t .22
.2 .1135
v31...

.9

vs

:9. .. 4

5...... ‘11er
.35?!
v. .. ,

. 1‘
A In.‘ .
I 1:. r: ..
. is... :1 C‘J. .. 3.
1:3,; nuns:
‘ .z. ._
a. o...
r...“
.. 2
..

>r‘i

q!!- .
.. ‘ .933
‘12:...-

»llix: \
.‘Ilc ,

.3 1
1.3.... ‘
I, push} .
31....
TI.

 

 

 

 

 

 

2 . y C
.1! . ‘3}. n (V

fifitrafi. . . . .
. . Qwfiflxum...»

13‘“.

 

 

,‘x , I
l ‘_l.
' ‘ rt?"

CHIGAN STATE UNIVERSITY LIBRARIES

Ill“ WWWWillmll‘lll mu

3 1293 01026 9862

 

l

 

 

 

 

 

 

 

This is to certify that the

dissertation entitled
NMR INVESTIGATIONS OF CHEMICAL SHIELDING,

STRUCTURE, AND MULTIPLE QUANTUM-DYNAMICS
OF APATITES

presented by

Gyunggoo Cho

has been accepted towards fulfillment
of the requirements for

Ph. D degree in Chemistry

 

 

Major professor /

Date &2‘%1923

MS U is an Affirmative Action/Equal Opportunity lnsn'tun'on 0-12771

 

 

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to move thII checkout from your record.
TO AVOID FINES return on or baton data duo.

DATE DUE DATE DUE DATE DUE

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU IoAn Atflnnutlvo ActionlEqunl Opportmlty InstItqun

 

 

 

 

—______Z——-———

 

NMFI INVESTIGATIONS OF CHEMICAL
SHIELDING, STRUCTURE, AND MULTIPLE-
QUANTUM DYNAMICS OF APATITES

by

Gyunggoo Cho

A DISSERTATION

Submitted to Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Chemistry

1 993

 

ABSTRACT

NMR INVESTIGATIONS OF CHEMICAL SHIELDING, STRUCTURE,
AND MULTIPLE-QUANTUM DYNAMICS OF APATITES

by

Gyunggoo Cho

The 19F MAS-NMR spectra of a series of fluorapatites, M5F(PO4)3, where M
a Ca“, Sr2+ and Ba“, and solid solutions of Ca/Sr fluorapatite have been ob-
tained. The crystallographic symmetry about the fluoride ions requires that the
chemical shielding tensors be axially symmetric. The principal components of
' the 19F shielding tensor of M5F(PO4)3 are obtained from 19F MAS-NMR spectra
using the moments method and Herzfeld and Berger graphical method. The use-
fulness of these two methods is demonstrated by using the comparison between
experimental spectra and simulated spectra obtained from the chemical shield-
ing tensors. The measured chemical shielding tensors enable us to separate the
contributions to the Ramsey paramagnetic shielding term from the sigma- and pi-
bonding between the alkaline earth metal ions and the fluoride ions. The values
of sigma- and pi-bonding contributions to 19F shielding for M5F(PO4)3 (M - Ca2+,
Sr’+ and 8a“) are 81.7 ppm and 24.6 ppm, 97.6 ppm and 26.0 ppm, and 138.4

ppm and 32.1 ppm respectively with respect to free fluoride ion.

The preference of Sr“ ions for the Ca(2) site for fluorapatite have

been studied using 19F MAS-NMR spectra of a solid solution of composition

Cag,97$r1,03F2(PO4)6. The peak intensities obtained from the centerband and
sidebands as well as the deconvolution peak indicate that Sr2+ ions have a 23%
preference for the Ca(2) site, which is adjacent to the fluoride ion. The assignment
of the spectra of Ca3,978r1.03F2(PO4)5 is aided by the existence of spin diffusion
performed by the SPARTAN pulse sequence.

The dimensionality of the distribution of spins in solids influences their multiple-
quantum NMR dynamics. We have studied these dynamics for the quasi-
one-dimensional distribution of uniformly spaced proton spins in hydroxyapatite,
Ca5(OH)(PO4)3, and related compounds, using a phase-incremented even order
selective MQ pulse sequence. The increase in effective size N with prepara-
tion times for stoichiometric monoclinic hydroxyapatite is linear at early times, in
agreement with calculations based on the incremental shell model; however, the
experimental slope is three times greater than the predicted slope. An upward
curvature observed at longer preparation times is qualitatively ascribed to the in-
complete isolation of linear chains. A slight deficiency of hydroxyl groups in a
sample of hydroxyapatite in the commonly-occurring hexagonal crystal form leads
to a measurable decrease in the slope of the linear portion of the curve. The
1H multiple-quantum dynamics of a series of fluorohydroxyapatite solid solutions,
Ca5(OH)xF1-x(PO4)3, exhibit decreased slopes for lower hydroxyl levels (smaller
x), and requires consideration of the different lengths of spin “clusters” in order
to model the behavior. The defect densities of apatites (clusters) are estimated

by using 1-D cluster model.

We have also studied the 19F multiple-quantum NMR of a single crystal of
mineral fluorapatite at different orientations with respect to the external magnetic

field. The observed oscillatory behavior of the multiple-quantum dynamics is

interpreted in terms of 1-D clusters of fluoride ions in the defect-containing sample.

 

To my mother, wife, and son

 

ACKNOWLEDGMENTS

I would like to sincerely thank Dr. James P. Yesinowski for his continuous
guidance and enthusiastic encouragement throughout my research. I would also
like to thank Dr. Dye, Dr. Cukier, and Dr. Schwendeman for helpful discussions.

My special thanks go to Liam B. Moran for helpful discussions of my research.
I would like to thank Mr. Kermit Johnson for helpful discussions of new NMR
techniques and aid with computers. Thanks are also extended to the members
of the NMR facility and the members of machine shop, in particular to Dr. L. Le,
Mr. D. Jablonski, and R. Geyer. I would also like to acknowledge GTE Electrical
Products Corporation for partial financial support of this work. i would also liked
to thanks Dr. Chung-Nin Chau of GTE Electrical Products Corp. for preparing
the fluorapatite samples and Dr. B. Fowler of National Institute of Standards and
National Institute of Dental Research for lending us the samples of monoclinic
hydroxyapatite and single-crystal fluorapatite.

I would like to thank my Korean friends at Michigan State University who
encouraged me when l was discouraged.

Finally, my special thanks are reserved for my family: my wife, Sukhee, my
son, Nammyoung, and my mother, who have prayed and waited patiently for my

success.

vi

TABLE OF CONTENTS

LIST OF TABLES ..................................... xi
LIST OF FIGURES ................................... xii

I. INTRODUCTION TO APATITE STRUCTURE AND CHEMISTRY. 1
References ..................................... 6
ll. PART 1
19F MAS-NMR Investigation of Alkaline Earth Fluorapatltes:
Measurement of Chemical Shielding Tensors and Characteriz-
ation of Sites In Ca/Sr Fluorapatlte Solid Solutions

1. Introduction ................................... 9
2. Nuclear Spin Interactions in Solids .................. 11
A. Zeeman Interaction ............................. 11
B. Radiofrequency Interaction ........................ 12
C. Chemical Shielding Interaction ...................... 12
D. Dipolar Interaction .............................. 14
3. Magic Angle Spinning (MAS) ....................... 17
4. Methods for Measuring the Chemical Shift Anisotropy
from MAS-NMR Spectra .......................... 20
A. Moments Method .............................. 20
B. Herzfeld and Berger Graphical Method ................. 21
5. Experimental .................................. 22
A. MAS-NMR Studies ............................. 22

B. Simulation of 19F MAS-NMR Spectra of Fluorapatlte Samples . . . 24
6. Results ...................................... 25

vii

A. Alkaline Earth Fluorapatites ........................ 25
B. Solid Solutions of Ca/Sr Fluorapatite .................. 34
7. Discussion ................................... 42
A. Measurement of the 19F Chemical Shift Anisotropies of M5F(PO4)3
Using Different Methods .......................... 42
B. Separation of Metal-Fluoride Sigma- and Pi-Bonding Contributions
to the 19F Shielding Tensor ........................ 44
C. Study of Site-Preference of Sr2+ in Ca/Sr Fluorapatite Solid

Solutions Using 19F MAS-NMR ....................... 53
8. Conclusions ................................... 63
9. References ................................... 65
III. PART 2

1H and ”F Multiple-Quantum NMR Dynamics of Quasi-One-
Dlmenslonal Spin Distributions In Apatltes
1. Introduction .................................. 69
2. Multiple-Quantum NMR Dynamics .................. 75
A. Density Operator Description of Multiple-Quantum NMR
Dynamics ................................... 75

B. Time Development of the Density Operator in the Rotating

Frame ..................................... 79
C. Pulse Sequence for Multiple-Quantum NMR .............. 81
D. Statistical Model of MO Coherence Intensities ............ 91
E. Simplified Models of Multiple-Quantum Dynamics .......... 92

a. Hopping Model .............................. 95

b. Incremental Shell Model ....................... 104

viii

3. Experimental ................................. 109

A. Multiple-Quantum NMR Studies .................... 109
B. Sample Preparation and Characterization ............... 113
4. Results ..................................... 114

A. 1H Multiple-Quantum NMR Study of Resonance-Offset Effects. . 114
B. 1H Multiple-Quantum NMR of Hydroxyapatite and Fluorapatite

Samples .................................. 121
C. 19F Multiple-Quantum NMR of a Single Crystal of Fluorapatite . 131

5. Discussion ................................... 131

A. Effect of Resonance Offsets in PI-MO NMR Pulse Sequence

on the Formation of Multiple-Quantum Coherence ......... 131
B. Dimensionality Effects in the Multiple-Quantum NMR of

hydroxyapatite ............................... 1 34
C. One-Dimensional Cluster Model .................... 136
D. Estimation of Defect Densities Using the 1-D Cluster Model . . . 139

E. 19F Multiple-Quantum NMR Dynamics of Single Crystal

Fluorapatite ................................. 1 44
6. Conclusions .................................. 149
7. References ................................... 154

Appendix A. Phase-incremented multiple-quantum pulse
sequence program for solids (Varian VXR
400 spectrometer) ..................... 163
Appendix 8. Selection rule of the 1-spin/2-quantum average
Hamiltonian ........................... 167
Appendix C. C program for calculation of the ratio of 40/20
intensities for 1-D chain with different defect

densities ............................ 1 70

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.
Table 6.

LIST OF TABLES

19F NMR chemical shift anisotropies and asymmetry parameters of
M5F(PO4)3 calculated using the moments method .......... 26
The 19F chemical shift anisotropy and asymmetry
parameter of M5F(PO4)3 obtained using the Herzfeld and Berger
graphical method ............................... 32
19F chemical shielding tensors of M5F(PO4)3, calculated from
the moments method using peak intensities obtained
from deconvolution ............................. 32
19F NMR paramagnetic shielding parameters a; and an for
M5F(PO4)3 .................................. 51
Calculated probabilities of configurations A-D in Ca/Sr solid solution . 58

Calculated probabilities of configurations H" in Ca/Sr solid solution . 58

LISTOF FIGURES

Figure 1. Structure of hydroxyapatite .......................... 5
Figure l-1. Chemical shielding powder pattern .................... 13
Figure l-2. Cartesian axis system and polar coordinates for the dipolar

coupling of two nuclei ........................... 16
Figure l-3. Orientation of magnetic field and spinning axis vectors for magic

angle spinning experiments ........................ 18
Figure l-4. SPARTAN pulse sequence for monitoring spectral spin diffusion . . 23
Figure l-5. Contour plots of Ca5F(PO4)3 spinning at 6.12 kHz and 10.18 kHz

using the Herzfeld and Berger graphic method ............ 29
Figure l-6. Contour plots of Sr5F(PO4)3 spinning at 6.26 kHz and 10.20 kHz

using the Herzfeld and Berger graphic method ............ 30

Figure l-7. Contour plots of Ba5F(PO4)3 spinning at 6.23 kHz and 10.18 kHz

1 using the Herzfeld and Berger graphic method ............ 31
Figure l-8. The experimental and simulated spectra of M5F(PO4)3 ....... 35
Figure I-9. 19F MAS-NMR center band HHLW vs spinning speed for

M5F(PO4)3 .................................. 36
Figure l-10 ‘9F MAS-NMR of Ca3,978r1,03F2(PO4)5 spinning at 8.23 kHz and

at 10.80 kHz ................................. 38
Figure l-11. Deconvolution spectrum of the 64 ppm peak in Fig. l-10b ..... 39
Figure l-12. 19F MAS-NMR spin diffusion experiment of Ca3,978r1,03F2(PO4)6

using the SPARTAN pulse sequence and spinning at 10.87 kHz . 40
Figure I-13. 19F MAS-NMR of CaSSr5F2(PO4)5 spinning at 8.25 kHz and

at 10.3 kHz ................................. 41

xii

Figure l-14.

Figure l-15.

Figure l-16.

Figure l-17.

Figure l-18.

Figure ”-1.

Figure "-2.

Figure "-3.

Figure ll-4.

Figure "-5.

Figure "-6.

Comparison of the simulated and experimental 19F MAS-NMR

stick spectra of M5F(PO4)3 at the same spinning speed ..... 45
a0 and an for M5F(PO4)3 vs. 19F isotropic chemical shift

with respect to free F' ion ........................ 52
Substitution of Sr2+ in the three nearest neighbor Ca(2) sites of
Ca/Sr fluorapatite solid solutions .................... 54
Possible configurations for single and double Sr2+ substitution

in the six next-nearest neighbor Ca(2) sites of Ca/Sr solid

solutions of fluorapatite .......................... 57
Variation in the instantaneous chemical shifts during a rotor

cycle of 64 ppm and 79 ppm peaks of the solid solution

under the MAS condition at the different crystallite orientations . . 62
Schematic idealized arrangement of linear columns of protons

in calcium hydroxyapatite (Ca50H(PO4)3) ............... 73
Random phases and correlated phases in an ensemble of
two level systems .............................. 77
General form of the pulse sequence of MO NMR experiments . . . 82
Pulse sequence for multiple-quantum NMR in solids ......... 85
Time-domain MO interferogram of hexamethylbenzene

using the TPPl ................................ 86
Frequency-domain 1H MO spectrum of hexamethylbenzene

using the TPPI ................................ e7

xiii

Figure “-7. Time-domain MO interferogram of hexamethylbenzene

using the phase-incremented method .................. 89
Figure "-8. Frequency-domain 1H MO spectrum of hexamethylbenzene

using the phase-incremented method .................. 90
Figure ”-9. Schematic energy level diagram for an N (odd) spin-1/2 system

in a Zeeman field, and the degeneracy number of each state . . . 93
Figure ”-10. Symbolic representation of the spreading of multiple spin corre-

lations in a coupling network with increasing preparation time . . . 94
Figure "-11. Projection of Liouville space onto a two-dimensional plane . . . . 97
Figure "-12. Pathways of the growth of MO coherences in Liouville space

for a 6-spin system under the 1-spin/2-quantum Hamiltonian . . . 98
Figure "-13. Evolution of n-quantum coherences predicted by the hopping

model in 6 and 20 spin systems ..................... 102
Figure "-14. Development of effective size with increasing preparation times

for a 20 spin system predicted by the hopping model ....... 103
Figure "-15. Diagram representing the average M spin operator and its

constituent operators of M a 4 in a 9 spin system ......... 106
Figure "-16. Schematic diagram of the growth of MO coherence of different

dimensionalities .............................. 1 10
Figure "-17. Values of effective size N predicted by the incremental shell

model for spin-1/2 nuclei evenly spaced on a line (1-D), square

(2-0). and cubic (3-0) lattice ...................... 111

xiv

Figure "-18.

Figure "-19.
Figure "-20.

Figure "-21.

Figure “-22.

Figure "-23.

Figure "-24.

Figure "-25.

Figure "-26.
Figure "-27.

Figure "-28.

Figure "-29.

1H Pl-MO-NMR spectra of hexamethylbenzene with different
transmitter carrier frequencies ..................... 115
1H 1-dimensional spectra of apatite samples ............ 116
1H MO NMR spectrum of HAP-M using a preparation time

of 864 [LS ................................. 118
Intensities of the 1H Pl NMR MO order of HAP-M for various
resonance offsets using a preparation time 01864 [IS ....... 119
Intensities of the 1H Pl NMR MO order of HAP-M for various
resonance offsets using a preparation time of 1872 p8 ...... 120
The normalized intensities of even-order MO coherences of

apatite samples as a function of preparation time .......... 122
Effective size N vs. MO preparation time for HAP-M ........ 125
In(N) vs. In(r/rc) for all of the samples of hydroxyapatite and
fluorohydroxyapatite ........................... 1 26
Decay of absolute MO coherences of apatites ........... 128
The normalized intensities of OO and ZO coherences of

a single crystal mineral sample of FAP with preparation times

at two orientations ............................. 129
In(absolute intensity of MO orders) vs. preparation time of

single crystal fluorapatite at two different orientation ....... 130
The effective magnetic field in the rotating frame in the presence

of an applied rf field .......................... 132

XV

Figure ”-30.

Figure "-31.

Figure "-32.

Figure "-33.

Figure "-34.

Figure "-35.

Figure "-36.

