Nuclear Spin Dynamics and Thermodynamics of Pulsed NMR in Solids - CaltechTHESIS

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Nuclear Spin Dynamics and Thermodynamics of Pulsed NMR in Solids
Citation
Burum, Douglas Peter
(1979)
Nuclear Spin Dynamics and Thermodynamics of Pulsed NMR in Solids.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/MF8E-RD37.
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

The investigations presented in this thesis deal with two basic topics of solid state NMR: coherent averaging of nuclear spin interactions by rf irradiation, and the application of thermodynamics in strongly time dependent interaction frames. In each case new theoretical insights are presented, and these lead to the development of new experiments.

After a general introduction, an extension of coherent averaging theory called the principle of pulse cycle decoupling is presented. This principle greatly facilitates the design and analysis of compound experiments, i.e. techniques which are combinations of several smaller experiments. A number of multiple pulse experiments are analyzed using pulse cycle decoupling and several new techniques are introduced, including 24-pulse and 52-pulse cycles which have a greater ability to resolve anisotropic chemical shifts in solids than any experiment previously developed. The 52-pulse cycle is used to measure proton chemical shift spectra for polycrystalline ice, C6H12, C5H10 and polyethylene. This new technique is also used to study proton chemical shifts in single crystals of gypsum, [...].

The second topic considered in this thesis is the application of thermodynamics to NMR experiments in which the amplitude of the applied rf irradiation is varied in a strongly non-adiabatic fashion. Sources of artificial spin heating are analyzed and methods of eliminating these effects are demonstrated. A calculation is presented of the spin-lattice relaxation time during one basic type of multiple pulse irradiation. Techniques based on this calculation are introduced which measure relaxation times in the laboratory frame, [...], and the rotating frame, [...]. A method for determining the first moment of an NTIR spectrum is also developed. These new techniques are demonstrated using a variety of materials, including CaF2 and C6F6.

A simple yet highly flexible pulse sequence generator which is capable of producing all the pulse sequences described in this thesis as well as many more complicated experiments is described in the Appendix.
Item Type:
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Subject Keywords:
(Applied Physics)
Degree Grantor:
California Institute of Technology
Division:
Chemistry and Chemical Engineering
Major Option:
Applied Physics
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Research Advisor(s):
Vaughan, Robert W.
Thesis Committee:
Unknown, Unknown
Defense Date:
26 March 1979
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CaltechETD:etd-04302007-153624
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NUCLEAR SPIN DYNAMICS AND

THERMODYNAMICS OF PULSED NMR IN SOLIDS

Thesis by

Douglas Peter Burum

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1979

(Submitted March 26, 1979)

ii.

To Belinda.

my wife, and my best friend.

iii.

Acknowledgments

I would like to express my sincerest gratitude to Dr. Won-Kyu Rhim.
Most of what I know about solid state NMR — indeed, most of what I know
about being a scientist -——- was learned from him. He has been a good friend
during the past five years, and our many discussions on all topics of life
have been both enlightening and enjoyable.

I would also like to thank my advisor, Dr. Robert Vaughan, for his
continuing faith in me and for always providing just the needed mixture of
guidance and freedom. It has been good to know that I could always call
on him when needed.

Much appreciation goes to my parents for their unfailing love and
support through the years and for always allowing me to be the person I was
meant to be.

I am grateful to my good friend Ginny Williamson, who saw me through
many difficult spots along the way.

Finally, my wife Belinda deserves special thanks for all her love and

encouragement, and for adding so much warmth and color to life.

iv.

Abstract

The investigations presented in this thesis deal with two basic topics
of solid state NMR: coherent averaging of nuclear spin interactions by rf
irradiation, and the application of thermodynamics in strongly time depen-—
dent interaction frames. In each case new theoretical insights are pre-
sented, and these lead to the development of new experiments.

After a general introduction, an extension of coherent averaging theory
called the principle of pulse cycle decoupling is presented. This principle
greatly facilitates the design and analysis of compound experiments, i.e.
techniques which are combinations of several smaller experiments. A number
of multiple pulse experiments are analyzed using pulse cycle decoupling and
several new techniques are introduced, including 24-pulse and 52-pulse cycles
which have a greater ability to resolve anisotropic chemical shifts in
solids than any experiment previously developed. The 52-pulse cycle is used
to measure proton chemical shift spectra for polycrystalline ice, CeHio>
CoHio and polyethylene. This new technique is also used to study proton
chemical shifts in single crystals of gypsum, CaSO, * 2H,0.

The second topic considered in this thesis is the application of thermo-
dynamics to NMR experiments in which the amplitude of the applied rf irradi-
ation is varied in a strongly non-adiabatic fashion. Sources of artificial
spin heating are analyzed and methods of eliminating these effects are
demonstrated. A calculation is presented of the spin-lattice relaxation time
during one basic type of multiple pulse irradiation. Techniques based on this
calculation are introduced which measure relaxation times in the laboratory

frame, T and the rotating frame, T° A method for determining the first

1?

moment of an NMR spectrum is also developed. These new techniques are demon-
strated using a variety of materials, including CaF, and CoFe-
A simple yet highly flexible pulse sequence generator which is capable

of producing all the pulse sequences described in this thesis as well as

many more complicated experiments is described in the Appendix.

vi.

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . 2. 1 6 6 © © © © © © © © © © © © © © 6 eo ee ee ew ew ww) (LEG
ABSTRACT . 2 1 1 6 0 ew we ee ee ew ee ew we ew hw tw he we wt iv
CHAPTER I, GENERAL INTRODUCTION. . .... 2... 6 2 ee ee eo ee ee l
Section 1, Theoretical Introduction, , ........ 0.0.8 02 see 2
Section 2, Instrumentation , . . . .. 0.2 ee ee ew et ww ww tl ke 7
CHAPTER II, NUCLEAR SPIN DYNAMICS IN SOLIDS DURING
MULTIPLE PULSE rf IRRADIATION . .. . 2 2. © 2 © © © © © e © we 12
Section 1, Introduction. . . 2. 2 2 6 © © © © © © © © © 8 ow ow ew ew we 13
Section 2, Average Hamiltonian Theory. . . « 2 2 6 6 «© +s «© © 2 © ew we 17
Section 3, Extraction of Quadrature Phase
Information from Multiple Pulse NMR Signals .......-. 23
Section 4, Pulse Cycle Decoupling Theory and the
Design of Compound Cycles . .... 2 «© «© © © © © @ © we 46
Section 5, A Chemical Shift Study of Gypsum,
CaSO, + 2H,0, Using the 52-Pulse Cycle. .... . 2.2. © ww 97
CHAPTER III, OBSERVATION AND UTILIZATION OF THERMODYNAMIC PHENOMENA
IN STRONGLY TIME DEPENDENT INTERACTION FRAMES. ....... . L107
Section 1, Introduction, . . . . 1. 6 1 «© © © © © © © © © © eo ew tw ew ew) «6ULOB
Section 2, Elimination of Spin Heating in Multiple
Pulse Experiments . . . 6. 21 2 © © © #© © © © © © wo ww we) LD
Section 3, Calculation of Spin-Lattice Relaxation During
Pulsed Spin Locking in Solids . .......2.4.4.24.2.6 +. 130
Section 4, A Multiple Pulse Technique for Accurately
Determining the First Moment of an NMR Spectrum , , , , ,. 147
Section 5, A Single Scan Technique for Measuring Spin-
Lattice Relaxation Times in Solids, . .,.......... 2170

APPENDIX, A SIMPLE, HIGHLY FLEXIBLE PULSE SEQUENCE GENERATOR ....... 202

Chapter I

GENERAL INTRODUCTION

Section 1

Theoretical Introduction

As indicated in the title, this thesis deals with the dynamics and
thermodynamics of nuclear spin systems in solids and their measurment by
pulsed NMR techniques. Specifically, it explores the physics of an ensemble
of quantum mechanical two level spin systems in contact with one another and
also with a classical thermodynamic bath called the lattice. Physically, the
term "lattice" refers to the electromagnetic spectrum generated by phonons in
the crystal lattice of which the nuclear spins areapart. However, the lattice
can be considered simply as a thermodynamic bath with a temperature equal to
the physical temperature of the solid. Under many circumstances, the nuclear
spin ensemble can also be described by thermodynamics. When this is the case,
the spin system can also be considered to have a temperature. At equilibrium
this spin temperature is equal to the lattice temperature.

The investigations in this thesis are limited almost exclusively to solids
which contain only a single species of nuclear magnetic dipole with gyromagnetic
ratio y and spin I = 4. The spin system is immersed in a strong, constant
magnetic field Em applied along the z-axis.

If it is assumed that the coupling between the spin system and the lattice
can be neglected, the Hamiltonian ‘in the laboratory reference frame can be
written! |

He = YHor, +H, +H (1)

where H, describes the direct dipole-dipole interaction between. the nuclei and
the chemical shift term, #2, characterizes the change in the magnetic field at
the nuciear site due to the diamagnetism of the surrounding electrons. There

are many other smaller Hamiltonian terms which have been neglected in equation

Ll.

In order to manipulate the spin system, rf irradiation of variable
amplitude and phase is applied at some frequency w near Wy = YH: By moving
to an interaction representation, or “rotating frame", with this characteristic
frequency w the Hamiltonian can be written!

y = _ 4 yp) a,
H(t) = (w wT +902 (t) +H +H - (2)

where ts” described the static, or truncated part of the dipole-dipole inter-
action in this frame. Because the rf irradiation, and thus 26s are under
experimental control, the form of M(t) can be manipulated in many ways. In

(z)

particular, it is often desirable to suppress. 4, so that the effects of
much weaker interactions such as chemical shift can be observed.

In chapter II, a coherent averaging theory, or “average Hamiltonian
theory", based on the Magnus expansion” which was first applied to NMR by
Eyans” is extended so that it can be more readily applied to long and complex
NMR experiments. This enlarged theory is applied in analyzing and developing
a number of experiments designed to eliminate the effects of the homonuclear
dipolar interaction in solids and allow the measurement of chemical shift
information. Several new experiments of this type are introduced which are
more successful than any technique developed previously. One new experiment,
a 52-pulse cycle, is analyzed in detail. In order to demonstrate its resolving
ability, this technique is used to measure chemical shift spectra for poly-
crystalline samples of ice, CoH o> CoH g and polyethylene. The 52-pulse cycle
is also used to determine the chemical shift tensor for protons in gypsum,

CaSO, °2H,0.

4 ~"2
As was stated earlier, equations (1) and (2) contain no information

regarding the interaction of the spin system with the lattice. This interaction

is thermodynamic in nature. Therefore one would expect that experiments

designed to measure characteristics of the spin-lattice coupling would involve
rf irradiation which was either constant in amplitude or at most varied
adiabatically. However, it is demonstrated in chapter III that this need not
always be the case.

In chapter ITI, circumstances are explored under which thermodynamics can
be applied to a nuclear spin system in a strongly time dependent interaction
frame. As a result, several new experiments are developed which are a valuable
contribution to NMR technology. These include methods for measuring thermo-
dynamic spin-lattice coupling parameters and also a new technique for deter-
mining the first moment of an NMR spectrum, The theoretical validity and
practical applicability of these new experiments are demonstrated using several
representative materials, such as CoFe> CeFio and CaF,.

The Appendix describes a simple pulse sequence generator which is capable
of producing all the pulse sequences described in the thesis as well as many
more complicated experiments. This instrument represents a practical demon-
stration that the complexity of the new experiments discussed in this thesis

is not an obstacle to their implementation.

References ~

1. A. Abragam, The Principles of Nuclear Magnetism, Oxford Univ. Press,
London (1961).

C. P. Slichter, Principles of Magnetic Resonance, 2nd Edition, Springer-—
Verlag, Berlin, Heidelberg and New York (1978).

2. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954).

3. W. A. B. Evans, Ann. Phys. 48, 72 (1968).

Section 2

Instrumentation

>" which was used to perform all the experiments

The NMR spectrometer?
discussed in this thesis is illustrated in Figure 1. RF irradiation at
frequencies near 56.4 MHz is applied to the sample in the probe coil either
in continuous bursts or as a series of short, nearly square pulses. Four
different phases of rf are available with relative shifts of 0°, 90°, 180°
and 270°. In the rotating frame these correspond to irradiation along the
X, y, ~x and -y coordinate axes. Pulse timing and phase channel selection
is determined through the use of rf switches, or "gates", which are controlled
by the pulse sequence generator described in the Appendix. This instrument is pro-
grammed by the PDP 11/10 computer but operates independently. It is capable of pro-
ducing a wide variety of experiements including sequences of up to 200 pulses.
The receiver is designed to detect the NMR signal between applications
of rf irradiation. Both the rf transmitter and the receiver utilize the same
coil in the probe. Despite various features designed to protect the receiver
from rf irradiation such as the use of crossed diodes and wave length coaxial
cables,” there is still a characteristic time interval, or receiver "dead"
time, following the end of an rf burst during which the receiver is saturated
and accurate data arenot available. This gives rise to a minimum sampling
time, or sampling "window", between the start of one rf pulse and the beginning

ls usec

of the next equal to the pulse width + the receiver dead time + about
during which the signal can be sampled. The receiver dead time for the spec-
trometer discussed in this thesis varied from 6 usec to 3 usec as improvements

were made in receiver protection. The typical rf pulse length required to

° . . : 4
produce a 90° rotation of the nuclear magnetization vectors was 1.5 usec.

References

A detailed description of this spectrometer is given in R. W. Vaughan,
D. D. Elleman, L. M. Stacey, W. K. Rhim and J. W. Lee, Rev. Sci.
Instrum. 43, 1356 (1972).

The spectrometer described in this thesis is representative of current

NMR technology. See for example, J. S. Waugh, Advances in Magnetic

Resonance, Academic Press, New York and London (1971).

I. J. Lowe and C. E. Tarr, J. Phys. E 1, 320 (1968).
I. J. Lowe and C. E. Tarr, J. Phys. E 1, 604 (1968).

See chapter II, section l.

10.

Figure Caption

1. Schematic diagram of the pulse spectrometer.

il.

| ATTEN

UATOR |

RY.

QUADRIPOLE NETWORK

FREQ. SYNTHESIZER |

>| > SHIFT |

~-X

nc

—<

—<
~=—GATES

A/

| DRIVER |
AMP |

7 TROMBONES

-— AMPLITUDE
TRIMMERS

\/4

To |x
GATES } y

ISOLATION
x AMPLIFIER

~me

~<0 oO

PULSE LENGTH
ADJUSTMENT

COMPUTER |
PROGRAMMED |
PULSE [be
SEQUENCE |
GENERATOR |

PULSE GEN |

(TRIGGER) |

FABRI-TEK & |
PDPIIVIO.

seem

INTEGRATE |

& HOLD

PROBE

Fig. l

“4

RECEIVER 1. _

i2.

Chapter IT

NUCLEAR SPIN DYNAMICS IN SOLIDS DURING |
MULTIPLE PULSE rf IRRADIATION

13.

Section 1

Introduction

14.

This chapter is concerned with understanding and developing experiments
designed to eliminate the effects of the homonuclear dipole-dipole inter-
actions in solids described py” to as high an order of magnitude as
possible while preserving information regarding weaker interactions such as
the chemical shift, J). Although determining the strength of”) provides
valuable information regarding the structure and dynamics of a solid, much
more insight is gained if the other interactions such as chemical shift can
also be measured. Unfortunately, in most solids the effects of these weaker
interactions on conventional NMR experiments are completely obscured ry”,

Recall from the first section of chapter I that if the interaction
between the spin system and the lattice is neglected, the Hamiltonian in the

rotating reference frame can be written

a _ _ (z)
Ht) = (Wp w)T, + H- +H +H (1)

where Ts Ty and I. are the spin angular momentum operators corresponding to
the total spin operator I. The experiments discussed in this chapter are
designed in such a way that KH - has the effect of eliminating”. These
experiments also invariably reduce, or "scale", the effective magnitude of
Hers but this reduced chemical shift term is observable if the dipolar term
wo” is sufficiently eliminated.

In principle, the amplitude of the rf irradiation in an NMR experiment
can be modulated in any way. However, all of the experiments described in
this chapter consist of sequences of short rf "pulses" applied along the
X, y, -x and -y coordinate axes in the rotating frame.

The Hamiltonian for an ideal, square rf pulse applied along the x-axis

can be expressed in the rotating frame as follows:

= YH, T, = a1 (2)

Hruise lox

15.

where the amplitude of the rf magnetic field in the laboratory frame is
2H, coswt. The time development operator which corresponds to an x-pulse can be

expressed as

Unulse = exp (iw, t I) . (3)

The rotation angle of a pulse is determined by its amplitude w= YH, and its
width to For example, a pulse for which wt, = 1/2 is referred to as a
(1/2) pulse or a 90° pulse. For purposes of analysis, the assumption is
often made that the pulses can be considered to have a 6-function shape. The
time development operator for a (1/2), pulse of this type is given by

Us cy = exp(i 1/2 a) . (4)

It is convenient to analyze the effects of WC. on the other Hamiltonian
terms in equation 1 by moving to the so called "toggling" frame, i.e. the
interaction frame determined by the time development operator Ue corresponding
to H.. In this reference frame the Hamiltonian terms acquire a time depen-

dence due to UL For example, the dipolar Hamiltonian in the toggling

fe
frame, 360 , is given by

yp (2) _ al Jz)
HO? = 0 APO. (5)
The effect of an rf pulse in this frame on a Hamiltonian term which is linear

in the spin operators Tes Ty and I, is to cause a simple rotation. For

example, a (1/2). pulse causes the following transformation of Tt
{4 i on
exp(-i, 1/2 Tot, exp(i 1/2 I) I, (6)
In section 2, a coherent averaging theory, or “average Hamiltonian"

theory, is reviewed. This theory is utilized in the remainder of the chapter

in analyzing the effect of Hg on the other terms in the Hamiltonian.

16.

Perhaps the most successful experiment currently in use which is
designed to eliminate the homonuclear dipolar interaction in solids is an
8-pulse sequence called "REV-8". Normally, only single phase information
is obtained using this experiment. However, in section 3 it is shown that
full quadrature phase information can be obtained using REV-8 by sampling
the data more often. This increases the effective data rate and eliminates the
need to use special quadrature phase detection apparatus.

Section 4 presents an extension of average Hamiltonian theory which is
' called the principle of "pulse cycle decoupling." This approach simplifies
the design and analysis of compound pulse sequences, i.e. experiments which
are combinations of smaller pulse groups. Pulse cycle decoupling is shown
to provide a method for the systematic improvement of pulsed NMR experiments.
This is demonstrated by the development of several new experiments which
resolve chemical shifts in solids more effectively than any techniques pre-
viously introduced. One of these experiments, a 52-pulse sequence, is
analyzed in detail. This pulse sequence is used to measure proton chemical
shifts in ice, Celio» CeHig and polyethylene at liquid N, temperature. None
of these spectra could be resolved as clearly using previously available
techniques such as REV-8.

The applicability of the 52-pulse cycle is further demonstrated in
section 5. This section presents the results of a proton chemical shift
study performed on single crystals of gypsum, CaSO, °2H,0, at ambient temper-

ature. The results are compared with a similar study which used the REV-8

sequence.

17.

Section 2

Average Hamiltonian Theory

(Most of this section is taken from an article by

D. P. Burum and W. K. Rhim, “Analysis of Multiple
Pulse NMR in Solids. III" which has been submitted
for publication to The Journal of Chemical Physics.)

18.

Average Hamiltonian theory is a formalism based on the Magnus
expansion! which describes the coherent averaging of terms in the Hamiltonian
caused by the applied rf irradiation. This theory was first applied to NMR
experiments by Evans“ and has been discussed in more detail by Haeberlen
et al. It was extended by Rhim et al.“ in their analysis of the REV-8

experiment to include the effects of rf pulse imperfections.

In the usual rotating frame the Hamiltonian describing the nuclear
spin system can be divided into terms describing the rf irradiation,

TO. ¢> and the contributions from internal interactions as follows:

H(t) =H g(t) +H, +O” (1)

where the off resonance and chemical shift term 4, and the truncated

dipolar Hamiltonian xO” can be written

Hy 7 ; , Ae + 97 221) Ta (2)
and
wt” = 28..G.), + ¥, - 31,1.) (3)
D ij ij i jj zi-zj
i

Hs can be divided into ideal and non-ideal parts as follows:

H(t) =H, (t) +) 6.00) (4)

19.

where IE, (t) describes the effect of ideal rf pulses of constant phase and
power and exactly the desired length, while the sum over k includes all

the error terms which complete the description of the true experimental
situation, such as k = P for phase misadjustments, T for phase transients,
6 for pulse length misadjustments and e« for rf inhomogeneity. For an

x-pulse these can be written as follows:

He, = -w, sin e,. My (5)
He. = W(t) Ny (6)
Ho = - = 1, (7)
He. ~~ tg (8)
~ Ww

Here te is the pulse width, so that Wyte = 7/2, $,. and 5. are the phase
angle and pulse size misadjustments of the x-pulse, Ey is the error in
rotation angle at the i'th nucleus caused by rf inhomogeneity and Wp Ct)
is the rf amplitude orthogonal to the x direction. It is assumed without

loss of generality that

dis = 0 (9)

and

wr(tde = 0 (10)

Similar definitions apply for pulses of different rf phases,
It is convenient to move an interaction represen-

tation, or “toggling frame", which

20.

is determined by HO «- In this frame the other terms in the Hamiltonian
acquire a time dependence from J ¢- For example, the dipolar Hamiltonian

in this frame, H,, is given by

me -1,
Hy = Up ee (11)
where Ue is the time development operator determined by HW --

Tf 4, (t) is assumed to be cyclic, with cycle time tas i.e.

Ug Nt.) = +1 (12)

then the usual rotating frame and the toggling frame coincide at times

Nt. Furthermore, if H - is also periodic,
Z + y my. -
HE Ct Nt.) HE. (t) (13)

then the time development operator in the toggling frame, which we call

Ua at? acquires the useful property

_ N
Vea Nt? ~ Wine ted (14)

This means that the behavior of the system for an arbitrary number of
pulse cycles can be deduced from the time development operator for a
single cycle.

1,2

The Magnus expansion ° can be used to expand Ue ar st? so that

it remains unitary no matter where the expansion is truncated:

U, ._(t_) = exp E (AO) + BO) + AY + J] (15)
int’ c cv int int int
where
BO
30) _ _-l ~
Hint Te Hey (Fpaty (16)

21.

=p 1) ~1

HE = (21) [° fl [Rene (ta)» Bne(ep| (17)
we.” -1

FH” = (6) [~ of af | } [ae Cey)

ney »)| | +4, aaa )s (Bae(ty)

%<)] (is)

(Hn pty) %

and, of course,

yp = If H 74 19
Cat He, +H, +’) Hy. (19)

By expressing H | according to equation (19), the various terms in the
Magnus expansion can be divided into contributions from each of the

Hamiltonian terms in a straightforward way, i.e.,

p00) = (0) 4 60) 3(0)

HO) = HO + HA? «SHH (20)

TO”? = FEY + HY + THEO + HO + 0, (21)
(2). pl2) 4 2 4 gl) ,

HO) = EO) + HE) + GO 4, (22)

and so forth.

