and
larger amplitude motion without limit. However, any real spring (including an
electron bound to a molecule) is hindered in its motion by frictional resistance, or
drag, expressed by a "damping" force (characterized by frictional coefficient, /)
proportional to the velocity of the moving mass.
dx
Damping force = - / -JT (1.11)
In the spectroscopic examples to be discussed later, the natural frequencies
of various springs correspond to the frequencies of power absorption (peaks) in
the spectrum and the damping constant of a given spring is related to the width of
a spectral line. The practical interest in these parameters is that the natural
frequencies indicate the strength of the spring of interest, and are related to
color, chemical bond strength or type, optical activity, and ionic mass-to-charge
ratio; spectral line widths reflect rates of molecular collisions, chemical reactions
and molecular motions.
1.2 Transient response to a sudden impulse: relaxation
The transient experiment consists of monitoring the displacement as a
function of time for a damped mass on a spring (damped harmonic oscillator), just
after the mass has been displaced by an initial distance, xo ♦ and then released.
Mathematically, the equation of motion becomes
d ^ x dx
Net force = m ~^T =~kx ~ f ~di (112)
Transient
Dispersion Absorption ι ρ τ
Hilbert
Figure 1.2 Interrelations between various scattering, spectroscopy, and transient
experiments, by analogy to the motion of either a sinusoidally driven or a suddenly
displaced damped weight on a spring. The weight has mass, m , bound to a spring
of force constant, k , immersed in a medium whose effect is represented by
frictional coefficient, / , and is subjected either to a sinusoidally time-varying
continuous force, F[t) = F cos ω t (steady-state response, left), or to a sudden
0
displacement (transient response, right) The massless limit corresponds to
scattering (steady-state) or relaxation (transient) experiments. The various results
are related by Fourier and Hubert transforms as shown.