Arrangement of hydroxyl groups separated by vacancies and/or
fluoride ion substitutions in apatite samples ............. 138
Run number distributions for various defect densities in a one-
dimensional chain . . . .. ......................... 141
1-D cluster model of MO growth for various defect densities in
apatite, compared to experimental data ............... 142
Comparison between experimental 40/ZO intensities and
calculated 40/ZO intensities for FOHAP samples after
excluding two hydroxyl groups adjacent to fluorine ions ...... 143
Simulated MO dynamics of 2, 3, 4, and 5 spin systems under
”Hg, for an oriented linear chain using ANTIOPE .......... 145
Mole fraction of spins in various run numbers and sum of the
mole fraction with different defect densities using percolation
theory .................................... 150
Comparison between experimental 20 intensities of single crystal
FAP and calculated 20 intensities of two assumed defect densities

at two different orientations ........................ 151

xvi

I. INTRODUCTION TO APATITE STRUCTURE AND CHEMISTRY

Although the chemical reactions leading to mineral formation in biological
systems are not fully understood, it is believed that a nonstoichiometric ”defect"
form of calcium hydroxyapatite, Ca5(OH)(PO4)3, is the primary mineral phase of
bone, dentin and dental enamel.” Apatites' are also important in the production
of fertilizer, in the lighting industry as a phosphor in fluorescent lamps, and in
chromatography.‘ At the pH values typically found in biological systems, the
“stability of calcium phosphates increases and the solubility decreases as the molar
Ca/P ratio increases. Thus, in vivo dicalcium phosphate dihydrate CaH PO4-2H20
(DCPD, CalP=1.00) hydrolyzes into octacalcium phosphate Ca3H2(PO4)6-5H20
(OCP, Ca/P=1.33), which hydrolyzes into hydroxyapatite (HAP) (Ca/P=1.67).5
Although amorphous calcium phosphate (ACP) is not found in bone and teeth,
it can occur in vivo and is transformed into crystalline apatite via an octacalcium

phosphate-like phase.6

The substitution of numerous impurities in the apatite lattice changes the
properties of apatites. For example, the fluoride ion is effective in prevents
the dental caries, since it increases the rate of remineralization and lessens
the acid demineralization.“ The surface of HAP reacts with fluoride ions to
yield calcium difluoride (Can), fluorohydroxyapatite (Ca5(OH)1-xe(PO4)3), and
fluorapatite (Ca5F(PO4)3), depending upon conditions. Although the fluoridation
of hydroxyapatite is not completely understood, several mechanisms have been
suggested, including the ionic exchange of F' for OH‘ in the apatite structure,9
direct precipitation of fluorapatite mineral, and dissolution of hydroxyapatite and
recrystallization of fluorapatite in the presence of F'.‘° Carbonate ions can also

replace hydroxyl groups or phosphate groups in the apatite lattice. A carbonate ion

1

can substitute for two hydroxyl groups in hydroxyapatite, forming type A carbonate
apatite. This reaction takes place at high temperature,11 and results in an increase
of the a axis length.12 The substitution of phosphate groups by carbonate ions
decreases the a axis and increases the c axis.“3114 The product is referred to as
type B carbonate apatite.11 Calcium ions in apatites can be replaced by strontium
ions. The presence of strontium ions in bone and teeth increases susceptibility
to caries. The incorporation of strontium ions increases the a axis and c axis

length.15

From X-ray crystallographic analysis, hydroxyapatites have either a hexagonal
crystal system with the space group P63/m or, for very stoichiometric samples, a
monoclinic system with the space group P21/b. In the hexagonal crystal system,
there are two Ca5(OH)(PO4)3 groups in a unit cell of dimensions a (=b) = 942
pm and c = 688 pm.5 The structure of hexagonal hydroxyapatite is shown in
Figure 1 projected down the c axis. There are two types of calcium ions in the
structure, Ca(1) and Ca(2). Hydroxyl groups are surrounded by three Ca(2) ions
in an equilateral triangle. Infinite linear chains of protons have a uniform spacing
of 344 pm. The position of the hydroxyl protons is located about 130 pm from the
plane of the triangle of Ca(2) ions. The central column of hydroxyl groups has
six hexagonalIy-situated neighboring columns at distances of 942 pm. Statistical
disorder of the hydroxyl groups result, on average, in protons in three of these
six columns being located about 130 pm below the Ca(2) triangle, and about 130
pm above the Ca(2) triangle in the omer three columns. When viewed down
the (hexagonal) c axis, the Ca(2) ions form two displaced equilateral triangles of
Ca(2) ions which are rotated by 60°. The distance between the two triangles is

344 pm. Columns of Ca(1) ions are parallel to the c axis. Each Ca(1) column

is located at the middle of a large equilateral triangle of three hydroxyl groups.
The unit cell dimensions of monoclinic apatite are a = 942 pm, b = 2a, c = 688
pm, and 7 = 120°.‘6'17 The space group of monoclinic hydroxyapatite, P21/b,
indicates that the monoclinic form is regarded as twins occurring at 120° rotations
about the c axis.“5v17 In the monoclinic crystal form, the protons in four of the
six neighboring columns are located about 130 pm below the plane of the three
Ca(2) ions, and those in the other two columns are located about 130 pm above
the plane of the three Ca(2) ions. Fluorapatite, which has a structure similar to
that of hydroxyapatite, has the same unit cell dimensions except that a = 937 pm.
Fluoride ions are exactly in the middle of the plane of the three Ca(2) ions, which

forms a mirror plane.18

Apatites have been investigated using various methods such as X-ray crys-
tallography, lR, Raman, NMR, etc. The chemical environment of fluorine ions in
alkaline earth fluorapatites and the preference of 812+ ions for Ca(2) sites of Sr/Ca
fluorapatite solids solutions are studied using 19F MAS-NMR. The high level of
defects found in naturally-occurring apatites is of interest in its own right, since
they presumably reflect the conditions of formation of the mineral. Fluoride ions of
fluorapatite are interrupted by charge-coupled vacancies, or substitution of fluoride
ions by carbonate groups, hydroxyl groups, or chlorine ions. Hydroxyl chains in
hydroxyapatite also have vacancies, or hydroxyl group substitutions by carbonate
groups, fluorine ions, or chloride ions. Vacancies of hydroxyl groups in hydrox-
yapatite have been quantitatively determined by using IR and 1H MAS-NMR.20
However, these methods cannot reveal the presence of microscopic heterogene-
ity of various types of anions. The 1H and 19F multiple-quantum NMR experiments

in this thesis provide for the defect in the hydroxyl or fluorine chains and make

it possible to calculate the defect densities of apatites. Furthermore, the quasi-
one-dimensional spin distributions in apatites represent a valuable model system
in which to study the effects of dimensionality upon multiple-quantum NMR dy-

namics.

11a.
o 3:, 3,) a

Figure 1. Structure of hydroxyapatite. Calcium ions (large open circle), hy-
droxyl groups (dotted circle), and phosphorus group (small open circles connected

by a line). See text. Taken from Ref. 5.

References

1. W. E. Brown, Clin. Orthop. 44, 205 (1966).

2. D. G. A. Nelson, J. D. B. Featherstone, Calcif Tissue Int. 34, (Suppl.2) 869
(1982).

3. R. Z. LeGeros and J. P. LeGeros, in ”Phosphate Minerals” (J. O. Niragu and
P. B. Moore, Eds), pp. 351-385, Springer-Verlag, Berlin, 1984.

4. S. J. Joris and C. H. Amberg, J. Phys. Chem. 75, 3167 (1971)

5. W. E. Brown, M. Mathew, and M. S. Tung, Prog. Cryst. Growth Charact. 4,
59 (1991).

6. J. L. Meyer and C. C. Weatherall, J. Colloid Sci, 89, 257 (1982).

7. L M. Silverstone, Caries Res. 11, 59 (1977).

8. D. J. White, Caries Res. 22, 27 (1988).

9. S. A. Leach, Brit. Dent. J. 106, 133 (1959)

10. M. A. Spinelli, G. Brudevold, and E. C. Moreno, Arch. Oral Biol. 16, 187
(1971).

11. G. Bonel and G. Montel, CR Acad. Sci. [0] (Paris) 258, 923 (1964).

12. J. C. Elliott, " The crystallographic structure of dental enamel and related
apatites", Ph.D. Thesis, University of London, 1964.

13. R. Z. LeGeros, Nature, 206, 403 (1965).

14. R. Z. LeGeros, J.P. LeGeros, O.R. Trautz, and E. Klein, J. Dent. Res. 43,
751 (1964).

15. H. J. M. Heijigers, F. C. M. Driessen, and R. M. H. Verbeeck,

Calcif Tissue Int. 29, 127 (1979).
16. J. C. Elliott, Nature Phys. Sci, 230, 72 (1971).
17. J. C. Elliot, P. E. Mackie, and R. A. Young, Science, 180, 1055 (1973).

18. K. Sudarsanan, P. E. Mackie and R. A. Young, Mater. Res. Bull, 7, 1331
(1972).

19. E. C. Moreno, M. Kresak, and R. T. Zaradnik, Caries Res. Supp/.1 142 (1977)

20. J. Arends, J. Christofferson, M. R. Christofferson, H. Eckert, B. O. Fowler,
J. C. Heughebaert, G. H. Nancollas, J. P. Yesinowski and S. J. Zawacki,
J. Cryst. Growth, 84, 515 (1997).

PART 1

19F MAS-NMR investigation or Alkaline Earth Fluorapatltes:
Measurement of Chemical Shielding Tensors and
Characterization of Sites in CalSr Fluorapatlte Solid Solutions

1. Introduction

The chemical shielding tensors of 19F in metal fluoride salts have been of
widespread experimental and theoretical interest."5 Line narrowing techniques
such as multipulse “dipolar decoupling,"2 magic-angle spinning (MAS),6'7 or the
combination of these two techniques (CRAMPS)8 have been employed to obtain
more accurate values for the isotropic chemical shift. However, knowledge of
the three components of an anisotropic chemical shielding tensor can provide a
more detailed understanding of the factors governing the chemical shielding, if the
structural environment of the fluorine atom is known. Although single-crystal NMR
studies are generally required to orient an arbitrary chemical shielding tensor in
the crystallographic axis system, if the shielding tensor is axially-symmetric, one
of its principal axes must necessarily be along the corresponding symmetry axis

of the crystal.

The fluoride ion in the alkaline earth fluorapatites (M5F(PO4)3, M = Ca, Sr, Ba)
resides on a crystallographic hexagonal screw axis,"'11 and therefore possesses
such an axially symmetric 19F chemical shielding tensor. The 19F chemical
shift anisotropy (CSA) for calcium fluorapatite has been determined from single-
crystal NMR measurements”:13 and an accurate isotropic chemical shift has
been obtained from 19F MAS-NMR studies (because of the special arrangement
of the fluoride ions in a linear chain, sharp spectra are obtained at modest spinning
speeds without using multiple-pulse techniques, as discussed in references 6
and 7). However, corresponding data for the strontium and barium analogues
have not been reported. We have therefore measured the 19F chemical shielding
tensors of these compounds, using three different methods that yield the CSA from

the spinning sideband patterns (spectral moments,” Herzfeld-Berger graphical

10

analysis,15 and spectral simulation), and will discuss briefly the relative merits of

these methods.

Knowing both the chemical shielding tensor components and the absolute
chemical shielding scale relative to the free fluoride ion enables one to separate
contributions to the shielding from the sigma— and the pi-bonding between metal
and fluoride ion for the three different metals investigated. This approach enables
us to predict chemical shielding tensors in other systems, such as solid solutions

of calcium/strontium fluorapatite.

The broad linewidth typical of powder samples arising from dipolar interactions
and CSA obscures identification of individual peaks having different chemical
environments in many solids of interest. Since MAS-NMR averages out these
anisotropies, and results in sharp peaks at each isotropic chemical shift, it is
- extremely useful for the structural study of solids having many components. For
example, the site preference of solid solutions of a semiconductor alloy was
studied by using 31P MAS-NMR and peak deconvolution,16 and ”F MAS-NMR
has been also used to quantitatively studying fluoride ions perturbed by antimony
ions in antimony-doped fluorapatites.20 Since the chemical shift of fluorine is
very sensitive to its chemical bonding environment, we have used high speed
19F MAS-NMR for the study of solid solutions of calcium/strontium fluorapatite.
The 19F NMR parameters measured for the pure alkaline earth fluorapatites
(M5F(PO4)3, M = Ca, Sr, Ba) are used in the interpretation of the data from
the solid solutions. Since the 19F chemical shift difference between strontium and
calcium fluorapatites is 33.2 ppm, the isotropic chemical shifts of samples having
different ratios of calcium to strontium can be distinguished by using high field/high

speed 19F MAS-NMR. The integrated intensity of the non-overlapped peaks and

11

the deconvolution of asymmetric broad peaks arising from the perturbation by
strontium substitution yield the ratios of each component. The results were used
to study the site preference of Sr2+ ion substitution in a solid solution having the

composition Cag,gysr1,o3F2(PO4)5.

2. Nuclear Spin Interactions in Solids
A typical nuclear spin Hamiltonian of a diamagnetic solid is given by

”H: H2+Hrf(t)+HCS+HD‘I-HQ+HJ (M)
where the various Hamiltonians on the right represent respectively the Zeeman,
radiofrequency, chemical shift, dipolar, quadrupolar, and scalar interactions of the
nuclei. The first two terms are determined by external static magnetic and applied
rf fields. Thus, they describe “external” interactions of the spins. The other terms
depend on the fundamental characteristics of the nucleus and its environment. In
the solids that we have studied, the quadrupolar and scalar interaction are not

relevant, and will not be discussed.
A. Zeeman Interaction

The interaction between a nuclear magnetic moment, it, and an applied static
field, 130 is represented by the Zeeman Hamiltonian
H, = -§L;Zflt'fio= -Z'rih1;;'fio 0'2)
where h is Planck’s constant divided by’21r, 7,- is the magnetogyric ratio of nucleus
i, and 1‘; is the angular momentum spin operator for nucleus i. The eigenvalues
of this term alone are
Ez/h = 2:.7IhmiHo 0‘3)

where m,- is the magnetic quantum number.

12

B. Radiofrequency Interaction
In NMR spectroscopy, transitions between energy levels are generally induced
by an applied rf field, which is applied perpendicular to the static field direction.
The Hamiltonian term for the radiofrequency (rf) field along the x direction is given
by
H,f(t) = 27rjl'3H1COS(2m/I) (l-4)
where ii, is the nuclear magnetic moment in the x direction, H1 is the magnitude

of the if field applied in the x direction, and u is the Larmor frequency.
C. Chemical Shielding Interaction

The screening of the nuclei from the external magnetic field by the surrounding
electrons slightly modifies the Zeeman interaction. The shielding generated at the
nuclei from the external magnetic field results in the Hamiltonian

”HOS .-. vilified, (15)
where 6 is the chemical shielding tensor, a dimensionless second rank tensor. In
solutions, the chemical shielding interactions are averaged out by rapid isotropic
tumbling. Thus, a single line in solution is observed at the isotropic average of
the shielding tensor (Tr{&}). In solids, since molecular motions are typically slow
or absent, a broad powder pattern is observed. The chemical shielding tensor 6
is symmetric in larger static magnetic field. In the principal axis system (PAS),
all off-diagonal elements of shielding tensor zero. The chemical shielding tensor
can ne described by the three principal values 011, 022, and 033, and three angles
which specify the orientation of the principal axis system. If ”as are the dominant
interactions, the three components of the chemical shielding tensor can be “read-
off' directly from the spectrum since a powder pattern is related the chemical

shielding. The theoretical powder patterns for a shielding tensor with different

I3

bl ' I

0'39

 

., - /

03

 

———J .\_h

 

 

‘ 'r'*~'i "r" " i I

140130120110 :00 9C 30 70 60 50 4C BC 20 ppm

Figure l-1. Calculated chemical shielding powder patterns (a,- =- 64 ppm): (a)
axially symmetric shielding tensor (CSA = 84 ppm and i) a 0) ; (b) non-axially
symmetric shielding tensor (CSA :1: 84 ppm and n = 0.3). The dashed line denotes
the isotropic chemical shift An exponential apodization function corresponding
to a 500 Hz line broadening was applied to the calculated FID before Fourier

transformation.

14

asymmetry parameters 77 are shown in Fig. H. The asymmetry parameter 7) is
defined as
n = FL“ (l-6)

Conventionally, the order of the principle values of the chemical shielding tensors
is 011 s 022 s 033. The chemical shift anisotropy is defined as

CSA = 033 - 1/2(011 + 022). (1-7)
The chemical shielding tensor is generally obtained from single crystal NMR
studies or as in the present work, from MAS-NMR spectra on polycrystalline

samples.
D. Dipolar Interaction
The dipolar interaction is the consequence of direct magnetic coupling of nuclei
through space. The dipolar Hamiltonian of two nuclear spins (spin 1 and spin 2)
is represented by
”D: (Vi/Thin? it 'iji M)

where r12 is distance between the nuclei and D12 is the dipolar coupling tensor.

In a Cartesian coordinate system x, y, 2 (see Fig. I-2),

(rfi - 3x2) —3xy -3xz .
D12 = l/rfi —3xy (1%? — 3y?) —3yz (I-9)
—3xz —3yz (rfi — 3Z2)

The trace of D12 is zero, and it is axially symmetric: D13 = 021, 023 = D32, and
0,3 = 03,. In the principal axis system, with r12 along the z axis, all off-diagonal

elements are zero. The dipolar tensor can then be rewritten as

. 1 0 0
D=1/ri’2 0 1 0 (HO)
0 0 —-2

It is useful to transform from Cartesian coordinates to polar coordinates (Fig. l-2).