By expressing the time development operator according to the Magnus
expansion the problem of observing certain interactions which are
normally obscured by other, stronger interactions in solids is reduced
to choosing a periodic and cyclic rf pulse sequence which preserves
the desired Hamiltonian terms while removing the unwanted terms to as

high an order as possible.

22.

REFERENCES

1. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954).

2. W. A. B. Evans, Ann. Phys. 48, 72 (1968).

3. U. Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968).

4, W. K. Rhim, D. D. Elleman, L. B. Schreiber and R. W. Vaughan, J. Chem.
Phys. 59, 3740 (1973).

5. Cases in which Ug (td = ~-l are of special interest. See, for example:

M. E. Stoll, A. J. Vega and R. W. Vaughan, Phys. Rev. Al6, 1521 (1977).

23.

Section 3

Extraction of Quadrature Phase Information

from Multiple Pulse NMR Signals

(This section is essentially an article by W. K. Rhim and

D. P. Burum, Rev. Sci. Instrum. 47, 720 (1976).)

24.

A. INTRODUCTION

Over the last decade, considerable progress has been achieved in the
field of high resolution solid state nmr, b73 Through the use of multiple
pulse techniques, the static dipolar interaction can be averaged out,
enabling one to observe relatively smaller interactions such as anisotropic
chemical shifts or indirect exchange interactions. Considerable attention
has been given to the REV-8 sequence for its simplicity, smaller chemical
shift scaling effect, high resolution, and better stability.>

In pulsed NMR experiments a quadrature phase detector is often used
to observe two components of the free induction decay signal, one in phase
and one out of phase with the rf pulse. In this scheme, when applied to
multiple pulse NMR experiments in solids, the signal is usually sampled
once per cycle and phase detected using the quadrature phase detector. ’
However, the multiple pulse sequence contains rf pulses whose phases are
mutually orthogonal so that one might expect the signal sampled at two
properly chosen windows within a cycle to furnish full quadrature infor-
mation without the use of a special detector.

In this section, the REV-8 cycle is further analyzed with the goal of
retrieving more information from the resultant NMR signal. Several im-
portant points which are emphasized are (1) the quadrature phase information
which can be obtained from a single phase detector, practically eliminating
the need for a conventional quadrature phase detector; (2) the increased
signal-to-noise ratio; and (3) the high frequency information which can be

made available due to the fact that the 8-pulse sequence allows one to

sample the signal at a higher rate.

B. THE 8-PULSE CYCLE

The 8~pulse cycle which will be considered in this section is shown
in Fig. 1. All the pulses, expressed by x, y, -x and -y are 90 degree
pulses whose phases are directed along the indicated axes in the rotating
frame. The spacings between adjacent pulses are T and 2T.

The usual receiver blocking time which follows the strong rf pulses
and the short values of te required for a better averaging effect prevent
sampling the NMR signal in the narrow windows. Therefore, only signal

detection at the 2T windows is considered.

25.

Cc. AVERAGE HAMILTONIANS

The interactions commonly encountered in solid state NMR can be expressed

in the rotating frame by the Hamiltonian
' = 4 ay agg? (z)°
Ht) =H Ct) +H) +H, +H, (1)

H(t) depends on the nature of the rf excitation, while the resonance offset

Heys the chemical shift %, and the truncated dipolar interaction # ) , have

the following forms:

H =~ (w= 0) I. (2)
Ho * = U¥ Cozi zi (3)
HO (2). ), j .f e
Dy L By (r5 3) (ty Ty 3t it) (4)

If the first few pulses are considered pre-pulses and the 8-pulse cycle
is redefined, average Hamiltonian theory can be applied to analyze the signal
appearing at each of the 2t windows shown tn figure 1. This analysis shows that the
average dipolar interaction vanishes at all 2T windows. However, the sum He, + He,
results in a different, non zero average Hamiltonian at each 27 window in the
8-pulse cycle when averaged over t+ These average Hamiltonians, HO, (a = 1,2,3 or 4)

at the a-th 2T window in each cycle are given by:

—(0
—(0)_ £ =
Hol st (Ao t+oo ) Gs + hp FE03 (Sa)
SE 02 se (Aw + &o 22) (li + Le (5b)

26.

—(0)_ 2 .
sh (Aw + 22) (los t

FEO i ) (Se)

yi

Since the chemical shift and resonance offset Hamiltonians have essentially

the same form, it is assumed for simplicity that Oot = 0 in order to describe the
development of the system in a pictorial way. Then eq. (5a)-(5c) can be

rewritten in the following form:

—(0) ak 2
Wo = ¥ Hex I (for @ = 1, 2, 3, and 4) (6)
wut,
where {1 = 4 = (x + y) =F (7a)
ad
He - 32 G+ (7b)
Ha * 3 "y (-y + 2) (7c)

AA A
and x, y, and z are unit vectors in the rotating frame. Of course, the vectors H.

will serve as the effective fields in the rotating frame. Therefore, given the
initial magnetization vectors in the various windows of the first 8-pulse cycle,
each magnetization vector will appear to precess around its own effective field
with angular frequency YE = teu (Fig. 2).

If 6-function rf pulses are assumed, the magnetization vector at each window
in the first 8-pulse cycle can be obtained by alternately applying 90° rotations
due to the rf pulses and appropriate precessions due to the resonance offset during
the windows. Beginning with the magnetization in thermal equilibrium along
z-axis, the initial vectors so obtained can be drawn as in Figure 2 when © ~ 0,
Therefore, if ‘the signal ia sampled once per cycle, the result is a stroboscopic view
in which the magnetization seems to precess around Hy Starting from ¥, as shown
in figure 2, 4 = 1, 2, 3, and 4.

Since the ff, vectors are all tilted by 45° from the z~axis, the projections
of the precessing magnetization vectors on the x-y plane will trace out ellipses
whose short axes measure 1N% times the long axes. When the NMR Signal is phase
detected along the y-axis, the My components which are expected at each window
are shown in Figure 3. The first four traces in Figure 3 are just those which
one would expect to obtain if the magnetization of a given isochromat was measured
at only one of the windows in each cycle. The last trace in this figure is the
composite of the first four traces which is observed when the signal is sampled

at all 2T windows.

27.

D. FHE NARROW SPECTRUM CASE

It is a well known property of the discrete, fast Fourier transform that
the bandwidth of the transformed spectrum depends directly upon the sampling
rate of the input data. Thus, if the time between sampling of the input points
is T, the total width of the transformed spectrum will be *, and signals of
frequency greater than + 1/T will be reflected back into the range =. When the
NMR signal is sampled at all 4 of the 2T windows, the time between successive
complex points is te/2, so that the transformed spectral band width is a

In this section, it is assumed that the NMR spectral width is much smaller than
a , 80 that the entire spectrum can be located near exact resonance. As was
described in the previous section, the magnetization M ey? stroboscopically
observed at the @-th window (Y= 1, 2, 3, 4) will appear to precess around Hy
with angular frequency MH = W. (w = £2u3/3) . In this case, it is assumed
that the y-component of the magnetization at each window, (M ) ot? satisfies the

following relations:

Ma = Moees (= (My), (8a)
M,
“Oe = pysin (Ht) =(My dg (8b)

The degree to which this assumption is valid will become clear when the more general
"Wide Spectrum Case" is analyzed in the next sub-section. The following are important
consequences of this situation: (i) relations (8a)=(8b) provide quadrature

phase information. If the amplitudes are matched by multiplying (M, dg by -[F

and (MDa by [7, the signals from the four windows combine to produce two traces
which can be treated as the real and imaginary parts of a complex video signal
proportional to My exp(iv t). Therefore, the complex Fourier transformed

spectrum will unambiguously exploit the direction in which each itsochromat is pre-
cessing. One should realize, however, that (8a) and (8b) are only orthogonal

when the detector phase is set either along the x- or y-axis in the rotating frame.
(At this point it may be worthwhile to mention that for the 4-pulse WAHUHA cycle’ the
signals at the two wide windows differ in phase by 120° when the detector phase

is set along the x- or y-axis. Also, the observation of a baseline shift is
inevitable for that cycle unless a special prepulse is applied.) (ii) Since

signal is coherent and noise is random, one can expect enhancement of the signal

28.

to noise ratio by a factor of {3 when utilizing the signals obtained at all
4 windows.

In order to demonstrate some of these effects experimentally, perfluorocyclo-—
hexane was chosen as a sample and the REV-8 sequence was applied at -60° c.®
In this case the signal was sampled only at the first two windows. Trace (A) in
Fig. 4 is the digitized signal so obtained, and traces (B) and (C) are the
computer separated signals from (A). Notice the predicted orthogonality in
phase as well as the amplitude difference. Inasmuch as the usual algorithms
are designed to work with complex quantities, the trace (A) becomes an ideal
input for a fast Fourier transform computation when the amplitude difference
between (B) and (C) is corrected.

The top trace of Fig. 5 is the real part of the Fourier transform of
trace (B) alone. As one would expect, this spectrum is symmetric about the
resonance point, since only single phase information was Fourier transformed.
The real part of the Fourier transform of trace (C) in Fig. 4 is an antisymmetric
spectrum, as shown by the second trace in Fig. 5. Note that the amplitude of this
spectrum is reduced compared to the top spectrum. Of course, the logical way to
handle the data of Fig. 4 trace (A) is to multiply trace (C) by {%, recombine
(B) and (C), and take the full complex Fourier transform of the resultant trace.
The lower two curves in Fig. 5 are the absorption and the dispersion spectra so
obtained. Now the images have disappeared and the direction in which each iso-
chromat is precessing is apparent. 4

The same procedure was repeated, but this time for frozen CsFsg and a different
resonance offset frequency (5 KHz above the liquid CsFe resonance point). Fig. 6
shows the real and imaginary parts of the familiar chemical shift powder pattern so
obtained. No additional correction was made, although the spectrum was nearly

9 KHz wide.

E. THE WIDE SPECTRUM CASE

In this sub-section the more general situation is considered in which the
spectral width may be comparable to 2n/t and may be located over any region within
the bandwidth + 2n/t- The types of errors which are introduced and schemes for
correcting these errors are discussed. For simplicity, the signal due to an

isochromat is analyzed.

29.

1. Sampling at 2 Windows Only

Gonsider first the case in which only the first and second windows are
sampled to form a complex input to the Fourier transformation.
The analysis of this case is relatively simple, and the symmetry properties of
the errors provide a very simple method for error correction. The signal
sampled at the first window (i.e. right after the first pulse) can be expressed
as A cos (ot) if the isochromat is off resonance by frequency wo /2m during the
rf burst. A is a constant which is proportional to the intensity of the undis-
torted absorption spectrum at this same frequency. In general the
precession of the »magnetization during the time between the
start of the sequence and the start of the second window in the first cycle
must be taken into account in determining the initial magnetization at the
second window. It is the deviation of this magnetization from its ideal value
which gives rise to the spectral distortions described below. The resulting

signal can be expressed as:
Ty. Ct) = A(dp cosh sinw t + Ao sind cos) t) + Cp

The amplitude 4;, phase change ¢2 and the constant shift G are all functions of
@. They can be expressed in the following way:

& =[U-ect)/2}*

gd = sin™ [(sin wr sin 2ur - Co) / bo ]
CG. = x[cosWt sin 2WT - ginwTsin2wT]
where W = %
r 3

Now, the complex input signal for the fast Fourier transform at frequency

0° fan can be expressed by

Ey (t) Rwy (t) + il, (t)
xr Yr xr

A cos ot + iAlAs cos & sin wit + & sin ¢cos wt] + iG

30.

Lot
= Ae + 4[2K sin (wt +) + Ce | (9a)
where
K = ; [ (& cos & - 1)7 + &® sin? d ]*
and

@ = tan™'[(4o sin %)/(f cost -1ltet

The first term in eqe (9a) is the ideal signal, while the second and third terms
represent the errors produced by the resonance offset effect in the first cycle.

The effect of these error terms on the spectrum is more easily seen in eq. (9b).

iw t -i(w t + d)
f(t) = Ae * Bell ~ Ke + iC, (9b)
where

[(1 + K cos 6)? + (K sin a2 7%

iss]
it)

tan Esin ¢ — (e Tif 1+ K cos ¢ < 0)

B and | in this equation indicate the amplitude and phase distortion of the main
spectrum caused by the errors. The second term produces an image at “Ds and the
third term resuits in a sharp spike at am = 0. Thus, if the spectrum extends
over o. = 0, it will be distorted by the central spike, and the image will be
superimposed on the spectrum. It is very difficult to correct for this. effect.
Fortunately, the error terms tend to be small near 4,= 0. In fact, after the

dispersion curve is multiplied by{2, the error terms in eq. (9a) will contribute

practical situations.

only ~ .06% at = +5 KH, and ~ 1.3% at +5 KH» which is negligible in most

If the spectrum does not overlap 4,= 0, however, correction of those errors
becomes rather simple. It is easy to see in this case that one can merely discard
the second term in eq. (9b), and correct the signal by rotating it by - and
multiplying it by 1/B. Fig. 7 shows 1/B and as functions of resonance offset
frequency when T = 5uUsec. While these are useful in estimating the amount of

expected error, there is yet an easier way to correct these errors.

31.

It can be shown that the Fourier transform of the error terms in (9a) is
antisymmetric in the absorption part and symmetric in the dispersion part about
° = 0. As long as the entire spectrum is located on one side of o. = 0, there-
fore, the error can be removed by adding U(-w ) to Uj) in the absorption part,
and subtracting V(-w ) from vio) in the dispersion part, where the spectrum can
be expressed as u@) + iv(w).

The two-window sampling case, then, provides full complex information
about the precessing magnetization, increases the signal to noise ratio by a
factor of [3/2, and provides a very simple way of correcting off resonance errors
even for wide spectra as long as the entire spectrum can be located on one

side of © =0,
ra

2. Sampling at All Four Windows

The basic procedure which is followed in analyzing the spectrum obtained
when all four windows are used for sampling is essentially identical to that
which was followed in the 2-window case. The actual expressions are much more
complicated, however, and will not be repeated here. Again, for simplicity,
the signal resulting from an isochromat is analyzed, and it is assumed that the
amplitude normalizations described above have been performed.

If the magnetization is taken to be initially along the z-axis and the
8-pulse sequence is applied as a series of classical rotations, the signals which

result at each of the four windows can be expressed in the following form:

Ag cos (ot + 6 +C, v= 1, 3
(10)
Ay sin (wit + by) + Cc, a= 2, 4

where the factors Ay by and Cy are functions of WT and are determined by the
excess precession in the first cycle.

When the detector phase is set along the y-axis, consideration of the
direction of the effective field for each window reveals that

C, = Cys = 0 and ¢, = 0

and A; can be set to unity for convenience.

Of course, in this case the data points sampled at windows 1 and 3, when
taken together, provide the real part of the complex Fourier transform input,
while those from windows 2 and 4 make up the imaginary part. This can be

expressed in the following way:

32.

R(nt .) = cos(W nt.)

RE(n + 3)t ] = dg cos [win #4)t + )]+ G
ce r c (11)

(nt) = & sin (ont. +h) +G@
1L(n + ‘tJ = A sin [w (a +a)t +a] +GQ

These discrete points can be replaced for analytical purposes by the following

pair of continuous curves:

R(t) = 3 [eos (wt) + Ag cos (wt + $3) ]

+ [cos (wt) - Ascos (wt + ¢3)] cos (ame /t .) (12)

T(t) = [a sin (t+) + & sin (Wt th) +e +Q)
+ [4 sin (Wt + de) - & sin (Wt + &) + - G] cos (am ~)
Cc

Through the use of trigonometric identities, the complex function represented

by equation (12) can be expressed in the following form:

f(t) R(t) + it(e)
iW ¢ it
=e * Be + ey exp(-i0 t) + & expli(w +2 ~ (13)

+ €3 exp Lic, = a del + €& exp [- i(w tr ie
> eo

271
+ €s exp [-1(o - = del + € exp (a2 ] + & (-i ) + &
c c c

B and } in the first term of equation (13) above indicate the amplitude and phase

distortion of the isochromat signal at frequency v » The eis are all error

factors. Namely, ¢, gives rise to a small mirror image of the spectrum, while
&2-€g give rise to sidebands at +. aa aT Finally, && and ¢7 produce sharp spikes

at the extreme edges of the transformed spectrum, while ¢a produces a similar spike

at WOW =Q,

33.

As was true in the 2 window case, these errors are difficult to correct

if they are superimposed on the "true" spectrum. Table 1 shows the size of
some of these errors at a few representative frequencies. However, if the

spectrum can be positioned such that it does not overlap the points o. = Q. and

* = » none of these errors overlap the spectrum, and the phase and amplitude
distortion (see the first term in eq. (13)) can be corrected by a simple rotation
and amplitude normalization. B and | can be rigorously calculated and can be
used to correct the main spectrum. Figure 8 shows 1/B and over the full
frequency range + 2m >» assuming T = 3h SOC. It is interesting to note that

+2 is just the fall spectral bandwidth for the single window sampling case.

34.

References

1. U. Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968).
2. P. Mansfield, J. Phys. C4, 1444 (1971).

3. W.-K. Rhim, D. D. Elleman and R. W. Vaughan, J. Chem. Phys. 9, 3740 (1973).

4. J.D. Ellett, Jvr., M. G. Gibby, U. Haeberlen, L. M. Huber, M. Mehring,
A. Pines, and J. S. Waugh, Adv. Magn. Reson. 5, 117 (1971).

5. R. W. Vaughan, D. D. Elleman, L. M. Stacey, W-K. Rhim and J. W. Lee,
Rev. Sci. Instr. 43, 1356 (1972).

6. J. D. Ellett, Jr., U. Haeberlen and J. S. Waugh, J. Amer. Chem. Soc. 92,
411 (1970).

7. M. Mehring, R. G. Griffin, and J. S. Waugh, J. Chem. Phys. a2, 746, (1971).

35.

Table I. Relative error magnitudes in eq. (13), with respect to the undistorted
signal amplitude, for representative offset frequencies (t, = 60 Usec).

w/27 (in KH) € (in%) < : Cin %)

25 35 38

15 13 23

5 5 4

4) ) fs)
~ 5 5 2
“15 26 7
25 87 37

36.

Figure Captions

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

The 8-pulse cycle discussed in this paper. x, y, and z indicate the
rf phase in the rotating frame. T is used to indicate the time

interval between pulses, and the 8-pulse cycle time te = 127.

cbt
The magnetization vectors My (w= 1, 2, 3, 4) as observed during the
lst 8-pulse cycle, and the effective field vectors i, around which they

will stroboscopically precess.

Stroboscopic views of the magnetization M, precessing around their
“a

effective fields He Each magnetization traces an ellipse on the

x-y plane. When the detector phase is set along the y-axis, the

top four curves are obtained if only one window is sampled in each

cycle. The last curve is the composite which is obtained if all

four windows are sampled.

NMR signal obtained for frozen CegFig at -60°C when the first two
windows in each cycle were sampled. (A) is the signal as it was
initially obtained. (B) and (C) are the two traces obtained from

(A) by computer separation.

FourLler transformations of various traces from Fig. 4. The top

two spectra are the real parts of the transformations of traces (B)
and (C) respectively. The other two spectra are the absorption and
dispersion parts of the transformation of a curve obtained by
multiplying (C) by {2 to normalize its amplitude, and recombining

it with (B). The frequency scale is in terms of actual changes of the

oscillator frequency away from resonance.

Ff? powder pattern obtained from frozen CeFsg. The first and second

2 T windows were sampled to form a complex input for Fourier transformation.
The scale is in terms of actual changes of the oscillator frequency away
from resonance (1 KH, /division). No error corrections were applied to

this spectrum.

Fig. 7

Fig. 8

37.

The quantities 1/B and vs. resonance offset frequency for the two
window sampling case, as used in eq. (9b). Note that these quantities
are functions of wT, and that this figure uses the particular value

T @ 5usec.

The quantities 1/B and ¥ vs. resonance offset frequency for the four
window sampling case, as expressed in eq. (13). Note that these
quantities are functions of WT, and that this figure uses the particular

value T = 5usec.

4 oS

38.

~~

2T

Tv 2

T 2T

ONE CYCLE

2T

Fig. 1

39,

Fig. 2

FIRST

wn Rem

40.

SECOND

THIRD
WINDOW

FOURTH

LL) a —_
WINDOW Y i a
“aN i _ Tea
ee a eee te “
YI M, ce.
Va a
winpow YPM /¥2_

NO Pe
WINDOWS a
1, 2, 3 AND 4 ra a rr

om. a i cee,
—> TIME

Fig. 3

41.

. +
be *
® +
@ *
Ce kL Ea ee Reh Eee ee HH HAH HHA HH HE ET EE
! ' 4 ‘
4 + - 4 + ot +
* r .
e . :
e i fe & i

ann a 9 os
g.
€,
fo

Fig. 4

42.

-5KHz

Fig. 5

te er er

Fig. 6

44.

1.3

1/B
nN)

5 10

20°
15° h

10°

-10°

Fig. 7

5 10

1/B

45.

9.0
5.0
-30 -20 -10 0) 10 20 30
100°
50°
0° i
| |
~30 -20 -10 0 10 20 30
Av (KHz)

Fig. 8

46.

Section 4

Pulse Cycle Decoupling Theory and the Design of Compound Cycles

(This section is taken from two articles by D. P. Burum and

W. K. Rhim, “Analysis of Multiple Pulse NMR in Solids. III",

and "An Improved NMR Technique for Homonuclear Dipolar

Decoupling in Solids: Application to Polycrystalline Ice",

to be published April 1, 1979 in The Journal of Chemical Physics.)

47.

A. INTRODUCTION

In this section, principles are introduced which greatly simplify the
process of designing and analyzing compound pulse cycles. These principles
are demonstrated through application in the design and analysis of several
cycles, including a new 52-pulse cycle which combines six different REV-8
cycles and has substantially more resolving power than previously available
techniques. Also, a new 24~-pulse cycle is introduced which combines three
different REV-8 cycles and has a resolving ability equivalent to that of the
52-pulse cycle.

