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THESIS
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MICHIGAN STATE UNIVERSITY LIBRARIES ‘

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This is to certify that the

dissertation entitled

NMR Studies of Dopants in Fluorapatite

presented by

Liam B. Moran

has been accepted towards fulfillment
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Ph.D. degree in Chemistry

 

 

 

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MSU I. An Affirmative Action/Emil Opporhmlly IWIOH
‘ W

m1

NMR STUDIES OF DOPANTS IN FLUORAPATITE

by

Liam B. Moran

A DISSERTATION

Submitted to Michigan State University in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1 996

ABSTRACT
NMR STUDIES OF DOPANTS IN FLUORAPATITE
BY

Liam B. Moran.

Phosphors based on calcium fluorapatite [(Ca5F(PO4)3] doped with small
amounts of Sb3+ as an activator are used in most fluorescent lamps. We have
used quantitative 19F and 31 P MAS-NMR to study a series of samples of calcium
fluorapatite containing 0.0 to 3.0 wt. % Sb3+ in order to determine the site of
antimony substitution. There were no peaks in 31 P MAS-NMR spectra of
fluorapatite related to antimony. The 31 P MAS-NMR spectra of the model
compounds SbPO4, Sr1.o3Ca3.g7F2(PO4)5, Sr5F(PO4)3 and Ba5F(PO4)3 were
also obtained. The 19F MAS-NMR spectra of the antimony doped samples exhibit
two peaks associated with antimony at 65.6 and 68.6 ppm. The SPARTAN and
SINK pulse sequences were used to measure spin diffusion between the antimony
related fluorine spins and those of unperturbed apatitic fluorine nUclei. These results
demonstrate that each Sb3+ ion is perturbing fluoride ions in at least two different
chains. An additional peak at 73.1 ppm observed in some samples is assigned to a
second type of antimony(|ll) substitution, with a single fluoride ion perturbed by
each antimony ion. The results in total provide detailed support for a substitution
model in which antimony(lll) occupies a phosphate site in the apatite lattice, with an
Sb033‘ group replacing a PO43‘ group. Two types of substitution at this site
appear to occur, depending upon which oxygen atom is replaced by the antimony
lone electron pair.

A method for obtaining multiple-quantum NMR spectra of a selected

homogeneously broadened portion of an inhomogeneously broadened powder

pattern is demonstrated with 19F results for polycrystalline calcium fluorapatite,
Ca5F(PO4)3, and doped fluoraptite systems. In addition to providing a means for
studying the anisotropic growth of multiple-quantum coherences, chemical shift
selective MQ NMR provides information about the correlation length of chemical
shielding tensors (i.e. the distance over which the shielding tensors of given
species maintain identical principal values and orientations). The growth of 19F
multiple quantum coherences in fluorapatite solid solution with 10% of the calcium
ions replaced with strontium ions is significantly slower compared to that of the
calcium fluorapatite at the same selected chemical shift. The slower growth rate is
interpreted in terms of a decrease in the average correlation length of the shielding
tensor in a one dimensional chain of fluoride ions in the solid solution, due to the
19F chemical-shift perturbations caused by strontium ion substitutions. The growth
of 19F multiple-quantum coherences in fluorapatite doped with 2.1 wt. % antimony
was the same as in stoichiometric fluorapatite. This result was used to rule out the
possibility that antimony substitution in fluorapatite is accompanied by the removal
of neighboring fluoride ions. The chemical shift selective MQ-NMR technique was
also used to study multiple-quantum growth dynamics as a function of dipolar
coupling strength in polycrystalline fluorapatite by recording the MO -NMR spectra

of different classes of crystal orientations.

iii

To my parents

iv

Acknowledgments

I would like to thank my advisor Dr. James P. Yesinowski for his guidance,
knowledge, and encouragement. I would also like to thank Dr. Dye and Dr. Cukier
for helpful discussions.

I would also like to thank Dr. Gyunggoo Cho for help with experiments and
mathematical problems. Many thanks to Kermit Johnson for help with NMR
hardware and computers. Thank you to the members of the NMR facility including
Dr. Evelyn Jackson, Denis Jablonski and Dr. Long Le. I would also like to thank Dr.

Jeffery Berkowitz and Dr. Barry DeBeor for obtaining doped apatite samples.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................................ ix
LIST OF FIGURES .......................................................... . ..................................... x
I. INTRODUCTION .............................................................................................. 1
1. Overview ............................................................................................... 1
2. Apatites ................................................................................................... 4
3. NMR Background ................................................................................. 7
3.1 Nuclear Spin Hamiltonian ............................................................... 7
3.1.1 Zeeman and Chemical Shielding .......................................... 8
3.1.2 Dipole Coupling ...................................................................... 12
3.1.3 Indirect Dipole Coupling ........................................................ 15
3.1.4 Quadrupole ........................................................................... 16
3.1.5 Radio Frequency Hamiltonian ............. 16
3.2 Density Matrix ................................................................................. 17
3.3 Multiple Quantum NMR ................................................................... 23
3.3.1 Detecting of Multiple Quantum Coherences ......................... 26

3.3.2 Statistical Model for Coherence Order and the Effective Size

Concept ................................................................................ 32

3.4 Magic Angle Sample Spinning (MAS) .......................................... 36
3.5 Spin Diffusion .................................................................................. 37

4. References ........................................................................................... 45

II. 19F and 31F MAS-NMR Magic-Angle Spinning Nuclear Magnetic
Resonance of Antimony(III)-Doped Calcium Fluorapatite Phosphors:
Dopant Sites and Spin Diffusion

vi

1. Introduction ............................................................................................. 46

2. Experimental ........................................................................................... 51
2.1. Sample Preparation and Characterization ................................. 51
2.2. NMR Equipment and Procedures ................................................... 52

3. Results ..................................................................................................... 56
3.1 31 P MAS-NMR ................................................................................ 56
3.2 19F MAS-NMR .............................................................................. 58

4. Discussion ............................................................................................. 70
4.1. Spin Diffusion ................................................................................ 70
4.2. Quantitative Aspects of 19F MAS-NMR Results and the

Extent of Antimony Incorporation ................................................. 74

4.3 Possible Antimony Substitution Sites ......................................... 80
4.3.1. Ca(1) Site ........................................................................... 82

4.3.2. Ca(2) Site ........................................................................... 83

4.3.3. Phosphate Site ............... 84

5. Conclusions ............................................................................................ 87
6. References ............................................................................................ 80

. A Method for Detection of Spectral Spin Diffusion from Minor Peaks and
Its Application to 19F MAS-NMR of Antimony(III)-Doped Calcium Fluorapatite

1. Introduction ............................................................................................. 94
2. Results and Discussion ............................................................................ 94
3. Conclusions ............................................................................................... 103
4. References ............................................................................................... 104

vii

lV. Chemical-Shift-Selective multiple Quantum NMR as a Probe of the

Correlation Length of Chemical Shielding Tensors

1 . Introduction ............................................................................................... 106
2. Experimental ............................................................................................ 1 12
3. Results and Discussion ........................................................................... 115
3.1 Chemical shift selection ............................................................... 115
3.2 Discussion of Sr/Ca fluorapatite MQ ............................................. 122
3.3 Discussion of Sr/Ca fluorapatite MQ ................................................ 128

4. Conclusions .............................................................................................. 130

V. Chemical-Shift-selective Multiple Quantum NMR Applied to the Study of the

Effect of Dimensionality on Growth Dynamics

1 . Introduction .................................................................................................. 131
2. Experimental .................... 134
3. Results and discussion ................................................................................ 134
3.1 Calculation of intraplanar and intracolumn couplings ........................ 134
3.2 Multiple quantum NMR of various orientations .................................... 141
4. Combined References for sections V and VI ......................................... 143

viii

List Of Tables

Table "-1. 19F MAS-NMR Chemical Shifts, Intensities, and Relaxation
Tlmes for Antimony-Doped Calcium Fluorapatite Samples ......... 60

LIst of Figures

Figure l-1 Schematic drawing of the calcium fluorapatite structure ............. 6
Figure I-2 Simulated 19F NMR chemical shielding powder patterns ............ 11
Figure l-3 Simulated NMR spectra of 19F dipolar coupled spins pairs .......... 14
Figure l-4 Energy level diagram for a dipolar coupled spin pair ...................... 25
Figure I-5 Even order selective Pl-MQ-NMR pulse sequence ......................... 27

Figure I-6 Schematic diagram of coherence order in density matrix for two

spin system ..................................................................................... 28
Figure I-7 Data from the Pl-MQ-NMR pulse sequence .................................... 31
Figure I-8 19F NMR chemical shielding powder pattern and MAS spectra ....... 37
Figure l-9 Schematic digram of the spin diffusion experiment ....................... 43
Figure I-10 Schematic diagram showing polarization transfer from

paramagnetic centers ................................................................... 44

Figure "-1 Schematic diagram of fluoride ions in fluorapatite ..................... 48
Figure "-2 SPARTAN pulse sequence for measuring spectral spin diffusion 55
Figure "-3 31 P MAS-NMR spectra of Sb-doped calcium

fluorapatite and SbPO4 ................................................................. 57
Figure "-4 19F MAS-NMR spectra of Sb-doped calcium fluorapatite ............. 59
Figure "-5 19F MAS-NMR difference spectra of Sb-doped

calcium fluorapatite .......................................................................... 63
Figure "-6 19F MAS-NMR spectra of Sb-doped calcium

fluorapatite samples ....................................................................... 64
Figure il-7 19F MAS-NMR spectra of Sb-doped calcium fluorapatite

obtained using the SPARTAN pulse sequence .............................. 67

Figure "-8 Plot of In ((rf + f)/r) vs. rm for the measurement of spin diffusion

rate constants ................................................................................... 69
Figure "-9 Schematic diagram of the arrangement of fluoride ions in the
fluorapatite lattice perturbed by antimony ....................................... 76
Figure "-10 Plot of the 19F MAS-NMR intensity of peak A vs. wt. %
antimony(lll) ..................................................................................... 79
Figure "-11 Plot of the 19F MAS-NMR intensity of peak A vs. mole fraction
of antimony be ........................................................................... 81
Figure "-12 ORTEP drawing of fluorapatite lattice with antimony in the

phosphate site ............................................................................... 85
Figure III-1 SINK pulse sequence .................................................................... 96
Figure III-2 Schematic arrangement of fluoride Ions in fluorapatite .................. 98

Figure III-3 19F MAS-NMR spectra of sb-doped calium fluorapatite
obtained using the SINK pulse sequence ...................................... 99
Figure III-4 Calculated and measured peak C intensities vs. the number of
saturation pulse trains (n) ................................................................. 102
Figure lV-1 Schematic diagram of the fluorine spin tensors in strontium doped

fluorapatite ....................................................................................... 108
Figure IV-2 Calculated chemical shielding powder patterns for the fluorine

spins in strontium doped calcium fluorapatite ................................... 110
Figure IV-3 Plot of the difference in chemical shift Ac between two types of

fluorine nuclei in as a function 8 and (ti ......................................... 111
Figure IV-4 Chemical shift selective PI-MQ—NMR pulse sequence .................. 114

Figure IV-5 Plot of the longitudinal magnetization Mz as a function of
resonance after varying numbers of saturation pulses ................. 118
Figure IV-6 Plot of the longitudinal magnetization Mz as a function of resonance
after saturation pulses of varying length ......................................... 119

Figure IV-7 Plot of calculated and experimental z-magnetization as a

xi

function of saturation pulse length ................................................. 120
Figure IV-8 Plot of calculated and experimental z-magnetization after a
saturation pulse as a function of resonance offset ...................... 121
Figure IV-9 Plot of the multiple-quantum coherence orders as a function of
preparation time for stoichiometric calcium fluorapatite and
strontium doped fluorapatite ......................................................... 124
Figure IV-10 Plot of effective size n vs. multiple-quantum preparation time ..... 125
Figure lV-11 ANTIOPE simulation of the triple-quantum intensity as a function
of multiple quantum preparation time ......................................... 126
Figure IV-12 Plot of the 4-quantumz2-quantum vs. preparation time for
stiochiometric fluorapatite and strontium-doped
fluorapatite ................................................................................... 129
Figure V-1 Schematic arrangements of 19F spins (black solids circles) in
calcium fluorapatite ....................................................................... 133
Figure V-2 Schematic diagram of linear and planar arrangements of spins in
fluorapatite ................... 136
Figure V-3 Plot of the D(linear):D(planar) as a funcion of 8 ............................ 140

Figure V-3 Effective size vs. preparation time in calcium fluorapatite ............. 142

xii

I. INTRODUCTION
I-1. Overview

The research presented in this thesis features the application of both
established and newly developed solid-state NMR techniques to some systems
based on calcium fluorapatite, including a phosphor used in the commercial lighting
industry. In this overview section (H) the different projects will be briefly described.
Then the chemistry and structure of the apatite system, M5X(YO4)3, will be
discussed in section I-2; in this work, M = Ca“, Sr“, Ba”, X = F', CI‘, OH' and
Y = P, (Si, S)1, As. Next, the portions of the nuclear spin Hamiltonian relevant to
solid state NMR will be described (section (I-3.1)). The principal tool for computing
the time evolution of a system of spins (hereinafter referred to as a ”spin system”)
under the influence of a particular spin Hamiltonian, the density matrix, will be
discussed in section l-3.2. Section I-3.3 will explain the concepts and experimental
techniques of multiple-quantum NMR that are used in Sections IV and V. In section
I-3.4 the experimental technique of Magic Angle Spinning (MAS) NMR will be
reviewed, and details of the dynamics of spins under MAS conditions required to
understand the newly developed MAS-based experiments in sections II and III will
be discussed. Finally, the concepts, experimental techniques, and uses of spin
diffusion experiments will be discussed in section I-3.5.

The common theme of the work in subsequent sections (ll-V) is the study of
dopants in calcium fluorapatite [Ca5F(PO4)3]. In section II a halophosphate
phosphor, antimony-doped fluorapatite, is examined using 31 P and 19F MAS-NMR
and spin diffusion techniques as an indirect probe of the antimony ion in the lattice.
Apatite based phosphors are of great commercial importance and are produced on
a kiloton per year scale for use in fluorescent lamps. In order to understand the
processes by which these apatite phosphors absorb and emit light, and how their

performance degrades over time, it is important to know the location of the activator

2

ion (antimony) in the apatite lattice. Based on the presence of several antimony-
associated peaks in the 19F MAS-NMR spectra and the lack of any such peaks in
the 31 P MAS-NMR spectra , we concluded that the antimony resides in a
phosphate site in the lattice. This conclusion is strengthened by the results of spin
diffusion measurements made between the antimony-associated peaks in the 19F
NMR spectra using the SPARTAN technique. In section III the spin diffusion
measurements between the antimony-related peaks in the 19F MAS-NMR
spectrum are extended with a technique called SINK that is especially well-adapted
to observing spin diffusion from minor to major peaks in a spectrum. This new
method, SINK, helps clarify some experimental observations of Section II and
strengthen some of the conclusions drawn there.

Sections IV and V feature a new multiple-quantum NMR technique applied
to the study of strontium-doped calcium fluorapatite and stoichiometric calcium
fluorapatite. Multiple-quantum NMR refers to a class of experiments designed to
create and detect multiple-quantum coherences among sets of coupled spins in
solids. When a single pulse of rf energy is applied to a sample there is a coherent
phase relationship between the magnetization vectors of each of the spins in the
sample and the collective behavior of all the spins results in the emission of
radiation. In such a sample, a single quantum coherence is said to have been
created. If there are groups of n tightly-coupled spins in a sample (for instance n
spins on the same molecule), up to n-quantum coherences can be created by the
appropriate excitation pulse sequence. The multiple quantum experiments featured
in sections IV and V are designed to examine multiple-quantum coherence growth
in spins in certain specific orientations of crystallites in a powder pattern, by
selecting specific frequencies in a chemical shielding powder pattern. We will refer
to such experiments as chemical-shift-selech've multiple-quantum NMR

experiments.

3

In section IV chemical-shift—selective MQ NMR is used to probe the distance
scale over which the chemical shielding tensors of spins in a lattice maintain the
same orientation and/or magnitude. This new experiment is demonstrated on a
sample of fluorapatite that has one of every ten calcium ions replaced by a
strontium ion. The strontium ions in the lattice have the effect of changing the
shielding tensors of neighboring 19F spins, creating, in effect, clusters of
unperturbed 19F spins separated by strontium-perturbed spins. The length of these
clusters of unperturbed 19F spins is probed by observing the growth of multiple
quantum coherence among sets of like spins. Since the strontium-perturbed spins
cannot enter the coherence because of their large frequency offsets, the limited
coherence growth reflects the finite size of the clusters of like spins.

In section IV the chemical-shift—selective MQ-NMR technique is used to
study the difference between multiple-quantum growth dynamics of crystallites
associated with different positions in the powder pattern. Chemical-shift-selective
MQ-NMR results on different peaks in the powder pattern are presented that
demonstrate that MQ coherence growth is faster in classes of cryStallites with larger
dipolar couplings between adjacent spins. Section IV also contains a description of
an experiment that should be able to show the difference between the MO growth
dynamics in linear (1 -D) and planar (2-D) arrangements of spins in fluorapatite. The
fluorine nuclei in fluorapatite can be viewed as linear columns of uniformly spaced
spinslthat extend the entire length of the crystal, neglecting the possible occurrence
of vacancies or substitutions. On the other hand, each column of fluorine nuclei is
surrounded by six other columns creating planes of spins perpendicular to the
columns with each spin in the plane having six nearest neighbors. By performing
chemical shift selective MQ experiments on peaks in the powder pattern associated

with crystallite orientations that have relatively different strengths of dipolar

4

couplings among planar spins or linear spins, differences in linear and planar MQ

growth dynamics might be observed.

L2 Apatlte Structure and Chemlstry

Apatites are a class of minerals having the general formula M10X2(YO4)5 ,
where M is an alkaline earth metal ion, X is a halogen or hydroxyl ion, and Y is an
element near the metal-nonmetal transition line on the periodic table. Apatites are
found in nature as calcium phosphates in mineralized tissue such as teeth and
bones, as well as calcium fluorapatite in the Earth's crust as a major phosphate ore.
Some important technological application of apatites include halophosphate
phosphors in fluorescent lamps, a gain medium in lasers, and calcium
hydroxyapatite chromatography materials. However, most apatite ore is converted
to other phosphates such as Ca(H2PO4)2.H20 and Ca2P04(OH).2H20, which are
in turn used for food processing or as fertilizers.2 In this thesis the variations of
apatites that will be examined include:
M = Ca+2, Sr+2, Ba“2
x = F'1, crl ,OH-l
Y = P+5, As+5, (Si+4, 3+6)1
Apatite solid solutions can be created by synthesizing an apatite containing more
than one of the cations and/or anions shown above. Four mixed apatite systems
that will be examined in this work include: strontium and calcium in the M positions;
silicon, sulfur, and phosphorus in the Y position; and arsenic and phosphorus in the
Y position. Apatites can be synthesized in the laboratory either by the firing of
mineral precursors in a quartz tube furnace or by aqueous precipitation. All of the
samples in this study were synthesized by the firing method.

In this thesis most of the sections focus on dopants in calcium fluorapatite

[Ca5F(PO4)3], so the details of its structure are important to the arguments that will

5

be presented. The apatite structure, obtained from x-ray crystallography,3 features
linear chains of uniformly-spaced fluoride ions parallel to the c axis and extending
the length of the crystal (Figure l-1). There are two types of calcium ions in the
structure, labeled Ca(1) and Ca(2), in the ratio of 4:6. Surrounding each fluoride ion
is an equilateral triangle of crystallographically equivalent Ca(2) ions that lie in a
plane perpendicular to the fluoride chain. Adjacent triangles of Ca(2) ions are
rotated by 60° about the chain axis, creating a hexagonal screw axis in the
structure. Adjacent to these Ca(2) sites are crystallographically equivalent
phosphate tetrahedra. Finally, there are columns of Ca(1) ions running parallel to
the c axis, each calcium column in the middle of an equilateral triangular prism

formed by three fluoride columns.

4.; la
a i: S; .

Figure l-1. Schematic drawing of the fluorapatite structure. Calcium ions (large open
circles), fluoride ions (black dots), and phosphate groups (small open circles

connected by lines). See text. Figure adapted from reference 4.

l-3 NMR Background
I-3.1 The Nuclear Spin Hamlltonlan

The nuclear spin Hamiltonian has a well known form, involving some
parameters that are generally measured rather than calculated quantum-
mechanically using electronic wavefunctions (viz. the chemical shielding, scalar
coupling, and quadrupole coupling). The Hamiltonian determines both the eigen
frequencies of the NMR spectrum and, through time-dependent quantum
mechanics, evolution of the spin system in time. The nuclear spin Hamiltonian is
complete enough, and at the same time often simple enough, that it can actually be
used to develop new NMR experiments before trying them in the laboratory. In this
section the important terms in the nuclear spin Hamiltonian will be described and
their effect on NMR lineshapes in solids will be illustrated through simulated
spectra

The nuclear spin Hamiltonian can be written as
31 = 5“ Zeeman + i‘RF + 3{s + 95D "’ ”Q T . J + 9{other (H)
where xZeeman is the term for the Zeeman interaction of spins with the large
applied static magnetic field, liar: is the term for the Zeeman interaction of the spins
with an applied linearly polarized magnetic field oscillating at rf frequencies, its is
the term for the chemical shielding at the nucleus, to is the dipolar coupling term
between pairs of spins, %Q is the nuclear electric quadrupole coupling term (for l >
1/2), and m is the indirect coupling (scalar) term between pairs of spins. The term
Kather represents other interactions such as spin-rotation coupling or coupling
between nuclear spins and electron spins. These terms will not be addressed due
to lack of relevance to this work. The first two terms in Eqn. H are often referred to
as the external Hamiltonian because they contain the applied fields that are under
experimental control. The remaining terms in Eqn. H are usually referred to as the

internal portion of the Hamiltonian because they reflect the local environment of the

8

spins and depend on the sample. The only internal interactions that not are
averaged out by rapid diffusion and tumbling in liquids are the (isotropic or scalar)
chemical shift and (scalar) J coupling terms; however, all of these interactions can
manifest themselves in solids as line broadenings.
I-3.1.1 Zeeman and Chemical Shielding Hamiltonian Interactions

it is convenient to treat the Zeeman and chemical shielding terms in Eqn. l-1
together since the shielding term is simply a correction to the applied magnetic field
at the nuclei that depends on the local electronic environment of the spin. The

combined Zeeman and shielding Hamiltonian has the form
Hz," +Hd=-7[-(l-O’)-B:o I-2)

where y is the magnetogyric ratio of the nuclei, 1 is an angular momentum vector

operator (ix, Iy, I2), 1.90 is the applied magnetic field. 9 is a second-rank Cartesian

tensor that describes the shift of the nuclei at various orientations of the molecule
with respect to the applied field. The shielding tensor is generally not symmetric but
the effects of the difference between elements on opposite sides of the diagonal (oil-
and Oji) are insignificant at high fields and can be neglected. Since the shielding
tensor can be approximated as being symmetric at high field, a reference system

can be chosen such that the shielding tensor takes the diagonal form
an, 0 0
a: 0 a" O . l-3)
0 0 022

This reference frame is called the principal axis system. There are a few definitions
and conventions for the values of the components of the shielding tensor. The
isotropic chemical shift can be defined as 1/3Tr c = 1/3( 0xx + oyy + 022) = 0150.
The isotropic shift is the average shift the spin would experience in the liquid state if
the molecular reference frame were subject to rapid isotropic tumbling motion. By

convention the principal axes of the shielding tensor are chosen such that

9

i022 -a,-so lzlam -—a,-SO |2|0yy ~aiso I.