15

The Cartesian coordinates x, y, and z are represented by the following
x = rsin 6 cos d)
y=rsin09in¢ (H1)
2 = r cos 6.

Eq. (l-8) can be rewritten in the polar coordinate system as

HD==7/rf,h(A-i-B+C+D+E+F) 0'12)

A = (1 — 3CO529)IlzI2z

9:...)(1 _ 3cos 2a) [ITIg + 171;] = 3(1 — aces 29) (11.12. — f1 - f2)
= _gsin9cos9e-‘¢ [1,ng +IT122] (I-13)

D = —-:2isin9c090t=3'm5 [11.212 +ITI22]

E =—%sin266'2i¢ITIl§

= —%sin29e+2‘¢I;I;
where I + and I ‘ are the ladder operators, I + = f; + if; and I “ = I; — if,
I+|a) = 0, I+|fl) = Id), I'|a) = Id), and I‘lfl) = 0. The various ladder operators
can change the nuclear spin quantum numbers if), and mg in a characteristic way.
Term ’A' does not shift the nuclear spin quantum number; term '3’ alters both spins
by t 1, but I TI 3 and I II *2" do not change the sum (11), + mg); the others change
(m + mg) by :t 1 or :I: 2. Since, at high magnetic fields, the perturbations due to the
off-diagonal elements C, D, E, and F are negligible compared to the terms ’A' and
'8’, Eq. (l-13) can be rewritten as the truncated homonuclear dipolar Hamiltonian

in terms of the A and B terms

16

 

 

Figure l-2. Cartesian axis system and polar coordinates for the dipolar coupling

of the two nuclei.

17

H0 = (ii/2ri.)n(1— 3cos 26001-11; — 311.12.). (l-14)
The truncated heteronuclear dipolar Hamiltonian remains only the term ’A' that is

replaced It and [2 by I and S.

3. Magic Angle Spinning (MAS)

In solids, anisotropic interactions such as the chemical shielding interaction,
dipolar couplings and electric quadrupole couplings can result in very broad NMR
spectra. These anisotropies can be efficiently removed by spinning the sample
about an axis making an angle of approximately 54.7° with respect to the external
magnetic field”19 (Fig. l-3) (see below). The rotation of the sample about an
angle 9 with respect to the external magnetic field makes the anisotropic terms
of the Hamiltonian time-dependent with the periodicity of the sample rotation
frequency 1.2,. The three anisotropic terms mentioned above are generally small
compared to a Zeeman term and are treated as perturbations. The Hamiltonian
is divided into two parts

H=H+Hh) am)
where 72 is the time-average of the Hamiltonian and ’H'(t) is time-dependent. In
this section, the effect of sample rotation on each of two anisotropic terms will
be discussed in turn.

Since the chemical shielding tensor is of small size relative to the Zeeman
term, it can be truncate it. We can rewrite Eq. (l-5) as

He, = 7hazzH0 (I-16)
where

0n = M21011+ A222022 + M23233- ("17)

18

 

 

Figure l-3. Orientation of magnetic field Ho and spinning axis vectors for magic

angle spinning experiments.

19

In these equation, the terms Uaa (a = 1, 2, 3) are the principal values of the
chemical shielding tensor and AW are the direction cosines of the principal axes
with respect to the external magnetic field. The rotation of the sample makes the
direction cosines time-dependent
A, = cos 8 cos X, + sin 8 sin X, cos (wrt + 8,) (I-18)
where ,8 is the angle between the rotation axis and the external magnetic field, X,
is the angle between the rotation axis and the p-th principal axis of the chemical
shielding tensor, and 18,, is the azimuthal angle of the p-th principal axis at t=0.
By substituting Eq. (I-18) into Eq. (H7) and taking the time-average, we obtain
the following equation
as: = %Sin2fiTr{&} + %(3cos2i’3 — 1) 20,, cosxp. (I-19)
Only the isotropic chemical shift 0,- (=1 /3Tr{ 6}) in Eq. (l-19) remains if [3 is 54.7°
[(3 cos28 — 1) = 0], the so-called “magic angle”.
Since the rotation of the solid sample makes cos 6 in the dipolar Hamiltonian
time-dependent, we can rewrite cos 6 as
cos 6(t) = cos 8 cos 8' + sin 8 sin 8’ cos (8.1 + 8’) (l-20)
where 8' is the angle between the axis of rotation and F12, and this the azimuthal
angle of £12 at 1:0. The time average of cos2 0(t) is
2637? = cos28cos28’ + %sin2fisin2 8' = F§.(3cos2 8 — 1) (3cos2 8’ -— 1) + I, (I-21)
Substituting Eq. (I-21) into Eq. (l-14), we can write the average (time-independent)
dipolar Hamiltonian as
Hp = guzcos2 8 — 1) (3 cos2 8’ — 1) (i, . i; — 311.12,). (1-22)
The time-independent dipolar interaction is averaged out at the magic angle.
Under the MAS condition, the averaged time-independent Hamiltonian yields

only isotropic chemical shifts, but time-dependent Hamiltonians are modulated by

20

the periodicity of the Spinning speed of a sample. From Eq. (H4) and (l-22), the

time-dependent part of the dipolar interaction is given by

Hm) = (37f/2r3)h(f1 - is -— 31,12) [sin 28 sin 28’ cos (wrt + 8) + sin2 8 sin2 8’ cos2(.c,.t + 18)
(l-23)

The time-dependent Hamiltonians of the other two interactions (chemical shift

and quadrupolar interactions) also show the same periodicity in w. and 28:, as

the time—dependent dipolar Hamiltonian. The modulations of w, and 2.2, in the

time-domain give rise to peaks in the frequency-domain, referred to as sidebands,

which appear at integral multiples of the spinning speed from a centerband. The

intensities of sidebands that are related to the CSA,“‘-15 and enable one to mea-

sure the chemical shielding tensors.

4. Methods for Measuring Chemical Shift Anisotropy (CSA) from
MAS-NMR Spectra

In this study, three methods were used to calculate the principal components
of the chemical shielding tensor: the moments method of Marlcq and Waugh,14 a
graphical procedure developed by Herzfeld and Berger that is based on spectral
simulation,15 and a MAS-NMR spectral simulation program.

A. Moments Method

The I-th moment of an NMR spectrum“ is obtained using the following

definition (integral) and approximation (summation):
M, = f _°:°w’g(w)d.a 2 WrIENng(Nwr) "'24)

with w the frequency of an isochromat (with respect to the isotropic chemical shift

position), g(w) the intensity of the area normalized to one at frequency u), N the

21

order of the N-th sideband (positive or negative integer, 0 for the centerband),
g(Nw,) the normalized area (or peak height) of the N-th sideband at frequency
N10,; and w, the spinning frequency. For a reasonably “sharp” experimental MAS-
NMR spectrum, the summation in Eq. I-24 yields moments very close to the true

moments defined by the integral expression
The moments method yields the components of the chemical shielding tensor

by relating them to the measured second and third moments of the MAS-NMR

spectrum with the following equations:“
M2 = (62/15)(3 + 772) (1-25)
M3 = (253/35)(1 — 7,2) (I-26)
where M2 and M3 are the second and third moments respectively, and 6 and the
asymmetry parameter 1; (Eq. l-6) are related to the shielding tensor components.
' 6 and an isotropic chemical shift are represented by the following equations:
5 = 033 - 0,- (1-27)
012%(‘711‘1'0221'030- (1'28)
Using experimentally determined values of the second and third moments, and of
the isotropic chemical shift (0,), we can determine the individual components of

the shielding tensor 011, 022, and 033. The chemical shift anisotropy (CSA) can

be calculated from Eq. l-7.
B. Herzfeld and Berger Graphical Method

The Herzfeld and Berger graphical method was used to obtain the chemical
shielding tensors by overlaying the various calculated contours corresponding to
the measured ratios lg/lo, where l,, is the intensity of the i-i-th sideband and I0

is the intensity of the centerband. The region of overlap provides both values

22

and error limits for the intermediate parameters 8 and p, which are then used to

calculate the chemical shift tensor from equations (I-29) and (I-30):15
it = (7H0) (033 - 0111/90 (1‘29)

P=(011 +033-20221/(033-0111- (1'30)

5. Experimental
A. MAS-NMR Studies

Alkaline earth fluorapatites were synthesized and provided by Dr. Chung-
Nin Chau of GTE Chemicals, Towanda, PA. The 19F MAS-NMR spectra were
recorded at 376MHz on a 9.4T Varian Associates VXR-400 spectrometer at the
Max T. Rogers NMR Facility at Michigan State University. The fluorine radiofre-
quency was amplified by an AMT model 3137/3900-2 amplifier. A high speed 19F
MAS-NMR probe from Doty Scientific with 5 mm o.d. Si3N4 rotors with Vespel
caps was used. The spinning speed was measured with a fiber optic detector,
and was constant to within :10 Hz during acquisitions. The magic angle was set
by minimizing the linewidth of calcium fluorapatite, which also provided a sec-
ondary chemical shift reference (64.0 ppm with respect to hexafluorobenzene at
0.0 ppm).6 The 1r/2 pulse length was 4.0-4.2 83, and spectral widths of 100 - 120
kHz were used. An exponential apodization corresponding to a line-broadening
of 37-80Hz was applied to the free-induction decay, which was the result of four
scans with a relaxation delay greater than five times the spin-lattice relaxation time
T1. The T1 values of centerbands were obtained by using an inversion-recovery
sequence, and are 85, 87, 88, 101, and 112 second for Cas F(PO4)3, Sr5F(PO4)3,
385 F(PO4)3, 083973133 F2(PO4)6, and Ca58r5F3(PO4)3 respectively. The SPAR-

23

TAN pulse sequence, shown in Fig. I-4, was employed for 19F MAS-NMR spectral
spin diffusion measurement.20 The centerband and assorted sidebands were se-
lectively inverted by a 180° DANTE pulse train21 consisting of twelve 15° (2 8s)

pulses given at the same point of each rotor cycle (rotor-gated synchronization).

DANTEPulses fit/2

lllllll—T’“

 

 

 

 

 

   
 

acquire

Figure I-4. SPARTAN pulse sequence for monitoring spectral spin diffusion.20

(see text)

24

The power levels and pulse lengths of the pulses in the DANTE train were
adjusted to make the excitation profile suitably selective. After the mixing period,
a nonselective 90° read pulse (12 89) was given with alternated phases to cancel
out imperfections in the DANTE pulse trains22 that resulted in incomplete inversion

of the magnetization at a specific frequency.

The peak intensities used to calculate the moments at the different spinning
speeds were obtained both from integration and from deconvolution of the indi-
vidual peaks using VNMR 3.2 software, and the peak intensities from integration
rather than peak heights were used for the Herzfeld and Berger analysis, since
the half-height linewidths (Al/1,2) of the centerband and sidebands can be some-
what different. All 19F NMR spectra were baseline-corrected prior to integration
and deconvolution to remove a “dip" presumedly due to receiver overload. Since
there are impurity peaks that overlap the peaks of strontium fluorapatite and bar-
ium fluorapatite in the 19F NMR spectra, the CSA values determined by using the
deconvolution data are assumed to be more accurate than those obtained using
integrated intensities. A Lorentzian shape was assumed for the deconvolutions,

and the half-height linewidth and frequency of each peak was allowed to vary.
8. Simulation of 19F MAS Spectra of Fluorapatite Samples

Simulations of the 19F MAS-NMR spectra that take into account of the CSA
but not dipolar couplings were performed by using the MAS-NMR simulation
routine provided with the Varian VNMR 3.2 software. The input parameters are
the chemical shift anisotropy, the asymmetry parameter, the isotropic chemical
shift, the Lorentzian linewidth, and the spinning speed. The peak widths of
the simulated spectra were chesen to agree with the experimental ones, and

8K complex points and 8K zero-filling were used. Alternatively, the PC-based

25

computer program ANTIOPE,23 which takes account for dipolar couplings, was
used to simulate the spectrum of five linear spins 3.44 Angstroms apart, with
a CSA tensor corresponding to that of Ca5F(PO4)3. The ANTIOPE simulation
was performed by observing magnetization of the middle spin (+3) of a 5 spin
system, since the homonuclear dipolar coupling pattern between the middle spin
and its neighbors better reflects the couplings of the infinite linear spin chains in
Ca5F(PO4)3. The 256 complex data points calculated were zero-filled to 8K to
avoid errors from inadequate digital resolution and apodized with a Iinebroadening

of 200Hz.

6. Results
A. Alkaline Earth Fluorapatites

Table 1 shows the measured 19F MAS-NMR spectral moments of Ca5F(PO4)3,
Sr5F(PO4)3 and Ba5F(PO4)3, and the calculated CSA and r) values at different
spinning speeds obtained from the moments method. The existence of imaginary
values of r), obviously lacking any physical significance, arises from experimental
errors in the moments measurements. From equations (l-7), (l-27), and (I-28), we
can rewrite the CSA in the following form;

CSA=%(033 —0,.) :38. (181)
Therefore, we can calculate the value of a CSA without knowing the value of n.

Theoretically, the second and third moments are independent of the spinning
speed.14 The changes in the second and third moments with spinning speed seen
in Table 1 are the result of experimental error; therefore, we use the average value

of the moments over all the spinning speeds to calculate an average CSA and n.

26

Table 1. 19F NMR chemical shift anisotropy and asymmetry parameter 1) of
M5F(PO4)3 (M = Ca, Sr, and Ba) calculated using the moments method, from

both integration and deconvolution. The numbers in parentheses are obtained

from the integration data.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u, M1 M2 M3 Measured Measured
(kHz) (ppm) ((ppm)2) ((ppm)3) 7) CSA (ppm)
4.12 -0.25 660 10690 0.092 36-0
(0.20) (684) (1 1740) (0.2191) (374)
5.12 -0.09 662 1 1340 0.167i 86.7
(0.14) (663) (11950) (0.2481) (87.3)
6.12 -0.13 651 10980 0.150 85.9
(0.20) (653) (11830) (0.2661) (86.8)
7.12 -0.08 646 10610 0.0951 85.4
,2 (-0.12) (649) (10480) (0.073) (85.4)
V
g 8.12 -0.03 636 10310 0.0681 84.6
‘3’ (-0.06) (638) (10470) (0.1 10) (84.9)
o 9.12 -0.01 638 104W __§—_0.081 W.
(0.18) (634) (10800) (0.1951) (85.0)
10.18 0.01 643 10440 0.0081 85.1
(0.10) (643) (10290) (0.062) (84.9)
= :l f 3
average 0081 648 1 0680 0.0951 85.5
0.08 1:12 1340 (0.1791) 10.7
(0.0901: (649: (11080 (85.6
0.08) 1 1) i660) 1:1 .0)
—
4.29 -0.48 20690 0.204 108.4
(0.05) (1°69) (22530) (0.08i) - (109.8)
5.63 -0.21 921 17700 0.077 101.7
(0.45) (917) (17090) (0.158) (101.2)

 

 

 

 

27

 

 

 

 

   

 

   

 

 

 

 

 

 

 

 

 

 

 

6.26 -0.37 1054 20630 0.194 108.2
(0.83) (1036) (19460) (0.243) (106.9)
7.00 -0.30 1077 23930 0.2011 1 10.8
(0.52) (1095) (24340) (0.1841) (1 1 1.6)
,3 8.20 -0.38 1036 19940 0.207 107.2
2' (0.54) (1025) (18770) (0.268) (106.1)
11; 10.20 0.11 1033 21180 0.026 107.8
:71 (0.16) (1047) (22110) (0.121 i) (108.2)
average -0.26 1030 20690 0.117 107.4
10.21 151 11820 (0.119) 12.8
(0.18 (1032 (20720 (107.5
10.46) 156) :I:2480) 13.3)
_
6.23 0.10 2314 69840 0.110 161.8
(0.74) (2304) (73820) (0.1701) (161.0)
g 8.19 0.02 2209 58000 0.267 157.2
g (0.53) (2248) (61030) (0.295) (155.4)
“a 10.10 0.03 2227 63410 0.190 157.3
3 (0.90) (2268) (70380) (0.150) (159.0)
1087 0.39 2289 70220 0.050 160.5
(0.71) (2326) (70470) (0.110) (161.5)
=#
average 0.00 2270 66130 0.170 159.0
10.33 134 14000 (0.1 12) :l:1.8
(0.59 (2277 (68170 (159.4126)
10.34) 145) 16030)
#

 

 

 

28

The values for the CSA and moments in parentheses In Tablet are those deter-
mined from integration data, and the standard deviations are larger than those

obtained from the deconvolution data.

The CSA values can be also obtained by using the graphical procedure of
Herzfeld-Berger, which involves measurement of the sideband intensities. Fig.
l-5 shows the two graphical plots of Herzfeld-Berger for Ca5F(PO4)3 spinning at
6.12 and 10.18 kHz. The measured intensity ratios Iii/Io of Ca5F(PO4)3 spinning
at 6.12 kHz do not overlap at any region of the plot. At the higher spinning speed
of 10.18kHz, a region of overlap is observed, centered around p = 0.95 i 0.05
and p = 3.19 :l: 0.19. Other plots using the Herzfeld and Berger graphical method
for Sr5F(PO4)3 spinning at 6.26 and 10.20 kHz are shown in Fig. l-6. From Fig.
l-6a, only four lines out of the ten lines overlap around p = 0.46 i 0.05 and 8 =
6.4 :i: 0.20, and the CSA and 17 value (92 1 4 and 0.47 1 5) are quite different
from those obtained at a 10.20 kHz spinning speed (see Table 2). However, the
overlap of the contour lines of Iii/l3 for Sr5F(PO4)3 spinning at 10.20 kHz occurs
at around p = 0.96 :t 0.04 and p = 4.0 :t 0.19.