It was suggested by Haeberlen! that REV-8 multiple pulse cycles? ?3°4
might be combined to form compound cycles which eliminate the effects of
the homonuclear dipolar interaction in solids to higher orders of magnitude.
Nevertheless, there has been no real attempt to do this in practice, pre-
sumably because of uncertainty regarding how to combine different pulse cycles
without reintroducing undesired Hamiltonian terms, especially those cross
terms between the dipolar interaction and various pulse imperfections. Also,
the need to analyze long and seemingly complicated pulse cycles appeared to

pose problems. There has been an attempt by Haeberlen to implement the
suggestion of Mansfield® that three WAHUHA cycles be combined to form a
compound cycle. However, the resulting 14-pulse cycle has shown little
improvement compared to REV-8 because it does not eliminate those first order
dipolar cross terms involving pulse imperfections which are removed by REV-8.
Average Hamiltonian theory is extended in this section through exploring
the conditions under which a given term in the Hamiltonian expansion for a

compound cycle is given by a sum of separate contributions from each of the

pulse groups making up the entire cycle. Under such conditions

48.

the cycle is said to "decouple" with regard to that Hamiltonian term,

and this principle of decoupling provides a method for systematically
combining pulse groups into compound cycles in order to achieve enhanced
performance. This method is illustrated by a logical development from

the 2-pulse solid echo sequence” to the wanna, ® the REV-8 and finally

the new 24-pulse and 52-pulse cycles. During this development the 14-pulse
cycle of Haeberlen??” is discussed, and it is shown that three WAHUHA cycles
can be combined in a different way to form an equivalent 12-pulse cycle.

A large number of average Hamiltonian terms are calculated for the

52-pulse cycle, and an experimental analysis follows with tentative
explanations of which terms are governing the performance of the cycle
under various experimental conditions. The resolution of the 52-pulse
cycle is compared with results obtained using REV-8. Finally, axially
symmetric proton chemical shift tensor components are reported without
detailed discussion for polycrystalline samples of ice, CEH» CoHig and

polyethylene, all measured near liquid Ny temperature. Several alternate

versions of the 52-pulse cycle are presented in the appendix.

B. Pulse Cycle Decoupling

One approach to the design of improved pulse cycles is to create
compound cycles which combine various pulse groups with known charac-
teristics in order to systematically eliminate undesired Hamiltonian
terms to as high an order as possible. However, this is difficult in
general since there is no simple way of predicting what the terms for
a compound cycle formed from these sub-groups will be. Nevertheless,
the process of designing a compound cycle can be greatly simplified
through the principle of pulse cycle decoupling. Whenever a given

Hamiltonian term in the Magnus expansion for a compound cycle is simply

49,

an algebraic sum of separate contributions from each of the pulse groups
which make up the entire compound cycle, the cycle is said to decouple
with regard to that Hamiltonian term. This is clearly the case for all

pulse cycles with regard to zero-order Hamiltonian terms, as can be seen

from equation (16) of section 2. In fact, zero order terms are a trivial
case of pulse cycle decoupling.

In this sub-section,. conditions are presented under which a compound cycle

will also decouple for higher order Hamiltonian terms. The principle

of pulse cycle decoupling will be shown to provide a method for combin-
ing pulse groups in such a way that additional unwanted Hamiltonian terms
are eliminated without reintroducing any of the undesired terms which
vanish for each of the pulse groups considered separately. This same
principle also greatly simplifies the analysis of compound cycles.

Consider a compound cycle composed of wm. sub-intervals,
O= Loy < hy wee < qo = toe It is straightforward to show by changing the limits
of integration in equation (17) of section 2 that for any Hamiltonian in the

toggling frame, #, the first order term in the Magnus expansion is

given by

m m j
ol) _ .-l >» (1), pos, yr! >? p00) (0)
He. t 057 a3 + (2it.) Capt ay Hr
j=l j=2 k=l |
(1)
where FO“ and EY) are simply GO and J calculated for the
Aj Aj A A
j'th sub-interval using equations (16) and (17), of section 2,
"4
wy) . -l ~
Hs; = tos H(t) )dty (2)

50.

he ty
Gov = (2it yvt dt dt (9 (t ) H(t )]
&, 1 ay
with
Ek, - &
“oj j j-l “

From this it is clear that if eS vanishes for each of the sub-intervals

0 < j < m the cycle decouples for H, i.e.,

Hy = t, \, ta; (5)
jel

In fact, even if RS only vanishes for each sub-interval when 6-function

pulses are assumed, i.e., Ks depends on pulse width tC to first order,

ZA0)

—.f wen
Aj HH”) in equation (23) will be second order in to Thus 5.” will

decouple to first order in toe i.e.,

go yl BY 4 (2

BY =e, BD + 000 (6)
j=l

From equation (18) of section 2, it can be shown in a similar way that

—(0
if Hy vanishes for each of the sub-intervals, the cycle also decouples for

RO, i.e.,

ml

yA?) 7h (2)
HE = t, ty Hy; (7)
j=l

51.

where #.)) is defined as HO) calculated for a single sub-interval.
Finite pulse width effects will be neglected for Hamiltonian
terms of second or higher order. Therefore, even if HS only vanishes
for each sub-interval under the é-function pulse assumption equation (7)
will still be considered valid.

Finally, if HS vanishes over each sub-interval then for any other

Hamiltonian He the cycle will also decouple with regard to TE”:

s3(1) _ -l zl)
Hy = ty toy any (8)

j=l
wal) , . . a1) - wf
where He xe is defined simply as Hp calculated for the j'th sub-
interval. Of course if a vanishes over each sub-interval only when
$-function pulses are assumed, the cycle may not decouple to first order
. . m1) . fey
int. with regard to Fs», » depending upon HE, «

These rules can be summarized as follows: if, for some Hamiltonian

Hy » cae vanishes over each sub-interval of a compound cycle, the cycle
will decouple with regard toHA”, HO?) and FD regardless of the
behavior of Hp. Furthermore, if a vanishes over each sub~interval
only when é-function pulses are assumed the cycle still decouples to
first order in to with regard to FE, and the cycle also decouples with
regard to HO) and TE” under the 6-function pulse assumption.

It is difficult to extend these results to Hamiltonian terms of
arbitrary order because no general expression is available for TO”,
However, there is a simple expression for HOM for at least one special

case. We make the following mathematical definition:

52.

=(n) n-l Be Ctl to
HE. = (-i) te dt dt. wee dt,
0 0 fe)

Hey (ty) F(t) +++ H(ty) (9)

Haeberlen et al. have shown? that if for a given cycle HS = Q for

all j of two sub-groups, 0 = Lg < an < Lo = tos then by changing the limits of
integration in equation (31) it is straightforward to obtain the

following equation:

zmm) _ .-l a(n) a(n) .
He, = ty He JE. a) FE x5 - it t

cl-c2
(10)
wA0) ~(n-1) ~A(1) (n-2) ag(n-1) a0)
(2° HE. + Hy Hy to... + HE oo FE.)
where
; at Coed oar)
KAI) FS (a) eT! dt dt dt
Al ~ ce n+] ni" 1
0 0 0
H(t) Hy (t,) “ee Hy (ty) (tl)

and a similar definition applies to HD).

From equation (11) it is clear that there are a number of circum-

stances under which the cycle will decouple. Perhaps the most important

case can be summarized as follows. If

HS) -0 , Wi

and

Aj) _ gli) _ . (n/2-1) for even n ;
Heyy = Hx = 0 5 Wis (n-1)/2 for odd n (13)

53.

then

—(n) _ gn) _ .-1 o(n) p(n)!
Hs ~ H 7 te eal + tag x9 j (14)

tea . . an) _ gn) Zin) _ wg(n)
but it is not necessarily true that Hy = Hoy or HE oo H vo . It
is simple to extend equation (14) to m sub-intervals. Equation (13) is
replaced by

pm) . gm . - pi) . .. (n/2-1) for even n
Ha = Hy Fre = FE an 0 > Wis (n-1)/2 for odd n (15)

and the result is then .

—(n)_ _-1 =(n)
Fey = Ee » €o5 ay (16)
j=l

Of course, in order for the compound cycle to decouple with regard to a

higher order cross term such as HO”, equations (12) , (15) and (16)

must be satisfied for Hb, + H,.

C. Reflection Symmetry

In the design of pulse cycles much use has previously been made! »6

of the principle of reflection symmetry. 10,11 If a Hamiltonian in the

toggling frame KH has reflection symmetry, that is if
He, (t) = Hy CE - t) for O< t

then He) vanishes for all odd j. Moreover, if H, also has reflection
P7200) zi) . ; .
symmetry, then B and HE > also vanish for all odd j.
This principle can be useful when analyzing the
properties of sub-cycles within compound cycles. However, it is more or

less accidental that sub-cycles within the 52-pulse cycle discussed below

54.

have reflection symmetry, and the improved resolution of the cycle does
not arise from this symmetry. In fact, the sub-intervals within the

24-pulse cycle introduced below do not all have reflection symmetry, yet
the principle of pulse cycle decoupling shows that this cycle is essen-
tially equivalent to the 52-pulse cycle, and indeed the two cycles have

been shown in practice to have comparable resolving abilities.

Cc. DESIGN OF COMPOUND CYCLES

Pulse cycle decoupling has obvious applications in simplifying the
calculation of average Hamiltonian terms for compound cycles. Perhaps
an even more important use of this concept is as an aid in the design
of pulse cycles. After the introduction of some useful notation, the
application of the decoupling principle to pulse cycle design will be
illustrated by several examples in which a series of compound cycles
designed to remove the effects of the homonuclear dipolar interaction®™

while preserving off resonance and chemical shift information will be
developed.

Notation

It is very helpful in designing pulse cycles to use a
notation similar to that of Mansfie1d® which focuses attention on the
state of the Hamiltonian in the toggling frame. Specifically, a pulse
cycle is expressed according to the state of T, =U LU... For

instance, one version of the solid echo sequence,

c a a -

Zz y? x (19)

55.

Only n/2 pulses of four different rf phases, (m/2) > (w/2) > (n/2). and
(w/2) will be considered. Therefore, a unique pulse sequence will
be determined by each series of tr states such as expression (19)

In order to illustrate the various features of this notation, the WAHUHA
4-pulse cycle will be considered as an example, one version of which can

be written

Pulse timing information is easily included in this notation. For
instance the pulse cycle in expression (20) can be written either as a

single pulse group,

(I,, IL, 21,, I, i) (21)
or as a combination of two solid echo pulse groups, i.e.,
Gt, tay tl, 1) (22)

There are usually many possible versions of a given pulse cycle.

For instance, the cycle

is also a version of WAHUHA, and is written in this notation as

I, -I.,tT1 I.,-1,1
(I, Ty» TIC - ts 1) (24)

In order to study the characteristics of a pulse cycle which do not

depend on which specific version of the cycle is used, the six possible

56.

states for To which are +1 at, and +I, will be written simply as
A, A, B, B, C and C, where of course A = -A etc. and A, B and C are
mutually orthogonal. In this notation the WAHUHA cycle is written

simply as
(ABC) (CBA) (25)

where the substitution A = lL B= le C = I, gives the version shown in

expression (20) and A = I B = “Ty. C = I gives expression (23).
Four important points should be remembered about this notation when
combining pulse groups to form compound cycles:
(i) An extra pulse will be required between two pulse groups when
the first state of the second pulse group is not the same as the last

state of the first pulse group. For example, the two WAHUHA cycles

which make up the following pulse group,
(ABC) (CBA) (CAB) (BAC) (26)

cannot be joined without inserting an extra (1/2) pulse between (CBA)

and (CAB). This extra pulse does not belong to either WAHUHA cycle.

Therefore: the entire compound cycle is not a simple combination of two

WAHUHA sub-cycles, and it may not decouple as one might otherwise expect.
(ii) A (1)-pulse will be required if two pulse groups are joined

which cause a pair of adjacent states such as AA. Therefore pulse cycles

like the following will be avoided in this section.

(ABC) (CBA) (ABC) (CBA) (27)

57.

(iii) The fact that a pulse sequence begins and ends with the same
T, state does not guarantee that it is a cycle. For a cycle the follow-

ing must be true,
Ug Ct) = +1 (28)

while the fact that a sequence starts and ends with the same Tr, state

only shows that
~ ~1
T(t) ~ Deg lt EU g(t) = 1, (29)

This is a necessary, but not a sufficient condition for the sequence to

be a cycle. For example, while

(ABC) (CBA) (30)
is a cycle,

(ABC) (CBA) . (31)

is not a cycle.

One can be sure, however, that if two cycles are joined side by
side or one complete cycle is inserted somewhere inside another one, the
resulting sequence is a cycle. Also, any pulse sequence such as (ABC)
(CBA) which has reflection symmetry is a cycle.

(iv) The arrangement of the pulse groups within a compound cycle
can be important even when the cycle decouples if the Hamiltonian term
being considered depends on To i, or on the specific rf pulses used

to generate the cycle. For example, assume that the following compound

eycle decouples with regard to both Ja” and H):

(ABC) (CAB) (BAC) (CBA) €32)

58.

Hy depends only on i but HE, also depends on the state of T, and Ty and
on the specific pulses used to generate the cycle. Therefore, if the
sub~groups in expression (32) are rearranged to form a different cycle,

such as
(CAB) (BAC) (CBA) (ABC) (33)

HA” will be unaffected, but HO) may be different for the two cases.
One important advantage of the notation introduced here

is that it makes the chemical shift scaling factor for any cycle

immediately available, assuming é-function pulses. Consider for

example the WAHUHA cycle given in expression (20). Since He. = 2 (au +

Wo rep ted it is apparent from the T, states shown in expression (22)

that the integral of KH, over a cycle is 2r(T, + I, + Ts and that the

cycle time is 61. Thus it can be seen from equation (16) of section 2 that

ZA0) 1
Ho = 3 Y (An + op Dg ta tL (34)
- |

Since the chemical shift scaling factor is given by He l(/1| eo? |, one
finds by computing the length of the vector in equation (34) that the
scaling factor in this case is v3. The same result could have been
obtained just as easily from expression (25). The feature of making

the scaling factor readily available is an important advantage of this

notation when designing a pulse cycle for measuring chemical shifts.

59.

The Solid Echo Pulse Group

The solid echo pulse sequence is a basic building
block. For this sequence, assuming 6-fynction pulses, FE) = HO)
+ HY) + 02) = 0, where HO) and 0? are obtained by substituting I, or
I for L. in equation (37) of section 2. Therefore it is clear from the principle
of pulse cycle decoupling that any compound cycle composed of solid echo
sequences will decouple assuming é6-function pulses with regard to IE,
HM? and all first order dipolar cross terms such as we”, TA» etc.
In fact, the cycle will decouple with regard to FA” when finite pulse
widths are considered, and with regard to IY wnen calculated to first
order in to Since these terms will be simple algebraic sums of the
contributions from the individual solid echo pulse groups, the objective
is to form a compound cycle by combining different versions of the
solid echo pulse sequence in such a way that the algebraic sums vanish.

It is convenient at this point to define two types of "compensated"
sets of pulse groups. Assume that a compound cycle is being designed
which will decouple with regard to some Hamiltonian term FO, Then a
set of pulse groups is a compensated set of the first kind with regard
to HO” if it will make no net contribution to FE as long as each
member of the set is included somewhere in the compound cycle. Compen-
sated sets of the first kind are usually associated with Hamiltonian
terms such as HO, He” etc. which depend only on the states of T.
Terms such as we, He”, etc. which also depend on qT; iy or on the
specific rf pulses are often associated with compensated sets of the
second kind. These are sets of pulse groups which must be included in
the compound cycle in a specific order, and the members of the set must

be either adjacent to each other or separated by complete subcycles.

60.

There are only two ways in which the members of a compensated set of
the second kind may be rearranged:

(i) The members of the set may be cyclically permuted, e.g.if
{ (ABC) (CAB) (BCA) } is a compensated set of the second kind then so are
{ (CAB) (BCA) (ABC) } and {(BCA) (ABC) (CAB) },

(ii) Adjacent cycles within the set may be interchanged. For
example, {(ABC) (CBA) (ABC) (CBA) (ACB)} can be rearranged to form
{ (ABC) (CBA) (ABC) (CBA) (ACB) }.

Table 1 summarizes the sets of solid echo pulse groups which are
compensated sets of the first kind with regard to HO, HO, HE”,
and FO), The generalized notation used in the table is a straight-
forward extension of the notation introduced earlier for representing
pulse sequences. Recall that A, B and C represent states of T> that

A = -A etc. and that A, B and C are mutually orthogonal. Since TO?) is

readily determined from I, by substitution in equation (3) of section 2, i.e.,
Zz) _ > Sal _
HH, Pius tT, 30 tay? (35)
i

it can be written in a generalized form according to T.. For example,

Yo OZ) _ la) We) | .
if 1 =A, #) HS where KH, is given by

ugla) _ > | coe]
He = 6, Gae Ty 3A,4,) (36)

i

Of course, HO is not affected by the sign of Ts i.e., if qT, = A,

~ — rol

HO) = H® Thus, for the pulse group (ABC) FO is given by the
D D D

following series of states:

6l.

SO?) = HO) QHE? = HO) THy) = HS ar

The compensated sets in Table 1 were determined in a very straight-
forward manner. First, a general result for each Hamiltonian term was
calculated using the above notation. This insured that the result for
any specific solid echo group could be obtained simply by substituting
the proper T, and TO states into the general result. Then, from
these general results it was determined which combinations of solid echo
groups would form compensated sets. Usually this could be done by
inspection, but if necessary the results for all possible solid echo
groups could be readily tabulated by substitution into the general

-5( 1
results and then compared to determine the compensated sets. For ras )

I ands”? the compensated sets were determined assuming 6~function
pulses. Notice however that {(ABC), (CBA)} is a compensated set for
FY) Lf it is calculated to first order in to
In order to illustrate the procedure followed in generating Table 1, the
ZAL)
Dd

tera KH will be considered as an example. If the formula for a first order

term given in equation (i7) of section 2 is applied to the solid echo group

(ABC), the following result is obtained assuming d-function pulses:

ps) _. faith flO) Ja) yAC) ala) , lb)
KH, = - (=| A 5 HS | + Ea 5 a + He. | (38)

Recalling that

yA) 1. yd) (c) _
KH t JE, + HH, = 0 (33)

62.

it is clear that the last term in equation (38) vanishes, giving the

general result shown in the table:
gAl) _ it (b) (4)
HY =~ (ZV [oe, 2] (40)

It is clear from this formula that the sign of qT) will not affect IY,
Thus equation (40) also applies to (ABC), (ABC), etc. The goal now is to
find solid echo groups for which the sign of HY) will be reversed.

One possibility is to switch the order of A and B, i.e., (BAC). Also,
because of equation (39) the desired result can be obtained by switching
either A and C or B and C, i.e., (CBA) and (ACB). Thus three simple
compensated sets for JA» are {(ABC), (BAC)}, {(ABC), (CBA)} and

{(ABC), (ACB)}. The other results in Table 1 were obtained in a similar

manner.

WAHUHA AND REV-8

It is clear from Table 1 that {(ABC),°(CBA)} is a compensated set
of the first kind with regard to both HEY) calculated to first order in
ty and also JES calculated assuming 6-function pulses. In fact, the

expression (ABC)(CBA) by itself represents the 4—pulse WAHUHA cycle.®

In order to eliminate HO for finite pulse widths a compound
cycle can be constructed by combining the four solid echo groups (ABC),
(CBA), (ABC) and (CBA). Notice from Table 1 that both {(ABC),
(CBA) } and { (ABC) , (CBA)} are compensated sets of the first kind with
regard to HOY and JA”, while {(ABC), (CBA)} and {(ABC)(CBA)} are
compensated sets for FO calculated for finite pulse widths. Thus

HO) for finite pulse widths, we? calculated to first order in to

63.

and Tw for 6-function pulses will all vanish for any compound cycle
composed of these four pulse groups. There are two ways to combine
these pulse groups so that extra pulses between the solid echo groups

are not required,namely

(ABC) (CBA) (ABC) (CBA) (41)
and
(ABC) (CBA) (ABC) (CBA) (42)

274

These expressions represent the two basic forms of the REV-8 cycle’.
An important property which is not obvious from Table 1 but which has
been demonstrated previously is that both expressions (41) and (42) are
compensated sets of the second kind with regard to all first order cross
terms between KH, and the various pulse imperfection Hamiltonians calcu-
lated assuming 6-function pulses. This means that all cross terms such
as H, HE, etc., vanish for REV-8. Expressions (41) and (42) are
actually the same compensated set. This can be seen by making a cyclic

permutation on (4]) to obtain
(CBA) (ABC) (CBA) (ABC) . (43)

and then making the substitution A' = C, B' = B, C' = A to obtain
expression (42).

An interesting comparison can be made at this point between the
usefulness of pulse cycle decoupling and of reflection symmetry in the
design of pulse cycles. Since the REV-8 cycles described by expressions
(41) and (42) both decouple with regard to HE” assuming 6-function
pulses, it is apparent from Table 1 that this term vanishes for both

forms of REV~8. However To and therefore JE, does not have

64,

reflection symmetry in expression (41). Thus the principle of reflection
symmetry cannot predict that we” vanishes for expression (41) , and on
this basis one might be misled regarding the effect of IY on the

performance of the two cycles.

The 14-Pulse and 12-Pulse Cycles

Clearly, the next step is to design a compound cycle which removes
HO), Pulse cycles have been suggested previously by Evans 12, Haeberlen
and Waugh, and Mansfield 6 which eliminate we?) as well as EO, FY)
and FE”. However, none of these cycles show substantial improvement
over the WAHUHA sequence because none of them remove the first order
dipolar cross terms which are eliminated by REV-8. For example, the
"72 t complementary doubly symmetrized" cycle proposed by Mansfield
bears some resemblance to the new 52-pulse cycle discussed below. How-
ever, it requires eighty 1/2 pulses and makes no attempt to remove the
dipolar cross terms involving such pulse imperfections as pulse length
and phase misadjustments and phase transients. More recently Haeberlen!»>

has used a 14-pulse cycle which generates a series of Tr, states originally

suggested by Mansfield°:
(ABC) (CBA) (BAC) (CAB) (ACB) (BCA) (44)

Since this cycle combines three WAHUHA cycles it is clear from the

og . gAD) _ zl) _ . .
principle of decoupling that H, = Fo = 0 assuming 6-function
pulses, and I) vanishes when computed to first order in toe Further-
more, expression (44) satisfies the requirement given in Table 1 for
HO? to vanish, namely that it be composed of equal numbers of solid

echo groups with A, B and C as the middle state. In order to generate

65.

expression (44) two extra pulses surrounding the center two solid echo
groups are required, making a total of 14 pulses.

Because the cycle described by expression (44) decouples, the
solid echo groups in the cycle can be rearranged so that extra pulses

are not needed:
CABC) (CBA) (ACB) (BAC) (CAB) (BCA) (45)

Since FO, HY), tA” and I? also vanish for this 12-pulse cycle,
it can be expected to have a resolving power equivalent to the 14-pulse
cycle.

Experimentally, the 14-pulse and 12-pulses cycles have been found
to be equivalent to each other, but in most cases
their resolving ability is less than for REV~8. Clearly this is because
the 12-pulse and 14-pulse cycles do not remove those dipolar cross terms

—f _
such as He and HY which are eliminated by REV-8.

The 24-Pulse and 52~Pulse Cycles

The principle of pulse cycle decoupling assures us that we may
combine different versions of REV-8 in order to eliminate J”) without
reintroducing any of the dipolar terms which vanish for. REV-8, including
dipolar cross terms such as HA”, we) etc. Following the example of

the 14~pulse cycle , consider the following series of r states:

[apc ][asc][Bac][Bac]facs][acs] (46)

where the square brackets indicate reflection symmetry, e.g., [ABC] =

(ABC) (CBA).