We can now define two useful quantities; the anisotropy of o , 5, which is given by
the formula

5 = 022 ' Giro

and the asymmetry parameter, 1], given by the equation

,,= 3:22.32.

The asymmetry parameter can vary from 0 (for axially symmetric tensors) to 1.

We can attach a physical meaning to the values on the diagonal of the
matrix in Eqn. I-3. The reference system in which Eqn. l-3 is diagonal is related to
the laboratory frame such that when the z axis of the principal axis system is
parallel to the applied field, the shift of the spin is given by on (furthest down field).
When the x or y axes of the principal axis system are coincident with 80 the shift of
the spin would be Gxx or oyy respectively. Furthermore, the shielding tensor in the

principal axis system can be related to the shielding tensor in the laboratory frame
by the orthogonal transformation R:

o-(LAB ) = R o-(PAS) R

where Rij is equal to the cosine of the angle between the i-axis of the lab frame and
the j-axis of the principal axis system. When all orientations of the chemical
shielding tensor with respect to 80 are present, such as is true in a polycrystalline
sample, the resulting distribution of shifts is usually referred to as the chemical
shielding powder pattern. As an illustration of these so-called "powder-pattems",
Figure I-2a is a simulated spectrum of a powder sample with an axially symmetric
chemical shielding tensor ( II = 0) with the same chemical shielding anisotropy and
isotropic chemical shift (w. r. t hexamethylbenzene) as the 19F spins in calcium

fluorapatite. The downfield edge of this pattern has the shift 022 = 120 ppm and the

10

intense edge has a shift of (SW = 36ppm, which is also equal to 0xx in this case.
Figure I-2b is an illustrative chemical shielding powder pattern for a spin with three

distinct principal shielding components located at the three singularities, as shown.

11

 

 

 

 

 

 

“w
“xx
022
01
A)
all I
IillliiiqiirlliiFiliihriTTIITITTIIIIIIIIITIIIITlIle'llilliiillllli]

140 120 100 90 80 70 60 50 40 30 ppm

Figure I-2. Simulated chemical shielding powder patterns: (a) axially symmetric
shielding tensor with CSA = 84 ppm and 11 = 0 ; (b) non-axially symmetric shielding
tensor with CSA = 84 ppm and fl = 0.24. An exponential apodization function
corresponding to a 400 Hz line broadening was applied to the simulated FID before

Fourier transformation.

12

I-3.1.2 Dipolar Coupling Hamiltonian
The dipole-dipole coupling term describes the energy of interaction between
the magnetic dipoles of spins. The energy of interaction between two interacting

magnetic moments is given by the expression
A! '9 30.1-00.1 r)
= l 2 _ I 2
3

r r5

where y is the vector from the first nucleus to the second nucleus, and r is the

l-4)

 

length of r . The magnetic moment vectors, yl and 4:2, are given by the equations

,1 = y»! and p = rill2 respectively. The quantum mechanical Hamiltonian for N
' I ' I ' 2 '

interacting spins can be written as
1 ~ ~ 939, 3(y.°r.,‘)(yk-r,) ,
110:; Z ’3 — J ’ 5 ’ (jack). I-5)

i=1 i=1 r jlt 1t

 

The magnitude of the small fields from other spins is typically on the order of
several Gauss, which is approximately 10'5 times the size of the field of a
commercial NMR magnet. Expressed in terms of Cartesian spin vectors and

tensors the dipole coupling Hamiltonian takes the form
N N

HD=§<2;r,-n 5-5-1, H»
where R is the dipolar coupling tensor given by
r2 - 3x2 -3xy —3x2

.15: -3xy rz-3y2 —3yz

-3xz -3yz r2 — 322

where r is the length of a vector t pointing from one spin to another and x, y and z
are the components of a unit vector parallel to f . When the Cartesian components

in l} are replaced by their equivalent expressions in polar coordinates:

x=rsin8cos¢
y=rsin8cos¢
z=rcos8

equation I-6 takes the form

13
h7137t

 

[A+B+C+D+E+F]

N
”o"- ‘33,?
A=(1-3cos: 601,4. I-6b)

1
Bz-z-(1—3C0526)(1jzlla —!j'1k)

The terms through C through F in Eqn. l-6b are given elsewhere 5 and are typically
neglected since their effects on the eigenfrequencies are negligible for high field

NMR. The term 8 in Eqn. l-6b is the angle between the internuclear vector and the
applied field _Bo. The Hamiltonian shown in Eqn. l-6b written with the "A" and "B”

terms takes the form:
HD =2 2_ h7j7k (l—3c052 0],)[Izjlz‘ “$419.11", +Ifllfi )] . I-7)
rjli
The ”dipolar coupling" Djk between a specific pair of nuclei is the quantity;

h:’:‘ (1- 3cos 6],).

jl

 

Djk=

The value of the dipolar coupling is 1.20067 x 105 Hz for two protons separated by
one angstrom with ejk = 90°.

The effect of the dipole-dipole coupling interaction on an 19F resonance for a
pair of fluorine spins is shown in Figure l-3. The pair of spins is separated by a
distance of 344 pm, the crystallographically determined distance between the
fluorine spins in the chains of fluorine found in fluorapatite. The strength of the
dipolar coupling between neighboring fluorine spins is relevant to the spin diffusion
measurements and multiple-quantum NMR experiments in subsequent sections.
Figure l-3a is a spectrum calculated for a pair of 19F spins having Bjk = 90°. The
splitting between the two peaks is equal to three halves of the coupling (DP-F).
which is equal to 3.9 (KHz). Figure l-3b is a powder spectrum of a pair of 19F spins
with a statistical distribution of orientations ejk present This powder pattern
distribution of spins is usually referred to as a Pake doublet. 5

14

 

 

 

 

 

 

 

 

 

B) l
l
-\_/
j
J
A)
- -JL. JL.
-6 -3 0 3 6

Figure I-3. Simulated spectra of a 19F spin pair separated by 344 pm at 9.4 T: (a)
internuclear vector oriented perpendicular to the applied field; (b) powder pattern

spectrum.

15

I-3.1.3 Indirect Dipole Coupling
The indirect dipole coupling, or "J" coupling as it is called, refers to coupling
to the magnetic dipole moment of a nearby spin transmitted through the Zeeman

order of valence electrons between the nuclei as shown in the diagram below.
electron spins

/ \

l l
\ /

nuclear spins
The nuclear spins are pictured in the antiparallel orientation but can also have a

parallel orientation. The indirect coupling Hamiltonian written in Cartesian form is
HD =$<$mlj~g~g l-8)
where ij is the indirect coupling tensor. Unlike the direct dipolar coupling tensor,
the J tensor is generally not traceless, which results in splittings of peaks in the
spectra of isotropic liquids.
If this Hamiltonian is recast eliminating nonessential terms for the high field

limit we get the following” form:
H =2 2 J},lj- 1 +2 2 J'”°,r[1y 1 ——(1g_ 1 +1 1 )] l-9)

where the two terms are the isotropic and anisotropic portions of the Hamiltonian
respectively. The magnitude of J can range in size from less than one Hz all the
way to many KHz. In Section II, an attempt is made to observe any "J" coupling
between the fluorine spins as a possible source of spin diffusion. Unlike the direct
dipolar coupling, the scalar coupling does not scale as 1/r3ij and does not depend
explicitly upon q]. Therefore, if the magnitude of "J" coupling is substantial

16

compared to the direct dipolar coupling, determination of internuclear distances

based on spin diffusion experiments would not be possible.

I-3.1.4 Quadrupole Hamiltonian

The quadrupolar term in the Hamiltonian describes the energy of interaction
of a nonspherical nucleus with an electric field gradient at the nucleus. All nuclei
with spin I > 1/2 (the majority of magnetic nuclei) are nonspherical, and are said to
be ”quadrupolar”. The quadrupole term in the nuclear spin Hamiltonian can be

written in Cartesian coordinates as follows:

———ZV¢ [% (1 1 W11) 6M2] l-10)

HQ:
2111(2Q1—1)wi=1

where V is a second rank electric field gradient tensor, e is the charge on the

electron, Q is the nuclear quadrupole moment, and l is the spin of the nucleus.

I-3.1.5 Radio Frequency Hamiltonian

In NMR, the radio frequency (rf) irradiation applied to the sample has an
oscillating (linearly polarized) magnetic field vector perpendicular to the static field.
The Hamiltonian for the Zeeman coupling of spins to this field can be described

semi-classically by the following time dependent equation:

Hrf = ~27H1cos(ax)1x l-11)
= -2w1cos(wt)1x

where lflll is the linearly polarized magnetic field vector rotating at a frequency (0

and 0:1 is the precession frequency of the spins about the If!l vector. Since it is

easier to treat a time-independent Hamiltonian, Eqn H1 is usually recast in a
reference frame rotating about the z axis at the rf frequency (0. In this rotating frame

the rf Hamiltonian simplifies to the following form;
H4 (rot) = -ct),l,r l-12)

17

A physical consequence of Eqn. H2 is that when the rf irradiafion is applied, the
magnetization vector of the spins on resonance precesses about the x axis of the
rotating frame at the frequency 031. This behavior is not true of the spins that are not
on resonance (not resonant at m). Instead, these spins process about the effective

field given by the equation
{‘14, =(Ho-év/7)+Hl I-lal

where the quantity (1510- (52/ 7) is the component of the effective field along the z

axis. The physical consequence of Eqn. l-13 will be discussed in section II in

reference to creating frequency selective excitation experiments.

i-3.2 The Density Matrix

As shown in examples given in the previous section, the nuclear
Hamiltonian can be used to compute the eigenfrequencies and therefore the NMR
spectra of spin systems. With a well-described Hamiltonian and complete set of
wave functions, the time development of a measurable property of a spin system
can be computed by using the Liouville equation. Such a computation can be
streamlined by placing all of the coefficients of the wave functions of a spin system
into an array called the density matrix (more generally, in a basis-independent form,
a density operator). This density operator contains all the information necessary to
describe any measurable property (observable) of the spin system. For a large
ensemble of spin systems an ensemble average of products of coefficients is used.
in this section (l-4) the equilibrium density matrix for spins in a large applied
magnetic field will be given and a connection between diagonal elements of the
density matrix and polarization (2 magnetization) will be made. Furthermore, a
connection between the off diagonal elements of the density matrix and other

observables a spin system can have, such as transverse magnetization and

18

multiple quantum coherences will be made. Portions of the following description of
the density matrix are adapted from reference 7 (chapter 5.4).

The procedure to compute an observable is to evaluate the inner product of
some operator corresponding to the desired observable with the wave function. For
instance, the expectation value of the x component of the spin angular momentum
would be
(Ix)=<'l’(t)|1x|V’(t)>- H4)

The time dependent wave functions are usually written as

W) = Zc,(t)¢,- l-15)

where in is an orthonormal set of basis functions with time dependent coefficients
Cj(t). Evaluating an expectation value as in Eqn. l-14 would be done by taking the

following sum;

(1x)=ZCI(t)C,(t)(¢r l1x|¢,). HS)
1.:

All of the information about the time dependence of the observable ( Ix ) is carried
in the products of coefficients of the matrix elements c'ij, since the matrix
elements (¢, |Ix| 43,.) themselves do not change with time. The products ckcj' can
be arranged into a matrix. It is convenient to think of the matrix ckcf as being the
representation of an operator P, the operator defined by its matrix elements:

(is, |P| ¢,) = cjc,‘ H7)
The expectation value of 1x can be written as in eqn I-16 with coefficients of the

matrix elements written as shown in Eqn. l-17:

(w; <¢ilP|¢,-)<¢. l1,|¢,). I-18)

The result of the operator P acting on a function (ilk may be written as
’3‘; =Zak¢k MS)
I:

19

since the dik's form a complete basis set. As usual we find the coefficients ak by

multiplying both sides from the left by (ilk. and integrating:
a, =14le¢j =(¢, |P|¢j) so that I-20)
P¢k =Z¢k(¢k|Pl¢j) "21)
k ,

Likewise we have

1x¢k =Z¢k<¢k I1x|¢j> l-22)
It

therefore

lx¢k =Z¢k<¢k |1x|¢j> I-23)
k

and

 

 

Plx¢h =ZP¢t<¢t ¢,)
= ZP¢..<¢.-IPI¢,)(¢,
J'Jl'

IX

l-24)
1

 

 

,¢i)

so that
<¢P lPlx

 

 

¢.)=; <¢.-|P|¢,-)(¢,

By using Eqn. l-23 , we find that

(1):}; <¢t|P|¢jX¢j 1; ¢t>

= Z (¢,-|PI. it.) l-25)
M
= Tr {P1,}.

11'

 

if.)

 

 

 

In words this simply states that the expectation value of an observable is the trace
of the product of the density operator and the operator corresponding to the
observable. Since (¢, |P| ¢,-) = <¢j |P| ¢, ) the density operator is Hermitian.

Until now we have been describing an operator P which is defined (as in
Eqn.I-17) by the wave functions of a single spin system and contains the
information required to compute observables of that spin system. In many NMR
problems (and all of those in this work) we are concerned with computing the

average expectation value of an ensemble of spin systems. This is generally the

20

case since the sample in most high field NMR studies contains a large number of
spins (ca. 1016 -1 023). However, some experiments 3 do detect the properties of
single spins. In a large collection of spin systems, the coefficients ckcf will vary

from system to system since each spin system has a unique set of wave functions,
but the matrix elements <¢j |1x| pk) will be the same. If we use a bar to denote the

ensemble average, we have, then,

<75=Zc;(t)c,<tl(¢.l1x|¢,). l-26)
M
The quantities c; (t)cj(t) form a matrix, and it is this matrix that we call the "density

matrix“. We shall consider it to be a matrix of an operator p, defined by the equation

<¢kH¢j>=Czcj =<¢lrlpl¢j>- 1-27)
Similar to equation I-25, p can be used to compute an ensemble average

expectation value in the following manner:

m=§<¢jH¢k><¢k 1. ¢,)=Tr{.01,}. I-28)

Having established how to compute an observable of the spin system, we

 

 

will now examine how to calculate the evolution of the density matrix under the

influegce of a Hamiltonian. By the Uouville equation:
2:3 = rip, H] . I“'29)
In theevent that H is independent of time, we may obtain a formal solution of Eqn.
l-29:
p(t) = e“”’p(0)e"”’ I-30)
If the Hamiltonian is time dependent, it can be approximated by a string of time-
independent H, operating for very short intervals:

21

1 t,+Ar
H, = (Z?) l H(t)dt. r31)
’1

In order to begin computing the evolution of the spin system we must first
know where it starts by describing the initial properties of the ensemble of spin
systems (i. e. the equilibrium density matrix). Off diagonal elements pjk = a; are
the average products of two different complex numbers. At equilibrium there is no
reason to expect a special phase relationship, and hence the off diagonal elements
would each have a value of zero (by the random-phase hypothesis) .7 Diagonal
elements describing populations should at equilibrium satisfy a Boltzman
distribution:

t 8
p(0) = cjcj =

 

451/er

 

l-32)
where Z is the partition function

—E
2:; fiT . I-33)

For concreteness, the equilibrium density matrix for 19F in a 14.1 Tesla field (500
MHz spectrometer) at 298 K evaluated in the in basis states { I a) , I B ) }would be:

1.00000453 0.0
0 = . I-
A ) [ 0.0 099999547] 34)
If we expand the exponential in eqn-32,
p(0)=(-;—)(1+ r001, /kT) l-35)

we notice the equilibrium density matrix can be decomposed into the unity matrix
and a term proportional to I2 (higher power terms of Iz can be dropped since they
are very small). Since the unity matrix does not change under unitary
transformations, it is simply dropped to simplify calculations; the resulting matrix is

referred to as the reduced density matrix.

22

To compute the z-magnetization from the equilibrium density matrix in Eqn.
l-35 we simply multiply by I2 and take the trace;
(12): Tr{plz}=pm-ppp. l-36)
Thus, the difference of the diagonal elements is proportional to the polarization.
Having shown that the diagonal components of the density matrix are related
to z-magnetization, we can now discuss which elements correspond to transverse
(x and y) magnetization. The elements that are immediately off the diagonal
correspond to transverse magnetization. For example the reduced density matrix
for a two spin system evaluated in the m-basis states after a single 90° (x) non-

selective pulse would take the form:
”co 't on on“

00 co no 00
co co 0o .
_uo co co oo_

p(t) = I-37)

 

 

with the asterisks denoting nonvanishing elements. Often the presence of such
density matrix elements are referred to as single quantum coherences.

They have the label single, because they connect states that vary by one unit of
total angular momentum. The word coherence is used because the spin systems in
the ensemble of spin systems have a coherent phase relationship. The question
that logically follows is: what do the other elements of the density matrix correspond
to? The first thing to observe is that they do not correspond to properties that are
directly observable in the laboratory such as transverse magnetization (immediate
off—diagonal elements). The other off diagonal elements are generally referred to as
multiple quantum or zero-quantum coherences and must be detected indirectly by
converting them into observable magnetization. The density matrix of a two spin

system;

23

q

”0.0 0.0 0.0 *

0.0 0.0 0.0 0.0
= l-38

p(t) 0.0 0.0 0.0 0.0 )

t 0.0 0.0 0.0)

 

 

has two elements associated with double quantum coherences indicated here by
asterisks. A physical interpretation of an ensemble of two-spin systems with non-
vanishing double-quantum elements is that the individual spin systems (spin pairs)
would be absorbing and emitting two quanta of angular momenta in a synchronous
fashion. The existence of nonvanishing double quantum matrix elements as in Eqn.
I-38 is referred to a double quantum coherence. More generally, the presence of an
n quantum matrix element is referred to as an n-quantum coherence. We will further
discuss the creation and detection of these "multiple quantum coherences” in the

next section.

24

I-3.3 Multiple Quantum NMR

Multiple quantum NMR experiments are used to detect transitions other than
"allowed“ (A M = i 1) transitions that correspond to the lines observed in NMR
spectra. The simplest example of a multiple quantum transition, or coherence,
would be the double-quantum coherence for a coupled spin pair. Figure I4 is a
diagram of the energy levels for a coupled pair of spins with the energy levels
labeled 1, 2, 3, 4 as shown. The double quantum transition between the energy
levels 1 and 3 is denoted with a dashed line. This transition is labeled a double
quantum transition because it corresponds to an event whereby the system is
absorbing two units of z angular momentum. As discussed in Section l-3 if a
double quantum coherence has been created the density operator would contain a
nonvanishing element between the states corresponding to energy levels 1 and 3.
From this example the utility of MQ—NMR spectroscopy is readily apparent: the
maximum MQ coherence is limited by the number of coupled spins in the spin
system. For the spin pair described in Figure I-4 a two quantum coherence can be
created. However, one can expect to be able to create both double and triple
quantum coherence for three spin clusters (such as in a CH3 group).

In this section (I-5) the double quantum selective NMR pulse sequence for
creating and detect MQ coherences will be discussed. Then, a model for
determining the size of clusters of spins involved in MO coherences will be
examined and explained in terms of the statistics of MO transitions available to a

spin system.

25

3

low

32Q

2 >+j3 >}
“QB Ia {laB>-|Ba>}

 

 

1 |l3l3>

Figure I-4. Energy level diagram for a pair of dipolar coupled spins. The double-

quantum transition between levels 1 and 3 is indicated with a dashed line.

26

I-3.3.1 Detecting MQ Coherences

The pulse sequence9 use to obtain even-order selective multiple quantum in
this work is pictured in Figure I-5. Although the pulse sequence used in this work is
actually designed to collect even order coherences, it can obtain odd order
coherences with a slight modification.1 0 This pulse sequence has four different
sections; the preparation period 1(U), the evolution period t1, the mixing period 1(V),
and the detection period t2. The preparation period has the effect of creating
multiple quantum coherence among neighboring sets of spins. The average

Hamiltonian during the basic eight pulse cycle is given by the formula:

17,, :31!” — Hm]
l-39)

l
= '2’ éDjk(Ij+lk+ + Ij—Ik-)
I

This Hamiltonian has the effect of transferring intensity from the diagonal elements
of the density matrix to those related to even order quantum coherences. This
transfer process is shown schematically in Figure I-6 for the reduced density matrix
of a spin system of two spin 1/2 nuclei. (Note: both even and odd orders are shown
to be populated in Figure I-6 for ease of visualization. This would actually only occur
under the influence of a non-selective MQ Hamiltonian.) Figure l-6a is the
equilibrium density matrix with the various m basis states explicitly given as row
and column lndices. The equilibrium density has intensity on the diagonal elements
which is related to population of spins aligned with the field or z-magnetization (see
Eqn. I-36). The statistical mechanical order is moved into the off-diagonal elements

of the density matrix corresponding to zero, one, and double quantum coherences

27

flu) T (V) (TE/2)x

ti

1/3 (11$) 1 -1/3 (Hd)

/ \

(rt/2)x (1l:/2)x (rt/2)x (rt/2))?(1t/2); (1t/2)—x(1t/2)-i(1t/2);

@IIA’IIAIIA’IIAIIA’llAflA’HA/z

 

 

 

 

 

 

 

Figure l-5. Even order selective PI-MQ—NMR pulse sequence.

28

 

 

 

 

 

 

 

 

 

 

 

 

1: (U) T (V)

 

 

 

 

 

 

 

 

 

 

 

 

Figure l-6. Bar graph plot showing the flow of multiple-quantum coherence order in
a schematic density matrix during the preparation and detection periods (U and V)

of the PI-MQ—NMR shown in Figure l-5.

29

during the preparation period as shown in Figure l-6b and I-6c. The coherence
order can be spread equally among the off diagonal elements by lengthening the
preparation period. This is done by increasing m, the number of repetitions of the
eight pulse cycle shown in Figure l-5. .