Fig. l-7 shows the Herzfeld and Berger plots of Ba5F(PO4)3 at two different
Spinning speeds (6.23 kHz and 10.18 kHz). Unlikely the other fluorapatite sam-
ples, all contours of Ba5F(PO4)3 at 6.23 kHz except for L3 intersect at one point
around p = 0.8 i 0.04 and 8 =- 10.4 1: 0.13, and those at a 10.18kHz spinning
speed intersect at around p = 0.78 :t 0.02 and 8 = 6.2 1 0.13. The p and 8 values
of the intersection points are used to obtain the chemical shielding tensors (CSA

and n) from Eq. (l-29) and (I-30).

29

 

1.0
b)

 

 

 

 

 

 

 

 

Figure l-5. Contour plots of Ca5F(PO4)3 spinning at 6.12 kHz (a) and 10.18 kHz
(b) using the Herzfeld and Berger graphical method.

30

 

 

1.0
b)

-O.5 ‘

 

 

 

-1.0 _
1.0

 

 

A

0.5 1

P 0.0 ‘

 

 

 

 

Figure l-6. Contour plots of Sr5F(PO4)3 spinning at 6.26 kHz (a) and 10.20 kHz
(0) using the Herzfeld and Berger graphical method.

3O

 

1.0
b)

0.5 '1

-O.5 '

 

 

 

-1.0, T '
1.0

 

a)

 

 

 

 

Figure l-6. Contour plots of Sr5F(PO4)3 spinning at 6.26 kHz (a) and 10.20 kHz

(b) using the Herzfeld and Berger graphical method.

31

 

1.0
b)

0.5 ‘

 

 

 

-1.0 , 1
1.0

 

0.5 ‘

 

-0.5

 

 

 

Figure. l-7. Contour plots of Ba5F(PO4)3 spinning at 6.23 kHz (a) and 10.18 kHz
(0) using the Herzfeld and Berger graphical method.

Table 2. The 19F chemical shift anisotropy (CSA) and asymmetry parameter

r; of M5F(PO4)3 obtained using the Herzfeld and Berger graphical method and

integrated peak intensities.

32

 

 

 

 

 

 

 

 

 

 

u,(kHz) Measured 77 Measured CSA
(ppm)
Ca5F(PO4)3 10.18 0.04 1 0.04 86 1 5
Sr5F(P04)3 10.20 0.06 1: 0.04 107 1 7
6.23 0.16 :l: 0.03 164 1 4
BagF(PO4)3 8.19 0.23 1: 0.04 157 1 7
10.10 0.18:l:0.01 15814
10.87 0.201: 0.04 160 i 5

 

 

Table 3. 19F chemical shielding tensors of M5F(PO4)3, calculated from the
moments method using peak intensities obtained from deconvolution. An average
value of the CSA at various spinning speeds was used, along with an assumed

17 value of zero, to obtain the principal components with respect to C3F3 and free

F' ion (parentheses).

 

 

 

 

 

 

 

011 (ppm) ( = 022) 033(ppm)
Ca5F(PO4)3 35.5 1 0.2 121.0 1 0.5

( 159.5 1 0.2 ) ( 245.0 1 0.5 )
Sr5F(PO4)3 61.4 1 1.1 168.8 1 1.9

(185.4:l:1.I) (292.8119)
Ba5F(PO4)3 131.8106 290.8 1 1.2

(255.9106) (414.9112)

 

 

 

33

Table 2 shows the CSA and 17 values obtained by the graphical procedure of
Herzfeld-Berger. The standard deviations are determined by the intersection area
(box size in Fig. I-5, I-6 and l-7). The contours of the experimental ratios Iii/Io
for Ca5F(PO4)3 and Sr5F(PO4)3 fail to overlap at spinning speeds below 9kHz
but they overlap at 10.18 and 10.20kHz respectively. Therefore, the CSA values
of CasF(PO4)3 and Sr5F(P04)3 are obtained at the high spinning speed. The
Iii/Io contours of Ba5F(P04)3 overlap within a small region, and the resultant CSA

values are close regardless of the spinning speed.

Even though the measured asymmetry parameters of M5F(PO4)3 are different
and non-zero at various spinning speeds, the chemical shift tensors of M5 F(PO4)3
must be axially symmetric on the basis of the X-ray crystal structure."'11 Therefore,
we constrain r) to be equal to 0 and use the average CSA determined using the
moments method to calculate the chemical shift tensors in Table 3. It is necessary
to know the chemical shift tensor values on an absolute chemical shielding scale
with respect to free fluoride ion in order to be able to calculate the contribution of
the sigma- and pi-bonding to the paramagnetic shielding. The isotropic chemical
shift of C3F3 has been calculated to be 124 ppm with respect to free fluoride ion.‘9
The chemical shift tensors of M5F(P04)3 with respect to both C3F3 and free F'
(parentheses), obtained using the average CSA from the deconvolution data in

Table 1 and assuming 1) = 0, are shown in Table 3.

Experimental and simulated 19F MAS-NMR spectra of M5 F(PO4)3 samples at
spinning speeds near 6kHz, using the chemical shift tensor components in Table
3, are shown in Fig. l-8. The isotropic chemical shifts of M5F(PO4)3 ( M = Ca,
Sr, and Ba) are 64.0, 97.2, and 184.8 ppm from hexafluorobenzene respectively.
The half-height linewidths of the centerbands of Ca5F(PO4)3, Sr5F(PO4)3, and

34

Ba5F(PO4)3 without line-broadening are 164 Hz, 385 Hz and 438 Hz respectively.

Brunner et. al. have used average Hamiltonian theory to provide a general
expression for the MAS-NMR linewidths of spins with axially symmetric shielding
tensors.24 Their expression takes into account homo- and heteronuclear dipolar
interactions and CSA, and predicts a half-height linewidth (HHLW) that is inversely
proportional to the spinning speed. Figure l-9 shows a plot of the 19F MAS-NMR
centerband HHLW vs. spinning speed for M5F(PO4)3 (M = Ca, Sr, and Ba). The
HHLW of Ca5F(PO4)3 decreases monotonically with increasing spinning speed
within experimental error, but those of Sr5F(PO4)3 and Ba5F(PO4)3 do not. The
dependence of the linewidth of M5F(PO4)3 upon spinning speed indicates that
the linewidth is broadened homogeneously. The “plateau” value of the HHLW
increases as the CSA increase (from Ca to Sr to Ba); this may simply reflect
_ larger effects of crystal imperfections upon the isotr0pic shifts of Sr5F(PO4)3 and
Ba5F(PO4)3.

8. Solid Solutions of Ca/Sr Fluorapatite

The 19F MAS-NMR spectrum of Ca3,g-,Sr1_o3F2(PO4)3 spinning at 8.23 kHz
is shown in Fig. l-10a. It is very difficult to unravel the isotropic chemical shifts
and their sidebands at a fixed spinning speed due to the overlap of the different
centerbands and sidebands. Since the sidebands are located at Integral multiples
of the spinning speed from the centerband, we can differentiate between the
centerband of one peak and a sideband from another peak with a different isotropic
chemical shift simply by increasing the spinning speed. In this way, three isotropic
chemical shifts of the peaks in Ca3_97$r1,33F2(P04)3 (Fig. l-10b) can be obtained,
at 64 ppm, 79.6 ppm, and 97 ppm. Overlap of centerband and sideband peaks

makes it difficult to obtain reliable integral intensities of individual peaks in the

35

_____._J__.LJ__LJL_JL_.__

 

 

c}. 1 LLJLJLL1__

 

 

 

LL_
“1.1111011.

 

 

 

 

 

a) e ‘—
V T V T T f r T r 1 1 r r I Y r r r r f T T r T T *1 v
300 280 200 150 100 50 pp-

Figure l-8. The experimental (lower) and simulated (upper) spectra of M5 F(PO4)3.
a) BagF(PO4)3 at 6.23kHz; b) Sr5F(PO4)3 at 6.26kHz; c) Ca5F(PO4)3 at 6.12kHz.
' indicates the centerband (0,). Simulation were based on the chemical shift tensor

components in Table 3, and neglect dipolar coupling.

Hall-linewidth (Hz)

36

 

 

 

 

600
n
500‘ a
:1
1.1 an
1::
4m'DAAA
1:1
,0 A D :1 0'3
a
300- °o a
1 A A A A A
°o
.. o
200 O O
4 o 00
100 1 1 . fir w - r -
2 4 6 8 10 12

Spinning speed (kHz)

Figure l-9. ‘9F MAS-NMR center band HHLW vs. spinning speed for M5F(P04)3
(M . Ca, Sr, Ba). CasF(PO4)3 (open circle), Sr5F(PO4)3 (open triangle), and
Ba5F(PO4)3 (open square).

37

spectrum. From Fig. I-10b, only the integrated intensity of the centerband and
sidebands of the 79.6 ppm peak can be measured since the other centerbands
and sidebands overlap. The ratio of the integrated intensity of the centerband and

sidebands of the 79.6 ppm peak to the total integral intensity in Fig. l-10b is 29 %.

The deconvoluted spectra of the 64 ppm peak and also the peak near 51 ppm
due to the -1 sideband of the 79.6 ppm peak of Fig. I-10b are shown in Fig.
l-11. Three peaks can be deconvoluted from the asymmetric 64 ppm peak. The
gaussian fractions of peaks l and II are 0.88, and that of peak III is 0.68. The
half-height linewidths of deconvoluted peaks I, II, and III are 454, 536, and 758
Hz respectively. The percentages of the integral intensity of deconvoluted peaks
I, II and III are 45.4, 37.3 and 17.4 %

Fig. l-12 shows the 19F MAS-NMR spectrum of Cag_978r1,33F2(PO4)3 obtained
using the SPARTAN pulse sequence at 10.87 kHz.20 The transmitter offset of
this experiment was set to invert the 64 ppm peak, leaving the other peaks
unperturbed. Since the pulses in the DANTE train are repeated every rotor period,
the sidebands of the 64 ppm peak are also inverted. The mixing times are 0.1,
1, 3, 6, 9, 12, 15, 18, 21, 24, and 30 seconds. As the mixing time increases,
the intensities of adjacent peaks decrease while those of the centerband (and
sidebands, not shown) of the 64 ppm peak recover quickly compared to the spin-

lattice relaxation time of Cag,gysr1,o3F2(PO4)3 (T 1 = 101 second).

Fig. I-13 show the 19F MAS-NMR spectra of Cassr5F2(PO4)3 spinning at
8.25 kHz and 10.30 kHz. The centerbands are resolved by spinning at 10.30
kHz. The values of the isotropic chemical shifts in CaSSr5F2(PO4)3 are 69.7ppm,
86.8 ppm, and 104.5 ppm, which are about 6 ppm downfield compared to those

in Ca3,97$r1,o3F2(PO4)3. The measurement of the integral intensity of peaks is

38

m l

 

@008me U

Q

 

 

 

 

 

 

 

I
I
a) i l
g \
: T'Tr'V'TrYFTTV'_'TTYTWTTWITTTTTVVf'rv—iriITrrYIYIYTTvvvvyvrrrrvvr'IV'vvi'YY'it‘V'iv'Y' rrVTTT"
180 160 140 120 100 80 60 40 20 0 ppm

figure I-10 19F MAS-NMR of Cag,g-,Sr1,o3F2(PO4)3 spinning at 8.23 kHz (3) and
at 10.80 kHz (b). ' indicates the centerband.

39

 

:5 '\/ \

I'T'VITYTrTTTV‘ITT'V r1
VTYYII'II'TT TIT
. 11111111'111111
‘r ITI'TTTTTTTTT‘TT
ITITTTI'TTIIIYT r
. (11111111111111
[1111] 1111711111
ITTTTITIVVTITIIVV
. I'VTTTTT

68 66 64 62 6
0 58 56 54 5
2 50 48 46 44
PP'“

 

 

figure I-11. Deconvolution spectrum 0164 ppm peak in fig. I-11b

 

‘fi—‘lij

60 55 50 ppm

[FTTfT

80 75 70 55

Figure l-12. 19F MAS-NMR spin diffusion experiment of Ca3_978r1_o3F2(PO4)6
using SPARTAN pulse sequence spinning at 10.87 kHz. Mixing times are 0.1, 1,
3, 6, 9, 12, 15, 18, 21, 24, and 30 seconds. (899 text)

I
l'l'l'Il'I‘I'Il

l l I l
140 100 30 60 40 20 O -20 ppm

, .
180

figure l-13. 19F MAS-NMR of Ca5$r5F2(PO4)3 spinning at 8.25 kHz (3) and at

10.30 kHz (b). ‘ indicates centerband.

42

hindered by the overlap of the centerbands and sidebands.

7. Discussion

A. Measurement of the 19F Chemical Shift Anisotropies of M5F(PO4)3 Using
Different Methods

The values for M5F(PO4)3 are obtained by using two different methods, the
moments method and the Herzfeld and Berger graphical method. In the moments
method, the accurate measurement of the second and third moments of MAS-
NMR spectra is difficult in general because of the low signalrnoise ratio of the
weak higher order sidebands that contribute significantly to the moments.25 The
homo- and heteronuclear dipolar interactions also contribute to the experimental
second moment, making it difficult to separate their contribution to the second
moment from that of the CSA alone. We will discuss how the dipolar interaction
influences the measurement of the CSA using the moments analysis method.
MAS-NMR simulations using ANTIOPE (Fig. l-14a) show that the including
homonuclear dipolar interactions result in a higher intensity of the centerband
relative to that of the sidebands. The contribution of the homo- and heteronuclear
dipolar interactions and of the CSA to the second moment is not simply additive.“
The values of the CSA obtained from the simulated 19F MAS-NMR spectra
calculated using the CSA alone (VNMR 3.2) and using the CSA along with dipolar
interactions (ANTIOPE) differ by approximately 2 ppm at the spinning speed of
6.12 kHz. Since the low signalznoise ratio of the weak higher order sidebands
gives rise to measurement errors in the experimental and since considering the
dipolar interaction can slightly change the values of the CSA, we use the moments

measured from the intensities of the sidebands of the experimental spectra to

43

obtain the values of the CSA in Table 1.

When we use the Herzfeld and Berger graphical method, the chemical shift
tensors are obtained from the coordinates of the overlapping contours of the
values of I,,,/lo ratios. Intensities of all peaks (a centerband and sidebands) in
a MAS spectrum are used to obtain a CSA with the moments method, whereas
those of a limited number of peaks are used to obtain a CSA with the Herzfeld
and Berger graphical method (lg/lo ratios for up to :I: 5 sidebands).15 Thus, the
measurement error of the CSA obtained from the Herzfeld and Berger graphical
method should beless severe than that from the moments method owing to the
neglect of weak higher order sidebands. The lfi/Io ratios of the samples with
a larger CSA and small dipolar interactions have a smaller relative contribution
from the dipolar interactions. The dipolar interaction decreases on going from
Ca5F(PO4-)3 to Ba5F(PO4)3 as a result of the expansion of the lattice,911 whereas
the CSA values increase going from Ca5F(PO4)3 to Ba5F(PO4)3. Therefore, it
is not surprising that the contours of lfi/lo for Ba5F(PO4)3 overlap at one point
or over a small region at all spinning speeds. Likewise, the contours of l,,,i/lo
for Ca5F(PO4)3 and Sr5F(PO4)3 fail to overlap below about 9kHz due to the
greater size ratio between the homonuclear dipolar interaction and the CSA. As
the spinning speed increases, the higher order sideband intensity contribution from
the dipolar interaction appear to decrease quickly. Thus, most of the intensity of
the higher order sidebands at high spinning speeds is due to the CSA. Figures
I-5 and 6 show that the value of 8 of the +1 sideband is insensitive to a change
of the spinning speed. The values of 8 of the other sidebands move to the value

of 8 of the +1 sideband with an increasing in the spinning speed.

The integrated intensity of each peak in the experimental spectrum does not

44

correspond to the peak height because of differences in the half-height linewidth of
the peaks. For example, the peak height of the centerband in Gas F(PO4)3 is larger
than that of the -1 sideband; however, the integrated intensity of the centerband
in Ca5F(PO4)3 is less than that of the -1 sideband since the half-height linewidth
of the centerband (164 Hz) is smaller than that of the -1 sideband (217Hz) (Fig.
MO). The ratio of the integrated intensity of the centerband to that of the -1
sideband is 1.00:1.13 in Fig. l-10. Thus, when CSA is obtained by using MAS-
NMR, the integrated peak intensity must be used rather than a peak height to

obtain a more accurate CSA value.

Since the linewidth of the sidebands in the experimental spectra vary for a
given sample, it is convenient to represent these spectra as stick spectra for
comparison with simulated spectra Stick spectra (Figure I-14) normalized to
the -1 sideband of M5F(PO4)3 were obtained from simulation (ANTIOPE and
VNMR 3.2) with the parameters of Table 3. The intensities of the centerband
and sideband of the simulated stick spectra taking into account only chemical
shift anisotropy (VNMR simulation) are different from those of the experimental
spectrum in Fig. I-14a. The simulated spectrum obtained using ANTIOPE (CSA
and dipolar interaction among uniformly-spaced five fluorine spins) more closely
resemble the experimental spectrum in Fig. l-14a. Fig. l-14 shows that the
intensity differences between the centerband and sidebands of the simulated
and experimental stick spectra become smaller and smaller less going on from
Ca5F(PO4)3 to Ba5F(PO4)3.