66.

Expression (46) represents three different versions of REV-8
combined in such a way that HB? vanishes for the cycle as a whole,
and this is what is desired. However, in order to generate the
pulse groups in the order shown in expression (46) it is necessary to
insert two extra 1/2 pulses surrounding the center REV-8 cycle. Since
these extra pulses are not part of the REV~8 cycles the possibility
arises that dipolar terms which vanish for REV-8 may not vanish for
this cycle.

One way to solve this problem is to rearrange the solid echo groups
in expression (46) so that extra pulses are no longer needed. One must
be more careful than in the case of the 14-pulse cycle because the REV-8
cycles are compensated sets of the second kind. Nevertheless, by
inserting one entire REV-8 cycle inside another one the compensated
properties of the REV-8 cycles are preserved and the following 24-pulse

cycle is produced:
{apc ][apc][acB ] (acs) [Bac ] [BAC] (BCA) . (47)

This cycle was discovered only recently and will be analyzed in detail
ina later paper. However, our initial measurements indicate that the
resolving ability of this 24~pulse cycle is equivalent to that of the
52-pulse cycle which is analyzed in this section. Of course, many.other

versions of this 24-pulse cycle are possible, the main criteria being:

(i) Three REV-8 cycles are combined in accordance with the rules
for compensated sets of the second kind, and
(ii) The resulting compound cycle contains equal numbers of solid

echo groups with A, B and C as the middle state.

67.

There is another way in which one can use the principle of pulse
cycle decoupling in order to correct for the effects of the two extra
pulses required by expression (46), namely we can combine different
versions of this cycle in order to eliminate the undesired Hamiltonian
terms created by the extra pulses. Fortunately, all of the first order
dipolar terms which vanish for REV-8 also vanish for the pulse cycle
described by expression (46). However, FO” no longer vanishes if
finite pulse widths are considered. For example if A = Ly B= -1

and C = I; one obtains

(0) _ vl Az)
KH, 7 t (18mit) LI, JE, 1 (48)

Since FO” would vanish if the extra pulses were not needed to generate
expression (46), it is natural to explore the behavior of HO as a
function of the rf phases of the extra pulses. If the rf phases of
these two pulses are reversed, with all other pulses remaining unchanged,

expression (46) becomes
[asc ][aBc ][Bac ][Bac ][acB ] [acs] (49)

and 90? for this expression is just the negative of 0° in equation
(48). Thus expressions (46) and (49) can be combined to form a com-

pound cycle for which #0?) = 0:

[ABC] [ABC] [BAC] [BAC] [ACB] [ACB] [ABC] [ABC] [BAC] [BAC] [ACB] [ACB] (50)

This is one version of the 52-pulse cycle.

68.

There are many other combinations of twelve 4~pulse groups which
are just as readily produced and can be expected to show similar capa-
bilities for homonuclear dipolar decoupling. The main criteria for
constructing these cycles can be summarized as follows:

(i) The compound cycle is formed by rearranging the 4-pulse groups
within six REV-8 cycles according to the rules discussed above for com-
pensated sets of the second kind.

(ii) Equal numbers of solid echo groups with A, B and C as the
middle state, disregarding bars over the states, are contained in the
compound cycle.

(iii) Four extra pulses are inserted between pulse groups in such
a way that the third and fourth extra pulses are reversed in phase
relative to the first and second pulses.

In practice, 52~pulse cycles which satisfy these three criteria
ean differ somewhat in performance due to their properties with regard
to “second averaging")? of non-vanishing Hamiltonian terms. This effect
is discussed in the next section. The version of the 52-pulse cycle
which has the most favorable characteristics of those tried so far can

be expressed as follows:
[ABC] [ACB] [BAC] [BAC] [ACB] [ABC] [ABC] [ACB] [BAC] [BAC] [ACB] [ABC] (51)

Notice that HO, Pi” and JO”? all vanish (for 6-function
pulses) over each quarter of this cycle. Also, expression (51) has
retlection symmetry for HO Therefore, the principle of reflection

symmetry tells one that = Q for the cycle as a whole.

69.

D. SPECIFIC rf PULSE CYCLES

The pulse cycles discussed in the previous sub-section were expressed

in a general notation which emphasizes the states of i. This sub-section

presents. specific versions of these cycles expressed in terms of

rf pulses. All of the cycles are constructed using solid echo pulse

groups such as

7 T
(Go (32)
x ~y

For simplicity, solid echo groups will be expressed according to the rf
phases of the two pulses. For instance, the solid echo sequence in
expression (52) will be written simply (XY). Four versions of WAHUHA,

which will be called la, 1b, 2a and 2b, can be expressed as follows:

la = (XY) (YX)
lb = (XY) (¥X)
2a = (YX) (XY) (53)
2b = (YX) (XY)

Of course, lalb = 1 and 2a2b = 2 form two versions of REV-8.
Using this notation, one version of the 14-pulse cycle discussed

in the previous section can be written

la (=) la (5) 2a
2 x 2 _x (54)

and the corresponding 12-pulse cycle is written

la(YX) 2a (XY) (55)

70.

These cycles produce the series of qT, states given in expressions (44)
and (45) if the substitution is made A = Io B = -T and C = TL. For

this same substitution, the following pulse cycle produces the series

of Tr states given in expression (47):
*la*1b*2a*(¥X) 2a2b (XY) (56)

This is a 24-pulse cycle with a cycle time of 36 t. The signal may be
sampled in each of the windows indicated by a * without distorting the
results unless the spectrum contains frequency components comparable to
or greater than (12 ool,

The 52-pulse cycle which is analyzed in the next section can be

written

*1a*2a($) 1a10(5) 2orib*lax2a(P) 1a10($) 1b*2b (57)
x -—x -—x x

This cycle contains 12 of the 4~pulse groups defined in equation (53)
along with 4 extra 7/2 pulses, and it produces the series of t states
given in expression (51) if the substitution A = Tos B= “ty and C = I.
is made. The cycle may be sampled in each of the windows indicated by
a * in expression (55).

It should be possible to obtain quadrature phase information from

the 24-pulse and 52-pulse cycles by sampling in other windows, as was

demonstrated in section 3 for the REV-8 cycle.

71.

E. ANALYSIS OF THE 52-PULSE CYCLE

Hamiltonian Terms

Table 2 Lists terms in the Magnus expansion for the 52-pulse cycle
given in expression (55). First order terms which vanished when 6-function
pulses were assumed were recalculated by including the effects of finite
pulse widths to first order in t except for 5) and @L"?. Cross
terms between pulse errors were assumed to have no effect and were not
included, and terms of second or higher order were calculated assuming
$-function pulses. The only important term which did not decouple and
was too complex to calculate for the cycle as a whole is WS”,

Because of the symmetry of the 52-pulse cycle and its decoupling
properties the calculation of the first non-vanishing pure dipolar term,
AY, was reduced to a summing of the contributions of only three
versions of the 2-pulse solid echo group. This allowed the term to be
obtained with relative ease, whereas the calculation would otherwise
have been extremely laborious.

The practical value of Table 2 will become more apparent in the
next section as the behavior of the 52-pulse cycle is experimentally

analyzed.

Experimental Analysis

Figure 1 shows the results from a measurement of the apparent off
resonance frequency Aw? as a function of the actual resonance offset
dw for the 52-pulse cycle applied to a water sample. The least squares
fit shown in the figure gives a scaling factor of 2.78, which is in
good agreement with the theoretical value of 2.81 calculated using #00

from Table 2.with t = 3.0 usec and t F 1.5 usec. It is not as simple

a task to predict the resolution of the 52-pulse cycle under various

72.

experimental conditions from the terms in Table 2. However, a
significant improvement compared to the REV-8 cycle is expected under
all conditions since #7? and f°? vanish for the 52~pulse cycle along
with all of those line broadening terms which vanish for REV-8.

A number of measurements have been made on a single crystal of
CaF, which give an overall picture of the cycle's performance and allow
‘some tentative conclusions to be drawn as to which of the Hamiltonian
terms in the Magnus expansion determine the resolution. An undoped,
spherical crystal of CaF, was chosen as the sample for the measurements
so that the resolution could not be limited by dopants, bulk suscepti-
bility, or effects due to molecular motion. Limitation of the resolu-
tion due to paramagnetic impurities has been observed in .52-pulse experiments

even for U-doped CaF, with T, = 3 sec. Substantial spectral broadening

due to bulk susceptibility has also been observed in CaF, powder
samples. One example is shown in Figure 2, which compares spectra

from a polycrystalline sample of undoped CaF, and a spherical crystal

of the same material oriented with the (100) axis parallel to Ho:

These spectra were obtained using the 52-pulse cycle with t = 2.8 usec
and Aw/2™ = 2.3 kHz. Part (a) of the figure agrees well with a value
for the spectral width of roughly 12 ppm which was predicted by compar-

ing two hypothetical CaF, crystallites assumed to be ellipsoidal with

axial ratios b/a = c/a = 0.1 for one and b/a = 1.0, c/a = 0.1 for the

= 14
other. The value ~28.0 x 10 6 cgs units for the bulk susceptibility

of CaF, and the tables of Osborn 5 were used in this calculation.

73.

Figure 3 illustrates typical behavior of the exponential decay time

. : + :
of the signal during the pulse train, T, , as a function of Aw. For

this measurement T = 2.8 usec, and the CaF, sphere was oriented with

the (100) crystal axis parallel to Ho:

Comparison with the dashed theoretical curve shows that below
Aw/27 = -3 kHz the resolution is determined by a term that is linear
in Aw, the most likely candidate being He”? (see Table 2). However,
this term apparently has little effect for 1 kHz < Aw/2m < 4 kHz. In

+ ; . .
this region of constant T, , which is called the high resolution "plateau",

qT, is obviously determined by terms in the Hamiltonian expansion which

do not depend on Aw. Further measurements at this crystal orientation
have shown that as t is increased the heighth and width of this plateau
tend to decrease, although the plateau is always found in the same
region of Aw/27, and the resolution becomes dependent on Aw.

Greater insight into which Hamiltonian terms are limiting Ty in

the high resolution plateau can be gained from Figure 4, which presents

: + . . :
experimental data for T as a function of crystal orientation for two

values of t, 2.0 usec and 7.0 usec, at two different values of Aw/2r7,
1 kHz and 2.5 kHz. From this figure the following observations can be
made:

(i) When the (100) crystal axis is parallel to Ho» which is the

: : ot -2
orientation of strongest dipolar coupling, T, depends on t for a

given Aw.

+ -1
(ii) For t = 7.0 usec T depends on Aw at all orientations.

74.

(iii) The more rounded shape of the two curves for Tt = 2.8 usec
as compared to the "propellor" shape of the curves for t = 7.0 usec

indicates a weaker dependence on the strength of the dipolar coupling

fi

for t = 2.8 usec than for T 7.0 usec.

(iv) The curves for t 2.8 wsec show only a slight dependence
on Aw.

From the first 3 observations it may be concluded that the two
curves for t = 7.0 usec are governed by that part of which is
proportional to awe | oh? |? Observations (iii) and (iv) indicate

that, except when HH. is nearly parallel to the (190) crystal axis, T,

is limited by one or more terms which do not depend on Aw and at most
vary linearly with the strength of the dipolar interaction. The most
likely candidates are the first order cross terms between the dipolar
Hamiltonian and the various pulse errors. All of these terms vanish
assuming 6-function puises and depend on ty rt oe? || when calcu-
lated to first order in toe and all of them are independent of Aw.

Thus, one would expect the resolution to improve as t is increased
whenever Ty is governed by these cross terms. This prediction is
confirmed by the results of additional measurements made in the plateau
region of Aw/27 for values of t between 2.8 and 5 usec and with the

(111) crystal axis parallel to Ho: Experience has shown that misadjust-
ment of the pulse lengths and faulty positioning of the sample in the

rf coil have the strongest effect on resolution. Thus it is most likely

+ — — =
that qT, is limited by and ¥')) when it is not governed by #”)

75.

The question of what causes the asymmetry with regard to Aw in
Figure 2 is not completely solved. It is most likely that those terms
in the average Hamiltonian expansion which do not depend on Aw and are
linear in the spin operators alter the direction, but not the magnitude,
of WO, causing it to have a more pronounced second averaging effect
on WL” above resonance than below resonance.

Experimental comparisons confirm that the 52-pulse cycle has
consistently better resolution than REV-8, as can be predicted from
Table 2. One example is shown in Figure 5, which compares proton
spectra from the two cycles for one orientation of a single crystal
of sypsum-© CaSO, ° 2H,0. These spectra were measured at ambient
temperature with t = 2.8 usec and Aw/27 = 2 kHz. This example is

typical of the improved resolution available using the 52-pulse and

24-pulse cycles, including those alternate versions discussed below.

F, EXPERIMENTAL RESULTS

As a further demonstration of the resolving power of the 52-pulse
cycle, axially symmetric proton chemical shift tensor components are
presented in this sub-section for polycrystalline samples_of several

organic compounds and for polycrystalline ice.

Figure 6 compares the resolution of the 52-pulse cycle with that
of REV-8 for a polycrystalline sample of cyclohexane, CoH o> at liquid
Ny temperature. Despite broadening due to bulk susceptibility effects,

an axially symmetric chemical shift tensor is clearly resolved which

76.

was not obtained using REV-8. For both spectra in the figure
t was 3.0 usec. Similar results have been obtained for cyclopentane,

CoH g> and polyethylene at liquid N, temperature. These results are

summarized in Table 3 along with the tensor components for ice which
are discussed below.

All of the measurements in Table 3 were made in the high
resolution plateau region of frequency, 1 kHz < Aw/2n < 4 kHz, and
with t = 3.0 usec.

An important example of a substance for which chemical shifts were
very difficult to resolve until now is polycrystalline ice.+/ Figure 7
contrasts the results from the 52-pulse sequence with those from the REV-8
cycle when applied to naturally abundant protons in ice near liquid N,
temperature. For both cycles T = 3.0 Usec. The powder spectrum appears to
be axially symmetric with 0 = 11.2 + 1 ppm and o = -17.5 +. 1 ppm relative
to TMS, and there also appears to be an isotropic shift relative to liquid
water of -5.3 + 1 ppm. Preliminary spectra have also been obtained near
-90°C which yield the same tensor components but also show spectral broadening
and shape distortions which are apparently due to molecular motion. These
results may be compared with a measurement made by Pines et _a1./® on 99%
deuterated ice at -90°C. They obtained o, = 15 + 2 ppm and oj = -19 + 2 ppm
relative to TMS, with an isotropic shift relative to liquid water of 2 + 1 ppm,
and their spectrum shows no sign of effects due to molecular motion. Thus

it appears that definite differences exist between the chemical shifts and

the dynamic behaviors of natural and deuterated ice.

77.

G. ALTERNATE VERSIONS OF THE 52-PULSE CYCLE

There are many other combinations of REV-8 cycles which satisfy the
requirements discussed in this section for eliminating #.” without
reintroducing any of the dipolar Hamiltonian terms eliminated by REV-8.
Most of those tried so far have shown a resolving power near resonance
similar to that of the 24~pulse and 52-pulse cycles discussed in this

section, but several have not shown a high resolution plateau which

extended over as wide a range of frequency as that shown in Figure 3.

gl)

One alternate version of the 52-pulse cycle eliminates #7), cal-
culated to first order in ty as well as all the other dipolar terms
that vanish for the 52-pulse cycle analyzed in thissection. This alter-

nate 52-pulse cycle combines two versions of REV-8 which can be written

as follows:

3 = 3a3b
(58)
4 = 4adb
where
3a = (XY) (YX)
3b = (XY) (YX)
_ (59)
4a = (YX) (XY)
4b = (YX) (XY)

The full cycle can be written

3236(5) 330 (5) ‘adbirsa(Z) 3a3n(5} 3a3b (60)
“TX ~ fx x

Xx

78.

Another 52-pulse cycle is designed to eliminate HE and W.

(assuming 6 = 6 = 6 = 6 )calculated to first order in t . It uses
x -x y -y Ww

the pulse groups given in equation (59) and can be written

2a(7) 3230 (5) 3230304a(5) 3a30(5) ba&db4b (61)
x -x -X x :

Another 52-pulse cycle which is composed of unseparated REV-8

cycles from equation (53) can be written as follows:
w\ ft 7 1
(220,208). :
x -x -x x

Experimental measurements indicate that this cycle has a narrower
plateau in the same region of Aw/27 as in Figure 3.

Many other combinations of REV-8 cycles are possible, for instance
1a(5 1a2azo10( 5) Ib 1a(5) 1242010 (5) lb (63)
x -X “XK x

and even a 26—pulse cycle,

13(5) ia10(3) (5) 2a20(5)] (64)
x ~-x —x x

which might be difficult to use in practice due to difficulties such as
a scarcity of sampling windows. Clearly the approach to pulse cycle

design discussed in this section provides a great deal of fhexibility which

tn

houlad lead to the development of pulse cycles with even greater resolv-

ing power than the 24-pulse and 52-pulse cycles.

79.

REFERENCES

1.

10.
li.
12.

13.

14,

15.

16.

U.

Haeberlen, High Resolution NMR’in Solids, Selective Averaging,

(Academic Press, New York, 1976).

W.

K. Rhim, D. D. Elleman and R. W. Vaughan, J. Chem. Phys. 59,

3740 (1973).

W.

J.

K. Rhim, D. D. Elleman, L. B. Schreiber and R. W. Vaughan,

Chem. Phys. 60, 4595 (1974).

Section 3.

U.

Haeberlen, private communication.

Mansfield, J. Phys. C: Solid State 4, 1444 (1971).

. G. Powles and P. Mansfield, Phys. Lett. 2, 58 (1962).

G. Powles and J. H. Strange, Proc. Phys. Soc. 82, 6 (1963).
Mansfield, Phys. Rev. A137, 961 (1965).

S. Waugh, L. Huber and U. Haeberlen, Phys. Rev. Lett. 20, 180 (1968).
Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968).

Mansfield, Phys. Lett. A32, 485 (1970).

H. Wang and J. D. Ramshaw, Phys. Rev. B6, 3253 (1972).

A. B. Evans, Ann. Phys. 48, 72 (1968).

Haeberlen, J. D. Ellet, Jr. and J. S. Waugh, J. Chem. Phys. 55,
(1971).

Pines and J. S. Waugh, J. Magn. Res. 8, 354 (1972).

Handbook of Chemistry and Physics, edited by R. C. Weast (Chemtcal

Rubber, Cleaveland, OH, 1974) 55th ed., p. E122.

J.

A. Osborn, Phys. Rev. 67, 351 (1945).

Section 5.

17.

17.

80.

A chemical shift powder spectrum for ice measured using the REV-~-8
experiment was reported by L. M. Ryan, R. C. Wilson and B. C.
Gerstein, Chem. Phys. Lett. 52, 341 (1977).

A. Pines, D. J. Ruben, S. Vega and M. Mehring, Phys. Rev. Lett. 36,

110 (1976).

8l.

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87.

Table 3
PROTON CHEMICAL SHIFT COMPONENTS FOR SEVERAL SOLIDS>

9 nT oO oH - 6

aL 0 aL
Ice -17.5 11.2 -10.6 28.7
CeHio> -5.0 3.9 -2.7 8.9
C.Hig -4.3 4.1 -1.1 8.4
Polyethylene ~3.5 2.4 -0.5 5.9

“Chemical shift values are in ppm relative to TMS.

bau measurements were made using the 52-pulse cycle at
liquid Nog temperature with t = 3.0 usec and
1 kHz < Aw/2n < 4 kHz.

88.

Figure Captions

lL.

Apparent off resonance, Awt/2n, as a function of actual resonance
offset, Aw/2n, for the 52~pulse cycle applied to a water sample.
The solid line is a least squares fit to the data.

Demonstration of spectral broadening due to bulk susceptibility:
(a) spectrum from a polycrystalline sample of undoped CaF. ;

(b) spectrum from a spherical single crystal of the same material

oriented with H (100). Both spectra were obtained using the

52-pulse cycle with t = 2.8 usec and Aw/2n = 2.3 kHz.

Exponential decay time, T,", as a function of off resonance for
the 52-pulse cycle applied to a spherical CaF, crystal oriented

with H (100). The dashed curve is a theoretical fit for

Aw/2n < -2.0 kHz which varies as 2r/Aw.

Decay time, T, , as a function of CaF, crystal orientation for two

values of t, 2.0 usec and 7.0 usec, at two different values of

Aw/27, 1 kHz and 2.5 kHz.

Comparison of the resolving power of REV-8 (part a) and the 52-pulse
cycle (part b) for a single crystal of gypsum at one orientation.

For both spectra t = 2.8 usec. The scale is in ppm relative to TMS.

Comparison of the resolving power of REV-8 and the 52-pulse cycle

for a polycrystalline sample of cyclohexane at liquid N, temperature.

For both spectra tT = 3.0 usec. The scale is in ppm relative to TMS

and the chemical shift tensor components shown in the figure are

given in Table 3.

89.

Comparison of the resolving power of REV~8 (part A) and the 52-pulse
cycle (part B) for naturally abundant protons in polycrystalline
ice at liquid N, temperature. For both spectra t = 3.0 usec. The
scale is in ppm relative to TMS and the dashed line shows the liquid
resonance position. The chemical shift tensor components for

part B are given in Table 3.

90.

Aw/2m (kHz)

Fig. 1

91.

Fig. 2

92,

AQ

30

20

TS (msec)

10

CaF. (100)
T= 2.8 psec

| |

-2

-] 0 1
Aw/2n (kHz)

Fig. 3

Aw/2r (kHz)

1.0

2.5

1.0

2.5

93.

Fig. 4

| | | \ ] 1
| L | | | |
-60 -40 -20 20 40

Fig.

95.

52=-PULSE

Fig. 6

96.

Z °STa

97.

Section 5

A Chemical Shift Study of Gypsum, CaSO, °2H,0,
Using the 52-Pulse Cycle

(This section is essentially an article by D. P. Burum
and W. K. Rhim, "Proton NMR Study of Gypsum, CaSO,*2H,0,
Using an Improved Technique for Homonuclear Dipolar
Decoupling in Solids" which has been submitted for

Publication to The Journal of Magnetic Resonance. )

98.

This section presents the results of a room temperature study of
proton chemical shift anisotropy in gypsum, CaSO, °2H,0, carried out
using the 52-pulse sequence analyzed in section 4. Gypsum is of funda-
mental importance because of the simple, planar geometry of the water
molecules and the presence of near linear O-H***°O bonds. It is also
appropriate because studies by neutron and X-ray diffraction!’* have made
precise structural information available. Furthermore, the power of the
new sequence to decouple the homonuclear dipolar interaction is demonstrated
by its ability to produce well resolved chemical shift spectra for gypsum
even at crystal orientations where the dipolar splitting is as great as
22 gauss.> The only other study of chemical shifts in gypsum was reported
by McKnett et al. who applied the REV-8 cycle at restricted crystal
orientations, and their results are in substantial disagreement with this
section.