After the preparation section of the PI-MQ sequence a short delay or
evolution period t1 is inserted to give the NMR spectrometer time to adjust the
phase of the rf pulses. This delay is made as small as the NMR hardware will allow
(typically 1 us in this work) so as to avoid decay of coherence order. Next, the
mixing period reverses the flow of order created during the preparation period
to the diagonal elements as shown in Figure I-6. This time reversal is implemented
by changing the x and 17 pulses in the preparation period to y and )7 in the mixing
period. The way this pulse sequence actually gains information about the multiple
quantum coherence orders that are created, is by incrementing iii, the phase of the
rf irradiation used for the pulses in the preparation period, while holding the phase
of the rf in the evolution period constant The z-magnetization represented by the
intensity of the diagonal elements of the density matrix immediately after the mixing
period can be written9 as:

S<¢. r): z 2: l¢j,t(T)l2 expiinA¢lexp(—iw,-,rtl

n j ,k I-40)

where ¢i,j(t) are elements of the reduced density matrix and cojk = coj - (0] is defined
as the M0 transition frequency between states j and k. We notice immediately that
Eqn. I-40 is a Fourier series in 4», leading us to two conclusions. First, a Fourier
transform of the data as a function of it; yields a ”multiple—quantum" spectrum that
will have a 5-function peak for each MQ coherence. Second, from the Nyquist

sampling theorem, one must sample at twice the ”frequency” of the highest order

30

peak to detect that frequency. This means that in order to obtain a maximum
coherence "max , 4) must be incremented by intervals of 1/2(21t/nmax ) = “mmax-

The final element in the Pl-MQ-NMR sequence shown in Figure l-5 is a
single nonselective 90° read pulse. The function of this pulse is to convert the iti-
modulated z-magnetization left after the evolution period V into detectable
transverse magnetization. Once transverse magnetization is created, a single
complex data pair is typically taken. However, the entire FID can be recorded and
transformed to yield information about the growth of MQ coherences in specific
peaks in the usual NMR frequency domain f2 (as will be shown in Section V).

As an illustration of what the data from a Pl-MQ-NMR experiment look like,
real points of a set of complex pairs as a function of it are shown in Figure I-7a. The
data points shown are often referred to as a multiple-quantum interferogram and in
this case were collected with (p incremented in intervals of 15° allowing for the
detection of up to 12-quantum coherences. The Fourier transform of the
interferogram yields the MQ—spectrum shown in Figure I-7 b. Each peak in this
"spectrum” corresponds to a different coherence order as labeled. As shown in
Figure l-7b the coherence orders have greatly differing magnitudes. In the next
section we will discuss how to determine the size of the clusters of spins giving rise

to such distributions of coherence orders.

31

3)

 

 

 

 

__._t_. Liable;

-100 -80 -60 -4Q -20 20 4Q 60 80 100

Figure I-7. Data from the Pl-MQ-NMR sequence shown in Figure l-5 with ill
incremented in intervals of 15° to obtain a maximum coherence order of 12; a) c-

domain interferogram; b) MQ spectrum .

32

I-3.3.2 Statistical Model for Coherence Order and the Effective Slze
Concept.

In the last section the actual pulse sequence for obtaining various coherence
order magnitudes was discussed. The end result of this experiment is a multiple-
quantum spectrum which consists of a series of peaks for each coherence, each
with a distinct magnitude. In this section a model which relates the MO peak
intensity distribution to the size of the clusters of spins in the spin system will be
given.11 Furthermore, the concept of the effective size of the spin system will be
developed. ”Effective size” is a phrase that refers to the size of the clusters of spins
that are participating in MO coherences in large, essentially infinite lattices of spins.
Finally, information that can be gained about geometrical arrangements of spins by
examining the growth of the effective size as a function of the preparation time will
be detailed. Understanding these MQ growth dynamics experiments is crucial to
understanding the studies in Sections V and VI of this thesis.

The magnitude profile of the peaks in a MQ spectrum can be related to the
size of the spin system N by considering the number of density matrix elements
associated with each coherence.1 1:12 This can be illustrated by the example of a
three spin system taken from reference 12. A system of three dipole—coupled spin-
1/2 nuclei (N=3) has 2N = 8 product states. These are explicitly listed as the row

and column indices of the schematic density matrix for the system given by:

33

 

 

aa> Iaafl> lafla> W060 13136» LBafl> laflfl> W516:

< m, '0 1 1 1 2 2 2 3

< earl 1 o o o 1 1 1 2

< aflaj 1 0 0 0 1 1 . 1 2

< final 1 0 o 0 1 1 1 2
p: <fi/3al 2 1 1 1 o o o 1
war: 2 1 1 1 o o o 1

< am 2 1 1 1 o o o 1
<me _3 2 2 2 1 1 1 o_

l-41)

where the number at each element is the multiple quantum coherence order
associated with that element. The order of a coherence, n, is simply the difference
of the 2 component of the angular momentum mz between two correlated states.
For instance, a transition from the mz = -3/2 state, 888, to the m2 = +1/2 state, (1043,
is a +2—quantum transition. The reverse process is a -2-quantum transition. The
number of density matrix elements Zn corresponding to an n-quantum coherence is

given by the following eqn:

2” 2N!
Z"‘[~-lnl]‘<2~-lnl>lnll "42’

 

where N is the size of the spin system. In the case of the three spin system (N=3)
the possible values of n are 0, :l:1, :t2, :3. The number of elements comprising each
order are 20 = 20, Z1 = 2.1 = 15, 22 = 12 = 6, 21 = 2.1 = 1. This is easily verified
by counting the elements in Eqn I-41 associated with each coherence order.
As we saw in Equation l-40 the actual intensity of an MQ order I(n,t) is
dependent on the preparation time 1. When I is sufficiently long, the coherence

34

order is spread equally among (or nearly equally) among the elements (see Figure
I-6) and the total intensity of the peaks in the MO spectrum will reflect the number of
elements corresponding to a given coherence order. For example, the MO
spectrum of the three spin-1/2 system would contain peaks for each order
proportional to the number of the elements Zn listed above. A schematic stick

spectrum of the three spin system at long ‘E values would appear as follows:

20
15

15
1

 

 

 

1 11

-3o -ZQ -1Q 00 1Q 29 3Q
with the intensities of the peaks indicated next to the lines.

 

From the practical standpoint, this model enables one to ”count" the number
of spins grouped together in well isolated clusters in a sample using MQ NMR, by
recording MQ spectra with increasing preparation times until the intensity profile of
the MO peaks ceases to change as a function of ‘t. Then the intensities of the MQ
peaks are fit to Eqn. M2 by varying N, thereby giving the size of the cluster of
coupled spins in the sample. The Pines group has shown that this ”counting"
technique can be used to count as many as 21 spins in a cluster in a sample.13

If a spin system is not characterized by finite clusters of spins but rather by a
large infinite lattice or network, the MO order will continue to shift toward peaks of
higher multiple quantum coherences with increasing 1 as larger groups of spins
enter coherences. This presents a problem for the statistical model: if the size of the
coherent groups of spins is increasing, the statistics of the number of correlated
states is changing at the same time. This necessarily obscures any information

about the size of coherence clusters of spins. Despite this limitation the peaks in the

35

MQ spectrum of a sample with large numbers of coupled spins can be fit to Eqn. l-
42 (or its Gaussian approximation). The size of the spins system, N, obtained from
this fit is usually called the ”effective size”. The effech've size parameter is roughly
equivalent to the average size cluster of spins that are in a coherence. ‘1 It has
been pointed out 14 that some deviation from Eqn. I-42 would be expected since
spins on the periphery of the cluster have neighbors outside of the coherence.

Some useful information about large spin systems can be gained from
measuring the effective size as a function of preparation time. The rate of growth of
coherences (and therefore the effective size) as a function of preparation time
scales with the dipole-dipole couplings between spins in the sample. Since dipolar
couplings depend upon the inverse cube of the internuclear distance, the MO
growth dynamics can yield information about whether spins are agglomerated or
dispersed in a sample. Furthermore, it has been shown recently15 that the
dimensionality of spin arrangements (i.e. linear, planar, solid) can be determined
from the shape of the MO growth curve. In this thesis ( section V) MQ growth is
examined in linear clusters of 19F spins in a doped fluorapatite sample at different
orientation with respect to the field. Obtaining orientation dependent MQ spectra is
achieved by using a newly-developed MQ-NMR technique that selectively records
MQ growth within a specific peak in a powder pattern. In section VI the frequency-
selective MQ-NMR technique given in Section V is used to test some of the
concepts concerning the dependence of coherence growth on dipolar coupling
strength and dimensionality.

36

I-3.4 Magic Angle Sample Spinning (MAS) NMR

The primary experimental technique employed to overcome the broadening
interactions in solids described in Section l-2 is Magic Angle Spinning (MAS) NMR.
MAS-NMR involves rotating powdered samples in a cylinder about an axis that
forms an angle of 54.70 with respect to the magneh'c field Ho. This technique can
eliminate the portions of second rank tensor interactions that transforms in real
space as P2(cose) = (1 ~300s28), where 8 is the angle between the principal axis of
the tensor and the applied field. Interactions governed by this form include chemical
shielding anisotropies (axially symmetric), the anisotropic portion of the indirect
dipolar coupling, the direct dipolar coupling, and quadrupolar coupling with axially
symmetric field gradient tensors.

An illustration of the effect of MAS is provided in Figure I-8. Figure I-8a is a
simulated static CSA pattern of the 19F spins in calcium fluorapatite in a 9.4 T field,
and is approximately 32 kHz wide. This simulation was performed using the
measured values of the principal components of the 19F shielding tensor in
fluorapatite and neglected the hetero- and homonuclear dipolar couplings. A
simulation of a spectrum for the same spin system subjected to MAS is shown in
Figure l-8b. This spectrum exhibits typical appearance, having a comb of peaks
separated by the sample rotation frequency a), with an intensity profile that roughly
mimics the intensity in the static spectrum. In a MAS spectrum there is always a
peak at the isotropic chemical shift which is referred to as the centerband. The other
peaks (called the sidebands) can be distinguished from the centerband since they
change position in the spectrum when the spinning speed is changed. When the
sample spinning speed is much greater than the breadth of the static spectrum, the
sidebands become vanishingly small and only the centerband remains. The utility
of MAS-NMR is easily appreciated when one considers a static spectrum

37

 

 

 

 

 

-B’ A ULJL

 

 

A)

 

ilTrITIIIIIIIIFITIIIIIIIIIIIIIIIIIIIIIIIllTITITY‘IIITTTIIIIIIIIITTTII

140 120 100 90 80 7O 60 50 4O 30 ppm

 

Figure I-8. Simulated chemical shielding powder pattern with an axially symmetric
shielding tensor with chemical sheilding anisotropy CSA = 84 ppm, c iso = 64 ppm
and an asymmetry parameter 11 = 0: (a) under static conditions and (b) under Magic
Angle Spinning conditions with v, = 8.0 kHz. The centerband is marked with an
asterisk. An exponential apodization function corresponding to a 400 Hz line

broadening was applied to the simulated FID before Fourier transformation.

38

influenced by more than one chemical shielding tensor present. In such a spectrum
it would in general be difficult to determine that there were two or more distinct
tensors present if the difference between the isotropic chemical shifts was less than
the CSA powder pattern linewidths. On the other hand, the MAS spectrum would
resolve the peaks corresponding to the different isotropic shifts of the different
shielding tensors. A striking example of how well MAS can resolve distinct
resonances is given in Section II.

The ability of the MAS process to create well resolved peaks in the spectra
of solids enables the use of selective excitation schemes to saturate or invert the
magnetization of single peak. Selective excitation can be carried out by a number of
different schemes typically used in solution state NMR such as a soft pulse,
DANTE,16 or SELDOM.17s In this work (Sections II and Ill) DANTE16 pulse
trains are use to selectively invert or saturate peaks under MAS conditions. A
DANTE pulse train is simply a series of small flip angle pulses separated by
uniform time delays. The DANTE pulse train has the effect of exciting peaks on
resonance with the rf. Under MAS conditions optimal selectivity is achieved by
synchronizing the small flip angle pulses in the DANTE train with the rotor rotation
period. In sections II and III pulse sequences which use rotor synchronized DANTE

trains are discussed.

l-3.5 Spin Diffusion .

This section will qualitatively explain the concept of spin diffusion and
describe some experiments used to measure spin diffusion. Spin diffusion is a term
used to describe the transfer of longitudinal (z) magnetization between different
(spatially distinct) regions of the sample via the dipole-dipole coupling of nuclear
spins. Spin diffusion has the effect of moving spin polarization from regions of high

polarization to regions of low polarization. Spin polarization can be transferred in a

39

qualitatively similar manner by atomic or molecular diffusion; however, in solid
materials where atoms are essentially locked in place, the transfer of polarization
can only arise from spins diffusion processes.

The actual process that moves polarization through space is the spin flip-flop
or zeroquantum transition. For homonuclear isolated spin pairs the frequency of

flip-flop transitions between states (i. e. lap) —> Ilia) or Ilia) —-) IaB)) can be

 

approximated by the formula.5
a) : ES -ET 2 72h (1"3COS2 6). I'43)
I 11 Zr3

This formula is only an approximation in the sense that these m-basis states are
not the eigenbasis of the total Hamiltonian (Zeeman and dipolar).5 The types of
transtions expected for a pair of coupled spins are zero ( I018) —> I Bot) ) , single (
I 018) -> I act) ) and double I act) -+ I 88) quantum transitions. However, in a
large magnetic field the zero-quantum transitions shown above are almost
exclusively the only type that occur; since they are the only type that conserve total
Zeeman energy. It has been pointed out by Abragam5 that the problem can be
thought of as one of an rf field of amplitude H1 =(o1/y inducing transition between
two states lap) and I Ba).

Spectral spin diffusion refers to spin diffusion between spins with different
resonant frequencies. In this case a flip-flop transition has the effect of transferring
magnetization from one peak to another and can be directly monitored through the
intensity of the resonances in the spectrum. Unlike the pair of like-spins mentioned
above, the flip-flop associated with spectral spin diffusion does not take place
spontaneously since it does not conserve total Zeeman energy. Spectral spin
diffusion is enabled when there is a source of energy to compensate for the energy

difference between lap) and I Bot) . The sources of this additional energy will be

4O

discussed in Section II, in reference to spectral spin diffusion in fluorapatite under
MAS conditions.

A schematic representation of a spectral spin diffusion experiment is given in
Figure l-9. The basic scheme to measure spectral spin diffusion involves saturating
the magnetization of a desired resonance and observing cross relaxation
(magnetization transfer) to other resonances in the spectrum. Various methods can
be used to selectively saturate a line. After a mixing period, any z-magnetization is
turned into transverse magnetization with a #2 pulse for detection. Competing
processes that affect the dynamics of z-magnetization of the resonances are
longitudinal (T 1) relaxation and spin exchange (by atomic or molecular diffusion).
Provided that the rate of magnetization recovery from longitudinal relaxation is on
the order of (or slower than) the spin diffusion rate, and spin exchange is very slow
compare to spin diffusion, spectral spin diffusion can be detected by a scheme
pictured in Figure I-9.

The presence of rapid spin diffusion in a sample can manifest itself as equal
(or nearly equal) longitudinal relaxation times T1 of various resonances in the
spectrum. Spins are relaxed by the fluctuations in local fields. When paramagnetic
centers are present in a sample, the nuclei near the fluctuating magnetic dipole of
an unpaired electron can be relaxed many orders of magnitude faster than they
would be in a sample lacking paramagnetic spins. The consequence of this is
shown schematically in Figure l-10. At first, the spins near the paramagnetic center
repolarize quickly, but then polarization is carried from these highly polarized
regions into the rest of the sample. If spin diffusion is rapid, the recovery rate of the
magnetization of all spins in the sample will be close to that of those near the
paramagnetic center. Furthermore, if there is a low level of paramagnetic centers in
the material, the rate of repolarization of spins distant from the paramagnetic spins

will be influenced by two processes: longitudinal relaxation and the transfer of

41

polarization from paramagnetic centers by spin diffusion. In this case the observed
recovery of the magnetization is nonexponential. These spin diffusion effects will be

considered in section II.

42

Figure l-9. Schematic of a spin diffusion experiment. Spectral spin diffusion
between two reservoirs of spins with distinct chemical shifts (A and B) can be
detected and monitored by saturating peak A by one of several methods (see text)
and observing the transfer of z magnetization to peak B as a function of mix time.
The 2 magnetization is typically obtained by placing the magnetization into the x-y

plane with a non-selective read pulse.

8

spectral "picture" spatial "picture"

L

A

 

 

 

[TI

3 3 s:

/\ /\

saturated _— polarizedl-fi-Tll—l—

ll

 

 

 

 

paramagnetic center

 

 

      
 

   
  

 

Figure l-10. Schematic diagram of polarization transfer from a paramagnetic
center. The lengths of the vertical line segments indicate polarization as in Figure l-
9. Spins in saturated systems near paramagnetic centers become polarized rapidly,

and this polarization is transferred to the rest of the sample.

45

I-8 References
1. Silicon and sulfur replace two phosphorus ions as a charged coupled pair.
2. N. N. Greenwood and A. Earnshaw. Chemistry of the Elements. (Pergamon,
Oxford, 1986).
3. s. Naray-Szabo. z. Krist.75, 387 (1930)
4. W. E. Brown, M. Mathew, M. S. Tung, Prog. Cryst. Growth Charact. 4, 59 (1981 ).
5. A. Abragam. The Principles of Nuclear Magnetism, (Oxford, London, 1961 ).
6. G. E. Pake. J. Chem. Phys. 16, 327 (1948)
7. C. P. Slichter. Principles of Magnetic Resonance. 3 rd Edition. (spri nger-Verlag,
Heidelberg, 1990).
8. J. A. Sidles. Phys. Rev. Left. 68, 1124 (1992).
9. D. N. Shykind, J. Baum, S.-B. Liu, A. Pines and A. N. Garroway, J. Magn. Reson.
76 (1988) 149.
10. D. Suter, S. B. Uu, J. Baum and A. Pines. Chemical Physics. 114, 103-109
(1987).
11. D. P. Weitekamp, in Advances in Magnetic Resonance, Vol. 11, ed. J. S.
Waugh (Academic Press, New York, 1983) p. 111.
12. K. K. Gleason, Conc. Magn. Reson. 5, 199 (1993) .
13. J. Baum, M. Munowitz, A. N. Garroway and A. Pines, J. Chem. Phys. 83, 2015
(1985) .
14. S. Lacelle, in Advances in Magnetic Resonance, Vol. 16, ed. J. S. Waugh
(Academic Press, New York, 1991) p. 173.
15. B. E. Scruggs and K. K. Gleason, Chem. Phys. 166, 367 (1992) .
16. G. Bodenhausen, R. Freeman, and G. A. Morris. J. Magn. Reson. 23, 171
(1976).
17. D. Boudot, D. Canet, J. Brondeau, and J. C. Boubel. J. Magn. Reson. 83, 428
(1989).

46

Section II

19F and 31 P MAS-NMR of Antlmony(lII)-doped Fluorapatite Phosphors:
Dopant Sites and Spin Diffusion

1. Introduction

The apatite structure,1 with its extensive and varied substitutional
chemistry,2 is the basis for a wide variety of inorganic solids important in
biomineralization, geology, and materials science. From the materials science
standpoint, one extremely important technological application of apatites is as
halophosphate phosphors in fluorescent lamps. The most commonly used
phosphor is based on calcium fluorapatite [Ca5F(PO4)3]. In commercial phosphors,
approximately 20% of the fluoride ions are replaced by chloride ions. Also, 0.5 - 1.0
wt % Sb3+ is present as an activator to absorb UV irradiation produced by the Hg
discharge present in the lamp. The antimony subsequently converts this excitation
energy into visible light by either of two processes. Direct Stokes shifted emission
from antimony can occur, or the excitation energy can be transferred to Mn2"',
present in 0.0 - 0.5 wt.% quantities, which also emits light in the visible region of
the spectrum. A detailed understanding of UV absorption, energy transfer, visible
emission, and defect structures that lead to photodegradation requires knowledge
of the lattice substitution site of the activator ion. Whereas the substitutional sites
and oxidation state for manganese have been well established, 3-4 this is not true
for antimony. Moessbauer results on antimony-doped calcium fluorapatite

phosphors5 have shown that the antimony in the lattice has an oxidation state of

47

+3. However, identification of the substitution site in the lattice is controversial, with
at least two different sites having been proposed.5'9

The fluorapatite structure features linear chains of crystallographically
equivalent fluoride ions parallel to the c axis and extending the length of the crystal
(figure lI-1).1-2 There are two types of calcium ions in the structure, labelled Ca(1)
and Ca(2), in the ratio of 4:6. Surrounding each fluoride ion is an equilateral triangle
of crystallographically equivalent Ca(2) ions that lie in a plane perpendicular to the
fluoride chain. Adjacent triangles are rotated by 600 about the chain axis, creating a
hexagonal screw axis in the structure. Adjacent to these Ca(2) sites are
crystallographically equivalent phosphate tetrahedra. finally, there are columns of
Ca(1) ions running parallel to the c axis, each calcium column in the middle of an
equilateral triangular prism formed by three fluoride columns.

Previous studies have considered three different possible substitution sites
for antimony in the fluorapatite lattice. The first site of antimony substitution that has
been considered is the Ca (I) site;5'7 however, no direct experimental evidence for
this model has been found. A second model, based on Rietveld analysis of x-ray
powder diffraction data, invokes the replacement of Ca2+ at the Ca(2) site with an
Sb3+ ion;8 however, charge compensation was not explicitly discussed. Studies
based on analyses of stoichiometry as well as spectroscopic data also support the
replacement of Ca2+ at the Ca(2) site by Sb3+, accompanied by the replacement of
the adjacent fluoride ion by 02' to achieve charge compensation.6s7r9 The third,
most recent model, involves the substitution of phosphorus(V) in the phosphate
tetrahedron by Sb3+, with the loss of one oxygen atom of the tetrahedron providing
charge compensation (the 55 electron lone pair of the antimony(lll) atom is directed
toward the missing oxygen atom site in this model). This phosphate substitution
model is based on measurements from Moessbauer data as well as theoretical

calculations of antimony nuclear electric quadrupole coupling constants.5

60°

344 pm

 

 

 

 

i1— 937 pm #

figure "-1. Schematic diagram of the arrangement of four linear columns of fluoride

ions (0) in the fluorapatite (Ca5F(PO4)3) lattice.