B. Separation of Metal-Fluoride Sigma- and Pi-Bonding Contributions to the
19F Shielding Tensor

The chemical shielding of a nucleus can be separated according to Ramsey’s

45

figure I-14. Comparison of simulated and experimental 19F MAS-NMR stick
spectra of M5F(PO4)3 at the same spinning speed. The asymmetry parameter
1) is forced to be zero and the average CSA values of M5F(PO4)3 obtained from

the moments method are used, and all peaks are normalized to the -1 sideband.

46

 

         
  

 
    
   

  
  
 

   
     

 

 

 

 

 

1.2
'2 .
31.0 E/
9 1 =/
.6 E/
08- =/
1'; 7‘/
/-/
9 1 8_é
N - =/—/
-.-_- =¢—/
to 1 E%—é
E 04- =/ E/_¢
3 ° =/ E?—/
. E/ =/—¢ '2
«E; -/E Eé—/=/
5 0.21 _é§ Eé=égé
'0.0' éE E%_%Eé

 

 

Order of sidebands

CI The intensity (from deconvolution) of the centerband and sidebands
of the experimental spectrum.

E The intensity of the centerband and sidebands due to the CSA only
(VNMR 3.2 simulation).

The intensity of the centerband and sidebands due to the CSA and
dipolar interactions among 5 fluorine spins (ANTIOPE simulation).

3) Simulated and experimental 19F stick spectra of CasF(PO4)3 Spinning at 6.12 kHz

47

1.2

 

 

 

 

 

 

 

 

 

 

 

E 10

3 .

Q

.9

f.’ 0.81

.0

E 0.6-

'5 ‘ =

E 0.4- E

5 ‘ E

g 0.2- E]:

s I 1.11 =11 s1 ..

0.0 , , es ,,

3 2

10-1-2-3-4 -5

Order of sidebands

CI The intensity (from deconvolution) of the centerband and sidebands
of the experimental spectrum.

E The intensity of the centerband and sidebands due to the CSA only
(VNMR 3.2 simulation).

b) Simulated and experimental 19F stick spectra of Sr5F(PO4)3 spinning at 8.20 kHz

 

2.0

 

 

1.0 'l

 

1

0.0 mafia-IE] EL“

876543210-1-2-3-4-5-6

Intensity normalized to -1 sideband

 

j
lllIIIllllllllllllllllllllllllI[IIIIIIIIIIIIII llllll

IllIIllllllllllllllllllllIIII
Ill[lllllllllllllllllllllllll I

llllllllllllllll I

El

f T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Order of sidebands

CI The intensity (from deconvolution) of the centerband and sidebands
of the experimental spectrum.

E The intensity of the centerband and sidebands due to the CSA only
(VNMR 3.2 simulation).

c) Simulated and experimental 19F stick spectra of Ba5F(PO4)3 spinning at 8.19kHz

49

formulation“27 into a diamagnetic term and a paramagnetic term. Lamb showed
that the diamagnetic contribution to shielding is proportional to the sum of in-
verse distances between the i-th electrons and the nucleus.28 The calculated
difference in the diamagnetic term for free fluoride ions in different ionic fluo-
rides is small since the distances of the i-th electrons to the nucleus for a free
F' ion are similar to those for different ionic fluorides.27 Therefore, differences
in the paramagnetic term are largely responsible for the fluorine chemical shift
changes observed in various ionic fluorides. The chemical shielding tensors of
the paramagnetic term, measured with respect to free fluoride ion are used to
separate metal-fluoride sigma— and pi-bonding contributions to the 19F shielding.
The paramagnetic contribution depends upon the electronic ground and excited
states. The paramagnetic contribution of electrons in s orbitals can be ignored,
since the angular momentum of s orbitals is zero. An asymmetric distribution of p
and d electrons near the nucleus, and low-lying excited states of these electrons,
can result in a large paramagnetic term.26'27 A superposition of the ground and
excited states of sigma-bonding orbitals arises when an external magnetic field
is applied. The angle between a single sigma-bond and the extemal magnetic
field determines the mixing of the states. No perturbation of the symmetry of the
electron cloud occurs when the external magnetic field is along the sigma-bond.
The shielding constant arising from the paramagnetic term in linear compounds
depends on the angle between the external magnetic field and the direction of the

sigma-bond in the following way.25'29
0(6) = aasinzél (I-32)
where 0(6) is the paramagnetic shielding constant at angle 0 and a0 represents

the contribution of the a-bond to the paramagnetic shielding. Since the pi orbital

50

is perpendicular to the bond direction, the shielding constant due to the pi-bond
is represented by

0(6) = awcoszo (l-33)
where a“ represents the contribution Of the pi-bond to the paramagnetic shielding,
and 0 is the angle between the bond and the external magnetic field. The
paramagnetic shielding term arising from both a sigma-bond and a pi-bond is
the sum of equations (I-32) and (l-33)

0(0) = aasinze + awcosze = (2/3)aa+ (1/3)a7r+ (1/3)(a7r - ao)(3cos29-1). (l-34)
The isotropic portion of the shielding 0,-(9 = 54.7°) is represented by (2/3)aa +
(1/3)a7r. The total paramagnetic shielding a is assumed to be the sum of the
contributions from the individual bonds;

0 (a): 2 0,-(éj) (I-35)
' Gagarinski els?9 separated the paramagnetic shielding components (a0 and a)
of CasF(PO4)3 by using equations l—34 and l-35, and experimental values for the
CSA and the isotropic chemical shift. When 033 is parallel to the external magnetic
field, the angle between the external magnetic field and the three Ca - F bonds is
perpendicular. From the equations I-34 and I-35, 033 (6 = 90°) and (Ii (0 = 54.7°)

can be represented by the following equations

033 = 3 [ 2/3 a, +1/3 a1r +1/3(a7r - a0) (3 cos290° - 1) 1 = sag (l-36);
ai = Zaa + a3. (I-37)
Table 4 shows the ac and a; values calculated with respect to free F‘ ion
at zero ppm (since the latter has no paramagnetic contribution to its shielding),
whose shift has been estimated theoretically in Ref. 29. We have assumed that

the Sr“ ions in Sr5F(PO4)3 form a plane containing the F. ion, since this is known

51

Table 4.19F paramagnetic shielding parameters a; and an for M5F(PO4)3 calcu-

lated from the chemical shielding tensors (absolute chemical shift scale) in Table 3.

 

 

 

 

 

 

 

 

30(ppm) a1r(ppm)
Ca5F(PO4)3 81.7 i 0.1 24.6 :1: 0.3
Sr5F(PO4)3 97.6 i 0.6 26.0 :t 1.2
Ba5F(PO4)3 138.4 :1: 0.4 32.1 i 0.8

 

52

 

  

 

 

 

 

140 "
‘ 385F(P04)a
100 '4
. S'sF(PO4)3
E Ca5F(PO.)3
so 1
« a.
____F a
20 P - 7" - r - . 1
180 220 260 300
01039“)

Figure l-15. a0 and a, for M5F(PO4)3 vs. ‘9F isotropic chemical shift with respect
to free F“ ion. The solid lines simply connect the data points; open circles and
open smares denote a0 and a“ respectively.

53

to be the case for Ca and Ba fluorapatitef’v11 The increase in ad and an from
Ca5F(PO4)3 to Ba5F(PO4)3 indicates more contributions to the paramagnetic
shielding, but not “increased bonding” necessary. The a0— and aw values for
Ca5F(PO4)3, Sr5F(PO4)3, and Ba5F(PO4)3 are plotted vs. the isotropic chemical
shift with respect to free F' ion in Fig. l-15. Fig. I-15 shows that the slope of a0
vs. the isotropic chemical shift is larger than that of a3 vs. the isotropic chemical
shift. This observation implies that the observed increase in the paramagnetic
shielding term as one goes from Ca to Sr to Ba in M5F(PO4)3 (M = Ca, Sr, Ba) is
due to primarily to an increase in the sigma-bonding parameter a0.

C. Study of Site-Preference of Sr2+ Ions in Ca/Sr Fluorapatite Solid Solutions

Using 19F MAS-NMR

The difference in the chemical bonds between a fluoride ion and either calcium
or strontium gives rise to the different 19F chemical shifts of fluoride ions. The flu-
oride ions of Ca3.37$r1,o3F2(P04)3 have the four different chemical environments
shown in Fig. l-16. Since the structures of A and D represent the local fluoride
ion environment in Ca5F(PO4)3 and Sr5F(PO4)3,9=‘° respectively, we assign the
isotropic chemical shifts of A and D in Fig. 3 to 64 and 97 ppm respectively. The
isotropic chemical shifts of B and C in Fig. l-16 are predicted using Eqs. l-34
and I-35, and are equal to the sum of the isotropic portions of the shielding of the
appropriate number of Ca - F and Sr — F bonds. The configurations B and C in
Fug. ~l-16 have two Ca - F bonds and one Sr - F bond, and one Ca - F bonds and
two Sr - F bond respectively, resulting in predicted isotropic chemical shifts of B
and C in Fig. l-16 of 75.1 ppm and 86.1 ppm respectively. The isotropic chemical
shift of either one is different from the isotropic chemical shift of the peak at 79.6

ppm in Fig. l-10b. There are no centerbands between the centerband at 64 ppm

54

Ca Sr
Ca Ca Ca Ca
A a
Sr Sr
Sr Ca Sr Sr
c

D

 

Figure l-16. Substitution of Sr2+ in the three nearest neighbor Ca(2) sites of

Ca/Sr fluorapatite solid solutions.

55

and the centerband at 79.6 ppm, and the isotropic chemical shift at 79.6 ppm is the
second isotropic chemical shift in the downfield direction in Fig. l-10b. Therefore,
we assign the isotropic chemical shift of the configuration B (Sr1Ca2F) in Fig.
MS to the 79.6 ppm peak in Fig. I-11b. The fact that the experimental shift of
the configuration B is 4.5 ppm downfield of predicted value is reasonable, since
the Sr3F configuration in the Ca58r5F2(PO4)6 solids solution is also downfield (by
7ppm) of the shift in Sr5F(PO4)3, presumably due to lattice distortion effect. The
isotropic chemical shift due to C in Fig. MS may be between 79.6 ppm and 97
ppm, and may be concealed by overlap with the downfield sideband of the 64

ppm peak in Fig. l-10b.

The degree of possible preference of Sr2+ ions for a Ca(2) site for Ca/Sr
fluorapatite solid solutions can be studied by using both the resolved peaks
obtained from 19F MAS-NMR and from peak deconvolution of overlapping peaks.
There are two different types of calcium ions in Ca1oF2(PO4)3. Since there
are 4 Ca(1) ions and 6 Ca(2) ions in two unit cells, we rewrite the formula of
Ca3,978r1,o3F2(P04)6 as Ca(2)6xSr(2)6yCa(1)4X,Sr(1)4), F2(PO4)6( 6x + 4x’ = 8.97
and 6y + 4y’ = 1.03, and x + y = x’ + y’ = 1). Since the fluorine ions are bonded to
Ca(2) and Sr(2) ions, the chemical shift of the fluorine depends on the numbers
of Ca(2) and Sr(2) ions to which it is bonded. The integrated areas of the peaks

at each chemical shift in Fig. l-16 are represented by the following equations

A = x3,
B = 3x2y, (l-38)
C = 3xy2,

D=y3

56

where(A+B+C+D)=(x+y)3=1.0.

If the strontium ions randomly substitute for the calcium ions in Ca3.378r1_33F3(PO4)3,
the calculated integrated intensity of B from Eq. (l-40) would be 24 %. The ex-
perimental value is 29 %, which implies values for x and y of 0.873 and 0.127
respectively. Thus, the experimental integrated intensity of the B peak indicates
a 23 % [(0.127 - 0.103)/0.103] site preference of a Sr2+ ion for the Ca(2) site in
Ca3,97$r1,o3F2(PO4)3. Table 5 shows the calculated probabilities of the various
configurations for both random substitution and a 23% preference of the Sr2+ ions
for the Ca(2) site using Eq. (l-38).

The 19F MAS-NMR peaks in Fig. l-11b are asymmetric and broad, presumably
due to perturbations from strontium ions which are substituted in Ca(1) site, or
from strontium ions in the next-nearest Ca(2) sites. We now consider what types
of strontium ions (strontium ions in Ca(1) or Ca(2) sites) mainly perturb the fluorine
peak at 64 ppm. From the crystal structure of calcium fluorapatite, when a
single Sr2+ ion substitutes in a Ca(1) site, it potentially perturbs 6 fluorine ions
in an equivalent fashion, but a Sr“ ion substituting for a Ca(2) site perturbs 2
fluorine ions. Thus, the Sr2+ ions in Ca(1) sites perturbs a fluorine ions more than
that in Ca(2) sites. The perturbation of the chemical shift of the 64 ppm peak
should depend on the number of the strontium ions at the next-nearest Ca(2)
sites. The schematic arrangements of the metal cations neighboring a fluorine
atom resonating near 64 ppm are shown in Fig. l-17. Since the probability of
substitution of more than 2 Sr2+ ions in Ca(2) sites in Ca3,978r1,o3F2(PO4)6 is low,
configuration III in Eq. (l-39) represents essentially the sum of the probabilities of
having 2 2 Sr2+ ions in Ca(2) sites. The integrated intensities of peaks arising from

the individual configurations in Fig. I-17 are then given by the following equations

57

C8 C8 C8

F F /F
' Ca

 

 

 

 

 

 

CL .4 C1 I’
Cl C. Ca
/ F\ F / F

C‘ .4 C‘ 9‘ Ca Ca
C! 0 Ca
/ F\ F / ,-

Ca .4 Ct ‘ C3 c3 3 .r

 

 

 

Figure l-17. Possible configurations for single (l) and double (l) Sr“ substitution
in the six next-nearest neighbor Ca(2) sites of Ca/Sr solid solutions of fluorapatite.
The middle equilateral triangle in each configuration represents the observed Ca3F
group resonating around 64 ppm, and the triangles above and below it (along '
the c-axis), although parallel, are drawn tilted for clarity. One Sr“ ion in the.
configuration If is either in a top triangle (as shown) or in a bottom triangle. Two
5,2. ions are either in a top triangle or a bottom triangle, or in both triangles (as

shown).

58

Table 5. Calculated probabilities of configurations AD in Ca3_37$r1_03F2(PO4)3
(Figure I-16) for both random substitution and a 23% preference of Sr2++ ions for

the Ca(2) sites, and comparison with measured integrated intensity of peak B.

 

 

 

 

 

 

 

 

 

Configuration Probability, Probability, Integrated
Random 23% Preference MAS-NMR Peak
substitution Ca(2) Site Intensity

A. 72.1 % 66.5% —

B. 24.9% 29.0% 29%

C. 2.9% 4.2% —

D. 0.1% 0.2% —
Total 1 00.0% 99.9% 1 00%

 

 

 

Table 6. Calculated probabilities of configurations H" in Ca3.37$r1.33F2(PO4)3
(Figure H?) for both random substitution and a 23% preference of Sr2++ ions

for the Ca(2) sites, and comparison with the experimental deconvolution data

obtained from the peak near 64 ppm.

 

 

 

 

 

 

Configuration Probability, Probability, Intensities of
Random 23% Preference Deconvoluted
substitution Ca(2) Site Peak at 64 ppm

I. 52.1 % 45.6% 45.4%

II. 35.9% 38.3% 37.3%

III. 12.0% 16.1% 17.4%
Total 100.0% 100.0% 100.1%

 

 

 

 

 

59

l = x6

II = 6x5y (l-39)

Ill = 1— l - II.
The comparison of the deconvoluted integrated intensities of the peaks near 64
ppm and the integrated intensities calculated for both random substitutions of Sr2+
ions and for a 23 % preference of strontium ions forth Ca(2) site is shown in Table
6. The predictions for a 23% site preference (x=0.877 and y=0.123) are closer to
the deconvolution data than those assuming random substitution.

The half-height linewidths of the deconvoluted peaks are broader than those
of calcium and strontium fluorapatite (130 Hz at 10.56 kHz, 365 Hz at 10.20 kHz
respectively). Since the perturbation effect of strontium ions substituted in the
Ca(1) sites on the peak near 64 ppm is small compared to that of strontium ions
substituted in the next-nearest Ca(2) sites on the same peak, we believe that the
main effect of strontium ions substituting in the Ca(1) site is a slight increase in

the half-linewidths of deconvoluted peaks.

To prove that the fluorine spins of the peaks at 64 and 79.6 ppm (see Fig.
I-10) are actually in the same phase and not in phase-segregated regions, a spin
diffusion experiment was carried out. Spin diffusion is the transfer of Zeeman
magnetization between two adjacent spins by means of a spin flip-flap?“31 Spin
diffusion requires both the existence of a dipole-dipole coupling between the
nuclear spins and the conservation of Zeeman energy. Spectral spin diffusion
between the peaks having different isotropic chemical shifts under magic-angle-
spinning can occur if the Zeeman energy level overlap during a rotor period,
the so—called “level-crossing”. The existence of level-crossing between two peaks

with different isotropic shifts can be demonstrated by calculating the instantaneous

60

frequencies of the pair of spins in a given crystallite during each part of a rotor

cycle.”