Two gypsum crystals were used in this work. One crystal was rotated
so that the external field remained in the plane perpendicular to the (010)
axis (crystal A) while the other was rotated such that the external field
swept out a plane containing the (010) axis (crystal B). The rotation
device allowed the angle to be set within 1°. ALL chemical shifts were
corrected for bulk susceptibility using a value of -74 x 10° (cgs units)
for the bulk susceptibility of gypsum and the tables of Osborn.” For this
purpose it was assumed that the samples were roughly ellipsoidal with the
following crystal axis ratios: for crystal A, (a,b,c) = (1.0,1.0,0.4);
for crystal B, (a,b,c) = (1.0,0.75,0.75).

Figure 1 shows part of the unit cell of gypsum. At room temperature

99.

: : 4
the protons in each water molecule rapidly exchange »6 so that only an
average tensor is observed, which can be characterized in its principal

coordinate frame by o and 0,,. Because of the high symmetry of

xx? Cyy 22.

the crystal, the 8 water molecules in the unit cell occupy only two
inequivalent positions in equal numbers, so that at room temperature one
expects to observe at most two lines.

The improved resolution of the 52-pulse sequence made it possible to
obtain complete and well defined chemical shift information for both
crystals. The chemical shifts as a function of crystal rotation angle,
along with theoretical fits to the data, are presented in figure 2 for
crystal A and in figure 3 for crystal B. As expected from the crystal
geometry, only one line was observed for crystal A, because the external
field remained in a plane of mirror symmetry for the water molecules. For
crystal B, comparison of the residual dipolar broadening of the two
spectral lines allowed the two chemical shift curves to be readily
assigned to the two inequivalent orientations of the water molecules.

Because all of the water molecules in gypsum lie in the plane con-
taining the (100) and (010) crystal axes, the Z-axis of the principal coor-
dinate frame can be taken to be perpendicular to this plane. With only this
assumption, it was determined from the theoretical fits to the data shown
in figures 2 and 3 that the X-axis makes an angle of 36.7° with the (010)
crystal axis, which agrees closely with the value of 38.8° obtained by
neutron diffraction’ for the proton-proton vector of the water molecule.
Of course, the Y-axis is perpendicular to the X-axis and lies in the plane

of the water molecules. The principal tensor values in this coordinate

100.

frame were determined to be Oxy = 0.1 + 0.5 ppm, Oy = -9.5 + 0.5 ppm

and Ong = -18.6 + 0.5 ppm relative to TMS. The angle between the water
molecule plane and the plane swept out by the external field in crystal
B was also determined from the theoretical fits shown in figures 2 and 3.
These chemical shift results can be compared with the study done by
McKnett et al.’. By using the REV-8 cycle, they were able to obtain data
corresponding to most of the points in figure 2. However, they were able
to obtain data corresponding to only a few of the points shown in figure 3.
They began by assuming that the Z-axis of the principal coordinate frame
was perpendicular to the water molecule plane and that the X-axis was along
the proton-proton vector in the water molecule, and obtained the following
results: O,..= ~10.9 + 1 ppm, Ovy = -5.9 + 1 ppm and On = -17.6 + 1 ppm

XX...
relative to TMS. The close agreement between the Ong values for the two
studies is not surprising, when one realizes that this value can be ob-
tained using only the data shown in figure 2. On the other hand, the reason
for the considerable descrepancy between the values obtained in the two
studies for Oxy and Ovy is not clear. It can only be conjectured that it

is due in part to the difficulty of determining values for Oyy and Oy
accurately and unambiguously when most of the information shown in figure 3
is not available, as was the case for McKnett et al.

If one is willing to make the additional assumptions that the tensors
for the two separate protons in the water molecule are equivalent and that
their principal axes are oriented with the X-axis along the O-H bond, the
Y-axis perpendicular to this in the plane of the water molecules and the

Z-axis perpendicular to this plane, then the tensor values given above can

be used to predict the tensor for an individual proton in the absence of

101.

exchange. Using the value 105.6° given by neutron diffraction! for the
H-O-H angle, the following principal tensor values are obtained:
o = 13.2 + 0.5 ppm, Oy = -22.5 + 0.5 ppm and O02 = -18.6 + 0.5 ppm
relative to TMS.

The two gypsum crystals used for this study were the same ones that

were used in the study made by McKnett et al.

102.

REFERENCES

1. M. Atoji and R. E. Rundle, J. Chem. Phys. 29, 1306 (1958).

2. W. F. Cole and C. J. Lancucki, Nature (London) Phys. Sci. 242,
104 (1973).
W. F. Cole and C. J. Lancucki, Acta Cryst. B30, 921 (1974).

3. 'G. E. Pake, J. Chem. Phys. 16, 327 (1948).

4. C. L. McKnett, C. R. Dybowski and R. W. Vaughan, J. Chem. Phys.
63, 4578 (1975).

5. J. A. Osborn, Phys. Rev. 67, 351 (1945).

6. D. C. Look and I. J. Lowe, J. Chem. Phys. 44, 2995 (1966).

103.

FIGURE CAPTIONS

I.

Portion of the unit cell of gypsum. The principal axis system for

a water molecule undergoing rapid proton exchange is indicated, and
a key to the nuclear species and the fractional coordinates (+ 0.05)
is given.

Chemical shift relative to TMS vs. crystal rotation angle for gypsum
crystal A. The crystal was rotated so that the external field
remained in the (010) plane. The solid line is the theoretical fit
to the data.

Chemical shifts relative to TMS vs. crystal rotation angle for gypsum
crystal B. The crystal was rotated so that the external field swept
out a plane containing the (010) axis and making an angle of 30.7°
with the plane containing the water molecules. ‘The solid lines show

the theoretical fit to the data.

104.

T ‘47a

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105.

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107.

Chapter III

Observation and Utilization of Thermodynamic Phenomena
in Strongly Time Dependent Interaction Frames

108.

Section 1

Introduction

109.

In this chapter, the thermodynamic relaxation of the nuclear spin system
due to coupling with the lattice is treated theoretically for several multiple
pulse experiments. The applicability of the thermodynamic spin temperature
hypothesis! in strongly time dependent interaction frames is demonstrated
through the development of several pulse techniques which yield results pre-
dictable by thermodynamic arguments. These new techniques are important
additions to NMR technology because they provide tremendously increased data
rates (> 10° or 10°) as compared to the cw techniques in use. This is because
the transient response of the system can be measured in between pulses through-
out the pulse train, whereas techniques involving cw irradiation only allow
the magnetization to be sampled at the end of the experiment. Thus the cw
experiment must be repeated many times, while the time development of the
system can be characterized by a single application of the multiple pulse
technique.

In Chapter II the time development of the nuclear spin system during
various multiple pulse experiments was predicted and interpreted through the
application of a coherent averaging theory based on the Magnus expansion
which was first applied to NMR by Evans.> This theory is very useful for
analyzing the behavior of the system under many circumstances, but it cannot
fully explain relaxation phenomena because it contains no information regarding
the thermodynamic coupling of the spin system with its environment. For
example, consider a train of identical, equally spaced (1/2), pulses applied
to a dipolar solid. Four pulses make a cycle for which, assuming 6-function

pulse shapes,

HO = — (1/2948? (1)

HE = 6 (2)

110.

This result is also obtained for continuous irradiation with the same average
amplitude. Therefore, one would expect similar behavior of the spin system

during either form of rf irradiation. In particular, since

764)
[1.9 l= 6 (3)

one would expect magnetization in the x-direction to remain "locked", i.e.

to decay very slowly, during both experiments. However, this is not what is
observed. Instead, the magnetization in the x direction decays much more
rapidly for the pulsed case than for the cw experiment. This effect is called
spin heating.

In the following sections the phenomenon of spin heating is shown to be
caused by an interaction between the first Fourier component of the pulse train
and the precession frequency of the magnetization in the rotating frame due to
the average rf field. It is shown that this effect can be made arbitrarily
small by reducing the neutation angle of the pulses. The thermodynamic
behavior of the spin system during these "small angle" experiments is demon-
strated by observing adiabatic demagnetization in the rotating frame (ADRF)
during an amplitude modulated rf pulse train. The decay time of the magne-
tization during pulsed spin locking is calculated, and this calculation is
verified by measurements performed on solid CoFe and CeF io: A pulsed version
of a method for determining the first moment of an NMR spectrum is introduced
and is used to measure shifts in the resonance positions of CaF, and BaF, as
a function of external pressure.

In the final section of this chapter, a single scan experiment for
measuring the spin-lattice relaxation time of a solid is introduced and compared
to the multiple scan method using a polycrystalline sample of frozen Coke:

This single scan technique utilizes the 2-pulse solid echo sequence” in

dil.

monitoring the time development: of the system, thus minimizing the loss of

magnitization during sampling. It is shown that the residual perturbation

of the system due to the measurement process can be considered a form of

thermodynamic spin heating, although caused by a different mechanism than

the spin heating discussed above. By measuring the strength of the spin

heating it is possible to obtain’ undistorted relaxation times using this

technique.

112.

References

1. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids,

Oxford Univ. Press (1970).
2. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954).
3. W. A. B. Evans, Ann. Phys. 48, 72 (1968).
4. See Chapter II, Section 2.
5. J. G. Powles and P. Mansfield, Phys. Lett. 2, 58 (1962).
J. G. Powles and J. H. Strange, Proc. Phys. Soc. 82, 6 (1963).

P, Mansfield, Phys. Rev. Al37, 961 (1965).

113.

~ Section 2

Elimination of Spin Heating in Multiple Pulse Experiments

(Most of the material in this section is drawn from the

following articles:

W. K.
Phys.

W. K.
Proc.

We. OK,
Phys.

Rhim, D. P. Burum and D. D. Elleman,
Rev. Lett. 37, 1764 (1976)

Rhim, D. P. Burum and D. D. Elleman,
XIXth Congress Ampere, 225 (1976)

Rhim, DB. P. Burum and D. D. Elleman,
Lett. 62A, 507 (1977).)

114.

This section is concerned with spin heating for the case in which a
dipolar solid is irradiated by a string of identical rf pulses near resonance.
In general the harmonics which are generated by such a pulse train will inter-
act with thespin system in a complicated way. However, if the fundamental
frequency Q is much larger than the average rf strength YH: resonance offset
frequency YAH and local field YAY oe? all the higher harmonics can be safely
neglected. The problem is then reduced to one in which the rf amplitude is
sinusoidally modulated. !

The Hamiltonian for this case is given by

H.=- bel -o VW
He. wr oO + Neos) I +H, 5 (1)
where W, = YH,, Aw is the resonance offset frequency and n in the modulation

1 1?

constant, which is x 1. The explicit time dependence of the Zeeman term in
eq. (1) can be transferred to the dipolar part if we move to an "oscillating

frame" by the transformation

nw,
U, = exp (i ll | sine (2)

The transformed Hamiltonian is given by

("a _ (2)
Horry = Awd 5 > I, - W I, tHE, + HE (t) (3)
- > z) a
= - OW. I +3 + HEAL) 5 (4)
where W. = AWS, >" +OD (5)

115.

rl

~~] 3

a| €
““_

ei

fe

ro)

et

Bb

eo)

cr

HAC) / (6)
+ 4J (ae - (GC, - G4)
fe) Q 2 2
2nw. ; .
1 i2kNt -i2kNt
4 > J | —) Leys + G_>5
and Gy, = (1/12) bd” ~#) ¥ i] (7)

In eq. (6), ©’ designates omission of the term for k=0 from the summation.

KH At) can be made arbitrarily small for large Q by reducing the pulse
angle, and thus reducing the arguments of the Bessel functions in eq. (6).

For instance, for 36° rf pulses, 2u,/2 = 2, Ji6-2) = .99, J, 62) = .1 and

the higher order Bessel functions are even smaller. A similar argument was
also given by Waugh and wang! in their analysis of the generalized Ostroff-
Waugh sequence, but they discussed only larger pulse angles. The major point
of this discussion is that H(t) can be made small enough to be treated by

perturbation theory even when the Zeeman term and HO in eq. (4)

are comparable in size. Fay in eq. (4) is
essentially identical in form to the continuous irradiation case if H(t) = QO.

For We < Wn” Ih the system obtains a common spin temperature within a time
of the order of Ty» »3 and the system can then be described by the spin

temperature Ts> with the semi-equilibrium density matrix given by
4 | 7 > 2?)
a — _- ° 43
Ony 1 KD On I (8)

3 : . :
The spin temperature hypothesis can thus be used to explain various experi-

ments performed during the time interval

; -1
tr, << t << |b (e)l (9)

116.

The above reasoning was tested experimentally using a CaF,
single crystal oriented with its (111) direction approximately along the
static magnetic field. Using discrete rf pulses, the spectrometer was
adjusted at exact resonance to produce a conventional Ostrof£-Waugh sequence >
(icee, (90°) Le (90°) q") with & = 10 gesec, and the decay time
constant Tie = 29.5 msec was observed, Under the same conditions, we
reduced the pulse angle 6. to (45°), the only effect of which was to
reduce the arguments of the Bessel functions in eq. (6). The result was a
considerable prolongation of Tie = 1.057 sec. Tie took on intermediate
values at other pulse angles between these two.

Now, suppose only the spacing between the pulses is changed.
Since 29, / 2. = 7O/% is independant of {l= #/®, increasing C1 will
simply increase the frequency of the sinusoidal oscillations in 6t),
resulting in a better averaging effect, Fig. 2 shows this effect for three
different pulse angles. The decay constant was recorded as a function of
average rf field; i.e., i, = 601/(2%Y), for three different values of
@. As can be seen in this figure, increasing Q has a very strong averaging
effect, increasingly so for smaller values of @. The saturation of The
above the 1 second level is caused by the spin lattice interaction, which
was confirmed by performing a separate cw locking experiment. Thus the
saturated Tie = Tie .

Having seen that Mie) can easily be made much smaller than the
inverse cross relaxation time between the Zeeman and dipolar reservoirs,
we can now test whether the system truly achieves a common spin temperature
‘after a few 8. pulses following the initiating (90°), pulse. If this is

the case, the magnetization mM of this semi-equilibrium state should obey

the formula

117.

yO
vA :
M =2,.2
fe) BAH) oc

(10)

which is identical in its form to the c.w. locking experiment with a 90°

prepulse, Here My is the equilibrium magnetization before the initiating

loc * re (M7?) 7/( ¥Tr(r2)). The actual

(90°), pulse is applied and H
variation of Hy was accomplished by changing the pulse angle as well as

the pulse spacing. The measured magnetization is shown in Fig. 3, together
with the theoretical curve from eq. (10) with Ho 1.1 gauss. Considering
the uncertainty in crystal orientation, this figure provides convincing
evidence that the spin temperature hypothesis is well justified for this
pulsed spin locking case.

A slow modulation can also be imposed on the values of o. in

the same pulse sequence, thus causing the spin system to follow Hy

isentropically, which is analogous to the c.w. case. By sampling the signal
between rf pulses, we observed the adiabatic demagnetization and inverse
remagnetization process almost instantly. Fig. 4a shows the slow amplitude
modulation of the rf burst, and Fig. 4b is the result obtained during such
an adiabatic process. In the remagnetization process the rf carrier

phase was changed by 180° compared to the demagnetization process, producing

inverse remagnetization. In this experiment the magnetization was propor-

ow 1
tional to B/G + ca) [2 as expected for an adiabatic process?

ADRF was also observed using pulse sequences for which 9 is gradually

varied past an integral multiple of 1.

ils.

Figure 5 shows the rf pulse sequence used. After the spin system reached
its thermal equilibrium polarization in a strong magnetic field, a GO. pulse
initiated the pulse sequence. This was followed by a string of near (nt)
pulses which were separated by a time t < T,- The amplitude of these pulses
was slowly varied in such a way that the pulses started with angles slightly
larger than nw and gradually decreased in angle to below nt.

The NMR signal from a single crystal of CaF, as observed between the

rf pulses is shown in Figure §. Starting from a Large initial polarization,
it decreased to zero as the pulse angle approached nt. The polarization then
continued to a negative value as the pulse angle decreased below nt. The
nutation angle of the pulses used to obtain Figure 6 was near 7. However,
a similar effect was also observed using pulses near 27.

The observed effect can be qualitatively explained using average

Hamiltonian theory. ° The Hamiltonian in the rotating frame is given by

Ae = - bul, +27, Pf. (t) (il)

where the first term is the off-resonance term, Af, is the secular part of
dipolar interaction and P(t) is due to the applied rf pulse sequence. Jf .(¢)

can be decomposed into two terms:

A) HP A Ow (12)

where oe P(e) is the term in which all the pulses are assumed to have nutation
(2) . ;
pulse angles exactly equal to nt and rf (t) is made up of the small differences
(1) . . . .
between. (t) and W . (t). By moving to an interaction representation,
assuming 6-function pulse shapes, we separate out the large time dependent term
a from the rest of the terms, and then calculate the average Hamiltonian

in this toggling frame. The average Hamiltonian so obtained is given by

119.

Par = - dol, - dw, 1, +, for n even (13)

- dwt, +, for n odd

where the off-resonance term disappears for n odd and du,

of AH , ) alone. In either case, the Hamiltonian has a form identical to that

is due to the average

which would be obtained if a continuous rf irradiation were applied. Therefore,
for Aw << do,; the nuclear polarization which was initially locked along the
effective field (YH eg z dw, x) can be adiabatically demagnetized as Au
gradually approaches zero and inversely repolarized as ‘Aw changes its sign.

This is exactly what is shown in Figure 6.

For solids with large dipolar interactions, the effect of n being even
or odd could not be easily distinguished, even though the off-resonance term
causes their effective Hamiltonians to be different. However, a definite distinc-
tion was easily observed in liquid samples due to the appearance of a rotary
spin echo! only when n was even.

The experiment was performed at slightly off-resonance using the pulse
sequence given in Figure 1 but without the initiating OD, pulse. In this case
the signal was observed along the x-direction instead of the y-direction. Since
rf inhomogeneity was present, we initially expected the signal to decay in an
ordinary way. In fact, for odd n, the signal decayed and no echo was observed.
However, when n was even, a strong rotary echo appeared. Figure 7 shows a

typical example of the echo phenomenon, where in this case n = 2. The explanation

for the appearance of an echo should be clear when one notes that the magneti-
zation changes its sign due to the adiabatic process, while the rf inhomogeneity

¥emains unchanged throughout the pulse string.

It is important to note that near nq pulse strings with In| > © provide

alternate methods of achieving the large values of ce which are required in

many 7) ) experiments.

120.

References

L. J. S. Waugh and C. H. Wang, Phys. Rev. 162, 209 (1967).

2. A. Abragam, The Principles of Nuclear Magnetism, Oxford Univ. Press
(1961).

3. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids,
Oxford Univ. Press (1970).

4. S. R. Hartman and E. L. Hahn, Phys. Rev. 128, 2042 (1962).

5. E. D. Ostroff and J. S. Waugh, Phys. Rev. Letters 16, 1097 (1966).
P. Mansfield and D. Ware, Phys. Lett. 22, 133 (1966).

6. Chapter TT, Section 2.

7. I. Solomon, Phys. Rev. Letr. 2, 301 (1959).

8. Section 3.

Figure 1.

Figure 2,

Figure 3.

Figure 4.

121.

FIGURE CAPTIONS

Decay curves for two different pulse angles. The pulse spacing

was the same for each and the decays showed good exponential be-
havior after a few T,'s. Note the large increase in decay constant
in going from 90° to 45°, The sample was a single crystal of

CaF, with its (111) axis approximately aligned along Ho:

The decay constant whe as a function of average rf field 4 for
three different values of pulse angle. The data points are smoothly
connected by dashed lines. A CaF, single crystal with its (111)

axis oriented approximately along Hy was used as the sample.

Initial semi-equilibrium values of the magnetization as a function
of average rf field strength. The dotted line is the theoretical
curve (eq. (10}),with Hoc = 1.1 gauss. Various Hy values were
achieved by changing both the spacing and the angles of the

pulses. A CaF, single crystal with its (111) axis oriented

approximately along H, was used as the sample.

Demonstration of adiabatic demagnetization in the rotating frame:
(a) the modulation envelope of the rf burst; (b) the corresponding
signal sampled between rf pulses. The maximum By a 9 gauaes and
the horizontal scale is § msec/cm. In the remagnetization
process the rf phase was changed by 180°, producing inverse

remagnetization.

122.

Figure Captions

Figure 5:

Figure 6:

Figure 7:

rf pulse sequence to observe ADRF using near nt pulses.

t = 20 usec, rf pulse width ~ 8 usec.

Adiabatic demagnetization and inverse repolarization process in a
CaF, crystal near the (111) direction. The maximum signal amplitude
is about half of the fid signal amplitude, and the maximum pulse
angle was ~ 210°, (Horizontal scale: 2 msec/cm).

Rotary spin echo signal observed in liquid CoFe when n = 2. The
pulse sequence in Figure 5, was used with the @, prepulse omitted

and the signal was detected along the x-direction. (Horizontal scale:

2 msec/cm).

NORMALIZED SIGNAL AMPLITUDE

0.6

0.5

0.4

123.

° PULSE ANGLE: 45°
PULSE SPACING: 20 usec

Np = 1057 msec

4 PULSE ANGLE; 90°
PULSE SPACING: 20 usec

Tip = 29.5 msec
| |
50 100 150 200 250 300
tin msec

Fig. 1

124,

PTT ET |

ome bbe eum ineenbiees amo amuw wm afiye mime

woe ee ce ee ee ee me

Lett tt

Pred dr |

SRST RY can) ees teeap

itt ff |

ITTY TT

LiEti | |

: 22,5°
: 60°
: 90°

It

peed

pat

90S Ul ef,

0.01

0.001

Ay in gauss

Fig. 2

125.

1.0 I r | |
S eee? pence genes ener eee eee
ao 8
—- o
a 5
Fe)
=! a0 |
—05b 7
} ° 36°
vO) .
a 6 » 23. 7°
bad
rs
fo)
fo)
oe | | |

Hy in gauss

Fig. 3

fe

126.

TT
—o
~—

Fig.

127.

G ‘Ota

NIVYL 3S 1Nd-A

THAT ASTAd -

fa
me wey
eens
ee eas

(4Lu)

tit}
rer

128.

Dell coheed

Sac adhd)
Le

fal
UY Oe

Bae!

pub Gg

Bh alco manll

o®

Fig. 6

| on

tot

ron)

130.

Section 3

Calculation of Spin-Lattice Relaxation During

Pulsed Spin Locking in Solids

(Most of this section is an article by W. K. Rhim, D. P. Burum

and D. D. Elleman, J. Chem. Phys. 68, 692 (1978)).

131.