49

The determination of which of these conflicting proposals is correct, and
whether there is exclusively one dopant substitution site, requires a technique
sensitive to local environments. High-resolution solid-state NMR, using the
technique of magic-angle spinning (MAS),10 is widely used to determine local
structure, but has rarely been applied to the study of dopants. The major difficulty in
studying dopant atoms at low concentrations in solids is limited sensitivity. Most
spin 1/2 nuclei have relatively low magnetogyric ratios and/or low natural
abundances, and quadrupolar nuclei (l>1/2) usually exhibit excessive NMR spectral
line broadening. However, both 19F, 31 P, and 1H nuclei have natural abundances
of 100 and large magnetogyric ratios, making them good prospects for studying
dopants (the high-resolution 31 P and 19F MAS-NMR spectra of undoped
fluorapatite have already been reported11'13). The large chemical shift range of
19F (several hundred ppm in metal fluorides“) makes it a particularly attractive
probe nucleus. However, strong homonuclear dipolar couplings can make it difficult
to obtain high resolution MAS-NMR spectra, particularly for 19F and 1H, unless
more involved multiple-pulse techniques are used.13r15 Fortunately, the availability
of MAS rotor designs having high maximum spinning speeds, combined with the
selection of samples having special structural arrangements, has made it possible
to obtain high-resolution 1H and 19F MAS-NMR spectra in a variety of interesting
systems. The special structural arrangements of homonuclear spins required
include relatively isolated spin pairs,1sa17 dilute spins, and linear chains of spins.18
The latter arrangement is characteristic of the fluorine atoms in fluorapatite, and
high-resolution 19F MAS-NMR spectra have been reported for calcium fluorapatite
and fluorohydroxyapatites.12

The use of MAS-NMR to probe local structure has generally involved the
measurement and interpretation of isotropic chemical shifts and possibly chemical

shift anisotropies. Special techniques to measure heteronuclear dipolar couplings

50

have also proven useful.19‘21 Homonuclear dipolar couplings of abundant spins in
MAS-NMR spectra manifest themselves in (at least) two ways: as ”dipolar
sidebands " in the spectra,15»17 and as the source of "spin diffusion” contact
among spins. Despite the great potential for obtaining local structhal information
from such spin diffusion measurements, since the dipolar coupling is inversely
related to the cube of the internuclear distance, few applications have been
reported.22‘25 In this study, we demonstrate that in an abundant homonuclear
dipolar system spectral spin diffusion between peaks with different isotropic
chemical shifts can be observed using a one-dimensional (1 -D) MAS-NMR
experiment referred to as selective population anti-z and rate of tranfer to adjacent
nuclei (SPARTAN).26 A rotor-synchronized DANTE pulse train is used to
selectively invert the magnetization of a 19F peak and its associated sidebands,
and all peaks are nonselectively observed after a mixing period during which spin
diffusion occurs. The change in intensity of a peak resulting from cross-relaxation to
other peaks as a function of the mixing time provides a quantitative measure of spin
diffusion and can aid in assignment of peaks.

In this study, we report quantitative 19F and 31 P MAS-NMR results for
calcium fluorapatite samples containing 0.0 to 3.0 total wt. Sb3+ as well as 31 P
MAS-NMR results for the model compounds SbPO4, Sr1_030a3_g7F2(PO4)5,
Sr5F(PO4)3 and Ba5F(PO4)3. The 19F and 31 P MAS-NMR results, including the
19F MAS-NMR spin diffusion experiments, will be discussed in relation to three
possible models for substitution of antimony into the fluorapatite lattice. We will
show that only the phosphate substitution site (the third model) is consistent with
the totality of experimental results.

51

2. Experimental
2.1. Sample preparation and characterization

The samples of antimony-doped fluorapatite that were prepared contained
0.00, 0.18, 1.3, 1.7, 2.1, and 3.0 wt.% antimony(lll) (determined by permanganate
titration). Samples containing 0.00, 1.7, and 3.0 wt% antimony were prepared by
combining Ca2P207, CaO, CaFg, and CaSb206 (GTE Chemical and Metals
Division, Towanda PA) and firing in a covered Coors porcelain crucible at 1200°C.
Both the 1.7 and 3.0 wt. % antimony samples were firings of identical raw material
blends. The difference in the final antimony levels in these two samples arises from
significant vapor transport taking place during firing as described in reference 8.
Because of this vapor transport process, it is difficult to control the final antimony
level in a given sample. The samples containing 1.3 and 2.1 wt. % antimony were
prepared from identical raw material blends, but in a different manner than the
samples containing 1.7 and 3.0 wt% antimony. The reagents CaHPO4, Ca003,
and CaFg (GTE Chemical and Metals Division, Towanda PA) were first fired in
stagnant air to create a fluorohydroxyapatite (with an indication of some carbonate
substitution for phosphate from IR spectroscopy). CaSb206 and a small amount of
Ca2P207 were then fired with the precursor at 1200°C in a covered Coors
porcelain boat. The 0.18 wt% sample was prepared by firing a combination of
Ca2P207, CaO, CaF2, and Sb203 at 1100°C in an open quartz boat with a flowing
nitrogen atmosphere. Finally, the 0.8 wt % sample was a commercial
halophosphate phosphor (GTE type 2440)

The specific surface areas of the samples containing 0.0 wt.% and 2.1 wt.
antimony were measured with a Quantachrome Monosorb Brunauer-Emmett-Teller
(951) apparatus and were determined to be 0.626 i 0.007 and 0.506 :I: 0.004 m2/g
respectively.

52

The SbPO4 sample was obtained from GTE Chemical and Metals Division,
Towanda PA, and was annealed in air at 600° C for 1/2 hour to improve its
crystallinity. Its identity was confirmed by using X-ray powder diffraction (two small
peaks attributable to Sb204, cervantite, were also present).

The Sr 1.03Cag.97F2(PO4)6 was prepared by combining the following
starting materials in the molar ratio: 6.0 moles CaHPO4, 2.982 moles Ca003,
0.994 moles SrF2, and firing for 2 hours at 1150°C under nitrogen in a Crystar

crucible.

2.2 NMR equipment and procedures

The 31 P and19F MAS-NMR spectra were recorded on a 9.4T Varian
Associates VXR-400 spectrometer at the Max T. Rogers NMR Facility at Michigan
State University, operating at frequencies of 161.76 MHz, and 376.2 MHz
respectively. The fluorine radiofrequency was amplified by an AMT model
3137/3900-2 amplifier. The phosphorus radiofrequency was amplified by an
Amplifier Research model 1000LPM10 amplifier. A fast switching PIN diode was
used to isolate the receiver during the pulse, and 1.0 its after the end of the pulse.
The 19F spectra were collected by using an ultra-high speed 19F MAS-NMR probe
from Doty Scientific (Columbia, SC) with 5 mm o.d. Si3N4 rotors with Vespel caps.
The probe was equipped with a fiber optic detector to measure both the speed and
the position of the rotor for the rotor-synchronized spin diffusion experiments. The 5
mm rotors contained approximately 160 mg of sample, and spinning speeds or
were typically 8.25 i 0.1 kHz. The 1t/2 pulses for 19F were 4.0-4.2 us. The magic
angle of the 19F MAS-NMR probe was set by minimizing the linewidth of the
fluorapatite peak. Chemical shift referencing of the 19F MAS-NMR spectra was
accomplished by using fluorapatite prepared as in ref. 12 as a secondary chemical

shift reference at 64.0 ppm (downfield) with respect to hexafluorobenzene at 0.0

53

ppm.12 The 19F MAS-NMR spectra were obtained with a 100kHz sweep width,
using 32,000 data points. An exponential weighting corresponding to a line
broadening of 37 Hz (0.1 ppm) was applied to the FID before Fourier
transformafion. Eight transients were collected for each Bloch decay 19F MAS-
NMR spectrum with relaxation delays of at least five times the spin lattice relaxation
time T1.

The 31 P MAS-NMR spectra were obtained by using a high speed
multinuclear MAS-NMR probe from Doty Scientific (Columbia, SC) with 7 mm o.d.
Si3N4 rotors with Vespel caps. The rotors contained approximately 500 mg of
sample. The probe was equipped with a fiber opfic detector to measure the
spinning speed of the rotor. Spinning speeds v, for the 31 P MAS-NMR experiments
were 6.15 :l: 0.1 kHz and the 1d2 pulses were 6 us. The magic angle was set by
observing the 79Br resonance in KBr. The chemical shifts of the 31F MAS-NMR
spectra were referenced to the peak of fluorapatite at 2.8 ppm downfield (with
respect to 85 % H3PO4).11 The 31 P MAS spectra were obtained with a 30kHz
sweep width, using 48,000 data points. An exponential apodization was applied to
the FID corresponding to a line broadening of 100 Hz.

The 31 P MAS-NMR spectra were obtained by collecting 80 transients using
1J4 (3 its) pulses and a delay time of 180 s between scans. Since 31 P MAS-NMR
spin lattice relaxation times can be extremely long in rigid solids (hours), the
spectrum of the 2.1 wt % Sb3+ fluorapatite sample was also obtained with a very
long relaxation delay of 3600 s and eight transients; no additional peaks due to a
slowly-relaxing component could be detected.

Values of the 19F spin lattice relaxation time T1 were obtained with an
inversion recovery pulse sequence with four transients for each value of the delay
time, and a relaxation delay at least 5 times the T1 of each sample. Thirteen delay

time values were used for each T1 experiment. The T1 values of the centerbands

54

and sidebands (equal within experimental error for a given sideband family) were
calculated by using peak heights and the VNMR 2.2 software. Values of the 19F
transverse relaxation times, T2, were obtained by using a rotor synchronized Hahn
spin echo experiment with four transients for each value of the delay. The value of
T2 was obtained by fitting the peak heights to an exponential using Varian VNMR
2.2 software. This sequence comprises a (1:12),, pulse, a delay, a (“)x pulse applied
at the same rotor orientation (i0.5°) as the (ft/2)x pulse, and a further delay before
acquisition. Synchronization was necessary to ensure that the second pulse comes
at the maximum of a rotational echo.18

19F MAS-NMR spectral spin diffusion was measured by selective inversion
of a specific resonance and its associated sidebands, followed by a cross-
relaxation delay and then a nonselective 11/2 pulse to monitor the longitudinal
magnetization (Figure "-2). Selective inversion of the centerband and sidebands
was achieved by a DANTE pulse train27 that consisted of fifteen 12 (1.6 us)
pulses applied once each rotor period at identical rotor orientations (to within
i 0.5°). These pulse lengths (and the corresponding power levels) were chosen to
give an optimal excitation profile for the peaks in the spectra. The nonselective 1t/2
observation pulse was 12 its, and its phase was alternated to cancel out the effects
of DANTE pulse trains that did not fully invert the magnetization of a specific
resonance.28 The cross-relaxation mixing period was varied from 1 ms to 12 sec,
with 8 transients collected for each mixing time.

Deconvolution of overlapping 19F MAS-NMR peaks was performed with the
standard deconvolution tools in VNMR 2.2, the software package on the Varian
VXR-400 spectrometer. This program operates by fitting the spectrum to a set of
user selected Lorentzian peaks by varying the line widths, chemical shifts, and

peak intensities.

55

DANTE (selective 180°) nonselective 90°
' |

 

 

 

 

 

 

 

 

 

 

figure "-2. SPARTAN pulse sequence for measuring spectral spin diffusion (see
text).

56

3. Results
3.1 31 P MAS-NMR .

The 31 P MAS-NMR spectrum of fluorapatite is shown in Figure Il-3a. It
contains a peak at the isotropic chemical shift of 2.8 ppm with a half-height
linewidth of 0.4 ppm and small sidebands arising from a small chemical shift
anisotropy.11 The spectrum of fluorapatite containing 2.1 wt. % antimony is shown
in Figure 3b. There are no additional peaks observed in the spectrum of antimony
doped fluorapatite with enough intensity to be attributable to the presence of
antimony (i.e. if each antimony ion perturbed five phosphorus atoms, eg. by
occupying a calcium site, the corresponding perturbed 31 P peak would have 15.1%
of the total spectral intensity). The small peaks shown in the expansion of Figure Il-
2b at approximately 9 ppm can be attributed to a very minor pyrophosphate impurity
phase. The 31 P MAS-NMR spectrum of antimony phosphate (SbPO4) is shown in
Figure ll-3c. A rather broad peak (Av1/2=4.0ppm) with an isotropic chemical shift of
-18.3ppm and weak sidebands is seen.

To test the effect of substituting different group II metal ions in the Ca sites,
31 P MAS-NMR spectra of Sr1.030a3.97F2(PO4)6, Sr5F(PO4)3, and Ba5F(PO4)3
were obtained. The spectrum of Sr1.030a3.g7F2(PO4)6 contains a peak with an
isotropic chemical shift of 3.3 ppm, with an upfield shoulder at 2.4 ppm. The
spectnim of Sr5F(PO4)3 contains a peak with an isotropic chemical shift of 3.3
ppm. The spectrum of Ba5F(PO4)3 contains a major peak at 1.4 ppm and two
weaker peaks, the first at 2.3 ppm with one twelfth the intensity of the main, and the

second (a weak shoulder) at 0.5 ppm, both presumably arising from impurities.

57

(a)

 

 

.2 JL -
(b)

JJ @ij

(C)

 

 

A

 

 

-"* — :— ‘Aufi—

IIWIIIIIIIITIllTiil‘iliiIllnllll[Iii]VIIIIIIIIIIIHIIITVIIIIll

50 30 O -20 -40 ppm

 

 

 

figure "-3. 31 P MAS-NMR spectra with W: 6.15 i 0.1 kHz; A. pure fluorapatite; B.
fluorapatite containing 2.1 wt.% antimony(lll) dopant, 20x vertical expansion above;
C. SbPO4.

58

3.2 19F MAS-NMR

The19F MAS-NMR spectrum of fluorapatite is shown in Figure ll-4a. The
isotropic chemical shift is 64.2 ppm and the Av1/2 value is 0.5 ppm. The sidebands
of fluorapatite in the 19F MAS-NMR spectrum arise principally from the axially
symmetric chemical shift tensor (the anisotropy has been reported as 84 and 85
ppm29u30), although the 19F-19F dipolar couplings31 and to a lesser extent 19F-
31 P dipolar couplings also contribute. The 19F MAS-NMR spectrum of fluorapatite
with 2.1 wt% antimony is shown in Figure II-4b. This spectrum contains an intense
resonance at 64.2 ppm and its associated sidebands, similar to those of
fluorapatite, as well as three new peaks at 73.1 ppm, 68.6 ppm, and 65.6 ppm. The
65.6 ppm peak is a shoulder of the main peak at 64.2 ppm but is resolved by
difference spectroscopy, as discussed below. Since peaks at (or near) 68.6 and
65.6 ppm are observed in the spectra of all of the samples containing antimony, we
will refer to them throughout this paper as peaks A and B respectively. Due to its
overlap with peak C, the chemical shift of peak B is less accurately determined than
those of peaks A and C. The large main peak at (or near) 64.0 ppm will be referred
to as peak C. The weak, broad peak at 73.1 ppm is present only in the 19F MAS-
NMR spectra of the samples containing 1.3, 1.7, and 2.1 wt.% antimony. All of the
spectra of fluorapatite containing antimony have a very small upfield shoulder on
peak C. Deconvolution indicates that the intensity of this shoulder is less than one-
hundredth the intensity of peaks A and B, implying that this peak does not arise
from the same source as peaks A and B. Since the shoulder is so weak, we will not
discuss it further. The chemical shifts, Iinewidths, spin lattice relaxation times (T 1 ),
and intensities (as a percentage of the total intensity) obtained from deconvoluh'on
of the peaks in each of the 19F MAS-NMR spectra are given in Table "-1.

59

(a)

 

 

 

 

 

 

 

 

 

aJLJU

140 100 80 60 4O 20 ppm

figure "-4. 19F MAS-NMR spectra with v, = 9.25 s 0.1 kHz: A. fluorapatite; B.
fluorapatite containing 2.1 wt.% antimony(lll); C. vertical expansion of spectrum in
B.

60

Table "-1. 19F MAS-NMR data for antimony(lll)odoped fluorapatite samples.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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a (ppm [w.r.t hexafluorobenzene], Hz, percentage of total)

61

It has been demonstrated32 that random molecular (or atomic motions) can
interfere with coherent NMR line narrowing techniques such as MAS, resulting in
broadening of lines if the MAS rotor period is comparable to the molecular
correlation time. In order to demonstrate that random atomic motion is not
broadening any of the peaks in the 19F MAS-NMR spectrum to the point where
they are undetectable, the spectrum of fluorapatite containing 1.7 wt. % antimony
was obtained at 223 K. Since there is no difference between the spectrum obtained
at 223 K and the one obtained at 295 K, random atomic motions do not appear to
be interfering with the MAS process and producing broadened peaks.

In order to obtain more accurate chemical shift, linewidth, and intensity
measurements of the shoulder corresponding to peak 8 in the spectra of doped
fluorapatite, difference spectroscopy was used for all samples. Figure "-5 is a 19F
MAS-NMR difference spectrum obtained from the spectrum of fluorapatite
containing 2.1 wt.% antimony and the spectrum of pure fluorapatite. Because the
chemical shift of peak C in the doped sample is 0.2 ppm upfield of the chemical
shift of fluorapatite, the difference spectrum was obtained by varying the frequency
offset, linewidth (exponential apodization), and intensity (scaling factor) of the pure
fluorapatite spectrum to provide the best null of the centerband of peak 0. The
amount of applied line broadening (apodization) required to make the linewidth of
the centerband in the fluorapatite spectrum equal to the linewidth of peak C in each
spectrum of fluorapatite containing antimony increased monotonically with the
amount of antimony incorporated into the lattice. This suggests that the presence of
antimony in the lattice slightly perturbs the apatitic fluoride ions which give rise to
peak C, resulting in a symmetrical linebroadening (the specific surface areas of the
0.0 and 2.1 wt.% Sb samples are similar, arguing against crystal size effects being
responsible). The additional linebroadening applied to the fluorapatite spectrum to

achieve nulling of peak C in the difference spectrum does not result in a significant

62

reduction of the intensity of peak B in the difference spectrum since the applied
broadening is small (between 0.1 and 0.8 ppm) compared to the chemical shift
difference between peaks B and C (1.6 ppm). The difference spectrum clearly
shows distinct peaks for A and B. The ratio of the intensity of peak A to peak B,
determined by integration, equals 2:1 to within 16% uncertainty for all samples
(except for the 0.18 wt. % sample). The deviation of this ratio from 2:1 becomes
less with increased dopant levels, presumably because the correspondingly greater
intensity of peaks A and B results in less uncertainty in their integrated areas.

The relative intensities of the sidebands of peak c in the 19 F MAS-NMR
spectra of the samples of fluorapatite containing antimony are similar to those in the
spectrum of fluorapatite, but the intensity of the third left sideband of peak C (in the
sample containing 2.1 wt.% antimony) is only 7O % of that of the corresponding
sideband in pure fluorapatite.

The vertically expanded centerband regions of the 19F MAS-NMR spectra of
each of the antimony doped fluorapatite samples as well as that of the undoped
sample are shown in Figure "-6. The heights (off scale) of peak C‘for each of these
spectra were set equal. The intensity of the peaks A and B associated with the
presence of antimony in the sample do not monotonically increase with wt. % of
total antimony, since not all of the antimony is present in the site responsible for

these peaks (see below).

63

 

 

 

 

7B 74 7O 66 62 58 ppm

Figure "-5. 19F MAS-NMR difference spectrum of fluorapatite containing 2.1 wt.%
antimony(lll) and undoped fluorapatite (see text).

64

wt.% Sb(|ll)

 

 

 

 

 

 

 

 

 

 

 

 

2.1

ML“:
jL
WWW-"mm

76 72 BB 64 60 56 ppm

Flgure "-6. 19F MAS-NMR spectra (centerband region only) of antimony(lll)-doped
fluorapatite samples, v, = 8.25 :t 0.1 kHz, wt.% antimony as indicated. The height of
the main apatite peak (off scale) is equal for all spectra.

65

The T1 values for peak C given in Table "-1 are the average of the (nearly
identical) T1 values for the centerband and the first right and first left sideband. The
T1 values for peak A are simply the T1 values of the centerband. The T1 values
given in Table 1 for peaks A and C vary by no more than 9% for a given sample.
The T1 values of peak B for the various samples are not reported, since this peak
severely overlaps peak C. Furthermore, the T1 values of the 73.1 ppm peak are not
reported since the intensity of this peak is too small. Nevertheless, the T1 values of
peak B and the 73.1 ppm peak are very close to that of peak C in a given sample,
since their intensities follow the intensity of the main peak C for various delay times
in the inversion recovery experiments. The spin lattice relaxation times of peaks A,
B, C, and the peak at 73.1 ppm are equal to within experimental error for each
sample; however, they vary irregularly from 129 s to 378 s, with no perceptible
relationship to the antimony content.

The measured 19F MAS-NMR transverse relaxation times (T 2) of peak C in
the samples containing 2.1 and 1.7 wt. % Sb are 32 ms and 40 ms respectively. A
T2 value of 32 ms would contribute 10 Hz (0.027 ppm) to the linewidth of peak C for
the sample containing 2.1 wt.% antimony, and 8 Hz (0.021 ppm) for the sample
containing 1.7 wt. % antimony. The external magnetic field was shimmed by
minimizing the proton line width of water in a spinning MAS rotor, yielding an
estimated contribution to the half-height linewidth from field inhomogeneity of about
20 Hz (0.053 ppm). The observed half-height linewidths of the samples are
considerably greater than the sum ofthe homogeneous broadening and the
contribution from field inhomogeneity (ca. 0.1 ppm); they vary from 0.5 to 1.0 ppm
(Table "-1). These half-height linewidths are significantly smaller than those
previously reported for fluorapatite,12:33 possibly because the fluorapatite in the

earlier study was synthesized by an aqueous precipitation method.

66

Figure "-7 shows the 19F MAS-NMR spectra of fluorapatite containing 1.7
wt. % antimony obtained with the SPARTAN pulse sequence. The frequency
selectivity of the DANTE train was adjusted so that peak A would be inverted and
the other peaks would remain unperturbed. The negative longitudinal magnetization
of the inverted peak A comes to quasi-equilibrium with the relatively large bath of
positive magnetization in the main apatite peak C in approximately 12 s (a time
much less than the measured spin-lattice relaxation time of 193 5). While the height
of the main peak decreases noticeably, the height of peak B remains constant. The
total integral of the centerband region containing peaks A, B, and C should remain
constant if spin diffusion is the only process affecting peaks A, B, and C.
Measurements of the integral of this region show a 1.5 °/o increase from short to
long mixing times, which is the amount expected from spin-lattice relaxation. To
obtain the spin diffusion time constant from the spectra in Figure "-7, we have
adapted the analysis in reference 34 for the present experiment. Although the data
could be treated by simply fitting the magnetizations of peaks A and C to simple
exponential curves, our analysis incorporates both data sets into one plot. The spin
diffusion time constant is obtained by plotting ln(rf + f)/r vs. tm (mixing time). The f in

this relationship is given by the equation

___ MAT,” = oo)—Ma(rm = 0)
M.(T..=°°)

 

f , "-1

where MA(‘Cm= op) and MA(tm= 0) are the intensities of peak A obtained from an

exponential fit. In this relationship r is given by the equation
,= MM»; , ll-2
MC(Trn)_ MC(TUI : 0)
where MA(‘f.‘m) is the integrated intensity of peak A, MC(‘tm) is the integrated
intensity of peak C, and MC(‘tm = 00) is the intensity of peak C at infinite mixing time

67

 

 

 

 

 

'"lT'V' Witt mvi '“T'lTl "W

74 72 70 68 66 64 62 6O PPm

Figure "-7. 19F MAS-NMR spectra (centerband region only) obtained using the
SPARTAN pulse sequence (Fig. "-2 and text) of fluorapatite containing 1.7 wt. °/e
antimony(lll). Spectra for cross-relaxation mixing periods of 0.1, 0.5, 1, 2, 4, and 12
seconds are superimposed. Peak A (at 68.6 ppm) is inverted at short mixing
periods, and peak C (at 64.0 ppm) steadily decreases in intensity for longer mixing
penods.