A MAS Hamiltonian can be transformed from the chemical shift principal axis
system (PAS) of each crystallite to a reference frame fixed on the rotor.32'33 The
MAS Hamiltonian for the chemical shift principal axis system becomes periodic
(cosine wave), and the offset of the cosine wave depends on the asymmetry
parameter and the crystallite orientation. Not only is the amplitude of the cosine
wave proportional to the CSA but it also depends on the asymmetry parameter

and crystallite orientation.”33

Fig. l-12 shows the existence of spin diffusion between the peak at 64 ppm and
the peak at 79 ppm. The CSA and asymmetry parameter of the peak at 64 ppm
are known.12 Thus, we have to know the CSA and asymmetry parameter of the
peak at 79 ppm in order to predict the level-crossing of the two peaks theoretically.
We assume that the configuration ll of Fig. I-16 is an equilateral triangle and that
the fluoride ion in the configuration If is in the middle of the equilateral triangle.
The largest chemical shielding tensor 033 is obtained when the external magnetic
field is perpendicular to the equilateral triangle in the configuration ll. Since the
shielding principal values 011, 022, 033 are orthogonal, 011 and 022 are on the
plane of the equilateral triangle. The smallest chemical shielding component 011
is obtained when the external magnetic field is parallel to a Sr - F bond. From
Eq. l-34 and 35, the chemical shielding tensor components 011, 022, 033 of
configuration II in Fig. I-16 can be represented by

011 = 0(0°) (Sr) + 0(120°) (Ca) + 0(120°) (Ca)
= 3/2 aa (Ca) + 1/2 an (Ca) + a3 (Sr) (|-40)
022 = 0(90°) (Sr) + 0(30°) (Ca) + 0(30°) (Ca)

61

= 1/2 a; (Ca) + 3/2 a (Ca) + a0 (Sr) (l-41)
033 = 0(90°) (Ca) + 0(90°) (Ca) + 0(90°) (Sr)

= 2 a0 (Ca) + a; (Sr). (I-42)
The values of the shielding tensor componentS011, 022, 033 for the peak at 79
ppm (configuration II in Fig. l-16) obtained using the ad and an values in Table
4 are equal to 160.9 ppm, 175.4 ppm and 261.0 ppm respectively. The CSA and
asymmetry parameter of the 79.6 ppm peak calculated using Eqs. l-8 and l-9 are
92.9 ppm and 0.23, and the CSA and asymmetry parameter of the 64 ppm peak
(calcium fluorapatite) are 84 ppm and 0.12 The calculated energy modulations
of the 64 ppm and 79.6 ppm peaks during one rotor cycle spinning for two
different crystallite orientations are shown in Fig. I-18. Fig. l-18a shows the
existence of an overlap of the energy levels between the peaks corresponding to
configurations l and II in Fig. l-16 during one rotor cycle, whereas the different
crystallite orientations of Fig. l-18b do not. From our calculations (not shown
here), most crystallite orientations of configurations l and II have an energy overlap
during one rotor cycle. Therefore, a powder sample of (Ca3,3-,Sr1,o3F2(PO4)3) will
have most crystallites experiencing level-crossing under MAS, resulting in the
observed spin diffusion between the 79.6 ppm and 64 ppm peaks. This spin
diffusion takes place through the 19F homonuclear dipolar interaction. This spin
diffusion between the 79.6 ppm and 64 ppm peaks in Fig. l-12 indicates that the
corresponding fluoride ions are close to each other and have equivalent chemical

shifts during the rotor cycle.

62

80

 

 

 

 

 

60 b)
40 A AAA AAA
A A A A A
20 . 0A Ao°°°oA Ao°°°oA
O A A O O A A O O A
o o o
o . oAA AA:3 0AA AAo 0A
.20 c,oAMoo 004531300 o
E . 063 0C9
0.
e '40 t v v
E 3° 8. an.
5 A A A
60 1 A a) A A
4 A A A
40 ‘ A A A
‘ A A A
20 -b 000 <90
00 A 00 c’o A oo 00 f
o A o o A o o
0 o A o o A o o
o A o o
-20 °009 9
-40 - . f - 1 - f r

 

 

 

0 0.2 0.4 0.6 0.8 1.0 1.2

Rotor cycle

Figure l-18. Variation in the instantaneous chemical shifts during a rotor cycle of
the 64 ppm and 79 ppm peaks of the Caa,978r1,33F2(PO4)3 solid solution under
MAS for at the different crystallite orientations, a = 0° and fl = 60°(a), and a = 0°
and fl - 90° (b): 64 ppm peak (open triangles) and 79 ppm peak (open circles).
a and 3 are Euler angles with respect to the principal axis system of tensors in

the rotor frame.

63

8. Conclusions

We have obtained the values of the CSA for M5F(PO4)3 (M = Ca, Sr, and Ba)
using the different methods. Even though the measured asymmetry parameter
77 of Ca5F(PO4)3 obtained by using the moments method and the Herzfeld and
Berger graphical method is not equal to zero, the value of the CSA of Ca5F(PO4)3
obtained from 19F MAS-NMR spectra by the two methods is close to that obtained
from a 19F single crystal NMR study.12 The accuracy of measuring the CSA of
M5F(PO4)3 from the moments remains the same for different spinning speeds.
The simulations using ANTIOPE show that the change of the CSA of Ca5F(PO4)3
due to the dipolar interaction is small. The change in the intensity of the center-
band and sidebands when the dipolar interaction is considered causes a failure
of the Herzfeld-Berger contour plots of Ca5F(PO4)3 and Sr5F(PO4)3 to overlap
at spinning speed below 9kHz. However, since the CSA values obtained from
the contour plots using high spinning speed data are close to the CSA values
measured from the moments method, the two methods are complementary under
these conditions.

There are two reasons that the intensities of the simulated spectra of
M5F(PO4)3 (M a Ca, Sr, and Ba) do not correspond to those of the experimental
spectra One reason is that dipolar interaction is neglected in the VNMR 3.2 sim-
ulatiOns. The other reason is that the different half-height linewidths of the peaks
can give rise to a difference between real and simulated spectra. The stick spec-
tra obtained from the integrated intensities of peaks are therefore more useful for
comparing the simulated and experimental spectra.

The separation of the ad and a3 parameters for from the chemical shift tensors

gives information about the contribution of the sigma- and pi-bonding components

54

of the chemical bonding between alkaline-earth metal ions and fluoride ions to
the 19F shielding in apatite samples. The values of a0 and a7; obtained from
19F shielding tensors for apatite samples make it possible to predict the CSA and
asymmetry parameter r; of configurations in the solid solutions of Ca/Sr fluorapatite
and the orientation of the shielding tensor in the molecular frame. This somewhat
novel approach to predicting the full shielding tensor in the molecular frame of solid
solution has proven valuable in studies of chemical-shift-selective MQ-NMR,34 and

may be useful in studying other solid solutions.

The shifts of the peaks in the 19F MAS-NMR spectrum of the solid solution
Ca8.978r1.o3 F3(PO4)5 were interpreted in terms of proximity of the fluorine spins to
strontium ions. The fact that the peaks of this spectrum arose from fluorine spins in
the same phase was established by demonstrating the existence of spin diffusion
. between them with the SPARTAN pulse sequence. The preference of Sr2+ ions
for the Ca(2) sites in Ca3,9-,Sr1,o3F2(PO4)6 was also studied. The observed 23 %
preference of Sr2+ ions for Ca(2) sites, using two different methods is in agreement
with an X-ray powder diffraction study of Ca/Sr hydroxyapatites, which determined
an approximately a 20 % site preference for Ca(2) site.35 A more recent EXAFS
study of Ca/Sr hydroxyapatite claimed a larger preference for the Ca(2) site, and
must be considered suspect.

65

9. References

u—b

. M. Mehring, A. Pine, W-K. Rhim, and J. S. Waugh, J. Chem. Phys. 54, 3239
(1971).

N

. R. W. Vaughan, D. D. Elleman, W-K. Rhim, and L. M. Stacey, J. Chem. Phys.
57, 5383 (1972).

3. R. J. Sears, J. Chem. Phys. 59, 5213 (1973).

4. R. J. Sears, J. Chem. Phys. 61, 4368 (1974).

5. J. Mason, J. C. S. Dalton, 1426 (1975).

6. J. P. Yesinowski and M. J. Mobley, J. Am. Chem. Soc. 105, 6191 (1983).

7. A. T. Kreinbrink., C. D. Sazavsky, J. W. Pyrz, D. G. A. Nelson, and R. S.
Honkonen, J. Magn. Reson. 88, 267 (1990).

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

9. K. Sudarsanan, P. E. Mackie, and R. A. Young, Res. Bull. 7, 1331 (1972)

10. K. Sudarsanan and R. A. Young, Acta Cryst. 828, 3668 (1972)

11. M. Mathew, I. Mayer, B. Dickens, and L. W. Schroeder, J. Solid State Chem.
23,79 (1979).

12. J. L. Carolan, Chem. Phys. Left. 12, 389 (1971).

13. AM. Vakhrameev, S. P. Gabuda, and R. G. Knubovet, J. Struct. Chem.
(USSR),19, 256 (1978).

14. M. M. Maricq and J. S. Waugh, J. Chem. Phys. 70(7), 3300 (1979).

15. J. Herzfeld and A. E. Berger, J. Chem. Phys. 73(12), 6021 (1980).

16. R. Tycko, G. Dabbagh, S. R. Kurtz, and J. P. Goral, Phys. Rev. B. 45, 13452
(1991).

17. E. R. Andrew, A. Bradbury, and R. G. Eades, Nature (London), 183 1802

(1959).

66

18. I. J. Lowe, Phys. Rev. Lett. 2, 285 (1959).

19. E. R. Andrew, Prog. Nucl. Magn. Reson. Spectrosc. 8, 1 (1972)

20. L. B. Moran, J. K. Berkowitz, and J. P. Yesinowski, Phys. Rev. B45, 5347
(1992)
21. G. Bodenhausen, R. Freeman, and G. A. Morrison, J. Magn. Reson. 23, 171
(1976); G. A. Morrison and R. Freeman, J. Magn. Reson. 29, 433 (1978).
22. A. Kubo and Charles A. McDowell, J. Chem. Soc. Faraday. Trans. 84, 3713
(1988).

23. F. S. de Bouregas and J. S. Waugh, J. Magn. Reson. 96, 280 (1992).

24. E. Brunner, D. Freude, B. C. Gerstein, and H. Pfeifer, J. Magn. Reson. 90,
90(1990).

25. N. J. Clayden, C. M. Dobson, L. Lian, and D. J. Smith, J. Magn. Reson. 69,
476 (1986).

26. N. F. Ramsey, Phys. Rev. 78, 699 (1950)

27. N. F. Ramsey, Phys. Rev. 86, 243 (1951)

28. W. E. Lamb, Phys. Rev, 60, 817 (1941).

29. Y. V. Gagarinskii and S. P. Gabuda, translated from Zhumal Struktumoi
Khimii, 11(5), 955 (1970).

30. N. Bloembergen, Physica15, 386 (1949).

31. A Abragam, The Principles of Nuclear Magnetism (Oxford University Press,
London, 1961), Chaps. V. and IX.

32. ET. Olejniczak, S.Vega, and R. G. Griffin, J. Chem. Phys. 81, 4804 (1984).

33. DP. Raleigh, E.T. Olejniczak, and R. G. Griffin, J. Chem. Phys. 89, 1333
(1988).

34. L. B. Moran and J. P. Yesinowski, to be published.

67

35. H. J. M. Heijigers, F. C. M. Driessen, and R. M. H. Verbeeck, Calcif Tissue
Int. 29, 127 (1979).

PART 2

1H and 19F Multiple-Quantum NMR Dynamics of
Quasi-One-Dlmenslonal Spin Distributions in Apatites

68

69

1. Introduction

Multiple-quantum NMR is a general term for experiments that observe nuclear
magnetic transitions forbidden by the standard selection rule Am = if. Although
multiple quantum transitions can be observed by using high power continuous-
wave (CW) spectrometers,"14 difficulties of interpretation and inconvenient instru-
mental requirements have limited the use of CW-observation of multiple-quantum
transitions. The advent of time-domain Fourier transform techniques15 made it
possible to detect the forbidden transitions. In the mid 1970s, Hashi16 and Ernst17
independently adapted a two-dimensional Fourier transform NMR technique to in-
directly observe multiple quantum transitions.

Since that time, the multiple quantum NMR technique has been mostly applied
to the liquid state, where the size of the spin systems is relatively small.“3'28 An 8-
. pulse sequence that creates an average-Hamiltonian operator for double quantum
NMR transitions of dipolar-coupled spin pairs, 29 combined with the use of a time-
reversal pulse sequence,30 increases the S:N ratio of multiple-quantum intensities
and has made the study of larger spin systems such as found in solids feasible.
1H MQ-NMR has been applied to the study of proton distributions in solids,3‘-32'36
imaging in solids,32"37 and adsorption of organic molecules in zeolites38'39. 1H
MQ-NMR has been also used to study nematic liquid crystalsgai‘M3 19F MQ-
NMR has been used to investigate fluorine distributions in polycrystalline“ and

photosensitive salts.45

The time-development of multiple-quantum coherences in the "infinite" dipolar-
coupled spin systems typical of many solids presents both a lure and a challenge.
The lure is the possibility of obtaining structural information about groups of cor-

related spins that would be otherwise unobtainable with conventional NMR spec-

70

troscopic techniques. The challenge resides in the development of theoretical
models describing multiple-quantum dynamics that are both computationally prac-
tical and experimentally realistic. The explicit calculation of the density-operator
for spin systems, widely used to describe modern NMR experiments,“"5"'8 can-
not describe many actual spin systems due to computational limitations (up to
nine spins is the current Iimit)."’9‘51 The actual spin systems present in strongly
dipolar-coupled solids consist of about 102° spins whose calculation time using
the density matrix is prohibitive. Therefore, simplifying approaches have been de-
veloped that make only statistical assumptions about the time-development of
the density operator, and neglect the detailed spin-dynamics arising from the
specific disposition of spins in space.“53 The earliest of these, the statistical
model,54 counts the combinatorial possibilities of having coherences of order n
("n-quantum coherences") in a spin system of ”effective size" N. For an "infinite”
Spin system, the effective size N increases monotonically with increasing prepa-
ration time allowed for creation of multiple-quantum coherences. The statistical
model predicts an approximately Gaussian distribution of intensities for the var-
ious ordersof multiple-quantum coherence. Although it tends to underestimate
the intensity of high order coherences, it does provide a measure of the effective
size of the spin system at a given preparation time.3"52'53v55-56 Since intermolec-
ular dipolar couplings of nematic liquid crystals are averaged to zero whereas
intramolecular dipolar couplings are not, “spin counting” by means of MQ—NMR
experiments of the known number of spins in molecules of nematic liquid crys-
tals demonstrates the usefulness of the statistical model.“55 The directed walk
through Liouville space (“hopping”) model57 predicts the intensities of various or-

ders of higher order coherences, at different preparation times, by counting the

71

combinatorial possibilities of allowed "transitions” or "hops” in Liouville space, and
assuming that any oscillatory behavior will be hidden by destructive interference
and resultant decay. Although the hopping model can successfully account for
features of the experimental data,57 Lacelle53 has pointed out aspects of the av-
eraging of multiplicative processes that, if properly taken into account, may lead
to significant differences from the predictions of the hopping model. Only very
recently has there been an attempt to develop a simplifying model, as opposed
to explicit density operator calculations?“51 that specifically considers the spa-
tial structure of the spin system (including its dimensionality). This "incremental
shell" model of Levy and Gleason36 describes multiple quantum dynamics dur-
ing the preparation period as a stepwise process: a given coherence can either
expand by incorporating one additional spin at the periphery of the spin cluster
involved in the coherence, or decrease in size by one spin. The rate of this
process is governed by the dipolar coupling between these neighboring spins and
by structure—dependent parameters. A set of differential equations for "average"
product operators of the density operator can then be solved numerically to yield
the effective size N vs. preparation time. Clear distinctions between the dynam-
ics of two- and three-dimensional spin systems are both predicted and observed
experimentally.36 A number of three-dimensional solids have been recently ob-
served to exhibit a universal growth behavior (when the time axis is scaled by the
strength of the square root of the second moments)36 that can be fit to theoretical
predictions of an ”incremental shell model; " the effective size increases as the
cube (or third power) of the preparation time. In case of a presumed surface film of
protons, the experimental results could be fit to the model with an approximately

quadratic dependence of the effective size upon preparation time, as predicted

72

for a two-dimensional spin distribution.36 An infinite one-dimensional distribution
of uniformly-spaced spins would be expected to provide the simplest experimen-
tal test of the various models. We report here results for a close approximation
to such a spin system: the 1H nuclei of hydroxyapatite, Ca5(OH)(PO4)3. Since
the multiple-quantum coherences are created by homonuclear dipolar couplings,
we need only consider the hydroxyl protons (or fluoride ions) in the structure of
hydroxyapatite (or fluorapatite). Figure "-1 shows the basic idealized geometry
of the ‘H spin system of hydroxyapatite: infinite linear chains of protons having
a uniform spacing of 344 pm, with each chain surrounded by six other chains
at a distance of 942 pm.“59 The largest intra-chain dipolar coupling is some 20
times greater than the largest inter-chain dipolar coupling, and should dominate
the multiple-quantum dynamics. The weak heteronuclear dipolar couplings to the

3‘P nuclei (<2 kHz) can safely be ignored.“4

In order to use apatites as a model for studying one-dimensional MO dynamics,
one must have knowledge about the occurrence of interruptions in the 1-D chain,
due to either vacancies or substitutions (collectively referred to here as "defects").
Such defects are commonly present at significant levels in both synthetic as well as
naturally-occuring apatites. The ability to obtain such information about defects
in apatites from MQ NMR experiments would thus provide a useful method for
investigating such systems, in addition to better defining the degree of ideality of

one-dimensional spin systems in powders and single crystals.