A, INTRODUCTION

The calculation made in the previous section assumed that the string of
rf pulses was applied with a repetition rate, or fundamental frequency Q,
which was much greater than the average rf strength yay. the resonance offset
yAH and the local field YA oe? The more general case is analyzed in this
section. It is assumed that the direct spin heating effect can be neglected,
and a calculation is presented of the spin lattice relaxation time due to
molecular motion in a dipolar solid when it is irradiated by a train of
identical rf pulses with arbitrary nutation angle and finite width. From the
resulting formula it is easy to obtain the results which have been previously
derived for special cases using 1/2 pulses. ! For heteronuclear spin systems
the relaxation time has been calculated only for 6-function pulses of
arbitrary angle. Finally, experimental confirmation is given for a homo-
nuclear dipolar system (CEP 5) which establishes the validity of the general

relaxation time formula.

B. THEORY

In this sub-section, a calculation is made of the spin lattice relaxation

time, T of a dipolar solid when it is irradiated by a train of identical,

le’
rectangular x-pulses. Each of the pulses has nutation angle 6 and pulse width
ti as shown in figure 1, and the time between pulses, 21, is assumed to be

much shorter than Ty. The procedure for our calculation is essentially similar

to that of Grinder et al.

132.

At exact resonance, the Hamiltonian of the spin system in the rotating

frame is given by

Abt) = Ay (t) - 41, 8(t) (1)

where g(t) is the rf amplitude modulation function,which alternates between
1 and O for the pulse sequence under consideration, and Ay is the secular

part of the homo- and heteronuclear dipolar interactions given by

= T.-T. - Soo. 2
Af 6) > B,(Cy-t, - 3 1,1,,) tPCT a 24 (2)

The normalized signal along the x-axis is given by

-] ;
Tr fU(t) L, U(t) TL} (3)
I(t) =
x Tr {17}
where U(t) can be expressed using the time ordering operator, T, as
u(t) = Texp{-1 Pf{(e')at'} . (4)
ie)

By separating out the coherent rf part of U(t), one obtains

U(t) = Ug (OU, Ce) (5)
where t
Ug (t) = Texpltfwja(e')1,de"} (6)
and t
Dear ft) = texp(-if; pit’ de" (7)
with ~ 4
Af p60) = ULAR (EU. (e) (8)

Since Ug) commutes with Io. eq. (3) can be rewritten as
=]
Te {U, (te) LU (et) ©
= int x int xt (9)
* Tr {12}

By applying the Magnus expansion to eq.(7), one obtains

72)
Uy ae 6) = exp{-iQ F(t) (10)

133.

where t
F(t) = (Aye par, (1)
0 ; t tl ow ~
Fi(t) = = z Jeri feedA yep Ay(ey! (12)
etc.

Taking only the first term in eq.(10), and assuming a Gauss—Markoff
process, eq.(9) can be written as

(13)

Tr =
vx 2Tr{Iz}

where < > ay designates the ensemble average.

Calculation of the spin
lattice relaxation time, T

le? is therefore reduced to the calculation of

the exponent in eq.(13). For many specialized pulse sequences, such as the

ew or 6 =1/2 cases, the calculation is simple and straightforward. For

the more generalized pulse sequence, however, one must go through some tedious
algebra.

(1) Homonuclear Case

Assuming an exponential correlation function defined by

= p2 witis tel
Be Bys Ey? = Big ere} To , (14)

eq.(13) can be expressed for t a as

= exp{-t/T,,} .

(15)
The full expression for The in this case is given by
1 Mt
—— = —- {[ 2a, - sinh(2a,) + D sinh(2a)]
T 2 1 1
le 20
++ [B+ 2 sinh(2a,) - 2D(sinh(2¢) - w, 7 sin(26))
aS
Aly. . -
- ~5l sinh (2¢,) - D(sinh@a}- 20, 7,sin (26) } (16 33
A a
where Mw is the second moment of the homonuclear dipolar interaction, and

the following definitions are made for other parameters:

134.

a = t/t.
a) 3 t/t,
8 = t/t,
2.2
= +
A, 1 Swit.
cosh (2a, ) - 1
and d= cosh(2a) - cos(26) (17)
Eq. (16a) can be expressed more compactly in the following way:
le 2a +
(20,0)
- 3 [A_(sinhs - sinh2a + sinh2a, cos26)
A, (cosh2a ~ cos26)
- 4w,t, (cosh2o, - 1)sin2o]} (16

A result which agrees with these expressions has also been obtained by Vega

and Vaughan using the stochastic Liouville equation’,

From eq.(16) the formulas for the following special cases can be immediately

deduced, ie.

(i) For the Ostroff Waugh > sequence, in which 0 = 1/2, eq. (16) reduces to

II
l My T
— = {{2a - 8 + sinh® - (1 + cosh8)tanhal]
T 2
le 20

+ + [8 - 2sinh8 + 2(1 + cosh) tanha]

- 5 [1 + cosh8)tanha - sinhé]} . (18)

Ay

135.

This is identical to the result obtained by Griinder et al.! It can be further

simplified if we assume 6-function pulses:

L wil. ir, _ tanha
7.72 tab a (19)

(ii) Eq.(16) approaches the more familiar cw case as t-—~0, which implies

that 2a-—-8, ie.
II
1 Mz Te
= z
Te Lt hole (20)

4 5

This is identical to the result obtained by Jones” and Look and Lowe”,

(iii) For the case of 6-function pulses of angle 6, which we will find has a

wider applicability, Ai > +”, and The is given by the following:

a wt, (1 sinh(20) (1 - cos 29) }.

Tre 2 c¢ ~ 2a (cosh (2a) — cos (28)

(21)

Figure 2 is a graph of TMT as a function of T /tywhich was obtained from

eq. (16) for several values of ® and t/t The overall behavior of equation

(16) can be seen more clearly if we observe the minimum values of TM) and
the corresponding values of t/t as a Function of t/t, for different values of

9, as shown in Figures 3 and 4. From these figures we note the following

behavior:

(i) For t/t, < wl, The is essentially given by the formula for a 6=function
pulse train, equation (21). Also, notice that the minimum values for @ and 1-0

converge to the same value in this region.

(ii) For t/t, > 10, The can be approximated by equation (20), which is valid
in the cw rf irradiation case. In this region the values for 6 and n-6 are

distinctly different, as expected.

136.

(iii) For .1 < t/t < 10, a smooth transition takes place between the (i)
and (ii) cases. For smaller values of 6 we observe smaller changes between

these two extremes. At 6 = 1/6, for instance, the overall difference in
IT

(T, Mo°t) , between t /t, = .01 and t /t, = 100 is about 9.1 %, and for

le 2 min wo oid wood
8 = 1/9 the difference is only 4.3%, For 6 = 1/2, however, the difference is
17.4%.

The above observations clearly show the advantage of using small values
of 8. Namely, for small 0 the direct spin heating effect is greatly reduced,

as was realized in the previous section, while the spin lattice relaxation time,

closely approaches the value T which would be obtained in a continuous

The? Lo

rf spin locking experiment.

(2) Heteronuclear Case

When a heteronuclear dipolar interaction is involved, equation (2) can be

written

_ il Is
Hp- My + KD. (22)

The contributions of the homo- and heteronuclear parts of XK to are

separable, so that one can write

_ II Is
= > (23)

where the II and IS terms are obtained from equation (13) when Xp in equation

(8)is replaced by Kr and Ke, respectively. The result is that

1 1 1
Te ll + Ts . (24)
le le

Is

le in the 6-function pulse limit. Our

We have calculated the expression for T

result is:

sinh 20 (1 - cos 2)

~ 2afcosh 2a - cos 9)! . (25)

137.

If 0 = 0/2,
1 _ Is tanh 20
ag 7 Mp te FE - GG (26)
le
1,6

which again agrees with the result obtained by Grinder et al.”

Cc. EXPERIMENT

The experimental verification of equation (16) was carried out using
commerical grade perfluorocyclohexane (C.F, 4) with unknown impurity content.
Instead of the graphical representation shown in Figures 3 and 4, it was found
to be more convenient for experimental verification to re-plot equation (16)
as a function of 6 for a number of different values of t/t» as shown in
Figures 5 and 6. It is clear from these figures that (TM) an is periodic

with period + for the 6-function case, and gradually becomes a straight line as
it approaches the cw case. Also, in Figure 5 we observe a region of @ in

which the minimum value for the discrete pulse case exceeds the cw value.

The experimental points in these figures were closely fit to the theoretical
curves by assuming a value of .40 ¢? for wy This agrees reasonably well
with the value .44 ee obtained by Fratiello and Douglass/ who assumed trans-
lational diffusion in addition to the fast rotational motion of the molecules.
The values of t/t which correspond to the The minimum points shown in Figure

5 are plotted in Figure 6. These experimental points were closely fit to the
theoretical curve by assuming an activation energy of 12.35 Keal/mole. This

agrees well with the value of 12.5 Keal/mole obtained by Roeder and Douglass.”

Tt should be clear from this section and the previous section that the

spin-lattice relaxation time in the rotating frame during the application of
the pulse sequence shown in figure | will be equal to the relaxation time for
cw spin locking, Tio? if Oo /n << 1. As an example, figure 6 compares values
of Tyo for frozen CeF, obtained using the multiple pulse technique with

measurements made by Albert et al. using cw spin locking. The differences

between the Tho curves at low temperature agree with the thermodynamic pre~-
diction according to the difference in average spin locking field strength.

For example, the ratio of the two minima at 170°K is 1.82, which compares

well with the predicted value 2.0. For the pulsed experiment o = 30° and

2t = 15 usec. It should be noted that the two T, experiments yielded

equivalent data, as can be seen in figure 6, but the data rate for the

multiple pulse technique was roughly 10° greater than for the cw method.

138.

REFERENCES

W. Grinder, H. Schmiedel and D. Freude, Ann. Phys. 27, 409 (1971).

W. Grunder, Wiss. Zeit Karl-Marx Univ., Leipzig 23, 466 (1974).

A. J. Vega and R. W. Vaughan, J. Chem. Phys. 68, 1958 (1968).

E. D. Ostroff and J. S. Waugh, Phys. Rev. Lett. 16, 1097 (1966).

P. Mansfield and D. Ware, Phys. Lett. 22, 133 (1966).

G. P. Jones, Phys. Rev. 148, 332 (1966).

D. C. Look and I. J. Lowe, J. Chem. Phys. 44, 2995 (1966).

Apparently, there is a typographical error in Grunder's expression for
§ = w/2 in reference 1 (1974). Our equation (26) agrees with his
limiting cases for Tv, >t and T< T

A. Fratiello and D. C. Douglass, J. Chem. Phys. 41, 974 (1964).

S. B. W. Roeder and D. C. Douglass, J. Chem. Phys. 52, 5525 (1970).

S. Albert,H. S. Gutowski and J. A. Ripmeester, J. Chem. Phys. 56, 2844

(1972).

139.

FIGURE CAPTIONS

1.

The rf pulse sequence analyzed in this paper. The initiating (W/2).
pulse is followed by a string of identical o pulses.
Graphical representation of equation (16) as a function of ty for

different values of 6 and t /T..
x wool

(T wit) .. as a function of t /t, for different values of @
le2 “min Il L x
t/t corresponding to (TM T) in as a function of t/t, for different

values of 6.

Il
(Ty aM
Experimental values obtained using solid C

t)... as a function of 6 for different values of t /t,.
min x w idl
6 12 are also indicated.

Theoretical and experimental values of t/t corresponding to (T) 1

2 D min
as a function of o. for different values of t/t.
Th as a function of temperature for frozen CoFs: Data obtained using

the pulsed spin locking technique shown in figure 1 are compared with
results obtained by Albert et al. using cw spin locking. The average
strength of the locking field is indicated for each curve. For the

pulsed experiment 0 = 30° and 2t = 15 Usee.

T °4ta

<— liz —» M, <<

141.

OOL

@ “STA

ey

Ol

ie

TTT TT ttt

FETE FETT P| JET |

(La /My 49)

LLL H

LiLLL I

LLL LI

Hil; PT |

EEL LL

— L0°0

OL

Aad |

142.

OOL

€ ‘hrz

Ol

L“O

LO°O

TTT TT
Si fit
b/it

E/1LZ
p/Le

G/LG
6/1L8
-—81/u/l= 6

HPT I

Mtr 7]

Mite t |

WATE |

WELLL LJ

Udi tte LLL LLL

143.

p °STa

by
ool Ol [ LO L0°0
TT TTT TT
b/ ue
— C/ lt Z
C/1
— C/1L
b/u
_ AT —
6/i = 6 —
ee TD TO DT TPP

144,

g *STa

145,

g ST

tt

or Oo Owe ON = ©

ujw oly 4D Lf

Ty pe

msec

146.

[o]

T, °K
80 90 100 120 140 160 200 280

| | | | | _]

@® PRESENT DATA: 1.4G @

O ALBERT et. al.: 2.86 O

100.0 -— _
— ® >
= le) =
a Le _
= J] 6 _

/% fe)
10.0 | —
= r _
—_. e —
_ @ —
oi ;
© \e fe)
1.0K— \ O —
-_ Os =
— O —
— @ / _
5 im
0.1 | | | | | | | | |

13 #12 WW 0.9 8 7 é 5 4 3

1/T, 10°/ °K

Fig. 7

147.

Section 4

A Multiple Pulse Technique for Accurately Determining

the First Moment of an NMR Spectrum

(This section is essentially an article by D. P. Burum,
D. D. Elleman and W. K. Rhim, J. Chem. Phys. 68, 1164
(1978)).

148.

A. INTRODUCTION

It was demonstrated in sections 2 and 3 that the spin temperature assumption
can be applied to a dipolar solid when excited by a string of identical,
discrete rf pulses, and that the Hamiltonian of a system
irradiated by such a pulse train rapidly converges to the Hamiltonian

describing continuous rf irradiation when the condition

w,/2<1 ()
is met. Here wy = Hy = O02 is the average precession frequency in
27

the rotating frame due to the rf pulses applied with repetition frequency
Q/2n, and 9 is the pulse angle. Hence, it is possible in principle to
replace cw irradiation by a string of identical, small angle pulses

without changing the physics involved as long as condition (1) is satisfied -

This was demonstrated for adiabatic demagnetization in the rotating frame

(ADRF), sudden spin locking and Th, measurements. The advantage of

using pulsed irradiation is that the signal can be observed between pulses,
thus allowing the time development of the system to be monitored during

a single pulse train.

In this section the validity of the multiple pulse correspondence principle
is further demonstrated by applying it to a technique for accurately determining
the resonance point of a dipolar broadened NMR line. This "zero crossing"
technique has long been known in soiid state wur.t >? However, because it
utilizes cw rf irradiation, this conventional technique only allows the
signal to be observed at the end of a scan. By replacing this cw irradia-

tion by a multiple pulse train, a technique is developed which,

149.

because of its extremely high data rate, is typically several orders of
magnitude more sensitive than the corresponding cw technique. (The
accuracy which can be expected is usually better than + 0.5 ppm.) The
multiple pulse method is also easy to apply, and is relatively insensi-
tive to pulse errors and probe detuning, which makes it particularly well
suited for use under varying conditions of temperature and pressure.
This is in contrast to multiple pulse line narrowing techniques, which are
highly susceptible to these sources of error. 3

In the following sub-sections the theoretical expressions which char-
acterize the multiple pulse zero crossing technique in its various forms, as
shown in Figure 1 are discussed and compared. The theoretical ex-
pressions which describe the cw method also describe the multiple pulse

technique when condition (1) is satisfied, as is required by the multiple

pulse correspondence principle. The more general case of a polycrystalline

sample is also considered, and conditions are presented under which the first

moment of the chemical shift powder pattern for such a sample can be
determined. Experimental procedures are discussed next, including
several modifications of the multiple pulse sequence which are useful

for determining the baseline of every scan. This allows one to eliminate
errors due to baseline fluctuations and distortions. As a demonstration
of the technique, the 19, chemical shifts at ambient temperature and
pressure are reported for a number of compounds, and are compared with
literature values where available. Finally, chemical shift data for CaF,

and BaF, as a function of pressure up to 5 kbar are presented and compared

with the results obtained by Lau and Vaughan using multiple pulse line

narrowing techniques. 4

150.

Be THEORY

In this .sub-section two versions of the zero crossing technique are
discussed, each of which can be applied in either a cw or a multiple pulse
form. Consider a system of N spins in a static field Hy which are
in thermal contact with each other through the dipolar interaction, so that
a single spin temperature applies. Figure 1A illustrates the cw form of
what is called the "sudden" version of the zero crossing technique. In
this approach, a burst of cw irradiation of frequency w and strength Hy

lasting several T, s is applied suddenly to the sample, and the initial

amplitude, M> of the free induction decay which immediately follows is

recorded. This initial amplitude is given by?

M hH

x = 1
M 2 2 2 (2)
a) sudden h* + Hy + Hy .

Here M, is the equilibrium magnetization, and Hy is the local dipolar

field defined by
2. (z),2 2 2
Ho Tr} 0)” / fy Tr(t)"). (3)
The off resonance term, h, is given by
2 woe - -
h=WN dla, Srey) 7 wv (4)

where aoe is the chemical shift of the j'th spin. Note that (2), which is

2 :
1 + Hs crosses zero at “exact resonance," which

linear in h for h’ < #
can be defined as the point where h = 0. When condition (1) is met,

equation (2) also applies to the multiple pulse form of the "sudden" tech-

nique (Figure 1B) because of the multiple pulse correspondence principle

151.

outlined in the introduction. This was confirmed experimentally, and
the comparison between theory and experiment is shown in Figure 2. The
details of the experiment are discussed in a later sub-section.

From equations (2)-(4) we see that the "sudden" zero crossing
technique will determine the average value of oe for a system of spins
in thermal contact with one another. For the special case in which all
the spins in a single crystal are equivalent, both chemically and crys-
tallographically, the full chemical shift tensor can be determined by
measuring oo. for various crystal orientations. If the same technique is
applied to a polycrystalline sample, then each of the crystallites will

obey equations (2)-(4) separately. The resulting signal is given by

M, asl h,H,N,
M | sudden 2 2 2 (5)
[e) hs + "Ls + Hy

where N, is the number of spins in the i‘ tw crystallite. Of course, if
h is independent of crystallite orientation, as is the case for CaF,,
then (5) will cross zero at h = 0.

The zero crossing point of equation (5) can be expressed rather sim-
ply if the dependence on the subscript i in the denominator can be

neglected. This is the case if the local field, He is isotropic, and

if Hy can be made large enough so that, near the zero crossing point,

2 25,2
A + HL Phe (6)
for all values of i. In this case (5) crosses zero when
Ay doh,N, = 0. (7)
a “7 it
NCH, + HL )

152.

If equation (4) is rewritten as

-1
h, = Ny 2, (a, - "22, ,) - w/y]
- yl . (8)
= Ni » [H,, ~ o/y]
then (7) becomes
wi Svan, = Ww ST H,,) - w/y = 0 (9)
eel fe MiG y
i i,j
or
we wi > Hi, 7 YH). (10)
i,j

Thus the zero crossing point determines the first moment of the chemical
shift pattern. Condition (6) can be clarified by writing hy in yet

another form:

tt

-1
(H, - w/y) Ne 2. H. "ea, (11)

il

a ~ #H, Cre) a .

Near the zero crossing point, the extreme values of Ho {9 are on the

zz) i

same order of magnitude as the largest h,. Therefore, condition (6) can be

written

2 2 2
q + Hy > (Hy Caz) > . (12)

In most cases He is orientation dependent. Nevertheless, Hy can

cften be made large enough so that the anisotropy of A can be neglected.
If the average of the local field over all crystallites, Ho» is large
compared to the anisotropic part, hy , for all i, condition (12) can be

replaced by

153.

2 2 2 =
H+ Hy > (Hy (n2)4) + aH . (13)
When this condition is satisfied, equations (7) - (10) are again valid.

Therefore, if H, can be made large enough to satisfy (13), the sudden

zero crossing technique can be used to determine the first moment of a
chemical shift powder pattern. One should note that (13) is particularly
easy to satisfy for proton spectra, because of the relatively small values
of GC Si that are encountered.

azpi

The multiple pulse correspondence principle can also be applied to
the version of the zero crossing technique shown in Figure 1C. This
approach, which we call the "adiabatic" technique, was introduced by

Kunitomo “and was used to measure chemical shifts by Terao and Hashi.6

Initially, a low level rf field, H is suddenly turned on. After thermal

li’
equilibrium is established, the rf field is adiabatically increased to

a much larger final value, Hye? in order to rotate the magnetization onto
the x-axis. The cw irradiation is then turned off, and the initial
magnitude, M of the free induction decay which follows is observed.

If this technique is applied to a sample in which all the spins are in

thermal contact, such as a single crystal, the result is given by

hi
M ; lf
. ay) 2 2.5 1», 2 2 2.35
M, adiabatic (h* + a + Hy) (h*~ + H+ Hs ¢) . (14)

Of course, (14) also applies te the multiple pulse form shown in Figure 1D
if condition (1) is met. Note that the "sudden" technique is a special
case of this "adiabatic" method, so that substituting Hig = dag = Hy

in (14) yields (2).

The sensitivities of the sudden and adiabatic techniques can be

154.

compared by comparing the slopes of (2) and (14) at h = 0. The slope of

(2) is maximum when Hq, = Hs and is given by

3 M

ian . (15)
gh M

o {sudden

The slope of (14) at h=0 increases as Hay > 0 and Hie >o, The

maximum value is given by

8h \M, jJadiabatic | h=0 H, " (16)
Ai +0
te 7°

Thus, at most, the sensitivity of the adiabatic method can be twice the
maximum sensitivity of the sudden technique.

When the adiabatic technique is applied to a polycrystalline
sample, each of the crystallites obeys equation (14) separately. In
order to determine the first moment of a chemical shift powder pattern,
however, it is Hay that must be large enough to satisfy the condition
that was imposed on Hy in equation (13). In practice, this reduces the
advantage in sensitivity of the adiabatic technique over the sudden
technique to a factor of roughly /2. Therefore, there is little reason to
apply the more complicated adiabatic technique to a polycrystalline
sample.

The high data rate of the multiple pulse zero crossing technique
gives it an advantage in signal-to-noise ratio over the corresponding cw
technique on the order of (Tf T5) 4. Care should be taken, however, to

avoid errors caused by anisotropy in T For relatively pure, rigid

lo*

crystals T 0 is usually long. However, in many molecular crystals T

1 lo

155.

changes substantially with temperature. In such cases a temperature should

be chosen, if possible, at which Tio is relatively long, and T, is short,

in order to attain higher sensitivity. It is also important to use a

value of Hy for which the signal amplitude will be greatest, taking care
that condition (1) is satisfied by a suitable choice of pulse angle and
repetition rate. It should be pointed out that if condition (1) is not
satisfied a strong spin heating occurs which can cause the magnetization
to decay much more rapidly than it otherwise would.’ Nevertheless, as long
as this effect does not cause the decay time to be shorter than the time
needed to establish thermal equilibrium, the expressions presented in

this paper remain valid and, in particular, the zero crossing point is

not altered. Therefore, the zero crossing technique can be used without

misleading results even when condition (1) is not met, but considerable

sensitivity may be sacrificed.