68

obtained from an exponential fit. Contributions to MC(tm) and MA(‘tm) from the
much slower longitudinal relaxation process can be safely neglected. The plot,
shown in Figure "-8, should be linear for a process governed by a single spin
diffusion rate, with the slope of the line equal to the reciprocal of the spin diffusion
time constant, 1/150. The slopes of the plots were determined by fitting the data to
a line, weighting points inversely proportional to their variances squared. The
values of 130 for the data sets shown in Figure "-8 are 0.68 :l: 0.04 s for v, =5.09
kHz, 1.79 :l: 0.11 s for v, =6.98 kHz, and 2.22 i 0.10 s for v, =9.23 kHz. The error
values given are one standard deviation of 1230 as determined by weighted
regression. To show convincingly that there is no spin diffusion between peaks A
and B, the SPARTAN pulse sequence was also used on the fluorapah'te sample
containing 0.8 wt.% antimony, since peak B was most well resolved for this sample.
As in the 1.7 wt.% sample, the height of the main peak C decreases noticeably as
peak A becomes positive and peak B remains constant within experimental error.
The spin diffusion time constant for this sample was 1.95 % i 0.31 s for v, =9.26
kHz.

To test the effect of temperature on the spin diffusion rate, the SPARTAN
experiment was performed on the sample containing 1.7
wt. % antimony at a temperature of 223 K and v, = 8.10 kHz. The measured rate of
spin diffusion between peaks A and C at the lower temperature was essentially
unchanged ( 130 = 1.89 i 0.12 s).

Attempts to monitor spin diffusion by inverting peak B and observing cross
relaxation to the other peaks were unsuccessful since no combination of pulse
lengths or power levels was found that could selectively invert peak B without

significantly exciting peak C.

69

 

[J
()r——-+ ——-t D
O

ln((rf+f)/r)
re

 

 

 

 

 

T
' I
1. 00+ 0
L +30
10
0 o
0 2 4 6
Time (sec)

Figure "-8. Measurement of spin diffusion rates between peaks A and C in the 19F
MAS-NMR spectra of fluorapatite containing 1.7 wt % antimony at three different
spinning speeds using the SPARTAN pulse sequence:

In ((rf + f)/r) vs. Tm (see text). vr= 9.23 kHz (0), 6.98 kHz (+), and 5.09 kHz (0) .
Representative error bars are included on two of the data points.

70

4. Discussion

The discussion portion of this paper is divided into three sections. The first
section discusses the basic principles of spectral spin diffusion, and shows how the
experimental results involving spin diffusion enable one to assign the 19F MAS-
NMR peaks to specific antimony-perturbed fluoride ions in the apatite Iath’ce. The
second section first demonstrates that potential effects upon the MAS-NMR spectra
arising from the presence of quadrupolar antimony nuclei do not appear, due to
rapid relaxation of the quadrupolar nuclei. It then discusses the quantitative aspects
of the 19F MAS-NMR spectra, and the agreement between the NMR results and IR
results regarding the amount of antimony actually incorporated into the apatite
lattice. In the third and final section the phosphate site is shown to be the only
location for antimony in the fluorapatite lattice that is consistent with all of the
experimental MAS-NMR results. Molecular modelling of the apatite lattice leads to a
plausible assignment of the 19F MAS-NMR spectra.
4.1. Spin Diffusion

Spin diffusion is a term used to describe the transfer of longitudinal(z)
magnetization within a homonuclear spin system between different regions of
sample via the dipole-dipole coupling of nuclear spins.35s36 The process is often
assumed to exhibit general Fickian diffusion behavior, although we need make no
such assumptions here, especially since our system is basically one-dimensional.
Spin-diffusion allows paramagnetic centers or mobile domains to act as relaxation
sinks for spins throughout the sample, and can manifest itself as a nonexponential
spin-lattice relaxation process.35r36 Spectral spin diffusion refers to the situation
where the spins have different resonance frequencies,37 and is usually observed in
single crystals or under MAS conditions. The transfer of polarization is

accomplished by the simultaneous exchange of Zeeman magnetization between

71

two adjacent spins, a process usually referred to as a spin flip-flop. One
manifestation of spectral spin diffusion is an equalization of the longitudinal
relaxation times of spectrally resolved spins.38 In a limited number of cases,
spectral spin diffusion in solids has been observed directly by selectively inverting
the magnetization of one peak and monitoring cross-relaxation to other peaks.
24,25,28,39-41

Spectral spin diffusion can occur provided inter-spin flip-flops conserve the
total Zeeman energy. Sources of energy that can balance any mismatch in the
Zeeman energies include large extraneous dipolar spin reservoirs, such as
abundant protons24r 28. 4043, external radio-frequency fields,44 and coupling
between the spin system and the rotation of the sample”: 38- 42- 45. 45 None of
the spectra reported here were obtained under such conditions.

Kubo and McDowell28 have pointed out that the scalar coupling can
facilitate spin diffusion under MAS conditions. Through space scalar couplings
between 19F nuclei have been reported to be as large as 217 H147»48 Since the
intensity of peak A in the Hahn spin echo experiments was observed to decay
monotonically with a time constant ('72) of 34 ms, any scalar coupling to a
neighboring C fluorine nucleus must be less than approximately 15 Hz. Since the
magnitude of the scalar coupling (15 Hz) is substantially less than the dipolar
coupling, spin diffusion must proceed predominantly through the dipolar interaction.

It is conceivable that atomic motions with spectral density at the frequency
difference between peaks A and C could provide energy to compensate for the
Zeeman mismatch between these peaks, and thus facilitate spin diffusion. The fact
that the spin diffusion rate is essentially unchanged at lower temperatures indicates
that if such a mechanism is responsible for the observed spin diffusion, the motion

must have an activation barrier less than kT.

72

Kubo and McDowell28 have pointed out that only if two identical chemical
shielding tensors of neighboring spins have the same orientation will there
generally be efficient spin diffusion. This situation certainly applies to the
unperturbed fluorine nuclei in the fluorapatite lattice, whose axially symmetric shift
tensors have their principal axes along the c-axis of the crystal. However, spin
diffusion in MAS-NMR spectra can occur, albeit less efficiently, even if the chemical
shielding tensors have a different orientation, or are unequal. For spin diffusion to
occur under such conditions, without other sources to provide for Zeeman
mismatch, level-crossing must take place.28 A similar enhancement of spin-
diffusion by levelocrossing during sample rotation for quadanolar powder patterns
of deuterium nuclei has been reported by Eckman.“9 Since the frequency of each
of the resonances varies during the rotor cycle, 50:51 level-crossing can occur
when the two frequencies coincide.

The effects of spectral spin diffusion provide information as to whether peaks
A, B, and the peak at 73.1 ppm in the 19F MAS-NMR spectra of antimony-
containing fluorapatite arise from fluorine atoms in the apatite lattice, or in
secondary, non-apatitic phases. Since the 19F T1 values vary greatly (and
irregularly) among the antimony containing fluorapatite samples (130 to 380 sec),
the primary source of longitudinal relaxation is probably paramagnetic impurities
introduced during synthesis, and not intrinsic lattice motions. When paramagnetic
species are present, the magnetization recovery is expected to be,35 and is
observed to be, nonexponential. However, the T1 values reported in Table 1 were
obtained by fitting the relaxation data to a single exponential, which appeared to
provide a good fit (no significant breaks in the logarithmic plot of the relaxation data
were obsenrable).

The fact that the T1 values of peaks A, B, C, and the 73.1 ppm peak are

identical within experimental error suggests that the spins corresponding to these

73

peaks are in spin diffusion contact with each other over the long time scale of T1
(but see below). This implies that the corresponding spins are in the same
crystallites, and not in different (secondary) phases. The rate of spin diffusion
depends upon, among other factors, the square of the dipolar coupling constant
between pairs of spins.38 Fluorapatite contains parallel linear chains of fluoride
ions with intra-chain fluorine atoms 344 pm apart, and with an inter-chain closest
fluorine-fluorine distance of 937 pm. The corresponding calculated maximum
homonuclear dipolar couplings (internuclear vector parallel to the magnetic field)
are 5234 Hz and 262 Hz respectively. The largest fluorine-phosphorus dipolar
coupling is ca 2200 Hz, and the largest phosphorus-phosphorus dipolar coupling is
ca. 640 Hz. The spin diffusion we observe in the present system is for a spinning
speed (up to 9.23 kHz) significantly exceeding the largest dipolar coupling in the
spin system. As far as we are aware, this is the first such case reported. Spin
diffusion has been reported to be quenched by MAS spinning speeds that exceed
the dipolar couplings,42 but the quantitative degree of reduction in rate or
quenching in this case has apparently not been studied experimentally.

Spectral spin diffusion among peaks in the 19F MAS-NMR spectrum is most
likely enabled by an instantaneous match of the frequencies of the spins of peaks
due to differences in the chemical shielding tensors of the spins (see above). The
isotropic chemical shifts of peaks A and C differ by 4.6 ppm, which is only a small
fraction of the total chemical shift anisotropy of 84 or 85 ppm29:30. Thus, it is
plausible that the frequencies could match during the rotor cycle. Although it was
not possible to accurately determine the components of the chemical shielding
tensors of peaks A, B, and C, the slight differences in sideband intensities of peaks
A and C suggest a difference in the chemical shift anisotropy of each peak.52 The
increase in the spin diffusion rates between peaks A and C at lower spinning

speeds is expected since the zero-quantum linewidth would be greater at lower

74

spinning speeds,28 thus increasing the time during which the zero-quantum
linewidths of peaks A and C overlap during a given rotor period.

The spectral spin diffusion results obtained using the SPARTAN sequence
allow assignments of peaks A, B, and C. Figure "-9 shows a geometrical
arrangement of the fluorine nuclei of peaks A and B in the fluorapatite lattice that
explains both the observed spin diffusion between peak A and peak C and the
considerably reduced spin diffusion between peak A and peak B. The model also
takes into account the 2/1 intensity ratio of peaks A and B discussed in section 4.2.
In this arrangement, the fluorine nuclei of peaks A and C are neighboring, with no
intervening peak B nuclei, thus permitting facile spin diffusion. Since peaks A and B
arise from fluorine nuclei in different chains, direct spin diffusion between these
peaks would be negligible; instead, spin diffusion would proceed indirectly and
more slowly through the nuclei of peak C. We should note that a model in which the
two fluorine nuclei of peak A are on different chains would also be consistent with
the spin-diffusion results; however, structurally this model is less plausible (see
section 4.3.1 ).

4.2. Quantitative aspects of 19F MAS-NMR results and
extent of antimony incorporation

In order to obtain quantitative information about the amount of antimony in
the fluorapatite lattice through its effect on the 19F MAS-NMR spectrum, it is
important to consider the effect that the quadrupolar antimony nucleus could have
on the spectrum of the spin-1/2 fluorine nucleus. Antimony has two quadrupolar
isotopes: 12186 with l = 5/2 and a natural abundance of 57.25%, and 12336 with l
= 7/2 and a natural abundance of 42.75%. Effects of half-integral quadrupolar nuclei
on the spectrum of a Spin-1/2 nucleus under MAS conditions have been observed
experimentally and calculated theoretically. 5355 Generally, magic angle sample

spinning fails to eliminate the effect of dipolar coupling on the spectrum of a spin-

75

1/2 nucleus dipolar coupled to a quadrupolar nucleus, because the axis of
quantization of the quadrupolar spin is not parallel to Ho. The MAS-NMR spectrum
of a spin-1/2 nucleus dipolar coupled to a quadrupolar nucleus is usually a
complicated powder pattern with singularities, whose total breadth is roughly on the
order of the dipolar coupling between the spin-1/2 and quadrupolar nuclei. The
details of the pattern are governed by the sign and magnitude of the nuclear
quadrupole coupling constant, the nuclear spin, and the angles between the
principal axes of the dipole and quadrupolar interaction tensors.53 The nuclear
quadrupole coupling constant of the 12‘ Sb nucleus in fluorapatite has been
measured to be approximately 700 MHz.5

For antimony nuclei in any of the proposed substitution sites, significant
dipolar couplings to both fluorine and phosphorus nuclei would be expected to
produce observable quadrupolar effects in the 19F and 31 P MAS-NMR spectra.
The calculated dipolar couplings between an 121 Sb (123Sb) antimony nucleus
occupying the Ca(2), Ca(1) site, or phosphate site and the nearest fluorine nucleus
in the fluorapatite lattice are 2.45 kHz (1.33 kHz), 161 Hz (87 Hz),and 630 Hz (340
Hz) respectively. If antimony nuclei were producing quadrupolar effects upon the
19F MAS-NMR spectra, peaks A and B would be much broader than peak 0;
however, only peak A is slightly broader, and we ascribe this to the presence of two
slightly different isotropic chemical shifts (see section 4.3.3). Furthermore, if peaks
A or B were influenced by dipolar coupling to antimony they would not be simple
Lorenztian peaks but rather complicated powder patterns.53 The peak at 73.1 ppm
and peaks A and B in the 19F MAS-NMR spectra of antimony doped fluorapatite
are not part of such a powder pattern, because the intensity of the 73.1 ppm peak
varies independently of the intensity of peaks A and B for the various samples.

Paralleling the 19F MAS-NMR results, the 31 P MAS-NMR spectra of
antimony doped fluorapatite samples fail to exhibit any quadrupolar effects. The

76

 

Figure "-9. Schematic diagram as in Fig. "-1, showing the arrangements of fluoride
ions giving rise to peaks A and B in the 19F MAS-NMR spectra. The antimony atom
(not shown) is between the fluoride ions labelled A and B (see also Fig. ll-12).

77

peak at the isotropic chemical shift of fluorapatite is not broadened at the base nor
is its half-height linewidth significantly greater than that of undoped fluorapatite. The
most probable explanation for the lack of an effect by the quadnlpolar antimony
nuclei on the 19F and the 31 P MAS-NMR spectra is "self-decoupling" due to rapid
spin-lattice relaxation (T 1 s 1 x 10‘4 s) of the antimony nuclei.53

In the 31 P MAS-NMR spectrum of SbPO4 the isotropic peak has a
Av1/2 of 4 ppm, about ten times greater than that of fluorapatite. The increased
linewidth in this case may arise from the above quadrupolar effect upon the
phosphorus nuclei dipolar coupled to antimony nuclei; presumably in this case the
antimony spin-lattice relaxation time is insufficiently short for complete ”self-
decoupling" to occur.

Having demonstrated the absence of quadrupolar effects upon the MAS-
NMR spectra of antimony-doped fluorapatite, we will now discuss the relative
quantitation of the 19F MAS-NMR peaks in relation to the amount of antimony
present Flgure "-10 is a plot, from the deconvolution data given in Table "-1, of the
intensity of peak A in the 19F MAS-NMR spectra vs. the total wt % antimony in the
samples. The lack of a meaningful correlation indicates that some antimony must
go into a secondary phase or a site in the fluorapatite lattice other than the one
giving rise to peak A. In the samples containing 1.3, 1.7, and 2.1 wt. % antimony the
antimony not associated with peak A is most likely in a site in the fluorapatite lattice
that gives rise to the peak at 73.1 ppm, as will be discussed in section 4.3.3. Since
the 19F and 19F MAS-NMR spectra of the sample containing 3.0 wt. % antimony
do not contain peaks with enough intensity to account for the antimony that is not
associated with peak A, the ”missing” antimony must be in a phase that does not
contain fluorine or phosphorus. X-ray powder diffraction data (of the identical 3.0 wt.
% antimony sample used in this study) suggest that this phase is calcium

metaantimonate.

78

The solid line in Figure "-10 is the theoretical relationship expected if two
fluorine nuclei are affected by each antimony atom in the lattice and all of the
antimony is incorporated into the lattice. Since two of the samples (0.18 wt. % and
0.8 wt %) have peak A intensities on or near this line, there are most likely two "A"
fluorine atoms affected per incorporated antimony atom. It is unlikely that peak A
arises from three or more fluorine atoms per incorporated antimony atom for several
reasons. The amount of incorporated antimony as determined by IR (see next
paragraph) for numerous samples containing less than 0.8 wt % antimony is
proportional to the total antimony content,56 suggesting that all of the antimony
present at these relatively low concentrations is incorporated into the apatite lattice.
Furthermore, [fluorine A]/[Sb] molar ratios greater than 2/1 are difficult to reconcile
with any of the models of antimony substitution discussed in section 4.3. Since the
ratio of the area of peak A to peak B is 2:1 from the difference spectroscopy results,
we can conclude that peak B arises from one fluorine nucleus perturbed by each
antimony atom in the lattice.

The fluorescence of anfimony doped fluorapatite has been shown to be
directly proportional to a specific peak in the IR spectrum at 686 cm'1 .3 The
intensity of this lR band can therefore be used to determine the mole fraction of
(fluorescence-active) antimony in the lattice of a given sample, using the 0.8 wt.%
sample as a calibration standard (assuming that all antimony in this sample is
incorporated into the lattice, as the above paragraph indicates). Flgure "-11 is a plot
from the deconvolution data given in Table "-1 of the intensity of peak A in the 19F
MAS-NMR spectra vs. the mole fraction of antimony in the lattice as determined by
IR. The mole fraction antimony is calculated using the defect formula XSb = [mol
SmeolCa5F(PO4)3-x(SbO3)x]. The linearity of this plot not only confirms that
peak A arises from antimony incorporated into the apatite lattice, but more
importantly demonstrates that this peak is related to the specific antimony site

79

 

 

 

 

0.15

. /

E I

2

< 0.1 l

e\° .

2; .

'76

5 r

g 0.05 » -

<

x

0

0

D.

0.0 t 1 1 1 L

0.0 1.0 2.0 3.0 ‘

Wt. % Antimony

Flgure ll-10. Plot of the19F MAS-NMR intensity of peak A obtained from
deconvolution (see Table "-1) expressed as the fraction of the total intensity vs. the
wt % antimony(lll). Data were obtained from spectra in R9. "-6. Solid line is
theoretical plot, with two assumptions: that peak A arises from two fluorine atoms
per antimony atom, and that all of the antimony is incorporated into the apatite

lattice.

80

responsible for fluorescence. The slope of the plot in Figure "-11 is 1.97 i: 0.18, or
approximately two, since there are two fluorine nuclei affected by each antimony

atom in the lattice.

4.3. Antimony substitution sites

In order to make specific assignments of peaks A and B in the 19F MAS-
NMR spectra of antimony doped fluorapatite, we must consider which of the
possible antimony substitution sites can account for all of the experimental results.
The antimony substitution site must be close enough to two apatitic fluorine nuclei
to significantly perturb their 19F chemical shifts, and must produce an equal or
nearly-equal perturbation (since peak A is a single peak 4.6 ppm from the main
apatite peak). It must also be close enough to a single fluorine nucleus to slightly
perturb its 19F chemical shift (since peak B is 1.6 ppm from the main apatite peak).
As discussed in the spin diffusion section, the fluorine nuclei giving rise to peaks A
and B must be in different fluorapatite chains. Finally, the substitution site of .
antimony must produce a negligible perturbation of the 31 P chemical shift

Simulations of 31 P MAS-NMR spectra of the 2.1 wt % sample show that if
each antimony atom were to perturb three phosphorus nuclei (the smallest number
of neighboring phosphorus atoms an antimony atom would have in any substitution
site), a chemical shift perturbation of 0.4 ppm would be easily detectable as a
shoulder on the main apatite peak. Although the effects upon the 31 P chemical shift
of replacing calcium in the fluorapatite lattice by strontium or barium are rather small
(2 ppm), unlike antimony, these ions have the same charge as calcium. Turner et
al.5'57 have shown an approximate empirical correlation of the 31 P chemical shift in
metal phosphates with the quantity Zr1/2, where Z is the formal cation charge and r
its ionic radius. Neglecting the stnictural differences between compounds, this

correlation rationalizes the much larger effect upon the 31 P chemical shift of

81

 

 

 

 

0.15 L
t
3 .
g 0.1 .
2 . -
s
<
x >
8 0.05 » .
a .
o t t L . 1 1 1 r
0 0.02 0.04 0.06 0.08

XSb (Mole Fraction Sb(lll) )

Flgure ll-11. Plot of 19F MAS-NMR intensity of peak as in Fig. "-10 vs. the mole

fraction of antimony incorporated into the apatite lattice as determined by IR (see
text).

82

replacing Ca2+ (2:2, r=99 pm) by Sb3+ (2:3, r=76 pm) rather than Sr2+ (Z=2,
r=112 pm) or Ba2+ (Z=2,r= 134 pm).58

The 31 P chemical shift of antimony phosphate, which has an antimony(lll)
atom sharing each oxygen atom of the phosphate tetrahedron,59 is -18.3 ppm. If an
antimony atom occupies a calcium site in the apafite lattice, it would be adjacent to
a phosphate tetrahedron. The Sb—O—P linkage would then be expected to produce a
detectable difference in the 31 P MAS-NMR spectrum, on the order of 5 ppm (1/4 of
the difference between the 31 P chemical shift of FAP and that of SbPO4).

In the following three sections we will discuss the three proposed sites for
antimony substitution in light of the combined 19F and 31 P MAS-NMR results.
4.3.1. Ca(1) site

Previous studies have provided evidence against the Ca(1) site as the
antimony substitution site.5r9 The lack of axial symmetry for the antimony nuclear
quadrupole coupling constant obtained from Mossbauer spectroscopy has been
used to argue against the Ca(1) site, which has threefold symmetry in the apatite
lattice. However, although not explicitly discussed, the mechanism of charge
compensation must be considered. If antimony occupied the Ca(1) site, charge
balance would most likely be accomplished either by removing two Ca(1) ions (or
alternatively one Ca(1) ion and one Ca(2) ion) and one fluoride ion, or by removing
one Ca(1) ion and replacing a fluoride ion with an 02' ion. In both cases, the
antimony atom would reside in the center of a triangular prism formed from three
columns of fluoride ions, but strict threefold symmetry would be absent. We should
note that the Mossbauer results provide an additional argument against the Ca(1)
site, since the theoretically calculated antimony nuclear quadrupole coupling
constant for 303+ replacing Ca(1) with no other lattice distortion is much smaller

than the measured value.