In this study we present experimental results on proton MQ NMR dynamics
in a stoichiometric hydroxyapatite sample (HAP-M), a hydroxyapatite sample
containing defects (HAP-N), and a series of fluorohydroxyapatite solid solutions,

Ca5(OH)(1-x)Fx(PO4)3, with the fluoride ion (replacing a hydroxyl group) forming

73

Figure "-1. Schematic idealized arrangement of linear columns of protons (black
circles) in calcium hydroxyapatite (Ca50H(PO4)3). The central column is sur-
rounded by six neighboring columns. The distance between columns is 942 pm
and the distance between intra-chain protons is 344pm. The position of protons
in four of the six neighboring columns is actually approximately 260pm below
the black circles in the monoclinic form, which exists only for very stoichiometric
samples. In the more commonly-occurring hexagonal form, the protons in three
of the six neighboring columns are located about 260pm below the black circles
due to statistical disorder. The geometry of the fluorine atoms in fluorapatite
(Ca5F(PO4)3) is similar to the arrangement of protons in hydroxyapatite, with a

distance between columns of 937pm and an identiml intra-chain distance.

74

 

 

 

 

 

 

 

 

 

 

a 1°
—> 942 pm
L - 120°
0 O O b
6,1
C5
63 1
Q.
i 344 pm
Q

 

 

75

a defect in the 1-D chain of spins. We model these results for a hydroxyapatite
sample with a slight hydroxyl deficiency in terms of a 1-D cluster model described
in the Discussion section. The 1-D cluster model considers randomly-distributed
defects in apatites as producing a distribution of 1-D clusters of varying lengths,
and uses the MO response of a stoichiometric hydroxyapatite sample as a ”cali-
bration". In addition, we report 19F MD NMR data on a single crystal of mineral
fluorapatite at several orientations in the magnetic field, thereby scaling the dipolar
interactions within the chains by a known amount. This approach, when applied
to a sample containing few defects, should eventually allow one to scale the rel-
ative contributions of one-dimensional and higher-dimensional MO growth, and

thus permit better isolation of the effects of differing dimensionality.

2. Multiple-Quantum NMR Dynamics
A. Density Operator Description of Multiple-Quantum NMR Dynamics

The description of NMR experiments can be started by first considering simple
two level systems. An isolated spin-1/2 particle in an external magnetic field has
the two eigenstates I15) and I—é), which represent the two allowed orientations
of its angular momentum.‘7'6° The state vector, a superposition of these basis
states, is given by a linear combination of these two states

' 1120)) = 1,2111%) +c-l/2(t)l — 1) (II-1)
where c1/2(t) and c_1/2(t) are the (in general) time-dependent complex coeffi-
cients. The state of the system can be described by the values of c1/2(t) and
c_1/2(t), which are solved by using the time-dependent Schroedinger equation

111(1)) = 4111111)). ("-2)

When the only Hamiltonian is the Zeeman Hamiltonian, the time-dependent coef-

76

ficients are given by
Cl/2(t) = a-erp(ia)ea:p(—iw0t) (ll—3a)
c-0211) = hemmexpawot) (II-3b)
where a2 + b2 = 1, and a and [3 are real numbers representing phases.47 The ex-
pectation value of any observable can be calculated from |¢:(t)). Expectation

values of the three components of magnetic moment are represented by

 

Mt) = (¢(t)lthzlw(t)) ("'48)
AN) = (1(t)|7thI1/2(t)) ("'45)
Mt) = (wlt)|7hlzlw(t)) (”'40)
which are solved by using the raising and lowering operators defined by
71 = fl(1+1)—m(m:l:1)|m:l:1). (ll-5)
The result is
1,“) = 7habcos (a — a + wot) (II-6a)
1,,(1) = 7habsin(a — 13 + 001) (ll-6b)
Mt) = 7h(02 — (12)/2. (ll-6c)

The expectation values of the magnetic moments of transverse components for an
individual spin precess about the external magnetic field with angular frequency (.00
and phase a — 5. However, the expectation value of the magnetic moment of the 2
component is constant, and, (at equilibrium) is proportional to a population differ-
ence of the two spin levels, which in turn is proportional to the energy difference in
the “high temperature approximation” (Boltzmann distribution). Since the behavior
of a single spin or group of spins cannot in general describe a macroscopic sys-
tem, we introduce an ensemble of independent two-level subsystems. Figure "-2
shows the mixed and pure states of this ensemble. The transverse components

of the subsystems have the same frequency, but need not have the same

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure "-2. Random phases (a) and correlated phases (b) in an ensemble of two

level systems. Taken from Ref. 61.

78

phase for a the mixed state (Fig. ll-2a). The average transverse components of
the subsystems over all values of the phase is equal to zero, and only the lon-
gitudinal component remains (see equation "-6). In a pure state (Fig. Il-2b), the
behavior of the macroscopic system is identical to that of each microscopic sub-
system, due to each subsystem having the same frequency and phase (phase
coherence)!“61 The term “coherence” is defined as the presence of some de-
gree of phase coherence between the basis states of the isolated subsystems

throughout the ensemble.61

An ensemble of spins can be described by the density operator, which is

represented by

p = Z pilzbt) (11%| = 111111. ("-7)
where p,- is the probability of each state 2/2, occurring in the superposition.62 The
equilibrium density operator can be represented by the M) (M,| basis whose

matrix formulation of Eq. "-2 is given by

p = iii/2C”? il/Qcil/Q . ("-8)
Cmaul/2 6—1/20-1/2
With the probabilities from the Boltzmann factor

 

p(M = 1.1,) = e$p(-tho/kT)/Z, (ll-9)
the equilibrium density operator is given by
PM = exp(—hw0Iz/kT)/Z ("'10)
after averaging over the phase differences. From the high temperature approx-
imation, the partition function Z is equal to 2I + 1. For the equilibrium density

operator of a spin 1/2 system, Eq. "-3 can be rewritten as

[p..]= 6$P(-hwo/2kT)/2 0
'1 0 exp(-fwo/2kT)/2 '

The random phase among spins of different subsystems in the ensemble averages

(ll-11)

the off-diagonal elements to zero, but an application of an appropriate rf pulse to

79

the ensemble results in non-zero off-diagonal elements in the matrix. The exis-
tence of the non-zero off-diagonal elements from a single pulse is described as
single-quantum coherence. The order n of a coherence between states Ir) and
Is) is defined as
n = AM = IM. - M,|, (II-12)

where M, represent the total (summed) M, values of all the spins in the state |r).
The dimension of the density matrix increases as size of the coupled spin system
increases. The presence of phase coherences in large spin systems produced by
a proper pulse sequence results in non-zero off-diagonal elements that represent
the multiple quantum coherences between states.

B. Time Development of the Density Operator in the Rotating Frame

The density operator determines the‘ state of a system at any time. The de-
. velopment of the density operator p is governed by the Liouville-von Neumann
equation.

31,42 = 2110.11]. (ll-13)

When the Hamiltonian H is time-independent, the formal solution of Eq. "-13 is

represented by
90‘) = exp{-(i/B)Htlp(0)eXP[(i/fi)71t]- (ll-14)
When p and ’H commute, no evolution of the density operator occurs
90) = p(0)expl-(i/hWtJeXPIG/fimt] = p(0)- (ll-15)

If ’H depends explicitly on time, we may satisfy both the Schroedinger equation
and the Liouville-von Neumann equation by replacing the propagator that makes
the evolution of one state to other state by a unitary transformation in Eq. "-13 with

U(t) = Tezp {—1 j 11(1) 41’] (ll-16)

where T is the Dyson “time-ordering operator”. Coherent averaging theory63v64 is

 

80

used to solve Eq. 15. p(t) is always related to p(0) by a unitary transformation in
Liouville space so that the length of the vector representing the density operator
remains constant as the system evolves. Since the expectation value of any op-
erator can be calculated from p,47

<A<t)) = 2: p31,,- = Triplt)Al (II-17)
the prediction and control of thje time development of p are always central prob-

lems for time domain NMR in general and multiple-quantum NMR in particular.

It is often convenient to go to the rotating frame when a system is acted on by
alternating magnetic fields. The transformation to a frame rotating at or near the
Larmor frequency in order to remove the fast precession due to the Zeeman inter-
action makes the solution of the Liouville-von Neumann equation easier. Defining
an operator R as

R = exp(—m,,,t) (ll-18)
where H m is the external Hamiltonian. We write the transformed density operator
as

pR = R'lpR; (ll-19)
and the transformed Hamiltonian as

HR = R‘1HR— 112-1% (II-20)

The transformed Uouville-von Neumann equation can be rewritten as
11:; = i[pR,’HR] (II-21)
Since the Liouville-von Neumann equation is of the same form after the transfor-
mation, we will omit the superscript on p and H with the understanding that the
laboratory frame has been replaced by a suitable rotating frame in which external

Zeeman interactions are absent.

81

C. Pulse Sequence for Multiple-Quantum NMR

Since multiple-quantum transitions are usually not directly observable with
an NMR coil, they must be detected indirectly by two-dimensional spectroscopic
methods. The general scheme of a two-dimensional multiple-quantum NMR ex-
periment is shown in Figure "-3. The pulse sequence creates a non-equilibrium
condition of multiple quantum coherence during a preparation period 1, allows the
coherence to respond during an evolution period t1 and then transfers the coher-
ence to z magnetization during a mixing time 7". The coherence is then detected
after a detection pulse during the detection period, which creates observable Iz
magnetization (corresponding to a single-quantum coherence).

The preparation period propagator

U(7') = exp(—th) (ll-22)
arises from a combination of pulses and proper delays. Under this propagator,
the density operator becomes

70(7) = U(T)p(0)U“(T) (ll-23)
by the end of the preparation period. The system is allowed to develop, freely
or otherwise, for an evolution period of length t1. During this time, the different
modes of coherence oscillate at the eigenfrequencies determined by the effective
Hamiltonian H1 (defining in Figure "-3). No signal is recorded during this interval.
After time development is halted at some time t1, the coherences are transferred
to detectable single-quantum modes during the mixing and detection periods and
the density operator becomes
p(1’,t17") = V(T')exp(—1H1t;)p(r)exp(iH1t1)V‘1(1") (II—24)
where V(r') = exp (—1H'T') is the mixing period propagator. Transverse com-

ponents of the total spin angular momentum I3 and I, are obtained immediately

82

Preparation Evolution Mixing Detection

 

 

Propagator U exp(-H1t1) V exp(-H2t2)

 

 

 

 

 

Time variable 1: t1 1' t2

Figure "-3. General form of the pulse sequence of MO NMR experiments. See

text.

83

after the mixing period
$00,137) = Tr [p(r,t.,-r’)1,,] a = 3:, y. (II-25)

In NMR experiments, a complex signal S = 5, +153, is detected with the combi-
nation of the transverse components in quadrature. From the high temperature
approximation, a reduced density matrix describing the initial equilibrium condition
of the spin system is represented by

p ~ 1,. (ll-26)
Even though the initial condition Iz is not measurable directly, it is related to I1B and
I3, by a simple 90° pulse. Thus we take I, as the observable operator. Defining
the coherence amplitude

z,,,, = (M,|Up(0)U-1|M,)(M,lv-lrzle,) (II-27)
we then express the trace as
S, (T, t), 7”) = g 2,33,.exp (—iw,(.,1)t1) (ll-28)

where the oscillation frequency c059,: w, -w, occur at the energy level differences
during the evolution period Hamiltonian. The full response during the evolution
period is recorded point-by-point by repetition of the experiment over a series
of regularly incremented values of t1. The signal detected during t2 shows the
modulation of the single—quantum signal with respect to the t1 domain. Fourier
transformation of the interferogram of the t1 time-domain provides the multiple-

quantum spectrum. _

In general the amplitude and the phase of each frequency component at t2 =
0 depend on the combined effects of preparation and mixing through the complex
factor Z. When U is equal to or is only different from V‘1 by a phase factor 0,
v-1 = exp( 4111,) Uexp(ic$I;) (ll-29)

the signal in Eq. 27 becomes

84

5411,) = 2; 5; (111,11,1.11,)‘2exp(1n¢)exp(—10£,"t,) (ll-30)
where all transitions within a given order n have the same phase, and all transitions
of two adjacent orders differ in phase by :l: a). The signal is detected by using
a simple 1r/2 pulse at t2 = 0. The-condition that the preparation and mixing
Hamiltonians be equal in magnitude but opposite in sign, results in time reversal
for the propagator. in solids, the time reversal sequences that refocus the

evolution due to dipolar coupling increase the signal to noise ratio by achieving a

constructive interference of the different transitions within an order.29

Multiple-quantum coherences can be created in dipolar-coupled solids under
the action of rf pulse sequences with an appropriate average Hamiltonian. Figure
"-4 shows a pulse sequence for multiple-quantum NMR experiments. Eight 90°
pulses of duration tp with spacings A’: 2A + tp generate the average dipolar
. Hamiltonian

Hp = 11,, = —5 aqua-+1“ + I,-_I,,-) (II-31)
where Iji = 1,, d: 11,-y, and 115,-), is the homonuclear dipolar coupling. The
average dipolar Hamiltonian, containing only the terms I j+Ik+ and I j_I,,-, excites
only even order multiple-quantum coherences. Two methods, time proportional
phase Incrementation (T PPl)29 and phase incrementation (PI)4°, can be used to
separate the different multiple-quantum orders. In TPPI, the overall phase 0 of
the preparation pulses is incremented in proportion to t1

Ad) = AwAtl (ll-32)
where q) is the phase of the rf pulses and Aw is a resonance offset frequency.
Substituting Eq. 31 into Eq. 29, we can rewrite Eq. 29 as
$47,111) = z: z: (M,|I,|M,)2exp(tnA0At,)exp(—10£1’t,). (II-33)

n r,s

Fourier Transformation of the signal with respect to t1 results in the separation of

85

(ft/2),,

T 11 T

I 1/3( Hd)¢ -1/3( Hd) H

 

 

 

 

 

(Ti/ZR ("mx (TE/2),, (Tr/2)x (ft/2);, (TI/2);, (Tc/2);, (It/2);,
mm A’ HAHA HAH A’ HAl—l A’ HM
— i e m

Hd : X Djk (Ij+Ik++ Ij-Ik-)
(Evenerder selective)

Figure "-4. Pulse sequence for multiple-quantum NMR in solids.”40 The time
reversal sequence is used in the mixing period. The transverse magnetization is
allowed to decay during the delay before detection, and spin locking (not shown)
is used after the detection pulse (final 90° pulse) to increase the S:N ratio. For
PI-MQ-NMR, the value of t1 is not stepped, but fixed at a short value, and the

phase 43 are incremented.

 

 

 

 

J

0 1r Zr 3! 4r Sr 61? 71r 8x 91! 101111r121r131r141ri§1r161r

 

Phil.

Figure "-5. Time-domain MO interferogram of hexamethylbenzene using TPPI

and the pulse sequence shown in Fig. "-4.

87

 

 

 

 

 

 

 

 

 

 

 

 

 

2 46 810121416182022242628

Mammotm)

Figure "-6. Frequency-domain 1H MD spectrum of hexamethylbenzene using the
TPPI and the pulse sequence shown in Fig. "-6. The experimental parameters
used are 90° pulse length a 3.8 ps, preparation time =- 504 ps (basic cycle time
=72pS), At1 . 200 ns and A<I> = 6°.

88

different MO orders by multiples of the resonance offset frequency Aw. The
spectral width of the MO spectrum is the inverse of the t1 increment. The number
of orders detected, nmaz, depends on the value of the phase increment Ag)
according to the relationship A45 = 27r/nm33. Figure "-5 shows the time-domain
MQ signal of hexamethylbenzene generated by the TPPI method. Figure "-6
shows the frequency-domain MQ spectrum of hexamethylbezene obtained using
the TPPI MO experiment. During the evolution time, the increase of t1 duration
results in a decay of the signal, which causes the frequency-domain peaks of the

MO spectrum to broaden for higher orders.

The phase-incremented MQ experiment proceeds just as described above,
but since the phases of the preparation pulse are incremented by A4), with a fixed
evolution period, the t1 domain MO signal does not decay and shows periodicity
(shown Fig. "-7). Fourier transformation of the time-domain MQ signal with
respect to 42 generates the series of 6-function peaks corresponding to the MO
order n. Figure "-8 shows the frequency-domain MQ signal of hexamethylbezene

obtained with the phase-incremented MQ experiment.