C. EXPERIMENTAL PROCEDURE

The "sudden" multiple pulse technique (Figure 1B) was used to
obtain all the data reported in this paper because it was more convenient,
and because the increase of roughly v2 in signal-to-noise ratio which
could have been gained in some cases by using the adiabatic method was
not needed. The rf pulse angle was usually about 30°, and the pulse
spacing was 20 usec. This provided an HL of about 1 gauss, which was
roughly equal to Hy. for the samples we used. The magnetic field was
stabilized during the experiment by using an NMR locking system with a
separate probe.

The signal observed between pulses during the rf irradiation was

integrated in order to improve the signal-to-noise ratio. However,

156.

any error in baseline level was also amplified by this procedure. There-
fore, it was important to determine the baseline of every scan very
accurately, so that it could be subtracted from the signal before
integration. If the shape of the baseline was not distorted, then only
its de level was needed. This was automatically determined if Tho was

so short that the signal decayed to baseline level before the end of the
sean. If Tho was long, the signal was forced to decay abruptly near

the end of the scan either by inserting a gap several T, s wide in the

pulse train, as illustrated in Figure 3A, or by using a (/2)., pulse to
rotate the magnetization away from the locking direction, as shown in
Figure 3B,
If it was necessary to correct for baseline shape imperfections,
the results from a separate baseline scan were subtracted from the data.
This baseline scan was identical in all respects to the scan used to

. q
obtain the data, except that a (1/2), pulse was applied several T, s

earlier than the eo. pulse train in order to remove the initial magaeti-
zation.

The direct result of applying the pulse sequence illustrated in
Figure 3A at 56.5 MHz to a CaF, erystal 1KHz above resonance is shown
in Figure 4A, and in Figure 4B these data have been integrated. Figure 5
compares the results obtained by this same procedure at exact resonance
and at +50 Hz.

Sufficient accuracy was achieved for all the data reported in
this paper by simply varying the synthesizer frequency until the zero
point was obtained. A more accurate method, however, would ve to plot

several points above and below resonance and determine the zero crossing

point graphically.

157.

When the resonance points of various compounds were being compared,
either Tip or the method of Figure 3A was used to set the electronic
baseline, and the signal was observed without integration.

For the high pressure measurements, the apparatus previously
described by Lau and Vaughan was used without modification’ and the
two scan method was used to correct the baseline. In order to insure
that there were no detectable effects due to probe detuning or changes
in the hydraulic oil as a function of pressure, the resonance point of a
CaF, crystal which was kept at one atmosphere pressure within a glass

capsule was monitored as a function of hydraulic pressure. The zero

crossing point remained constant to within 0.1 ppm.

D. EXPERIMENTAL RESULTS

The experimental points shown in Figure 2 were obtained by applying
the "sudden" multiple pulse zero crossing technique with a pulse spacing
of 15 sec. and a pulse angle of 29° toa single crystal of CaF, oriented
with its (111) axis parallel to Ho: They are.in close agreement with

the theoretical curve, which was obtained from equation (2) with Hy = 1.0

gauss. Because equation (2) was originally derived from the cw technique

shown in Figure 1A, Figure 2 also demonstrates the validity of the multiple
pulse correspondence principle and the validity of the theruodynamic spin
temperature assumption.

Table I presents the results of 19, chemical shift measurements made

on a number of compounds at room temperature and pressure, along with
previously reported data where available. With the exception. of CdF,,

the data presented here agree very well with the literatur- tn parti-
cular, our value for KPF, is indistinguishable from that obtained by Sears!"
using Kunitomo's approach, and our value for LiPF, agrees closely with the
curve given by Sears for chemical shift as a function of cation radius

fe)
if the radius of Lat is taken to be 0.60 A. Single crystals of CaF

2?

158.

CdF, and BaF, were used, although powder samples would have yielded the
same results. The rest of the measurements were made on powder samples.
The error estimate is smaller for the three single crystals because the
data for these samples were integrated, whereas the data for the other
samples were not.

Figure 6 presents the results obtained in measuring the chemical
shifts of CaF, and BaF, as a function of applied pressure. The measured
value of -0.29 + .02 ppm/kbar for CaF, is in good agreement with the
theoretical prediction of -0.46 ppm/kbar given by Lau and Vaughan. 4
However, it is somewhat at variance with the experimental value of -1.7 +1
ppm/kbar which they obtained by using an 8-pulse cycle to remove the
dipolar boadening of the CaF, resonance line. Although line narrowing
techniques are very useful for determining the full structure of a
complicated resonance pattern, they are very susceptible to pulse errors

and probe detuning.?

Therefore, the multiple pulse zero crossing tech-
nique is often a better choice when it is only necessary to accurately
determine the first moment of a simple resonance spectrum, especially when

this is tc be done as a function of temperature or pressure.

Our result of -0.62 + .05 ppm/kbar for BaF, which is just over twice that

for CaF., igs consistent with the larger radius of the Bat? jon. The

error bars for the BaF, data are larger than those for CaF, because T)
was longer for the BaF, crystal. This allowed the long term electronic
instabilities of our system to become more noticeable in the BaF, case.
Also, the apparent hysteresis of the BaF, data at low pressures was

actually caused by a slight drift in the magnetic field near the end cf

the data run. These last few points were therefore ignored in computing

159.

the slope of the chemical shift vs. pressure curve.

Finally, note that the pressure data presented here for BaF, could
not have been obtained using a standard multiple pulse line narrowing
technique, because the pulse sequence would not have removed the hetero~

nuclear dipolar broadening of the BaF, resonance line.

160.

REFERENCES
i. I. Solomon and J. Ezratty, Phys. Rev. 127, 78 (1962).
2. M. Kunitomo, J. Phys. Soc. Japan 30, 1059 (1971).
3. W.-K. Rhim, D. D. Elleman, L. B. Schreiber and R. W. Vaughan, J. Chem.
Phys. 60, 4595 (1974).
4, K.~-F. Lau and R. W. Vaughan, J. Chem. Phys. 65, 4825 (1976).
5. A. Abragam, The Principles of Nuclear Magnetism, Oxford University
Press, (1961), Chapt. 12.
6. T. Terao and T. Hashi, J. Phys. Soc. Japan 36, 989 (1974).
7. Section 2.
8 R. W. Vaughan, D. D. Elleman, W.-K. Rhim and L. M. Stacey, J. Chem.
Phys. 57, 5383 (1972).
9. J. S. Waugh, L. M. Huber and U. Haeberlen, Phys. Rev. Lett. 20, 180
(1968).
10. LL.M. Akhutsky, Yu. V. Gararinsky and S. A. Polyshchuk, Spectry. Letters
2, 75 (1969).
ll. C. D. Cornwell, J. Chem. Phys. 44, 874 (1966).
12. Wang Yi-ch'iu, Dokl. Akad. Nauk SSSR 2, 317 (1961) [Soviet Phys. Doklady
6 39 (1961) 1].
13. F. I. Skripov and I-Chu Wang, Wu Li Hsueh Pao 20, 41 (1964).
14. R. E. Sears, J. Chem. Phys. 59, 973 (1973).

19

F CHEMICAL SHIFTS

TABL

161.

ET.

OF VARIOUS COMPOUNDS AT AMBIENT TEMPERATURE AND PRESSURET

COMPOUND o,, (ppm) PREVIOUSLY REPORTED DATA
NaF 62.0 #1,
KF ~36.6 +1.
CaF, -56.6 £0.5 -55.6 +0.5°
~61 t1 ?
“58 +2 ©
-53 +3 4
-64 ©
StF, -78. #1. -32 +1 ?
67 +3 4
CaF, 24.8 40.5 33.0 #1 °
39 «#8
BaF, ~153.5 40.5 -154 +1 ?
-138 +3 4
~108 +15 *
-113 +20 &
TiF, -13.1 41,
LiBF, -1.1 #1.
NaBF -3.2 f1,
K BF, ~13.1 41.
RbBF, -16.3 #1.
KSiF, -32.5 +1.
BaSiF, -57. £5,
LiPF, -74.2 +1. .
K PF, ~91.5 41. “91.3 41

+t All values are relative to
a.
b.

Ce

Reference 6
Reference 8

Reference 9

CoFe-

Reference 10

~~ fF Mm ®@ A

Reference 14

From Figure 1 of reference 1}
From Figure 1 of reference 12

From Figure 3 of reference 13

162,

FIGURE CAPTIONS

1.

The "sudden" (A and B), and "adiabatic" (C and D) versions of the
zero crossing technique in both their cw and multiple pulse forms.
Result of applying the sudden multiple pulse zero crossing technique
2 Single crystal with H) az 1.35 gauss. The

theoretical curve (equation 2) was fit to the data by assuming a

(Figure 1B) to a CaF

value of Hy = 1.0 gauss. The pulse angle was 29° and the spacing
between pulses was 15 usec. Part B is a magnification of part A
near the zero crossing point.

Two modifications of Figure 1B which can be used te determine the
baseline level of the scan. In part A, a gap in the pulse train

allows the magnetization to decay to zero in a few T In part 8,

a (n/2) pulse rotates the magnetization away from the spin locking
direction, so that MY = 0.

Result of applying the pulse sequence shown in Figure 3A to a

CaF, erystal 1 KHz above resonance. Part A shows the signal as it

was observed between pulses. Note the effectiveness of the baseline
determining method.In part B the data from part A have been integrated
for the purpose of signal averaging.

The procedure of Figure 4B applied at three frequencies near resonance:
(A) +50 Hz; (B) exact resonance; (Cc) -50 Hz.

19, chemical shift vs. pressure curves for CaF, and BaF. The slopes

of the curves are CaF: -0.29 £.02 ppm/kbar, and BaF: ~0.62 +.05 ppm/kbar.
Arrows pointing up or down indicate data taken as the pressure was

being raised or lowered, respectively. The apparent hysteresis in the

BaF, curve near zero pressure was caused by a drift in the magnetic field

near the end of the data run. The chemical shifts for each curve are

163.

given relative to the value at zero pressure. The shift between CaF,

and BaF, is not shown.

T “Sra

(a)

(D)
DiLvaviay

Ft

ix

St

164.

(a)

(Vv)
Naqans

Ix

381d Jd LINW

MS

165.

M 0.4
% o34 ¢ “Yes,
°e
ant Se
0.1 ~2
Zl | | | l |
| | | l T
-30, -20, -10, | 10, 20. 30.
et -0.1
| h (KHz)
ay
es @ [7 7002
¥, |
®, e + -0.3
o6, d
*ee® Log
mM. 0.06 ~
Mo /
0.04 — Sf
0.02 4 a
ss oe | 4 i |
i se rs 1
-0.4 -0,2 / 0,2 0.4
Ce)
/ I =0,02 h (KHz)
/ °
/ — =0,04
a“
- =0,06

Fig. 2

166.

eoeaeo

S@Ge080

<=

> 2
ae

o2
id

iin

=—~5

eGo
) 2.
Las

@2eea0e
>>

ia

oo

eT

tT

Fig. 3

167.

Tv
a:
ean
Sead

bry

weer

ty

J cd eared

vo

bd ead mad
a ee

vo
4 ceudbecand eal
en

1 ed nada
PER?
J han an

eeypeat

axsipane
LL
a on

~—

“—

wei

mii

a“

——

-—

extn

Perv ed "4 POT
ahi
Fig. 4

vor

ry

At

jf
UU G

tj

168.

‘i re ae!

Bel coel ee

f nd,

Rhee

Li

Fig. 5

LEii

Lf

it

169.

Q.
Q.
cy -!
<=
“n
ond
ud
QO
on
| i 1 |

0 1 2 3 4 5
PRESSURE, kbar

Fig. 6

170.

Section 5

A Single Scan Technique for Measuring

Spin-Lattice Relaxation Times in Solids

(This section is essentially an article by D. P. Burum,

D. D. Elleman and W. K. Rhim, Rev. Sci. Instrum. 49,
1169 (1978).)

i171.

A. INTRODUCTION

In the previous three sections it was demonstrated that thermodynamic
properties ofaspin system in a solid could be measured using one kind of
multiple pulse experiment, namely spin locking by a train of identical oY
pulses. In this section it is shown that thermodynamics can also be applied
in analyzing a pulsed experiment applicable to solids for measuring the spin-

lattice relaxation time in the laboratory frame, T in a single data scan.

1°
The only difference between the two thermodynamic spin-lattice relaxation
times, T

in the rotating frame and T, in the laboratory frame, is the

lo 1

difference in the strength of the spin locking fields. As was seen in section
3, Th involves spin locking of an applied rf field which is typically

between 0.5 and 10 gauss. On the other hand, T, is the characteristic relax-
ation time for magnetization locked along the Zeeman field, Ho» which can
easily range from 10 to 75 kilogauss.

The usual method for measuring T, is to use one of the following pulse

sequences: {(r/2) -T- (n/2) 3 or (Gn), -T- (1/2) }. In either case,
the first pulse perturbs the equilibrium state of the spin system, and the
second pulse allows one to observe the progress that the system has made
toward returning to equilibrium during the time interval T. Thus the exper-
iment must be repeated many times with various values of Tt in order to fully
characterize the time development of the magnetization.

The technique discussed in this section allows the magnetization of a
solid to be observed many times as it returns to its equilibrium state. It
is shown that the perturbation of the system by the measurement process can

be treated thermodynamically as a source of spin heating, which should not be

confused with the spin heating effect discussed in section 2. By measuring

172.

the strength of the spin heating, it is possible to extract Ty from the
relaxation curve during the experiment. The assumption is made throughout
this section that the entire system can be described at all times by a
common spin temperature.

There have been several other attempts to develop techniques for

; : : : . 1
measuring Ty in a single scan. For example, the z-restoring pulse sequence

{1 -Tt- L(m/2) = To (m), - t - (m/2)_- t! -13 has been particularly

successful form measuring Ty in liquids. this technique
procuceg- a vipsaasly undisterted spin-lattice degay curve by

utilizing Hahn's inhomogeneity echo to refocus the magnetization before restoring
it parallel to the He field. However, the single scan Ty measuring techniques
for solids have not been as successful in avoiding signal decay caused by the

sampling process. Some of these methods, such as the small angle pulse sequence

oe

(7-7 = (Q@-7'-) 3 8 <3) introduced by Look and Locker,2 can reduce this

effect only by sacrificing signal amplitude. Others, such as the "flip-flop"

sequence (7+ T = (=) -t- (2) - t’ -)_) proposed by Demco et al.,? attempt
2° 2° -x n

to avoid this signal decay by returning the magnetization to its alignment with

the Ho field immediately after sampling. However, the sampling window must be

larger than the recovery time of the receiver, and in solids a significant
loss of magnetization usually occurs during this time. None of the single
scan techniques for solids which have been proposed so far have made any
attempt to refocus the magnetization after sampling, in analogy with the

z-restoring technique for liquids.

173.

This section introduces a pulse sequence for single scan Ty
Measurements in solids (see Figure 1) which utilizes the so-called "solid
echo" #76 to refocus most of the magnetization which would otherwise be lost
during sampling. The effective spin heating is thereby considerably reduced,
allowing more sampling during a scan with a corresponding increase in sensi-

tivity. The residual spin heating which is caused by the incompleteness of

the solid echoes is taken into account in determining the actual value of T

2,7,8

by applying the analysis of Look and Locker. This analysis does not
require a detailed knowledge of the various sources of spin heating. Therefore, the
technique tends to be insensitive to spectrometer misadjustments since these
errors appear in the form of spin heating, and are automatically corrected
along with the spin heating due to incomplete echoes. The only exception is
misadjustment of the initiating m-pulse (if it is used), which causes an error
in the initial amplitude of the decay curve. The data rate of the technique
can be further increased by repeating the scans in a time much less than the
6 Ty required for the magnetization to fully recover.

In the following sections the theoretical expressions which describe
the solid echo and the method developed by Look and Locker for taking spin
heating into account are reviewed. The analysis of Look and Locker is then

extended to apply to scans of any length which are repeated at an arbitrary

rate. Experimental procedures and methods of data analysis are next discussed,

and various experimental results using a single crystal of CaF, are reported
which demonstrate our technique and compare it with more conventional tech-
niques. The applicability of our method to liquids is then illustrated using
CoFe. Finally, several other possible "echo" single scan Ty techniques for
solids are discussed, along with our reasons for concluding that the sequence

presented in this paper is the most advantageous.

174~.

B. THEORY

The dipolar "solid echo" which is utilized in this paper was
analyzed for the case of an isolated dipole pair by Powles and Mansfield *
and then for the more general case by Powles and Strange and by Mansfield &
The expression for the echo amplitude, Mos generated by the pulse sequence

(G), -~ T = Oy - Tt) applied to a sample with initial magnetization M, is

M {1 6 yy } 1)
° “4! ute (

ui

My

where
Me = 7, (ta, ‘*? * By”, (Hy, oar
ta, ‘*?, cH, ‘2, 1 JVr, (1,7)
By ~ 42; 444,73 Lahp
B ~ 2, Aas - 3 Leaky
and Ai; => + Pah -3 cos”, 5)/ 1,3 : (2)

It is clear from equation (1) that M, approaches M, very rapidly as T/T, ~ 0,
In fact, for small + the factor governing the incompleteness of the echo

will be???

4a, 4
(t /T, ).
The Ty measuring sequence which we wish to present in this paper is
illustrated in Figure 1. The scan begins with a 7 pre-pulse which inverts
the magnetization. (This pulse is omitted in some cases. See the discussion

in the Experimental Procedure section.) Then the relaxation of the spin

system is observed periodically using a group of three pulses which generates

175.

a solid echo and then returns the refocused magnetization to the z-axis.
In solids it is impossible to avoid a slight loss of magnetization due

to the last term in equation (1). However, this can be taken into account
in calculating T, by applying the analysis of Look and Locker, 7°7°8

Tf M and Mm are the magnetizations before and after the n'th sampling

pulse group, respectively, n= 0, 1, 2,..., then the loss of magnetization

during sampling can be described in terms of the invariance factor, X, by

ML o=Ml-x) . (3)

For all n, assuming exponential decay between samplings,

M td (MCX) = Mi exp (-T /T,) + Mog

ui

M_(1-X) exp (-1 /T,) + Mog (1 - exp (-t'/T,)), (4)

‘where Mog is the value of the magnetization when it is in thermal equilibrium
with the lattice. After many samplings, the magnetization will arrive at a

steady state value, M: which can be obtained from equation (4):
— - ~ a ~ - é
M. = MCL xX} exp (-T /T,) + Mag (1 = exp (-7T /T,)) . (5)
This yields

‘ é -1
M_= Mott - exp (-T /T, ditt - (1-X) exp (-T /T,)] . (6)

176.

Subtracting (5) from (4) gives

Miyy 7 My = (4, - ML) CL-X) exp (-1//T,) . (7)

From this recursion relation one obtains

M > M, = Qt, - M,)CC-X) exp (-1//T,) ry, (8)

Substituting (5) in (8) gives

it

eo

ii

(4, ~ M,) exp (+ t/T,) , (9)

where the last term expresses the fact that the observed decay shape will

be a single exponential with decay constant T,. Since t = n(r‘’ +27) = nt’,

this implies that
M,
exp (-7’/T,) = 1 - <4 ((1 ~ exp (-7’/T,)], (10)
oO
which yields
M,
a =. - ~ - é motte
T /T, = - gn [1 - (1 - exp (-t /T,,) mt (11)

Thus, the parameters which mist be determined experimentally in order
to solve for Ty are Ty Mand Mog: Note that equation (11) places no
restrictions on the initial state of the magnetization, Mo- As can be

seen from equation (9), the three parameters which are determined experi-

mentally are T,,, M, and M,- Therefore, it is necessary to perform the

177.

experiment in a way that will allow Mog to be determined from My and the
other measurable parameters. The most straightforward way of accomplishing
this is to allow the system to fully recover before the start of each scan,
so that M, =o Mog or + Mag depending on whether or not the initiating 1-pulse
is included.

It is possible to determine qT even if the measuring scans are being
repeated so frequently that the magnetization cannot fully recover in between,
as long as all the scans are identical and the time between the end of one scan and
the beginning of the next, T, is held constant. After the first few scans are applied,
the system will reach a steady state, and the value of Mo will be the same for all the
decay curves which are obtained thereafter. The approach of My to its steady state
value can be characterized by a time constant, T.. If each scan contains N sampling
groups and the initial magnetization of the j'th scan is mu? (j = 0, 1, 2,...);
then the time between uD and 4 St) will be Nr’ +. If we assume that the

initiating t-pulse is not being applied, and that

(1-X) >> exp (- Nr “/T,,) > (12)

(j+1) (}) which is similar to

then we obtain an equation relating Mo to M,

equation (4):
‘ 2 ée é
yt) - tu, 3 et /Ty Mm (1 - ooNt [Tay 4 en /Ty

+M (- en T/T1y

eq : (13)

If the steady state value of M, is given by 4, then by replacing uP
and 4 GD) by ra in equation (13) an expression can be obtained which

is similar to equation (5). Then, in analogy with equations (8) and (9)

178.

we obtain

am 6 M6 te7NT /TH -T/T 55

ce) ° Lo)
: 2
acm 6) . y ()y iG + D/T, (14)
° [o)
This yields
T, T
T = —+H (Nr +T) . (15)
o ONT T, + TT,

A similar analysis shows that (15) also applies when the initiating T-pulse
is included in each scan, Table 1 gives several representative examples
of the approach of 9? to its steady state value. As can be seen from
the Table, only a few scans are required to reach steady state when Nr’ >
Th and T 2 Ty.

Once the system has reached steady state, signal averaging can begin.
It is then possible to determine Ty, by considering the detailed behavior of

the magnetization. The magnetization at the end of each scan, M, will be

given by

M=M., (1-x)

where equation (9) has been used to expand Mya: By comparing equations
(8) and (9), we see that

aQexy eet /T1 ot /TH (17)

179.

which can be combined with (6) to give
& é
_ -t /Ty ot /Tyanl
Mo M4@ - @ )[l - e 7 ° (18)

At this point, the analysis must be divided into two cases. If the scan
is not initiated by a t-pulse, M, is given in terms of M by

Mo = (M My e + Mog . (19a)

If the tepulse is included, it imverts the sign, giving

“TITY Ly (19b)

M = -(M = Mag? e oq

Equations (1) and (16)-(19) can be combined to give the following results:

For the case of no tepulse,

é ; M
eo ‘Ty _ L-(G-e7 ‘Ty € i - evT/Tl 4
-Nr //T. -7'/Ty 7 /T
A. - G-e Hy(1 - e He Ly . (20a)

(1 - en /THy (4 . eft eo Nt /Ty et /T1y

if the rm-pulse is included,

: ; ML .
et Tye ae et” (Hy 7 fel te7t/T1 A]
fe)

é é & ~ é rf _
_a- eo Nt /Tyy 4 . en? /Ty eo” /T}) +2. Nr '/Ty e IT gy e t'/Ty)

Ge ot Taya ee T/T1 oN /TH 1/71)
(20b)

In either case the last term, A, is clearly a correction term, and equation

(20) can be solved by iteration. In all the practical examples we encountered

180.