83

The 19F MAS-NMR results do not explicitly rule out the Ca(1) site as a
substitution site for antimony atoms. If either of these charge compensation
schemes above were invoked, peak B in the 19F MAS-NMR spectra would have to
be assigned to a fluoride ion in the column of fluorides containing the vacancy (or
oxygen ion), and peak A would be assigned to two identical fluoride ions in the
other two chains of the triangular prism. This model offers no explanation for the
greater linewidth of peak A compared to peak C (Table "-1). Finally, the presence
of only one fluoride ion corresponding to peak B, rather than two essentially
equivalent fluoride ions above and below the vacancy, is difficult to rationalize in the
above model. Perhaps the most definitive experiment to rule out this model solely
on the basis of 19F MAS-NMR would be to demonstrate with a type of REDOR
experiment19 that peak A arises from two adjacent fluoride ions in the same chain.

The 31 P MAS-NMR NMR results provide the strongest evidence against the
Ca(1) site. This conclusion is based on the lack of a distinct antimony-related
resonance in the 31 P MAS-NMR spectra, despite the expectation that such a peak
should be readily observable (see above).

4.3.2. Ca(2) site

The combined MAS-NMR results seem to rule out this possibility. As
discussed in section 4.3.1, an antimony atom occupying a Ca(2) site would be
expected to produce a detectable additional peak in the 31 P MAS-NMR spectra,
which is not observed.

The 19F MAS-NMR results provide additional evidence against the Ca(2)
site. If an antimony atom were in the Ca(2) site, a fluoride ion adjacent to the
antimony atom would be replaced by an 02' ion for charge compensation.
Assuming that the two fluoride ions adjacent to the 02' ion were responsible for
peak A, the spin diffusion results require that the fluoride nucleus giving rise to peak

B must be in an adjacent fluoride chain (Figure "-9). However, this fluoride nucleus

84

would be distant (700 pm away) from the antimony atom and separated from it by
five chemical bonds (Sb-O-P-O-Ca-F); the perturbation of the fluorine chemical shift
observed for peak B is more difficult to rationalize in this case than for the
phosphate model below.
4.3.3. Phosphate site

Both the 31 P and 19F MAS-NMR results support the phosphate site as the
location of antimony in the apatite lattice. If an antimonate tetrahedron (three
oxygen atoms and a lone electron pair surrounding antimony) occupied the site of a
phosphate tetrahedron, and the lone electron pair on the antimony atom occupied
the site of an oxygen atom nearest the fluoride chain (Figure ll-12), the environment
of two Ca(2) ions normally bonded to this oxygen atom would be perturbed. The
two fluoride ions adjacent to each of these perturbed Ca(2) ions can therefore be
assigned to peak A. Since these two fluoride ions do not have an identical bonding
geometry to the antimony ion, they would be expected to have slightly different
chemical shifts. Such a small chemical shift difference may be responsible for the
approximately 120 Hz (0.32 ppm) greater linewidth of peak A compared to peak C.
The downfield shift of peak A may be qualitatively ascribed to an increased
paramagnetic contribution in the Ramsey theory of chemical shifts,60 due to a
greater tendency of the Ca‘*2 ion missing an 02' ligand to withdraw electrons from
the fluoride ions. To satisfy the spin diffusion results (Figure "-9), peak B must be
assigned to a fluoride ion in the closest linear chain of fluoride ions, with a three-
bond bonding network of Sb—O-Ca-F linking the antimony ion to the fluoride ion
(Figure ll-12). The chemical shift difference between peak B and peak C (1.6 ppm)
is expected to be less than that between peak A and peak C, since the perturbation
is less. Although the fluoride ions adjacent to the peak A fluoride ions have the
same bonding network to the antimony atom (Sb-O-Ca-F) as the fluoride ion

assigned to peak 8, we must assume that their chemical shift is undetectably

85

 

 

 

 

Figure "-12. ORTEP drawing of fluorapatite lattice, with antimony(lll) and its lone
pair placed exactly at the phosphate site. The assignment of fluoride ions to peaks
A and B in the 19 F MAS-NMR spectra is indicated (see

text). ‘

86

different from that of the peak C fluoride ions.

The lack of an antimony related peak in the 31 P MAS-NMR spectra is also
readily explicable if antimony occupies the phosphate site. The bonding network to
the nearest phosphorus atom would be: Sb-O-Ca-O-P. The presence of four bonds
between the antimony and phosphorus atoms, rather than two bonds as for the
Ca(1) and Ca(2) site models, would plausibly result in the absence of a significant
perturbation of the 31 P chemical shift.

Having shown that peaks A and B arise from fluorine nuclei near an
antimony atom with a distinct orientation in the phosphate site, we will now attempt
to assign the 73.1 ppm peak. As shown in section 4.1 the fluorine nuclei giving rise
to the weak peak at 73.1 ppm must be in the fluorapatite lattice. The peak at 73.1
ppm does not arise from fluoride ions at the surface of the crystallites, since the
specific surface area measurements indicate that only approximately one millionth
of the fluoride ions are at or near the surface of the crystallites, whereas the
intensity of the 73.1 ppm peak is as high as 4.6% of the total intensity. The 73.1
ppm peak in the 19F MAS-NMR spectra of the fluorapatite samples containing 2.1,
1.7, and 1.3 wt. % antimony cannot be due to the antimony atom in the phosphate
site that gives rise to peaks A and B, since its intensity does not correlate with that
of these latter peaks. The 73.1 ppm peak is not due to an antimony atom that
occupies one of the Ca sites, based on the 31 P NMR arguments in sections 4.3.1
and 4.3.2.

It is possible that the 73.1 ppm peak arises from an apatitic fluoride ion near
an antimony atom in the phosphate site whose electron lone pair occupies the site
of a missing oxygen atom directed away from the nearest adjacent fluoride chain
(unlike the antimony associated with A and B). Supporting this assignment is the
correlation observed (except for the 3.0 wt% sample) of the intensity of the 73.1
ppm peak with the amount of antimony not associated with peaks A and B.

87

Specifically, when the 19F MAS-NMR intensity of the 73.1 ppm peak (expressed as
the fraction of the total intensity) is plotted versus the mole fraction of antimony
atoms in the sample not associated with peaks A and B, the three points can be fit
to a line with a slope of 1.01 and a correlation coefficient of 0.93, indicating that
there is one antimony atom associated with each fluorine atom giving rise to the
73.1 ppm peak. The increased linewidth of this peak may be due to an observable
quadrupolar effect from the antimony nuclei. One fact is not convincingly
rationalized by the assignment of the 73.1 ppm peak to a single fluorine nucleus
perturbed by an antimony atom in the phosphate substitution site with a different so
lone pair orientation. The chemical shift difference between the 73.1 ppm peak and
the main peak at 64.0ppm is twice as large as that between peak A and the main
peak, even though the distance and bonding linkage from the antimony atom to the
nearest fluorine nuclei are nearly the same for the proposed assignments.

5. Conclusion

We have shown by 31 P and 19F MAS-NMR of antimony-doped fluorapatite
samples that antimony(lll) occupies the phosphate site in the apatite lattice rather
than one of the cationic (calcium) sites. Our data strongly support the model of
Mishra and coworkers,5 in which the PO43' group is replaced by a (slightly
displaced) 80033‘ moiety, with the antimony 5s electron lone pair directed towards
the missing oxygen atom site. The MAS-NMR results also show that these samples
contain a maximum of 1.6 wt. Sb3+ in this well-defined lattice site, with most of the
excess antimony going into a secondary site.

Since we are not aware of previous studies of antimony(lll) dopant sites in
ionic lattices, it is difficult to draw conclusions regarding the generality of the
behavior we observe. At least in this one instance, we have shown that 803+,
rather than substituting for a divalent cation of larger ionic radius (99pm vs.

76pm),58 instead essentially replaces a formal P5+-02' moiety. In this sense,

88

antimony(lll) could be said to resemble a non-metal more than an alkaline earth ion.
The oxygen coordination polyhedra may also be influential in determining the
bonding mode, with the three oxygen atoms of a phosphate group providing the
hemihedral coordination common in crystal structures of antimony(lll) compounds.

High-field 19F MAS-NMR is a useful technique for quantifying the amount of
antimony actually incorporated into the fluorapatite lattice. It can readily detect
antimony levels as low as 0.2 wt.% in a single scan. This study demonstrates
several advantages to studying substitution by dopants through their effect on
magnetically-active nuclei rather than by direct observation of the dopant nuclear
resonance, even assuming the latter is observable. First, it is easier to disfinguish
actual incorporation of the dopant into the host lattice from possible formation of
secondary phases. Also, a single dopant site may result in unequal perturbations of
a number of atoms of the host, thus providing more detailed structural information.
The influence of spin diffusion upon the 19F MAS-NMR spin-lattice relaxation times
was important in this study in establishing that the perturbed peaks arise from
fluoride ions in the apatitic lattice. We have shown that the rate of spin diffusion
between spectrally-resolved peaks can be quantitatively measured by using the
selective-excitation SPARTAN pulse sequence in a one-dimensional NMR
experiment. The spectral spin diffusion rates measured using the SPARTAN
sequence have enabled us to assign the two peaks perturbed by antimony to
fluoride ions on different linear chains. Difference spectra have shown that these
two peaks are present in a 2/1 intensity ratio, consistent with expectations based on
molecular models.

The observation of spectral spin diffusion for MAS spinning speeds greater
than the maximum dipolar coupling in the spin network is an interesting
experimental fact; we are aware of only one other comparable result that has been

briefly alluded to.61 If this behavior is fairly general (occuring either via level

89

crossing or atomic motion) spin diffusion measurements obtained using the
SPARTAN sequence may prove very useful for assigning MAS-NMR peaks. It
would also be worthwhile to investigate theoretically as well as experimentally the

"quenching" of spin diffusion by MAS at high spinning speeds.23s42

90

6. References

1. S. Naray-Szabo. Z. Krist.75, 387 (1930). q

2. D. McConnell. Applied Mineralogy. edited by A. V. D. Frechette, H. Kirsch, L B.
Sand, and F. Trojer. (Springer-Verlag, New York, 1973).

3. R. W. Warren. Phys. Rev. 8.2.4383 (1970).

4. R. W. Warren and R. Mazelsky. Phys. Rev. B. 10, 19 (1974).

5. K. C. Mishra, R. J. Patton, E. A. Dale, and T. P. Das. Phys. Rev. B. 35,1512
(1987).

6. T. S. Davis, E. R. Kreidler, J. A. Parodi and T. F. Soules. J. Lumin. 4, 48 (1971 ).
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J. Electrochem. Soc. 122, 752 (1975).

8. B. G. DeBoer, A. Sakthivel, J. R. Cagle , and R. A. Young.

Acta Crystallog. in press.

9. J. G. Rabatin, G. R. Gillooly, and J. W. Hunter. J. Electrochem. Soc. 114,956
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10. E. R. Andrew. Progress in Nuclear Magnetic Resonance

Spectroscopy, edited by J. W. Emsley, J. Feeney, and L. H. Sutcliffe (Pergamon,
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11. W. P. Rothwell, J. S. Waugh, and J. P. Yesinowski. J. Am. Chem. Soc.102,
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12. J. P. Yesinowski and M. J. Mobley. J. Am. Chem.

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13. K. A. Smith and D. P. Burum. J. Magn. Reson. 84, 85 (1989).

14. M. Mehring, in NMR Basic Principles and Progress Volume 11 edited by P.
Diehl, E. Fluck, and R. Kosfeld. (Springer-Verlag, Berlin, 1976).

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15. B. C. Gerstein, C. Chow, R. G. Pembleton, and R. C. Wilson, J. Phys. Chem.
81, 565 (1977).

16. J. P. Yesinowski and H. Eckert, J. Am. Chem.Soc.109, 6274 (1987).

17. J. P. Yesinowski, H. Eckert, and G. R. Rossman. J. Am. Chem. Soc. 110, 1376
(1988).

18. M. M. Maricq and J. S. Waugh. J. Phys. Chem. 70, 3300 (1979).

19. T. Gullion and J. Schaefer. J. Magn. Reson. 81, 196 (1989).

20. M. Munowitz, W. P. Aue, and R. G. Griffin. J.

Chem. Phys. 77, 1686 (1982).

21 . J. E. Roberts, et. al. J. Am. Chem. Soc. 109, 4163 (1987).

22. C. E. Bronnimann, R. C. Zeigler, and G. E. Maciel. J. Am. Chem. Soc. 110,
2023 (1988).

23. P. Caravatti, J. A. Deli, G. Bodenhausen, and R. R Ernst J. Am. Chem. Soc.
104, 5506 (1982).

24. A. Kubo and C. A. McDowell. J. Chem. Phys. 89, 63 (1988).

25. V. Bork and J. Schaefer. J. Magn. Reson. 81, 196 (1989).

26. L B. Moran, J. K. Berkowitz, and J. P. Yesinowski. submitted to J. Am. Chem.
Soc.

27. G. Bodenhausen, R. Freeman, and G. A. Morris. J. Magn. Reson. 23, 171
(1976)

28. A. Kubo, and Charles A. McDowell. J. Chem. Soc.,Faraday Trans. 84, 3713
(1988). J. L. Carolan. Chem. Phys. Let. 12, 389 (1971).

29. A. M. Vakhrameev, S. P. Gabuda, and R. G. Knubovets. J. Struct. Chem.
(USSR). 19, 256 (1978).

30. W. Van der Lugt and W. J. Caspers. Physica 30, 1658 (1964).

31. W. P. Rothwell and J. S. Waugh J. Chem. Phys.74, 2721 (1981).

32. A. T. Kreinbrink, et. al. J. Magn. Reson. 88, 267 (1990).

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33. C. Connor, A. Naito, K. Takegoshi, and C. A. McDowell. Chem. Phys. Lett.
113(2), 123 (1984).

34. N. Bloembergen. Physica, 15, 386 (1949).

35. A. Abragam. The Principles of Nuclear Magnetism, (Oxford, London, 1961), Ch.
v and IX. '

36.D. Suter and R. R. Ernst. Phys. Rev. B. 25, 6038 (1982).

37.E. R. Andrew, A. Bradbury, R. G. Eades, and T. Wynn.

Phys.Lett 4, 99 (1963).

38.P. Caravatti, M. H. Levitt, and R. R. Ernst. J. Magn. Reson. 68, 323 (1986).

39. D. L Vanderhart J. Magn. Reson. 72, 13 (1987).

C. A. Fyfe, H. Gies, and Y. Feng. J. Am. Chem. Soc. 111, 7702 (1989).

40. D. Suter and R. R. Ernst Phys Rev. B. 32, 5068 (1985).

41. P. M. Hendrichs, M. Linder, and J. M. Hewitt J. Chem. Phys. 85, 7077 (1986).
42. P. Robyr, B. H. Meier, and R. R. Ernst. Chem. Phys. Lett. 162, 417 (1989).

43. M. G. Colombo, B. H. Meier, and R. R. Ernst Chem. Phys. Lett. 146, 189
(1988).

44. D. P. Raleigh, M. H. Levitt, and R. G. Griffin. Chem. Phys. Lett 146,71 (1988).
45. J. Kowalewski. Prog. in NMR Spec. 11, 1-74 (1976).

46. K. Hirao, H. Nakatsuji, and H. Kato. J. Am. Chem.

Soc. 95, 31 (1973).

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50. J. Herzfeld and A. E. Berger. J. Chem. Phys. 73, 6021 (1980).

51. N. Zumbulyadis, P. M. Hendrichs, and R. H. Young. J. Chem. Phys. 74, 1603
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93

52. J. Bohm, D. Fenzke, and H. Pfeifer. J. Magn. Reson. 55, 197 (1983).
53. W. W. Fleming, et. al. Macromolecules. 13, 460 (1980); E. M. Menger and W.S.
Veeman. J. Magn. Reson. 46, 257 (1982).

54.E. Dale. Private communication. ,

55. G. L. Turner, K. A. Smith, R. J. Kirkpatrick, and E. Oldfield.

56. J. Magn. Reson. 70, 408 (1986).

57. Handbook of Chemistry and Physics. edited by R. C. Weast (CRC Press,
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94

Section III

A Method for Detection of Spectral Spln Diffusion from Minor Peaks and Its
Application to 19F MAS-NMR of Antimony(lll)-Doped Fluorapatite

1. Introduction

Spectral spin diffusion refers to the transfer of longitudinal (z-) magnetization
within a homonuclear spin system between nuclear spins having different chemical
shifts. Since the process requires a dipolar (or scalar) coupling, structural
information is implicit in spin diffusion experiments.1 '10 Although the expectation
has been that spin diffusion will be quenched at spinning speeds significantly
exceeding the homonuclear dipolar couplings in a spin system,1 we and others
have recently demonstrated the existence of measurable spin diffusion under such
conditions.7'10 Our previous experiments have used the SPARTAN pulse
sequence to measure spin diffusion in a one-dimensional 19F MAS-NMR
experiment.7 We present here an improved technique for more accurately
measuring spectral spin diffusion under MAS condih'ons that is particularly suitable

when the two peaks involved have different intensities.
2. Results and Discussion

The new technique involves repeatedly saturating a peak and its associated
sidebands with selective trains of pulses, and subsequently detecting the rate at
which the saturation is transferred to other peaks in the spectrum. We refer to the
sequence used (see Fig. Ill-1) as SINK (Saturation Inter-Nuclear Kinetics). It
contains a series of DANTE trains11 (small flip-angle pulses spaced at intervals of

the rotor period, which have the effect of selectively exciting the on-resonance peak

95

and its associated sidebands), each of which is equivalent to a selective 90°
(saturation) pulse. Alternatively, a shaped DANTE train could be used to improve
selectivity.12 Spectral spin diffusion (cross-relaxation) occurs during the mixing
period following each DANTE train, and a non-selective 90° observe pulse is given
after the last mixing period. The SINK experiment is conceptually related to
saturab'on transfer experiments in solution-state NMR, which use low power
continuous rf irradiation to selectively saturate a given peak, and monitor the loss of
magneb'zation in other peaks due to chemical exchange.13

We have used the SINK pulse sequence to study calcium fluorapatite
(CasF(PO4)3) doped with small levels of 6b+3 "activator, which is used as a
phosphor material in fluorescent lamps. Since the nature of the Sb+3 ion
substitution site is in dispute,14-18 we have used 19F (and 31 P) MAS-NMR as an
indirect probe of the antimony substitution site.7 Figure Ill-2a shows the structural
arrangement of the identical fluoride ions (black solid circles) in the parent
fluorapatite structure. The fluoride ions are separated by a distance of 344 pm, and
form parallel linear chains along the (vertical) c-axis of the crystal.19 The presence
of Sb3+ in the lattice renders the nearby fluorine nuclei inequivalent, as suggested

in Figure III-2b (the bases for these assignments are given in reference 7).

Figure Ill-3 shows superimposed 19F MAS-NMR spectra of a fluorapatite
sample doped with 1.7 wt. % Sb+3 (sample preparation and characterization and
NMR conditions are given in reference 7; since the chemical shifts of the main
apatite resonance (C) and the antimony related peaks in the spectra of antimony
doped fluorapatites exhibit slight variations,7 we have set the chemical shift of peak
C in Figure 3 to 64.0 ppm). The spectra of Figure ill-3 were obtained using the SINK
pulse sequence with varying numbers of selective saturation DANTE pulse trains

applied to peak A. The selectivity of the DANTE train was verified by performing the

96

selective 90° DANTE nonselective 90°

f 4——Tm

Illlllllll
/t.\

 

 

 

\

 

Ill-1. SINK (Saturation Inter-Nuclear Kinetics) pulse sequence for measuring spin
diffusion in MAS-NMR. The rotor period (t,) is indicated. Rotor-synchronization is
carried out using an optical detector to gate the rf pulses. The mixing period for spin

diffusion rm is fixed and the number of saturation pulse trains n is varied.

97

SINK experiment with the transmitter frequency on the upfield side of peak C with
an offset from peak C equivalent to the frequency difference between peaks A and
C. The mixing period (rm) should be more than five times T2. in order to allow all
tranverse magnetization to decay; it was chosen to be 0.1 s for the current
experiment. The intensity of the main peak (C) decreases with the number of
saturation pulse trains (n) since the spins of peak C are in spin diffusion contact
with the spins of peak A. The intensity of peak A in Figure Ill-3 decreases with
increasing n until it = 20 since less magnetization is transferred to peak A in the
final mixing period once the magnetization of peak C becomes depleted. After
multiple saturation pulse trains that greatly reduce the intensity of peak C, two much
weaker peaks that are less strongly affected become resolved. These are the
previously-discussed7 peak B at 65.6 ppm, and an even less-resolved shoulder on
the upfield side of peak C at 62.9 ppm that we refer to as peak D. The results of the
SINK experiment in Figure III-3 demonstrate even more clearly than the SPARTAN
experiment that spin diffusion from A to C nuclei is faster than from A to B nuclei.
This observation strengthens our previous conclusion7 that A and B fluorine nuclei
are in different chains (as shown in Fig. Ill-2b). Similarly, peak D may also arise
from apatitic fluoride nuclei near an Sb3+ ion in a fluoride chain other than the one
containing the A fluorine nuclei.

Spin diffusion time constants (130 's) were obtained by fitting the behavior of
the magnetization to a simple model of two site magnetization transfer. The
intensities of peak C were calculated numerically using the kinetic equations for a
reversible process, where A and C represent the magnetization of the A and C
peaks respectively, and kfor and krev are the rate constants for the process;

det = krevC - kforA III-1
dC/dt = kforA - krevC

98

 

III-2. a) Schematic arrangement of fluoride ions (0) in calcium fluorapatite
(Ca5F(PO4)3);19 b) Assignments based on SPARTAN results7 of peaks A, B and
c in the 19F MAS-NMR spectra of calcium fluorapatite doped with Sb3+; the Sb3+
ion is located near the middle of the rhombus drawn involving fluoride ions A and B.
The solid black circles represent fluoride ions that give rise to the main peak C in
the 19F MAS-NMR spectrum.

99

 

Figure Ill-3. 19F MAS-NMR (376.29 MHz) superimposed spectra of a calcium
fluorapatite sample containing 1.7 wt. % Sb+3 dopant obtained using the SINK
pulse sequence (Figure III-1). The location of the transmitter rf (at peak A) is
indicated on the spectrum with an arrow. The spectra shown were obtained with the
number of saturation pulse trains (n) equal to (top to bottom) 1, 10, 20, 30, 40, 60,
80, 100, and 140, with tm set to 0.1 6. Seven 1.6 ms pulses (nominally 13° each)
comprise the DANTE train, which has a total length of 0.75 ms; the spinning speed
vr is 8.01 kHz.

100

The basic scheme for calculating the intensity of peaks C and A at various times t
[C(t) and A(t)] involves recalculating the magnetization of the peaks after small
increments of time At using the following piecewise solutions (including T1

relaxation) of Eqn (III-1);

A(t + At) = A(t) + (1/tSD)C(t) At - (1/ISD)A(t) At + (1/T1A)[Ao - A(t)]At
Ill-28

C(t + At) = C(t) + (1/th)A(t) At - (1/th)C(t) At + (1rr1c)[co - C(t)] At

Ill-2b
The rate constants kfor and krev have been replaced by (1/tSD), since the flip-flop
cross-relaxation mechanism requires that km, = krev- The last terms in Eqns. Ill-2a
and III-2b correct for longitudinal relaxation. The time increment At was chosen to
be 0.01 s, a value short enough to accurately calculate the magnetization. The T1
values (>100 s) used in Eqn. Ill-2 are reported in reference 7. Equilibrium
magnetizations A0 and Co were set equal to values obtained from a single non-
selective 90° pulse after allowing for complete relaxation. The effect of the
saturation pulse train is incorporated into the fitting procedure by lowering the value
of A(t) to a fraction of its previous value each time the sum of the time increments
equals a multiple of rm. The value of A(t) was not decreased to zero at these times
since the DANTE train gives an effective 90° pulse only to the middle of peak A,
leaving some positive side-lobes. The amount by which the DANTE train
decreased the total A magnetization was determined by integrating peak A after a
single DANTE train immediately followed by a non-selective read pulse. Cross-

relaxation of the system during the selective DANTE trains has been neglected

101

since the train duration (0.75 ms) is small compared to the mixing period (Tm =
100ms).

Since the present system features linear chains of C spins interspersed with
A spins, longitudinal magnetization from most of the C spins must be transferred to
A spins via other C spins. A thorough treatment of the cross-relaxation dynamics in
this system would explicitly include cross-relaxation between C spins; however, we
have treated the C spins as a "pool”, based on the assumption that C spin
polarization redistributes itself rapidly compared to the time scale of cross-
relaxation between the A and C spins. This assumption is valid since the two
samples studied have the same tsp despite having different C spin chain lengths
(caused by different levels of antimony doping).

Figure III-4 shows the peak C intensities calculated using Eqns. III-2 and the
measured values plotted versus the total number of saturation pulse trains for the
fluorapatite sample doped with 1.7 wt. % Sb+3. The calculated C(t) values were
obtained by varying ‘tSD until the best visual fit was obtained. The spin diffusion
time constants ‘tSD (single exponenfial) measured from the SINK data for samples
containing 1.7 % and 0.8 wt. % Sb+3 at a spinning speed v, = 8.0 kHz are 2410.1
and 22:01 s respectively, which are close to the values obtained using the
SPARTAN pulse sequence.7

We note that the final (n > 60) data points could not be accurately fit to the
calculated curve regardless of the 130 value used. The fact that the measured
values of C magnetization are greater than the calculated values after many
saturation pulse trains (11) suggests that some of the C spins are not in spin
diffusion contact with the A spins. The fit of Figure III-4 is substantially improved
over the entire range of n if only 95 % of the C spins are allowed to cross-relax with
the A spins; the resultant tSD values remain essentially unchanged. One possible

reason that 5 % of the C spins are not in spin diffusion contact with the A spins is

102

 

 

63* 3°"

9:: lb

g 70‘"

550..

6 so.»

CU

: 4°:

(3 8

.fi 8

'85 20“ I

5; A -

(13 10‘" I

2' o .A . Z 1..

0 r 1 fl
0 50 100 150

Number of Saturation Pulse Trains (n)

Figure Ill-4. Calculated (A) and measured (I) peak C intensities (of spectra from
Figure Ill-3) vs. the number of saturation pulse trains (n) applied to peak A. The
calculated points were obtained using Eqns. Ill-2 as described in the text, with a
130 of 2.4 s.

103

physical isolation, either due to fluorine vacancies or sample inhomogeneity. A
second possible explanation is an orientational dependence of the spin diffusion
rate.9 We have attributed the existence of spin diffusion between peaks A and C to
level crossing,7 the momentary equality at some point in the rotor cycle of the
chemical shifts of two adjacent spins with different chemical shielding tensors.5 For
those crystallite orientations that do not lead to level crossing, spin diffusion will be
greatly slowed.
3.Conclusions

The SINK pulse sequence has several advantages over the spin diffusion
measurement technique (SPARTAN) previously used7s9 and other spin diffusion
experiments.4s3s1o The SINK experiment provides more accurate spin diffusion
fime constants (tsp), particularly for spin diffusion from minor spectral features to
major components. The SINK pulse sequence can also be used to clarify crowded
regions of the spectrum and reveal "hidden peaks" by saturation of an overlapping
peak through saturation transfer. Finally, the SINK pulse sequence is well suited for
detecting long range spin diffusion in spatially separated phases. such as polymer
blends or epitaxially-related crystal phases, since a spectrally resolved resonance
of one phase may be saturated for lengthy periods while magnetization from the
spins of the other phase slowly disappears. We believe that detecting spectral spin
diffusion via the SINK sequence will be useful for making MAS-NMR assignments
of such widely abundant spins as 19F, 1 H, and 31 P and for studying impurity sites
in disordered or glassy solids. For surface studies, this approach should also
provide useful information about clustering, spatial proximity, and epitaxial

relationships.

104

REFERENCES )

1. D. Suter and R. R. Ernst, Physical Review B, 32 (1985) 5068.

2. P. Caravatti, G. Bodenhausen and R. R. Ernst, Journal of Magnetic Resonance,
55 (1983) 88.

3. P. Caravatti, M. H. Levitt and R. R. Ernst Journal of Magnetic Resonance, 68
(1986) 323.

4. N. J. Clayden, Journal of Magnetic Resonance, 68 (1986) 360.

5. A. Kubo and C. A. McDowell, Journal of the Chemical Society. Faraday
Transacfions, 84 (1988) 3713.

6. C. A. Fyfe, H. Gies and Y. Feng. Journal of the American Chemical Society, 111
(1989) 7702.

7. L B. Moran, J. K. Berkowitz and J. P. Yesinowski, Physical Review 8.45
(1992) 5347.

8. W. Kolodziejski, H. He and J. Klinowski, Chemical Physics Letters, 191 (1992)
1 17.

9. Z. Zhang, J. H. Kennedy and H. Eckert, Journal of the American Chemical
Society, 114 (1992) 5775.

10. K. Takegoshi, I. Tanaka, K. Hikichi and S. Higashida, Macromolecules, 25
(1992) 3392.

11.G. Bodenhausen, R. Freeman and G. A. Morris, Journal of Magnetic
Resonance, 23 (1976) 17.

12. w. S. Warren, Journal of Chemical Physics, 81 (1984) 5437.

13. S. Forsen and R. A. Hoffman, Joumal of Chemical Physics, 39 (1963) 1963.

14. K. C. Mishra, R. J. Patton, E. A. Dale and T. P. Das, Physical Review B, 35
(1987) 1512.

105

15. T. S. Davis, E. R. Kreidler, J. A. Parodi and T. F. Soules. Journal of
Luminescence, 4 (1971) 48.

16. R. W. Warren, F. M. Ryan, R. H. Hopkins and J. Van Broekhoven. Journal of
the Electrochemical Society, 1 22 (1975) 752.

17. B. G. Deboer, A. Sakthivel, J. R. Cagle and R. A. Young, Acta.
Crystallographica, Secfion B, 47 (1991) 683.

18. J. G. Rabatin, G. R. Gillooly and J. W. Hunter, Journal of the Electrochemical
Society, 114 (1967) 956.

19. S. Naray-Szabo, Zeitschrift feur Kristallographie, 75 (1930) 387.

106

Section IV.
Chemlcal-Shift Selective Multiple Quantum NMR as 8 Probe of
Dopants In Calcium Fluorapatite

1. Introduction

Multiple-quantum NMR has been used previouSly to count the number of
spins in clusters of spins in tightly dipolar-coulped systems."4 Spin counting is
done by creating multiple-quantum coherences among sets of spins in a sample
and observing the relative magnitudes of the coherences. The coherence order
profile can be related to the number of coupled spins in a cluster by the statistical
model developed Pines and coworkers.5 This technique has been used to count
clusters of spin in several systems including the number of protons on a single
molecule6 and the number of protons in the silica lattice.5 Gleason and Scruggs7
have extended this technique to study the relative fraction of fluorine spins that are
randomly dispersed in polymers mixes versus the amount in agglomerated salts.
Furthermore, Cho and Yesinowski have fit the coherence orders of the MQ-NMR
spectra of distributions of spin cluster sizes in apatitic mineral phases to probe the
nonrandomness of dopants and vacancies in the lattice.8

In this section (IV) we will examine distributions of clusters of 19F spins in
families of crystal orientations in doped, polycrystalline fluorapatite using chemical-
shift-selective MO-NMR. Chemical shift selective MQ-NMR is a technique designed
to obtain the MO spectrum of the spins giving rise to a single peak in a chemical
shielding powder pattern. The reason chemical shift selection must be used on
fluorapatite is because the total width of the powder pattern is to large for the Pl-MQ
sequence to effectively create coherences in the entire powder pattern. Chemical-
shift-selection is achieved by burning away all but a single peak in the chemical
shielding powder pattern with a series of weak stepped-frequency rf pulses. The PI-

MO portion of the sequence is then performed with the rf frequency on resonance

107

with the single remaining peak. We have employed this technique on two doped
calcium fluorapatite samples to look for evidence of clustering of spins.

The first system is a calcium flourapatite sample with one in every ten
calcium ions replaced by a strontium ion. The strontium ions have the effect of
changing the chemical shielding tensors of approximately one third of the 19F
spins in the apatite latfice.9 Anisotropic shielding tensors can be represented
geometrically by ellipsoids; the distance between the origin and a point on the
surface is related to the chemical shift.10 The 19F spins in the strontium doped
system can be represented by the schematic diagram in Figure lV-1. Figure lV-1 a
shows a single column of four unperturbed fluorapatite spins bounded by strontium
perturbed spins and Figure lV-1b shows a collection of columns of unperturbed
spins punctuated by perturbed spins. When the PI-MQ sequence is used to record
the MO spectrum of unperturbed spins pictured in Figure lV-1, coherence growth is
stopped at the perturbed spins because they are to far off resonance (see
discussion section below). Therefore, magnitude of the coherence orders in the
multiple quantum spectrum will reflect coherences that were created among linear
cluster of like spins. In this study we have analyzed the coherence orders in terms
. of a linear cluster model 9 to compute the mean distance over which the tensors of
like spins remain the same. We have deemed this distance the ”correlation length”
of the tensor.

Two criteria were used to determine which peak in the powder pattern was
observed using chemical-shift-selective MQ—NMR experiments. First, a peak was
chosen that corresponded to a family of orientations of crystallites where the
chemical shifts of the unperturbed spins were most different from those perturbed
by strontium ions. The large chemical shift difference ensures that the perturbed
spins have resonance offsets large enough so that they cannot enter MO

coherences with the unperturbed spins.

108

 

Figure lV-1. Schematic diagram of the fluorine spin tensors in strontium doped

fluorapatite; a) a one-dimensional arrangement , b) three dimensional array.

109

The second criteria is that the peak should be of great enough intensity to allow for
detection. A detailed analysis 9 of the MAS-NMR spectra of this compound yielded
the principle components of the shielding tensor of the strontium perturbed fluoride
spin of 136.1, 57.3, and 43.7 ppm, compared to the values of 120, 36, and 36 ppm
for the unperturbed sample. Chemical shielding powder pattern simulations
corresponding to these two sets of tensor components are given in Figure lV-2.

To identify the peak in the pattern where the pertubred and unperturbed
spins have the largest chemical shift differences, the chemical shift difference
between the two types of spins can be calculated for all crystal orientation. The
difference in the chemical shift AO’ as a function of the crystallite orientation is given

by the formula

A0= 0......(6’.¢)- 0,...(0.¢)

= —{E+ a,,[(3cos2 0- I) + 77st 06092¢]}W IV-1)

+(E+ cr,,[(3cos2 6—1)+ nsin2 00092¢]}m
where 6 and (b are the usual spherical coordinates and 3 and a, are the isotropic
chemical shift and the characteristic chemical shift anisotropy respectively. Values
for Ao are plotted as a function 6 and o in Figure lV-3. Figure lV-3 also contains a
schematic drawing of the chemical shielding powder pattern of the unperturbed
fluorapatite spins to show which peaks in the pattern have the highest intensity as a
function of 0. Although Ao averaged over 0 has the largest value at 0 = 0° (Ao =
16.1 ppm) this is the least intense portion of the powder pattern. Since the intensity
of the powder pattern at 6 = 90° is the greatest the chemical shift selective MQ-
NMR experiment was performed on the spins in the crystallites at this orientation.

The second system on which the chemical-shift—selective MQ—NMR pulse

sequence was used to gain clustering information was a fluorapatite sample doped

with antimony. This compound is a phosphor and ,Iike several other metal-doped

110

Sr1CagF

083 F

 

L.

 

 

1

150 100 50 ppm

Figure lV-2. Calculated chemical shielding powder patterns for the two main types
of fluorine environments found in strontium doped calcium fluorapatite. An
exponential function corresponding to a line broadening of 500 Hz was applied to

the calculated F ID before transformation.

111

 

 

 

 

Figure lV-3. Plot of the difference in chemical shift Ac of the two main types of
fluorine environments (shown in Figure lV-2.) as a function of the spherical

coordinates angles lb and 6 .

112

 

fluorapatite compounds, is used in commercial fluorescent lamps. Antimony is
believed to be responsible for giving this material its phosphor qualities, that is, the
ability to adsorb uv photons and emit in the visible region of the spectrum. It has
also been proposed that the phosphor properties in this materials arise from a
series of photo centers which are simply fluoride vacancies present in the apatite
lattice which compensate for the -2 charge missing when antimonate group
(S0043') replaces a phosphate group (PO4'3) in the lattice.11 It has also been
proposed that the charge compensation is achieved by simply removing an oxygen
atom from the anti monate group as the means of charge compensation instead of
two fluorides.11 Chemical shift selective multiple quantum NMR is well suited
ansWer the question as to whether fluoride vacancies are present in the lattice on a
level similar to the antimony(lll). If such vacancies were present, they would slow
the MO growth dynamics. The multiple-quantum NMR result we present does not
indicate that defects are present strengthening our original conclusion that an
Sb03'3 group replace a phosphate group, with the lack of the oxide ligand
responsible for charge compensation.
2. Experimental

The 19F NMR experiments were carried out on a Varian VXR-400S
spectrometer equipped with a 100 watt fluorine amplifier (376 MHz) and a high-
power probe with a 5 mm o. d. solenoid coil. The M2 pulse length for the high
power portion of the experiment was measured before each experiment, and was
typically 3.7 us. The phase incremented multiple-quantum portion of the sequence
has been used previously12 with the t1 value fixed at 2 us and the cycle time at
70.8 us. Twenty-four saturation pulses with yB1 = 330 Hz and separated by 0.5 ms
were applied to the powder pattern so as to saturate the magnetization of all but the
upfield edge in the following manner: the frequency offsets of these saturation

pulses from the upfield edge were, in order of application 25, 24, 23,22, 6 KHz ,

113

with five saturation pulses repeated at only the last offset. The spin-lattice relaxation
time T1 of the calcium fluorapatite sample was measured to be 91 s, and that of the
calcium/strontium fluorapatite is 859; such long, nearly-equal, values argue against
the presence of motions of the type mentioned above, which would have to occur
on the time scale of hundreds of microseconds. Because of these long spin-lattice
relaxation times, it was useful to employ short relaxation delays 0.6 T1 and four
sets of "dummy scans" (i.e. repetitions of the sequence without data acquisition), a
strategy widely used in 2-D solution NMR experiments. The full 20 version of the
experiment, used to obtain a frequency slice in the f2 dimension for comparison
with the chemical-shift selective experiment, involved 256 complex points in the
phase-incremented domain, and sampling 256 complex points in the t2 domain (a 2
ms delay after the mixing sequence was followed by a (n/2)x pulse, 100 us spin
lock (y) pulse, a Hahn echo sequence with a (1t)x pulse between two 35 us delays,
and then detection). The psuedo 1-D chemical-shift-selective experiment involved
collecting 64 complex points in the phase-incremented domain, with each complex
point in the t2 domain (obtained as immediately above, but without the Hahn echo
sequence) resulting from integrating the FID (summing the first 50 points using a 10
us dwell time). In order to characterize the width of the MO excitation profile, and to
enable comparison with other reported MO studies, we obtain the effective size N
by least-squares fitting of the equally-weighted orders of coherence (up to n=12 at

longer preparation times) to a Gausssian function.

114

STEPPED- FREQUENCY PI-MQ SEQUENCE

 

SATURATION PULSES
U

 

 

——T

V

 

 

)

 

 

 

 

 

 

 

 

 

1M0 TRANSMITTER ‘

 

Figure lV-4. Pulse sequence for chemical-shift—selective phase-incremented

multiple-quantum NMR (see text for details). The periods labeled ”U" and 'V" refer
to the phase-incremented eight-pulse double-quantum selective MO preparation
period and the “time-reversal” portion of the MO sequence, respectively. The 731
of the frequency-stepped saturation pulses is weak compared to the "hard 90 "

pulse in the Pl-MO portion of the sequence, and Is in the present case less than the

local fields.

115

3. Results and Discussion
3.1 . Chemlcal Shift Selection

The basic concept behind the chemical shift selective MQ- NMR experiment
is to saturate all of the magnetization in a powder pattern with the exception of a
specific homogeneously broadened peak and perform the MO experiment on the
remaining peak. When the homogeneous broadenings of the peak within a powder
pattern are small compared to the total width of the pattern, strategies such as
DANTE13 and SELDOM14 have proven useful for selection a specific peak in the
powder pattern. The relatively broad homogeneous linewidths of fluorapatite
severely limit these schemes as an effective means of suppressing off resonance
magnetization. For this reason we have chosen soft pulses applied to the powder
pattern to selectively eliminate all but the desired peak followed by a standard
multiple quantum detection scheme as shown in Figure lV-4.

To determine the most efficient way of placing the saturation pulses to
minimize magnetization from of resonance peaks, we computed the time
development of magnetization under the influence of saturation pulses using
saturation theory of solids.15 Furthermore, we have obtained experimental data on
saniration of a single crystal of fluorapatite using weak rf pulses. Provotorov15 has
derived an expression for the time development of magnetization under the
influence of prolonged, weak rf fields applied to tightly coupled solids. This

expression takes the form:

M2: ho LHL[15+Hiexp(- nyzle-h°—2—;Hzg(t an.)]] M1)

The terms in this Eqn IV-1 include HL the local field which is equal to J1 / 3 the root
second moment; H1 the amplitude of the applied high frequency field; ho , the

116

difference in frequency between the frequency of an applied rf filed and the
resonant frequency yHo ; y, the magnetogyric ratio, and 9(0); the normalized
lineshape function. Equation lV-1 is valid under the condition that the rf fields are
"week” or applied H12 << HL2 . Furthermore, saturation must be achieved rapidly
compare to the longitudinal relaxation time T1 of the system. This condition can be

describe by the following inequality:

2 2
7:72le ilififi-gfiufll' >> 1 . lV-2)

L
Although the lineshape function of in fluorapatite is triplet1 6. we have used the
normalize Gaussian to approximate the lineshape of fluorapatite which is given by

the formula:

_ 1 -h§ _
g(w) — HLmexp(2Hi ]. IV 3)

The terms in Eqn. lV-3 are the same as those in Eqn. lV-1.

 

 

Eqn lV-1 has been used to compute the selectivity attainable using soft
pulse on the fluorapatite powder pattern when a saturation pulse is applied 6.0 kHz
down field (at 20.4 ppm) of the left edge of the pattern. The results of this
computation are shown in Figure IV-5 which is a plot of the remaining 2-
magnetization after 4 millisecond saturation pulses. The highest trace is the
magnetization remaining after a single pulse and each lower trace corresponds to
increasing numbers of saturation pulses. After a single saturafion pulse the
remaining magnetization profile is about 4 kHz which is approximately equal to 9
degrees of variance in azimuthal angle 0 of the crystallites orientation. After six
saturation pulses, the half-height linewidth of the magnetization profile is 1500 hz
corresponding to a the variance in 6 of 3 degrees.

The time development of the z-magnetation for the same upfield portion of

the fluorapatite powder pattern shown in Figure lV-5 is given in Figure lV—6;

117

however, the length of the saturation pulses is varied from 4 to 80 ms instead of the
number of pulses (as in Figure lV-5). The highest trace in Figure lV-6 corresponds
to the magnetization profile after a 4 ms saturation pulse and the lower traces
correspond to longer saturation pulses. One notable feature of this plot is the
selectivity of the magnetization profile increase with increasing pulse width, but the
desirable magnetization (at 36 ppm) is decreased concomitantly. The experimental
parameters for removing unwanted magnetization from the powder pattern such as
the number of saturation pulses, pulse widths, and offsets must be chosen to
minimize off-resonant magnetization and preserve enough magnetization in the
peak on which the multiple quantum experiment will be performed.

To test the accuracy of the theory of saturation of solids (Eqn. IV-1) at
predicting the saturation behavior of fluorapatite, two saturation experiments were
performed on a single crystal of fluorapatite oriented at 0 = 90°. The first
experiment involved applying saturation pulses on resonance to the single crystal of
fluorapatite with pulse width ranging from 4 to 80 ms. Figure lV-7 contains the
results of this experiment and the computed values predicted by the saturation
theory (Eqn IV-1). The experimental points in Figure IV-7 are less then the
computed values for short pulse width, but begin to converge for longer saturation
pulses.

The second saturation experiment was performed to test the ability of Eqn.
lV-1 to predict saturation behavior when rf irraditation is applied off resonance.
Figure lV-8 is a plot of the normalized values of z-magnetization obtained
experimentally and the values calculated using Eqn. lV-1. The experimental points
are lower than the calculated points for all regions of this plot with the greatest
difference occurring when the pulses are on resonance. Although the values for z-
magnetization predicted by Eqn. IV-1 were qualitatively similar to the experimental

values show in Figures lV-7 and lV-8, the quantitative agreement is poor. Since

118

    
   
 

 

 

1 i
0.8 Single Saturation Pulse -----
F X
.2
g 0.6
‘2'
9 0.4 1- K
it Six Saturation Pulses
02 db
0 .........
20.4 ppm 36

Figure IV-5. Plot of the longitudinal magnetization Mz as a function of resonance
offset for crystallites in the upfield portion of the fluorapatite powder pattern
calculated using the Provotorov 1 theory of saturation of solids by weak pulses.
The plot represents that remaining z-magnetization left in the fluorapatite powder
pattern between oi (36 ppm) and 6 kHz down field (20.4 ppm) after the application

of various numbers of saturation pulses (7H1 = 300 Hz) at 20.4 ppm.

119

 

 

 

Resonant Offset (KHz) 0 A

Figure IV-6. Plot of the longitudinal magnetization M; as a function of resonance
offset for crystallites in the upfield portion of the fluorapatite powder pattern
calculated using the Provotorov ‘5 theory of saturation of solids by weak pulses.
The plot represents that remaining z-magnetization in the same portion of the
fluorapatite powder pattern; however, the traces correspond to different pulse width

of a single saturation pulse ( yH1 = 300 Hz).