(T= T,, Nr’ 2 @,) this iteration process converged to within .1% in less
1 H
than 6 cycles.

In the limiting case

Nr’ >> Ty > (21)

only one scan is required for Mo to reach steady state, and equations

(20a) and (20b) reduce to a common expression,

an! at M - ~ “7 Ty t'/T
eT Tere a-e7 MH 2 - e/T Lie sy, (22)
[M, | 1 -e77 /Ty

which agrees with the expression derived by Look and Locker, & Of course,
if T >> T,, then [M | = Mo and equation (22) reduces to equation (10). It

is interesting to note that under the added condition
T ‘/T <1, (23)

which is equivalent to assuming that the spin heating process is evenly
distributed over the entire scan, equation (10) reduces to a much simpler

form:

(24)

If, for example, M,_/M,, = 2, the fractional error generated by this dis-
tributed heating assumption is given by 1'/Ty. Therefore, when this level
of accuracy is acceptable, the analysis of the data can be greatly

simplified by using equation (24).

181.

c. EXPERIMENTAL PROCEDURE

In order to extract all of the parameters from a single scan which
are required for calculating Th» it is necessary to record the baseline
along with the exponential decay of the magnetization, This usually presents
no problems, because the baseline can be sampled between the triplet pulse
groups. Sometimes it is more convenient, however, to employ a two-scan
approach in order to avoid the need to measure the baseline directly. In
this scheme the magnetization is allowed to return to equilibrium before
the start of each scan. The first scan is applied as shown in Figure 1,
yielding

M(t) = GM, - ML) exp (-t/Ty) +M (25)

while in the second scan the initiating m-pulse is omitted, giving

M(t) = QL, - MD exp (-t/Ty) +M, (26)

One can then add and subtract the data from these two scans to obtain

M, = (- 2M) eT /TH 4 2 M, (27)
and
= -t/Ty
M= (2 M,,) e . (28)

By extracting the amplitudes and decay constants of these two curves, the
required parameters can be determined without directly measuring the elec-

tronic baseline level.

182.

Because of the design of the pulse spectrometer, it was con-
venient to direct the initial t-pulse along the -y direction. Also,
the recovery time of our receiver was about 8 wsec, so T was set to 10 wsec
and the signal was sampled near the peaks of the solid echoes, i.e., just
before the @) pulses,

For analyzing exponential decays, the exponential least squares fir? was
found to be much more satisfactory than the logarithmic fit which is
more commonly used. The logarithmic fit involves taking the logarithm of
each point and then extracting the slope and y-intercept from the resulting
Straight line. This method requires that the baseline, M,, be determined
very accurately before the logarithms of the points are calculated. Also,
taking the logarithms of the points in the exponential tail tends to amplify
the noise in that region tremendously, thereby reducing the accuracy of the
fit.

The least squares fit, on the other hand, requires no independent
determination of the baseline. In fact, it automatically yields the best
fit to the baseline as well as the amplitude and decay constant, and it
utilizes the full curve, Although the calculations for this least squares
fit are more complicated than those for the logarithmic method, and although
some iteration is involved in their solution, they are easily handled by a
minicomputer or programmable calculator, In our case the data were analyzed
by a DEC PDP 11/10 minicomputer interfaced to a Fabri-Tek 1070 Signal aver-

aging computer.

183.

D. EXPERIMENTAL ANALYSIS

In order to test the solid echo technique and compare it with other
single scan and multiple scan techniques, a number of measurements
of Ty were made at ambient temperature and pressure using single crystals
of uranium-doped CaF,.

The result of a typical T, measurement is shown in Figure 2. The

lower trace in the figure was obtained using the full sequence shown in
Figure 1, while the t-pulse was omitted for the upper trace.
Also, the magnetization was allowed to return to thermal equilibrium
before the start of each scan so that Mog could be obtained directly
from Mi:

The data shown in figure 2 are analyzed in Table 2. The Tt, values
shown in the first column of the table were calculated using equation (11),
while the results in the second column are from equation (24). A comparison
of the Ty calculations therefore measures the validity of the distributed

spin heating assumption. The ratio Mag /Mos is also listed in each case

as a measure of the strength of the spin heating.

A measurement of qT, was also made using the inversion~recovery multiple
scan technique under conditions identical with those of Table 2. The result
obtained, T,=2-48 sec., is indistinguishable from the results in the table.
Also, the flip-flop method was compared with our single scan technique. The
results of the two methods agreed well, but the uncertainty for the flip-flop
method was greater by a factor of 3, and the spin-heating ratio was larger by a
factor of 3.6. This illustrates the clear advantage of our technique over
the flip-flop method.

The effect of varying 1t' in our solid echo pulse sequence is shown in

Table 3. There is good agreement between all the T, values in the table, and the

ratios Meq/Mo show the decreasing trend of the spin heating with increasing _'.

The two scan method was used to obtain all the data in the table.

184.

The validity of equation (20) for determining Ty when the scans are
repeated rapidly was demonstrated experimentally with and without the initiating
w-pulse for values of Nt?/Ty between 2 and 7 and T/T, between 1 and 6.

(The CaF, single crystal which was used to obtain the data in Tables 2 and 3
was also used in this case, but the crystal orientation was different than

it was for either table.) The result, T,_=2.35 sec, was obtained consistently,

but the repeatability of the measurements was somewhat reduced for the smaller
values of Nr ?/Ty and T/T, because in these cases the noise led to greater
errors in exponential curve fitting. In order to characterize the repeatability
which can be expected when applying our experimental procedure, over 5,000
computer simulated decay curves were analyzed for a wide range of noise levels
and other experimental conditions. The results are discussed in sub-section F
along with formulae and characteristic curves which relate the experimental
parameters to the expected uncertainty of the results.

As a more practical test of our method, Ty was measured for a different
CaF, single crystal with a relatively long spin lattice relaxation time.
A single scan with an initiating t-pulse and with 1* = 5 sec was used.
The resulting yalues of Ty = 107.3 sec and Mag! Mo = 1.75 were found to be
repeatable within 1.0%. Only about 20 min were required to obtain a 250 point
decay curve using our method, while a curve of only 50 points using a multiple
scan technique such as inversion~recovery would have required almost 9 hours.

The solid echo method was also used to measure T for powder samples of

10's and CF io: The repeatability

of these measurements and the spin heating ratios were found to be comparable

several other solids, including NaF, KBF » C

to the CaF, case.

Figure 3 compares Ty measurements for frozen CoP. made using the

technique described in this section with measurements made by Boden et ai, 19

185,

using the (1/2). -TtT- (n/2) } sequence. The differences between the two
curves agree well with the thermodynamic prediction according to the Zeeman
field strengths. For example, the ratios of the qT) values for the two curves
at the minima are 1.13 and 1.14, which can be compared with the predicted
value 1.06. Notice also that the overall shapes of these curves are the

same as the shapes of the T curves for CoP, given in figure 6 of section 3,

ip

except that the T, curves are shifted considerably toward higher temperature.

This is expected, since the Zeeman field Hy is much stronger than the rf spin-~
locking field, Hy» used in measuring Tip
Although the technique described in this section was designed for use
with solid samples, it is also applicable to liquids. This is illustrated in
Figure 4, which was obtained using CoFes As in Figure 2, the lower trace was
generated using an initiating 1-pulse while in the upper trace this puise was
omitted, and the magnetization was allowed to recover completely before each
scan. The value of qT, obtained from the
lower trace is 1.88 sec, and the spin heating ratio, Mag! Me = 1,09, shows
that there was almost no spin heating. Even thoughthe pulse sequence does
not refocus the magnetization in the liquid case, there is still very little
spin heating because, for liquids, t can be made much smaller than T,.
When our technique is applied to liquids, however, all of the effects due to
spectrometer misadjustments will not be in the form of spin heating, as they
are in the case of solids. Therefore, accurate determination of qT, in the
liquid case wequires more careful tuning of the spectrometer. There are
methods which haye a slight adyantage over our technique in the liquid
case, such as the z-restoring sequence, which was discussed in the Introduction,

but they are restricted more or less to liquids. Our technique eliminates

the need to use separate pulse sequences for solids and liquids.

186.

E. OTHER POSSIBLE SEQUENCES

Several other pulse sequences were considered as possible alternatives
to the solid echo technique presented in this section. One idea was to use a
“magic” echo Ilhan place of the solid echo. Such a sequence might be expected
to produce even less spin heating than the solid echo method, because the
"magic" echo causes a more complete refocussing of the magnetization, especially
when the initial free induction decay and its echo are separated by more than

T However, the solid echo is nearly as efficient as the magic echo for the

2°
small @ which are used in the Ty sequence. Also, the magic echo involves a
more complicated pulse sequence, which means that a magic echo qT, sequence

would be more complicated to produce than the solid echo sequence, and would

be more sensitive to pulse imperfections.

Another possibility was to use a single cycle of either the 4- or 8-

12

pulse line narrowing sequence as the sampling pulse group. However, it

was found that these Ty sequences are more sensitive to resonance misadjustment

than the present method. In one experiment the 4-pulse T, sequence showed

a strong systematic error in T, at 1 KHz off resonance and the 8-pulse T

1 1

sequence showed a similar effect at 2 KHz while the present solid echo technique
gave accurate T, results under the same conditions, although the resonance
misadjustments caused some increases in spin heating.

Ty measuring sequences which utilize the 4- or 8-pulse cycle may be more
convenient to produce on some spectrometers which have already implemented
these cycles for measuring the high resolution chemical shift spectra of

solids. Otherwise, the basic solid echo technique introduced in this paper

is clearly a better choice.

187.

F. REPEATABILITY OF MEASUREMENTS

In order to determine the repeatability which can be expected from the exper-
imental procedure outlined in this section, over 5,000 computer generated decay
curves representing a wide range of experimental conditions were analyzed. In
order to simulate the results of an experimental measurement, M, was
obtained in each case from equation (18) and MS was calculated using
equation (20).- For convenience, it was assumed that T,, = 1/2 Ty. Random

noise with a gaussian envelope of the form

f(r) =exp(-$ r’/a’) (29)

was superimposed on each decay curve, and the simulated spectrum was
analyzed using the exponential least squares fit as described in the
Experimental Procedure section, Equation (20) was then solved by
iteration to obtain T)>

It was found that y, the root-mean-square of the error, in percent,
of the T determinations, was directly proportional to the inverse noise
ratio, x, defined by

X= a/M, (30)

where a is the standard deviation of the noise as defined in equation (29),

for any given values of Nt“ /T,, and T/T, : i.e.,

y/x=B : (31)

188,

Furthermore, the dependence of B on Nt°/Tiy and T/T, was found to

be of a simple, exponential form:

e Nt Ty + C. et/Ty + C,

-Nt“/T,, el/t (32)

BeC, +¢C

1+ & 1

where c,-C, are constants. Thus it is possible to characterize the dependence

of the RMS error in qT on the noise ratio under all experimental conditions

by specifying the four constants C, through C The values which were extracted

1 4°
from the simulated data are given in Table 4 for both when the 7f-pulse is included
and when it is not.

The characteristic curves in Figures 5 and 6, which were generated using

equations (29) - (32) and Table 4, can be very useful for selecting experimental

parameters which will provide the required accuracy in T For example, assume

that the noise ratio (1/x) is 100, and it is required that Ty for a
certain sample be determined within 2%. Then y/x=200, and it can be seen
from Figure 5 that if the initiating 1™-pulse is included in the measuring
sequence, and if the signal is allowed to decay for 2 time constants

(Nt /T,, = 2), then the time between scans need only be about 2T,-
However, it can be seen from Figure @ that if the initiating npulse is

not included, then a measurement using these same parameters (T/T, =2,

Nt“/T,=2) will produce data which areonly accurate within 10% (y/x=1000).

189.

REFERENCES
Ll. A. Csaki and G. Bene, Compt. Rend. Acad. Sei. 251, 228 (1960).

10.

i.

12.

R. L. Streever and H. Y. Carr, Phys. Rev. 121, 20 (1961).
D. C. Look and D. R. Locker, J. Chem. Phys. 50, 2269, (1969).
D. E. Demco and V. Simplaceanu, Rev. Roum. Phys. 18, 623 (1973).

D. E. Demco, V. Simplaceanu and TI. Ursu, J. Magn. Res. 15, 166
(1974).

J. G. Powles and P. Mansfield, Phys. Lett. 2, 58 (1962).

J. G. Powles and J. H. Strange, Proc. Phys. Soc. 82, 6 (1963).
P. Mansfield, Phys. Rev. A 137, 961 (1965).

D. C. Look and D. R. Locker, Phys. Rev. Lett. 20, 987, (1968).
D. C. Look and D. R. Locker, Rev. Sci. Instrum. 41, 250 (1970).

M. Sass and D. Ziessow, J. Magn. Res. 25, 263 (1977).

N. Boden, P. P. Davis, C. H. Stam and G. A. Wesslink, Mol. Phys.
25, 87 (1973).

W. K. Rhim, A. Pines and J. S. Waugh, Phys. Rev. Lett. 25, 218
(1970.

W. K. Rhim, A. Pines and J. W. Waugh, Phys. Rev. B 3, 684 (1971).
U. Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968).

W. K. Rhim, D. D. Elleman and R. W. Vaughan, J. Chem. Phys. 59,

3740 (1973).

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TABLE IV

CONSTANTS FOR EQUATION (32)

With t-pulse Without t-pulse
24, 100.
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30690. 32,700.

194.

FIGURE CAPTIONS

1.

The solid echo pulse sequence for single scan qT, measurements in
solids. The initiating t-pulse may be applied along any direction,

and is sometimes omitted.

Spin~Lattice decay curves obtained by applying the pulse sequence

of Figure 1 to a CaF, single crystal. For the upper trace the

initiating t~pulse was omitted. The data rate was 80 msec./point.

Ty as a function of temperature for frozen CoFe: Results obtained using the
solid echo technique are compared with measurements made by Boden et al. using

a multiple scan method. The strength of the Zeeman field is indicated for

each curve.

Spin-Lattice decay curves obtained from a liquid sample (CoFe)

by applying the pulse sequence of Figure 1. The initiating
w~pulse was omitted for the upper trace. The data rate was

40 msec. /point.

The anticipated RMS error in qT due to inaccurate exponential curve
fitting. It is assumed that the initiating 1-pulse is included

in the pulse sequence, and that the scans are of length Nt’ and
are repeated with separation time T. It is also assumed that the
noise is gaussian with standard deviation = a, and that the non-linear
least squares method is used to fit the data. x= a/Mog? and y is
the root-mean~square of the error in the experimentally determined

value of Tt:

195.

The anticipated RMS error in Ty due to inaccurate exponential curve
fitting. It is assumed that the initiating t~pulse is not included
in the pulse sequence, and that the scans are of length Nt and are
repeated with separation time T. It is also assumed that the noise
is gaussian with standard deviation = a, and that the non-linear
least squares method is used to fit the data. x= aM? and y is
the root-mean-square of the error in the experimentally determined

value of Th:

196.

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202.

APPENDIX

A SIMPLE, HIGHLY FLEXIBLE PULSE SEQUENCE GENERATOR

203.

This appendix describes a simple yet highly flexible pulse sequence
generator which was entirely designed a build in the laboratory using
readily available components. Not only is this pulse generator a useful
instrument, it also demonstrates that the seemingly complex experiments de-
scribed inchapters 2 and 3canbe easily implemented using current technology.

A block diagram of the puise sequence generator is shown in
Figure 1. Although this device is essentially a special purpose computer,
it does not utilize a micro-processor. Instead, it was found that a
faster and more efficient instrument could be constructed using a high
speed 256 x 16 bit memory and simple logic circuitry composed of TTL com-
patible gates, counters and multiplexers. A wide variety of experiments
can be performed using this device, including single and double resonance
experiments involving both cw and pulsed rf irradiation.

The pulse sequence generator can be controlled either from the front
panel or via its interface to a PDP 11/10 computer. It has three types
of output: Five primary outputs Xl, Yl, -Xl, -Yl and Zl; five secondary
outputs X2, Y2, -X2, -Y2 and 22; and eight triggering outputs A-H. One
primary output, one secondary output and any number of trigger outputs
can be altered simultaneously. For example, alwusec. pulse might appear
at Xl at the same time that Y2 is turned on and trigger pulses appear at
outputs A, E and H.

The block diagram of the pulse sequence generator given in figure 1
illustrates its basic operation. Programmed instructions are loaded into
the 256 word x 16 bit (RAM)-Memory either from the front panel or from the

PDP 11/10 computer. The instruction cycle time, and thus the minimum time

204.

between adjacent pulses, is 1.2 usec. After being loaded, the pulse
generator operates independently of the PDP 11/10. The Memory is
addressed by a Program Counter which can be preset with a value from
the Data Bus. The Program Counter is advanced by pulses from the Master
Clock when the P.C. Flip-Flop enables the buffer between them.

Tables 1 and 2 summarize the pulse program instruction set. All
timing is controlled by Function 1 instructions. When one of these
instructions is executed a signal from the Output Multiplexer shown in
figure 1 causes the value of T given in bits 0-7 of the Data Bus to be
loaded into the Tt-Counter, which immediately begins counting pulses from
the Master Clock. Simultaneously, the P.C. Flip-Flop is reset by the
Output Multiplexer and the Master Clock begins advancing the Program
Counter. Instructions in memory are executed until the next Function 1
appears on the Data Bus, at which point the Command Decoder sets the
P.C. Flip-Flop and halts the Program Counter. Meanwhile, the t-Counter
continues to measure the elapsed time since the previous Function 1
instruction was executed. When it has received the specified number of
clock pulses, the t-Counter sends a Puise Enable signal to the Output
Multiplexer. The Function 1 instruction currently being addressed is
still not executed until the Output Multiplexer subsequently receives a
Pulse Fire signal from the Master Clock. If the Function 1 command does
not require an external trigger, this will occur with the next Master
Clock pulse. If a trigger is required, the Command Decoder will have
set the Trigger Flip-Flop, isolating the Master Clock from the Output
Multiplexer. In this case, the Function | instruction is not executed

until an external trigger pulse resets the Trigger Flip-Flop and allows

205.

the Pulse Fire signal to be transmitted to the Output Multiplexer. In
either case all output is synchronized with the Master Clock. Note that
the value of t specified in a Function 1 command determines the time
separating the execution of that command from the execution of the next
Function 1, assuming that the second Function 1 does not require an
external trigger.

All the other instructions shown in Table 1 are executed as rapidly
as possible, the required time per instruction being 1.2 usec. When a
Function 2 command appears on the Data Bus, the Command Decoder causes
bits 0-11 to be loaded into a latch in the Output Multiplexer. The
appropriate secondary and trigger outputs then occur simultaneously with
the next Function 1 execution.

Conditional program loops are generated using the Loop Counter.

The number of loops is preset using the SET N command. When this command
appears on the Data Bus, the Command Decoder causes bits 0-11 to be
loaded into the Loop Counter. Whenever a LOOP command subsequently
appears the Command Decoder signals the Loop Counter and also causes the
Program Counter to jump to the address specified by bits 0-7 in the LOOP
command. After the specified number of loops the Loop Counter disables
the "Load" input of the Program Counter, andthe next LOOP instruction is
effectively ignored.

A BRANCH command of course has the same effect as a LOOP command,
except that it does not interact with the loop counter and therefore is
unconditional.

Table 2 summarizes the function codes for bits 8-11 of Function 1

and Function 2 commands. Note that output pulse lengths can be controlled

206.

in two ways. For a "pulse" command such as X PULSE, the output pulse
width is controlled by a one-shot and a timing potentiometer on the
front panel. Alternatively, the pulse width can be set under program
control by using separate X ON and X OFF commands. Of course, the pulse
length must then be longer than the 1.2 usec instruction cycle time.
The use of a 10 MHz clock (Figure 1) allows values of t to be specified
in multiples of 0.1 usec.

A pulse programming example is given in Table 3, which presents a
set of instructions to generate the REV-8 pulse cycle shown in Figure 1
of chapter 2, section 3. The first command sets the loop counter to
cause 1024 loops. The next two instructions cause the device to halt
and wait for an external trigger pulse. When the pulse generator is
triggered, a pulse is produced at output A. This can be used to trigger
an oscilloscope or other data recording instruments. After 10 usec. an
X pulse is generated as a pre-pulse, and simultaneously a triggering
pulse appears at output B to indicate a sampling window. Instructions
5-13 (octal) proceed to generate all but the last pulse in the REV-8
cycle. This final pulse is generated by looping back to instruction 3,
so that the X pre-pulse becomes part of the cycle. Thus the following
pulse sequence is generated by the series of instructions given in Table 3:
A- tT -[(X + B)- 2T-X- T-Y-2T-Y-T-X- 27

-X-t-Y-2t-Y-t-] (1)

1024

where T' = 10.0 Usec., T = 3.0 Usec. and X =- X, etc. The final
instruction in the pulse program causes the entire procedure to repeat,

so that the next external trigger pulse will initate another 1024 cycles.

In this example all pulse lengths are set by means of the front panel

one-shot controls.

207.

Table 1

INSTRUCTION SET

INSTRUCTION BINARY CODE
15 14 13 12 11109 8 76543214
FUNCTION 1 6 § @ @ FUNCTION T.

(NO TRIGGER)

FUNCTION 1:
TRIGGERED

FUNCTION 2

SET N
LOOP
BRANCH

CODE?

6 6 @ 1 FUNCTION
CODE?

1 1 FUNCTION
CODE?

aan

HGFEDCBA

es ts
Sass a

® NOT USED
1 NOT USED

Soar

+——-—- ADDRESS -——>

ADDRESS —>

3+ See Table 2.

208.

Table 2

FUNCTION CODES

FUNCTION FUNCTION
CODE
No op. GOODG
X PULSE 64661
Y PULSE 6d16
-X PULSE OGi1t
-Y PULSE 9199
Z PULSE O1G1
Z ON 61146
Z OFF d1iii
X ON 14666
X OFF iddl
Y ON igid
Y OFF 1é11
~X ON 1166
-X OFF 1161
-Y ON 1116
-~Y OFF l1ii1il

ADDRESS (OCTAL)

10
il
12
13
14

15

Program

209.

Table 3

Example: REV-8

Instruction
SET N = 1024,
FN. 2 (NO OP) A
FN. 1: TRIGGERED (NO OP) t = 10.0
FN. 2 (NO OP) B
FN. 1 (X PULSE) Tt = 6.0
FN. 1 (X PULSE) t = 3.0
FN. 1 (Y PULSE) t = 6.0
FN. 1 (-Y PULSE) T= 3.0
FN. 1 (-X PULSE) T= 6.0
FN. 1 (-X PULSE) T= 3.0
FN. 1 (Y¥ PULSE) T= 6.0
FN. 1 (-Y PULSE) t= 3.0
LOOP TO 3

BRANCH TO 0

210.

Figure Caption

Fig. 1. Block diagram of the pulse generator.

211.

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