120

 

 

0.81: .
€0.61: -
S 0.41" . I .
.
.. I
0.2“ ‘ ' I .
‘ 2 H _ I I . . . ' .
0 7; " fl
0 10 m 30 40 50 00 70 N

Figure lV-7. Plot of calculated (I) and experimental (D) z-magnetization as a

function of time that a single saturation pulse ( yH1 = 330 Hz) is applied to a crystal

of fluorapatite oriented at ( 8 = 90°).

121

 

 

0 : i : : .g .
0 1000 2000 3000 4000 5000 0000
Resonance Offset (Hz)

 

Figure lV-8. Plot of calculated (I) and experimental (D) z-magnetization as a
function of resonance offset when a single saturation pulse (30 ms, 7H1 = 330 Hz)

is applied to a crystal of fluorapatite oriented at 0 = 90°.

122

these plots show a lack of quantitative agreement between theory and experiment
the conditions for the saturation scheme such as pulse widths, offsets, and
repetitions were based solely on experimental observations.

The lack of agreement between the experimental and computed z-
magnetization values shown in Figures lV-7 and IV-8 may arise in part from the
from inappropriate lineshape functions (9(a)) in Eqn. lV-1. As previously mentioned,
the a Gaussian function was used to approximate the line shape function of
fluorapatite, whereas the actual lineshape function a fluorapatite crystal oriented at
0 = 90° is a triplet structure.16 If the true lineshape function is used the difference
between experimental and computed points furthest from resonance shown in

Figure lV-7 becomes less.

3.2. Discussion of the Sr/Ca Fluorapatite MO NMR results

The results of the chemical shift selective PI-MQ NMR experiment with the
transmitter place at the up field edge of the powder pattern (36 ppm) are given in
Figure IV-9. Figure lV-9 is a bar graph showing the magnitude of each coherence
orders normalized to the double quantum coherence as a function of preparation
time for the pure and strontium doped fluorapatite samples. The magnitudes of
higher coherences orders are larger for the pure fluorapatite as compared to those
for the strontium deped fluorapatite. This clearly demonstrates that the growth of
multiple quantum coherences are slowed by the presence of the strontium-
perturbed fluorine spins.

The coherence orders in Figure lV-‘9 can be fit to a Gaussian function to
obtain the effective size n at each preparation time. A plot of the effective size
values versus preparation time is given in Figure IV-10. There is a monotonic
increase in the effective size in the data for each sample with the exception of one

of the points in the strontium doped sample which deviates from the upward rise by

123

an amount within experimental error. At all preparation times the effective size of
the doped sample is less than those for stoichiometric fluorapatite and the
difference between the two data sets increases with preparation time. This indicates
that the coherence growth is being limited, presumably by the presence fluorine
spins perturbed by strontium atoms.

Our interpretation of the relatively slow growth dynamics observed in the
strontium doped sample involves viewing the spin system as clusters of like spins
bounded by perturbed spins which cannot enter multiple-quantum coherence.
Although the Pl-MQ—NMR scheme employed is designed to create a double-
quantum selective Hamiltonian to first order”, even for spin systems involving
spins with different chemical shifts, it fails when these shifts become to large. At the
orientation which was selected (8 = 90 degrees) the chemical shift difference Ao
(depending on (it ) between unperturbed fluorine spins and strontium perturbed
spins varied from 8 ppm to 21 ppm (see Figure IV-3). To demonstrate how shift
differences of ca. 20 ppm block coherences, we have computed the time
development of the triple-quantum coherence for linear arrays of four spins
separated by 3.44 Angstroms (as in fluorapatite) with chemically shifted spins. The
results of this computation are given in Figure lV-11. Figure IV-1Ia is a plot of the
triple quantum coherence as a function preparation time for a system of four like
spins and a system with one of the middle spins shifted by 10 ppm. Each of the two
traces show a rise to approximately the same level and have deep oscillations;
however, there is a substantial difference in the values after approximately 200 us.
Figure IV-1 10 contains the calculated triple quantum-coherence for a linear array of
four spins with one of the middle spins shift by 50 ppm in addition to the triple
quantum coherence fora set of spins with equivalent shifts. On the time scale
shown the trace corresponding to the cluster of like spins rises quickly and

oscillates about a values of approximately 6X10'4. The trace corresponding to a

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Preparation Time (us) 1132.8

Figure lV-9. Plot of the multiple—quantum coherence orders as a function of
preparation time for stoichiometric fluorapatite and strontium doped fluorapatite. The
upfield edge of the pattern 36 ppm was ”selected” corresponding to crystallites 6 =

90°. All coherence orders are normalized to the 2-quantum intensities.

125

 

 

8-
I
20..
00‘“ I
.5
(D
0
.g I
0 I
0
fie~~ . P”
U
I. U
DO
U
o 4 t
0 500 1000 1500

Preparation Time (us)

Figure lV-10. Plot of effective size n (obtained from the coherence order in figure
lV-9.) vs. multiple-quantum preparation time. Results from the two samples are
shown, calcium fluorapatite (I) and 9/1 calcium/strontium solids solution (CI) .

Experimental errors, including both reproducibility as well as fitting errors, are less

than 1.5x the width of the squares.

126

0.003 1
0.0025 *-

.9

0.0015 “
0.001 ..

 

3Q Coherence Intensity

 

 

 

0.0005 . .

 

 

 

 

 

 

 

 

 

0 3 6 9 12 15
Preparation Time ms

2.10603 1

B)

i

 

 

 

 

 

 

 

 

30 Coherence Intensity

 

0.6
Preparation Time ms
figure lV-11. ANTIOPE simulation of the triple-quantum intensity as a function of

 

multiple quantum preparation time for the double quantum selective Pl -MQ NMR
pulse sequence. The spin system is a linear cluster of 4 fluorine spins separated by
3.44 angstroms initially in the Ix state. 11a) trace corresponding to four spins with
equivalent shifts ( solid line) and trace corresponding to four spins with a single
middle spins shifted by 10 ppm. 11b) trace corresponding to four spins with
equivalent shifts ( solid line) and trace corresponding to four spins with a single

middle spins shifted by 50 ppm.

127

cluster with a chemically shifted spin only rises to oscillate about a level of
approximately 1X10'4 and does so near the end of the preparation time shown.
Figure lV-11 demonstrates that higher coherence orders do not develop in the
same manner in spin clusters with equivalent shifts as compared to those
containing shifted spins. Furthermore, higher coherence are severely limited when
the chemical shift perurbations are only several times greater than the homonuclear
couplings. This ability of shifted spins to block growth of higher coherence is what
we attribute the slow growth dynamics shown in Figure lV-9 and lV-10. That is,
fluorines spins perturbed by strontium ions in the strontium-doped sample create
boundaries that block coherence growth and the observed coherences ( in figure
lV-9) simply reflect the growth dynamics among clusters of like spins.

We have tested the assumption that the growth dynamics for the stronfium
doped sample can be modeled as a distribution of clusters of like spins created by
the random placement of perturbed spins. The basic scheme to extract information
about the size of linear clusters from MO coherence orders was first proposed by
Cho and Yesinowski.8 This model predicts that the observed coherence orders are
the sum of the contributions of the coherence order magnitudes expected for each
cluster weighted by the mole fraction of spins in cluster of a specific size. A
comparison of the values of the ratio of the 4>quantum intensity to 2-quantum
intensity predicted by the linear cluster model and the experimental values of this
ratio is given in Figure lV-12. The agreement between the cluster model and the
observed values for the strontium doped sample implies that the MQ—NMR
dynamics of the strontium doped spins system can be simply viewed as clusters
bounded by perturbed spins.

Since this technique has allowed us to gain information about the distance
over which the tensors of unperturbed spins remain the same, it should be capable

of yielding similar information on other spin systems. We have previously defined a

128

correlation length for spins with like tensors as the distance between the outermost
spins in a cluster of like spins. The average (mean) run length of the cluster of
unpertubed spins in this sample was 3.5 giving a correlation length of 12
angstroms.
3.3. Antimony Doped Fluorapatite

In this section the chemical shift selective MQ—NMR technique is used on a
practical problem; the search for fluoride vacancies in anti mony-doped fluorapatite
which are thought to give this material its phosphor qualities. As in the strontium
doped system the chemical-shift-selective Pl MQ—NMR experiment was set up to
record the MO spectrum of the upfield edge of the pattern (crystals with the
orientation at 6 = 90°). The profile of the coherence orders (not shown) matched
that of the stoichiemtric fluorapatite within experimental error. Specifically, the 4-
quantum peaks were 0.32(Sb doped apatite)and 0.330 (pure fluorapatite) the height
of the 2-quantum peaks at a preparation time of 991 us. The experimental error on
the 4020 intensity rations is about 0.03. Furthermore, the linear cluster model
predicts that if fluoride vacancies were present at the level of antimony in the lattice
(13.6 mole %) the 4-quantum peaks would be only 0.22 as large as the 2—quantum
peaks. Since the normalized 4-quantum intensity in the antimony doped sample is
within experimental error of the value for stoichiometric fluorapatite we conclude
that there are no fluoride vacancies in the lattice. This strengthens the previously
proposed scheme of charge compensation for the antimonate ion whereby SbOg'3

replaces a PO4’3 proposed in section II.

129

40:20

 

 

 

 

 

425 566 708 850 991

Prepartion time (us)

figure lV-12. Plot of the 4-quantum'2-quantum vs. preparation time for
stiochiometric fluorapatite (D) and strontium-doped fluorapatite (O). The solids
squares (I) represent the values calculated using the linear cluster model (see

text).

130

4. Conclusion.

The result of doing chemical shift selective MO-NMR experiment on
fluorapatite systems succeeded in gaining orientation specific M0—NMR data of a
specific class of crystallite orientations. In the stronb'um doped sample we were
able to show that spins with large enough shift perurbations could effective block
coherence growth much the same way a vacancy would. The multiple quantum
dynamics in this sample reflect coherence growth among clusters of spins with the
same chemical shift bounded by strontium-perturbed spins with different shielding
tensors. Using this technique to obtain correlation lengths in other types of systems
could yield useful information such as the curvature or tacticity of polymers. Finally,
this technique was able to clearly show that fluoride vacancies are not responsible
for charge compensation in antimony doped fluorapatite. The implication of this
finding is that antimony alone is responsible for giving this material its phosphor

qualifies.

1 31
Section V.

Chemlcal-Shift Selective Multiple Quantum NMR of Fluorapatite:
Study of the Effect of Dimensionality on M0 Dynamics

1. Introduction .

In section IV the chemical shift selective multiple-quantum NMR technique
was use to obtain clustering information about clusters that were present in
fluorapatite samples. To expedite data collection, the peak in the powder pattern at
the upfield edge was observed since it is the most intense portion of the powder
pattern. In this section (V) we will look at other orientations of crystallites by
examining different peaks in the powder pattern with the goal learning how the
spatial arrangements of spins and the dipolar coupling strength between spins
effects growth dynamics. The rate of coherence growth is proportional to the
strength of dipolar couplings 1‘4; however, this is difficult to demonstrate
experimentally since the couplings between neighboring spins in even simple
systems have different dependence upon the crystal orientation. In fluorapatite the
dipolar coupling between fluorine spins found in columns have-a uniform
dependence on the angle between the c axis of the crystal and the applied field.
Using the chemical-shift-selective M0 technique to record the M0 spectra of
various peaks in the fluorapatite powder pattern we will show that coherence growth
among columns of spin is proportional to the dipolar couplings.

The second aspect of growth dynamics examined in this section is the effect
of dimensionality of spin arrangements on M0 growth rates. Levy and Gleason
have developed a model for the growth dynamics of spin coherences that
incorporates spatial information about the arrangements of spins. The Levy-
Gleason18 model predicts that the growth of coherences as a function of

preparation time is linear, quadratic and cubic for one, two and three dimensional

132

arrangements of spins with the absolute rate of growth scale by the diplolar
couplings between spins. This prediction has been experimentally verified for
coherence growth in two and three dimensional spin arrangements of the protons
on the surface of CVD diamond and in calcium hydride.4 Cho and Yesinowski
have shown that effective size growth in the quasi-one-dlmensional arrangement of
spins in apatites is linear with preparation time; however, the rate three time that
predicted by the Levy-Gleason model.8

In this section ideas about how to use the chemical-shift—selective M0 NMR
technique to study the growth dynamics among planes and columns of spins in
fluorapatite are given. The basic idea is to record the M0 growth dynamics of peaks
in the fluorapatite powder pattern that correspond to crystallite orientations that
would either favor growth among columns of spins or planes of spins in fluorapatite.
To illustrate these columns and planes of spins, a schematic arrangement of the
fluoride spins in fluorapatite is given in Figure V-1. Each of the spins in the middle
columns are surrounded by six nearest neighbors in a plane perpendicular to the
columns. We have computed the intra-planar and intra-column dipolar couplings as
a function of crystallite orientations (see below) to determine peaks in the powder
pattern that would optimally favor one-dimensional or two-dimensional growth

coherence growth.

133

 

 

 

 

 

 

 

 

 

 

figure V-1 Schematic arrangements of 19F spins (black solids circles) in calcium

fluorapatite.

134

2 . Experimental

The details of the chemical-shift—selective Pl-MQ-NMR experiment are given
in section lV-2. The frequencies of the twenty-four saturation pulses were
adjusted to remove different portions of powder pattern. One experiment
involved removing all of the magnetization except for the peak at 57.4 ppm
which corresponds to crystals oriented 0=67°. The offsets of the saturation
pulses from the upfield edge were, in order of application 25, 24, 23,....12,
0, 12, 0, 12 ,0 ,12, 0, 12, and 12 KHz. In a different experiment all of the
magnetization except for the peak at the downfield edge of the pattern at
120 ppm which corresponds to crystals oriented 0=0° was removed. The
offsets of the saturation pulses from the downfield edge were, in order of
application, 25, 24, 23,22, 6 KHz , with five saturation pulses repeated
at only the last offset.
3. Results and Discussion
3.1 Computation of intra-planar and intra-column couplings

In order to determine the crystal orientations that will favor coherence growth
in either the planes of spins or the columns of spins in fluorapatite, the dipolar
couplings among the spins in both of these types of arrangement must be
computed for all crystal orientations. The Levy-Gleason18 model for the effect of
dimensionality upon M0 coherence growth dynamics predicts that the growth rate
of coherences is proportional the dipolar couplings between the spins and the
number of spins immediately outside a coherent cluster of spins that can enter that
coherence. The fact that the number of spins on the periphery of a coherent cluster
of spins increases from; chains of spins, to planes of spins, to solid blocks of spins

is responsible for the linear, quadratic, and cubic growth of coherences for each of

135

these arrangements respectively. In the case of fluorapatite each spin in the column
has two nearest neighbors and each spin in a plane has six nearest neighbors.
These planes and columns of spins are shown clearly in Figure V-2. The relative
growth rate of coherences among spins in the planes vs. spins in the columns will
be proportional to the dipolar couplings between the spins in these arrangements
and the number of nearest neighboring spins.

Expressions for the relative M0 coherence growth rates in the planes and
columns of spins can be created by summing the dipolar couplings between each
spin pair in these arrangements. These summed dipolar coupling expressions can
be written as functions of crystallite orientation by explicitly considering how each of
the internuclear vectors transform as a function of crystallite orientation.
Furthermore, these expression should only be a function of angle between the c
axis of the fluorapatite crystal and the applied field, since this is the orientation that
can be ”selected" from the powder pattern using the chemical-shift-selective M0-
NMR technique. For clarity in this discussion we will refer the angle between the c
axis of the crystal and the applied field as 6FAp. The dipolar coupling between a

pair of spins has the form:

D(r, 6) = Ci3(1—3cos2 9) V-1
r

where r is the distance between spins, 6 is the angle between the internuclear
vector and the applied field, and C represents product of several factors (see
section l-3.2 for details). Since the value of C is the same for all of the 19F-19F
spins pairs we are considering, we only need to consider r and 6 in the
development of an expression for the dipolar couplings between chains and planes
of spins. In order to relate the 6 of each spin pair pictured in Figure V-2 to the angle

OFAP we must examine how each of the inter nuclear vectors transforms in the axis

 

136

 

 

 

 

 

 

 

Figure V-2 Schematic diagram of linear and planar arrangements of spins in

fluorapatite. The coordinate system relates the molecular frame to the applied field.

137

system of Figure V-2. This can be done by employing a pair of rotation matrices.
The first matrix describes the new coordinates of a vector after the axis system has
been rotated about the z axis so that the angle between the original y axis and the
new y' is y as shown in Figure V-2 . The second matrix describes the vector after
rotation about the y' axis so that the angle between the original 2 axis and the new
2' is 6FAp as shown in Figure V-2. These two matrices R2 and Ry acting on vector

(Vx’ Vy, Vz)T and transforming it to a vector V‘ take the following form:

cos 7 sin 7 0 cos 617A}: 0 - sin 9MP VX
RZRy I” = - sin 7 cos 7 0 0 1 0 Vy
0 0 1 sin 91:)”: 0 COS aFAP VZ
V-3
VX
= V),
V2
When the two rotation matrices are multiplied together their product is
cos 0“}: cos 7 cos QFAP sin 7 — sin 6”}:
RZRy = — sin 7 cos 7 0 . V-4

sin OFAP cos 7 sin 9MP sin 7 cos 6FAP

We see from Eqn. V-4 that when a general vector (Vx, Vy, VZ)T is subjected this
double rotation transformation the 2 component of the new transformed vector is

VZ = VJr sin 9MP cos 7+ Vy sin 6”}: sin 7 + cos 934p. V-5
This can be shown by multiplying (vx, vy, v2)T by RzRy. By knowing the 2
component of the rotated vector one can then know the angle between the rotated
vector and the applied field since the coordinate system is defined with the applied
field vector Ho coincident with the z axis. The 2 component of a vector of unit length
coincident to the rotated vector V' is equal to the c096 where 6 is the angle between

the rotated vector and the applied field.

138

The development of an expression for the dipolar couplings between the
linear and planar arrangements of spins picture in Figure V-2 can be completed by
installing the vectors that described each of the spins in the axis system into
equation V5. The position of the planar hexagon of spins can be described by the
following vectors of unit length: (0,1 ,0)T, (0.866, 0.5, 0)T. (0.866, -0.5, 0)T. (0,-1,0)T,
(-0.866, 0.5, 0)T (-0.866, -0.5, 0)T. The position of the two spins in the chain of
spins above and below the x-y plane can be described by the following vectors of
unit length: (0,1,0)T, (0.866, 0.5, 0)T. Since the Vz' component given in Eqn. V5 is
equal to c096, we can simply install the right side of Eqn. V-5 into equation Eqn. V-1
to get an expression for the dipolar coupling as a function of crystallite orientation.

For the planar array of spins this expression takes the form:

1- 3(sin an”) sin y)2]+
1- 3(O.866sin 9m. cos{7}+.5 sin em. sin 7)2]+
3 72h I— 3(O.866sin 19m. cos{7} — 0.5sin 19m. sin 7)2] +

 

' . . 2
rmter 1_3(-Sm6FAPsm7) 1+

0 ' . 2
l— 3(-0.866s1n 9MP cos{7} + 0.58m 6MP sm 7) ]+

 

 

 

I — 3(—0.866 sin 9MP cos{ 7} + —0.5 sin 9MP sin 7)2]

V-6
where rinter is the distance between the spin at the origin in Figure V-2 and the
spins on the vertices of the hexagon. A similar expression can be derived for the
dipolar couplings between the column of spins:

-3 7211 2
D(ld)-Z 3[2[1-3cos 9]]. v-7

rmtra

 

139

With the expressions for the summed dipolar couplings for the linear and
planar arrangements of spins in fluorapatite given in Eqn. V-6 and V—7 we can
compare the MO coherence growth rates in either of these two arrangements as a
function of crystal orientation. Since only BFAp can be experimentally controlled (by
selecting different peaks in the powder pattern), the expression for the dipolar
couplings among spin pairs in the plane (Eqn V-6) is averaged over values of y at
each value of BFAp. This results in an expression for the intra-planar summed
dipolar couplings which is soley a function of OH“). Figure V-3 is a plot of the ratio
of the summed dipolar couplings between spin pairs in the column to the summed
couplings in the plane as a function OFAP. Regions of the plot where the ratio of the
”linear" to the "planar" couplings is large corresponds to orientations that would
favor coherence growth among the columns of spins in fluorapatite. When OFAP is
equal to the magic angle (54.7°) the intra-chain couplings are the smallest and
planar growth would dominate. The most favorable orientation for linear coherence

growth is at 9 = 0°.

140

 

 

 

 

Figure V-3 Plot of the ratio of the summed dipolar couplings of linear spins D(linear)

to the summed dipoar couplings of planar spins D(planar) as a funcion of 9.

141

3.2 Multiple quantum NMR of various orientations

To demonstrate that the growth of MO coherence does scale as the dipolar
coupling between spins the chemical-shift-selective MQ-NMR was used to obtain
the MO growth dynamics of crystals at different orientations. In the first orientation,
6 = 67.5°, the strength of the dipolar coupling between the spins in the columns is
half of the dipolar coupling at O = 90°. Figure V-4 is a plot of the effective size as a
function of preparation time for crystals oriented at 9 = 67.5° and 9 = 90°. The slope
of the lower trace over the first 5 points in Figure V-4 corresponding 9 = 67.5° is
approximately half that of the upper trace. The points beyond the fifth data point
cannot be compared since the MO growth deviates from linear (toward quadratic)
and this region of the plot represents the onset of growth among planes of spins.
Figure V-4 supports the prediction that coherence growth is proportional to the
coupling between spins.

An attempt was made to record the MO growth dynamics of crystallites
oriented at 9 = 0 °. At this orientation the dipolar couplings between spins in the
chains is strongest relative to the spins in the planar arrangements. The relatively
stronger ”linear couplings” should have the effect of increasing the linear portion of
a plot of the effective size vs. preparation time since growth among the one-
dimensional arrays would be favored. Unfortunately the MO spectra were not able
to be recorded. This presumably resulted from the relatively low intensity of the
portion of the powder pattern corresponding to orientations at 9 = 0 °; however the

real cause has yet to be determined.

142

301'

:20»

15 i» , _
g 10 ‘J’ n ‘

 

 

300 500 700 900 1100 1300

Figure V-4 Effective size vs preparation time in fluorapatite. Results from two

orientations are shown; 9 = 90° (I) and 6 = 90° (Cl).

143

References for Sections IV and V

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4. K. K. Gleason, Conc. Magn. Reson. 5 (1993) 199.

5. J. Baum, M. Munowitz, A. N. Garroway and A. Pines, J. Chem. Phys. 83 (1985)
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6. J. Baum and A. Pines, J. Am. Chem. Soc. 108 (1986) 7447.

7. B. E. Scruggs and K. K. Gleason, Chem. Phys. 166 (1992) 367.

8

9

OD

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12. D. N. Shykind, J. Baum, S.-B. Liu, A. Pines and A. N. Garroway, J. Magn.
Reson. 76 (1988) 149.
13. P. Caravatti, M. H. Levitt and RR. Ernst, J. Magn. Reson. 68 (1986) 323.
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