Most MO NMR dynamics in solids are interpreted in terms of the effective
sizes that are measured from the integrated intensity of MO orders. The spectrum
from the phase-incremented method has a higher signal-to-noise ratio than that
from the TPPI method, which reduces the number of transients required. The
integrated intensity obtained by simply measuring the peak height also makes
the data analysis simpler. Since enough digital resolution can be obtained by
replicating the time-domain MQ signal from the phase-incremented method, due
to the periodicity of the time-domain of MO signal, the number of time-domain

points required in the phase-incremented method is much less than that of the

89

a)

0 7f 27..

b)

I

 

1
I
l

I . i I) i w
I l ,' / 1 i . J

101v 111r 12x 13x 14x 15x 161r

 

 

 

 

 

 

 

 

 

 

 

—.-—«_.__
-. —-—- __

 

 

 

 

 

 

 

 

 

 

 

 

0' 1r 21r Sr 41 511‘ Sr 71r 8x91
PM”

Figure "-7. Time-domain MO interferogram of hexamethylbenzene using the
phase-incremented method and the pulse sequence shown in Fig. "-6. The
60 complex points (21r) in (a) were experimentally obtained, and were replicated

up to 480 complex points (160).

90

 

 

 

 

 

 

 

 

 

1le
)1, 1.1

JUL?

 

 

 

 

 

 

N U @1ij JUL/ wLJLDLJL.

8 18 20 22 24 26
n(nurrbsrofquanta)

Figure "-8. Frequency-domain 1H MO spectrum of hexamethylbenzene using

the phase-incremented method and the pulse sequence shown in Fig. "-6. All

parameters are the same as in Fig. 8, except the fixed t1 interval was set to 200 ns.

 

91

TPPI method. However, the 6-function peaks of the phase-incremented method do
not yield the spectral information about the different frequencies occurring within
each order.32 When only the information from the integrated intensity is needed,
the phase-incremented method is preferable due to the lower time required to
ecllect the data.

The pulse programs for the MO pulse sequences used on the VXR 400
spectrometer were kindly provided by Dr. T. Barbara of Varian Associates. We
have modified the MO pulse sequences for our experiments, as shown in Appendix
A.

D. Statistical Model of MO Coherence Intensities

In MO NMR experiments, the intensity of each order in the MO spectrum
depends on the number of correlated spins (effective size). The dimension of
the density matrix is 2N for an N spin-1/2 systems. The number of spins in
a macroscopic sample is about 102°. It is impossible to calculate the density
matrix of such a macroscopic sample in solids. However, in the statistical
mwel31-52-53'55'“, we assume that the intensity of each order in a MO spectrum
is proportional to the number of possible MO transitions, and that the transition
probability of each order is the same. The effective size can be measured by
counting the number of such MQ transitions. '

In a strong Zeeman field, an N spin-1/2 system has 2N stationary states, which
can be classified according to the total magnetic quantum number M,, given by

M, = z; m,, (II-34)
where m,, is the eigenvalue (m,, =‘t1/2) of the i-th spin in the system. The energy
eigenvalues of the Zeeman term are represented by (see Eq. l-3)

E, = —7hH0M,. (ll-35)

92

The energy levels can be sorted by the number of spins in the la) or |+ {-5) state,
no, and in the [13) or | — 1), 123 state, since we can rewrite Eq. "-34 as

M, = (no, — n3)/2. (ll-36)
The degeneracy S2 of states having a particular energy level E, is represented by

n = NI/(naing!) (II-37)
Substituting Eq. "-36 and, using the fact that no, + 71.3 = N, we can rewrite Eq.
"-37 as

12 = N!/[(N/2 + M,)!(N/2 — M,)I]. (II-38)

The energy levels of the Zeeman term and the total magnetic quantum number M,
are shown in Figure "-9. Degenerate in the Zeeman energy, the levels within a
manifold are shifted and split by the internal interactions, making possible a large
number of spectroscopic transitions or coherences. The dashed lines indicate the
MO transitions, in which several spins flip together subject to the general rule,
72 = |AM|. The number of all possible transitions as a function of their order, n,
can be calculated by combinatorial arguments. The number of n-quantum transi-

tions is represented by

2()t) = (1:2) ‘ «139)

where (3) = a!/[(a — b)!b!]. The number of zero quantum transitions between

pairs of states in the same Zeeman manifold is given by
-N/2+l

2° = 2 (11,111) [(m’iM) ‘ 1] = [(211v ) " 2N1 ("'39)

M1=N /2-1
For nonzero orders, Eq. "-39 is well approximated by the Gaussian distribution

using Stilring's approximation
I(n,N) = 4N/\/—N-7r_*exp(—%:) (N 2 6). (II-40)
E. Simplified Models of Multiple-Quantum Dynamics

The time-dependent behavior of multiple quantum coherences can be

-N/ 2

 

-N/2+1 —

 

. - won/2] 2

 

EZ o
+YhH0N/2 1

+Yf1HO[N/2-1] N

+Yf‘lHo[N/2-2] N(N-1)/2
2(N+1I2)
+‘Yf1Ho[1/2] (n N)"2

(NH/2)
(TC N)1/2

 

-, Yleo [N/2-2] N(N-1)/2
- YhHo[N/2-1] N

- YhHo W2 1

Figure "-9. Schematic energy level diagram for an N (odd) spin-1/2 system in a

Zeeman field, and the degeneracy number for $2 of each state. Taken from Ref 53.

94

Increasing Preparation Time (m) —>

-> '74;
{7.9
,/

"‘\'

"Vr

     
  

 
 
 
 

   
  

Figure "-10. Symbolic representation of the spreading of multiple spin correlations
in a coupling network with increasing preparation time. The circle denote spins

and the lines indicate the link of MO coherences through dipolar couplings

95

calculated explicitly using the Liouville-von Neuman equation applied to the den-
sity matrix. Since the multiple-quantum coherences in solids are created through
homonuclear dipolar coupling, we can rewrite Eq. "-14 as
p(t) = eXPl-(i/fimutlm0>6Xp[(i/E)Hnt] (ll-41)
where Hg is defined in Eq.lI-31. For short times, an explicit form of the solution
of Eq. "-41 can be written as the power series
W) = p(0) + (i/h)t[p(0),’Hn] + (i/h)2(t2/2!)[lp(0),7101.710]

+(i/h)3(t3/3!)[111401710], 71011710] + '°°- ("'42)
The contribution of high order terms in Eq. "-42 becomes more and more impor-
tant with increasing time. Figure "-10 shows symbolically the development of
multiple-spin correlations by way of the homonuclear dipolar coupling. The num-

ber of correlated spins increases with increasing preparation time.

The explicit calculation of the MO dynamics using the Liouville-von Neuman
equation is limited to small spin systems (3 9 spins) because it is computationally
demanding. Reduction of the size of the operator space can increase the size of
the spin systems that can be treated. Since classes of coherences are detected
in an M0 experiment, as opposed to individual coherences, a simplified, albeit
inexact, calculation of MO dynamics is possible by use of an average operator.
The following sections will discuss two simplified calculations of the MO dynamics:
a directed walk through Liouville space (hopping model)57 and a incremental shell
model3‘3 making using of the concept of an average operator.

a. Hopping Model

The density operator can be represented by a vector in Liouville space,

w» = 1:50 fiKggK..(t)lKnp) (II-43)

where g Knp(t) is the component of the Liouville states, K is the number of single-

96

spin operators involved in forming the product operator |Knp) of the Liouville state,
72 is the order of the coherence, p labels the different states having the same val-
ues of K and n. The symbol for Liouville states, In), is used to distinguish these
from the symbol Hilbert states, |---). The equation of motion in Liouville space is
represented by a vector equation

1w» = 481711)) (II-44)
where ”H is the superoperator [H, ....]. Eq. "-43 can be expressed in terms of the

components 9K“,

21179113110“) : —i g: 2 Z QKnP;K’n’p’9K'n’P’(t) ("'45)
I n! pl
where
Kn
me) = H411“: 2;, (II-46)
and

Knp 'H K'n'p’
QIfitp;1("n’p’ = (KanKnp) ("'47)

 

The different oscillatory behaviors of the various components of the density op-
erator in Eq. "-44 are hidden by destructive interference and resultant decay,
which decrease the number of degrees of freedom. Since Eq. "-45 is a kind of
first order kinetic equation, Munowitz etc. assumed that the motion of the density
operator can be solved by using the following equation 1

g; = By (II-48)
This equation represents a generalized hopping model, in which the elements of
R give the rate of change from one component of g(t) to another. R is a matrix of
real numbers. For this model, we need to define the space over which the coher-

ences hop and to develop the rates and selection rules that govern the motion.

The size of the Liouville space can be reduced by grouping together modes

97

 

 

 

 

 

 

 

n-quantum coherence

 

 

 

 

Figure "-11. Projection of Liouville space onto a two-dimensional plane. Each

point corresponds to a family of K-spin/n-quantum operators. Taken from Ref. 57

98

 

 

 

 

 

 

 

 

 

n-quantum coherence
.p

 

 

 

 

Figure "-12. Pathways of the growth of MO coherences in Liouville space for a

6-spin system under the 1-spin/2-quantum Hamiltonian. Taken from Ref. 57.

99

of the quantum numbers K and n. In general, the components of the vector 9
are labeled as 9.3,. The projection of the Liouville space is shown in Fig. "-11.
Each point on a two-dimensional grid corresponds to a family of coherences (
|Kn) operator basis). K runs from 1 through N and Inl runs from 0 through K.
With the assumption that the coherences are of equal magnitude, the number of

cperators ggndepends on K, n,and N.

The selection rules and hopping rates from one site to the other site through
Liouvlle space must derive from the Hamiltonian. The MO pulse sequences in
Fig. "-6 used in our experiments create the 1-spin/2—quantum dipolar Hamiltonian
which adds one (or subtracts) spins, and changes the order by 2 quanta at a
time to a multiple-spin mode. Under this Hamiltonian, the selection rules in the
Liouville space are

AK = i1, An = 1:2. (ll-49)
The detailed proof is in Appendix B. The pathways through the Liouville space
may be constructed on the basis of the selection rules in terms of the specified
starting point, that the reduced density operator p(0) at thermal equilibrium is
proportional to I, (K=1, n=0). For example, Fig. "-12 shows the allowed changes

of the Liouville states of a six-spin system under a 1-spin/2-quantum operator.

The time development of p an be solved by using the rate equation (Eq. ll-
48). With the assumption that all coherences have equal magnitudes, the hopping
rates depend on the degeneracies of the coupled states and the strength of the
dipolar interaction. Each element of the rate matrix is given by

RKn;K'n’ = WKn;It"n’SI (”'50)
where WK”; K1,.) is the generic hopping rate between 9m. and gran). and S, is the

lattice parameter, which reflects the strength of the dipolar couplings. The general

 

100

behavior represented by the statistical factors (the degeneracies of the Liouville
state) as well as the individual details represented by the structural factor (dipolar
couplings) are of interest in this model. As a first approximation, we will con-
centrate on the universal trends, leaving 5', as an adjustable parameter to fit the
behavior of arbitrary systems of N spins to specific experimental examples. The
lattice parameter, which depends on the structure of the material, may be given

by a lattice sum of the coupling constants

51 = 7%, 231193). (ll-51)
t<J
as an approximation, we may simply take 5) as proportional to the dipolar linewidth

or the square root of the second moment.

The matrix elements (hopping rates) are constructed from the degeneracies of
the Liouville states |Kn) and |K’n’). Let C... be the number of raising operators, 0.
the number of lowering operators, and co the number of zero-quantum operators
so that

c.—c_ = n (ll-52)
and
c... + c- + co = K (ll-53)
For a given choice of c). and 0-, the number of ways to choose c+ operators out
of K spins is given by the combinatorial coefficient
(if) = firfiw. (ll-54)
Similarly, the number of different ways of choosing c- spins out of the remaining
K - 0+ is represented by
(ff-fa) (II-55)
Therefore, the total number of ways to choose 0+ and c- is the product of the

binomial coefficients of Eqs. (ll-48) and (ll-49). The degeneracy of the Liouville

101

state is obtained by summing over all admissible values of c+, beginning with
0+ = n, and multiplying the total combinatorial factor to account for the number of

ways that K spins can be selected from N. The result is

AN)... = (1)91. (II-56)
where
0,... = 25;, (’5) (£15.)- ("-50

The degeneracy of the Liouville state is used to calculate the hopping rates.
For 2-quantum/1-spin operator (H,,), the forward rate in which K increases is
given by

WKn;K+l,n:l:2 = K8951” ' QK‘I’"+3,,K'I’"*1 ("‘53)
and the reverse rate in which K decreases is given by
WK+1,ui2;Kn = K1981!) - 94"5}t3$“"*—‘. ("'59)

Given the rate matrix (Eq. ll-48) we can compute all the coherence amplitudes
gKn(t) by solving Eq. "-47. Then summing over all K gives the amplitudes of n-
quantum coherences

gn(t) = é: 9Kn(t)- ("'60)

Figure "-13 shows the theoretical development of the n-quantum intensities for
systems of both 6 and 20 spins evolving under H”. The matrix dimensions for 6
and 20 spin systems are 6 and 60 respectively. The effective sizes are obtained
from the theoretical calculation by fitting the orders n to a Gaussian distribution
(Eq. ll-40) for each preparation time. The effective size of the 20 spin system
increases with increasing preparation time, finally reaching an equilibrium at the
longer preparation time (see Fig. Il-14).

The hopping model makes it possible to understand the evolution of the MO

dynamics in systems too large or too complicated to be treated exactly. The

102

 

 

 

 

 

 

 

0.4
)o
0000 0000
0.310
o b)
0.210 A AAAA
A
A
.2. 0.1 A
g a” noun
D
3; D sat sun:
E04
2 ‘oaamooooooo
031°
0 a)
0.2
0.1
AAAAAAA
0.0
0 10 20

Preparation time (ID)
Figure "-13. Evolution of n-quantum coherences predicted by the hopping model

in 6 (a) and 20 (b) spin systems. 20 (open circle), 40 (open triangle), 60 (open

square), and 80 (cross).

 

103

 

 

 

 

20- a a a a a
c
2
0
N
'0 a
3
1; 1o-
?) a
. a
o - r 1
0 10 20

Preparation time (1/D)

Figure "-14. Development of effective size with increasing preparation times for
a 20 spin system predicted by the hopping model. The effective size for each

preparation time is fitted by a Gaussian function.

 

104

replacement of the Liouville—von Neumann equation by a set of rate equations with
an exponential solution is based on the severe damping that accompanies the
superposition of a large number of different frequency components. The equation
in the hopping model can be solved straightfonNardly for groups of spins of various
sizes. However, it cannot directly incorporate the effects of dimensionality on
multiple quantum dynamics, and is limited to approximately 40 spins.

b. Incremental Shell Model

As mentioned above, the density operator needs to be simplified in order
to understand the MO dynamics in larger spin systems. In the incremental
shell model,36 it is assumed that the density operator can be approximated by
a weighted sum of time independent angular momentum operators, Pm, where
both an and a are vectors of dimension M

p(t) = E) g Cma(t)Pma- (ll-61)
Since the operator products Pm are time-independent, all information on the time-
evolution of MO coherences is contained in the coefficients cma(t). The angular
momentum operator products can be given by
Pma = 3le.11.... (ll-62)
where m), is the index of all M spins of the vector m, and a), is the angular
momentum operator ( + = 1, - = -1, and z=0) for spin k.
We rewrite Eq. "-31 for calculations using the incremental shell model as

11,. = 12 g Dijon- (II-63)
where D3 is the dipolar coupling between the pair of nuclei i and j (2,,- is a
condensed notation for raising and lowering operators (Ii+I,-+ + I,_I,-_). From
Eqs. (II-61) and (ll-62), the Liouville-van Neumann equation can be represented

in the form of a summation of operator products

105

§ gdc t pm = 5,; E0; cm (2:, 2D,,[Pmm 011]) (II-64)
where the factor i is the square root of -1 (sings there is an i representing nucleus
i). The MO coherence growth is contained in the time-dependent coefficients
(cm). In large spin systems, since the large number of terms in the summation
makes explicit calculation of Eq. "-58 impracticable, we need to simplify Eq. "-58
and find the selection rules resulting from the commutator [Pmeij]. In order
to simplify the Eq. "-58, we follow Levy and Gleason36 and assume that the
individual operators Pm can be replaced by an average operator PM, and set D,,-
to zero for all non-neighboring nuclear pairs. All combinations of individual product
operators having the same number of individual spin operators in their product are
represented by the one average operator. Figure "-16 shows an average 4 spin
operator schematically formed from a cOmbination of individual 4 spin operators
. of 9 spins in real space. We can rewrite Eq. "-61 with the average operator

pa) = ggcmxtwm 5: §0M(t)PM (II-65)
where CM is the summation of the coefficients cm over all values of m and (1.

Then, Eq. "-64 can be rewritten as

g gf‘PM = 1§CM(§Z§DMPM.Q11])- (ll-66)
The commutators [Pm Qij] can be evaluated according to the selection rules for
M0 evolution. Since the dipolar coupling is proportional to the inverse cube of
the distance, only adjacent i, j pairs are used to calculate the commutator. The
MO coherence growth depends on the position of spins. First, when both spins
i,j of the pair are outside of the existing multi-spin coherence,the coherences do

not grow since the commutator is zero. Secondly, when one spin is inside of the

existing coherence and the other is outside, PM and Q3 do not commute since

..I u A. {Uh'l'lfl

 

106

 

Multiple Operators Average Operator

 

 

 

 

 

 

 

 

 

 

‘ L3; C o s s
5. 06 ma ‘—'
8 9 ‘CM
1. 2. 2.
4. 5. 6. Cmnan
7 3 O

 

 

 

 

Figure "-15. Diagram representing the average M spin operator and its con-
stituent operators for M - 4 in a 9 spin system. The individual components cm
are simplified to an average coefficient for the M spin operator CM. Taken from

Ref. 36.

’1'.“

 

107

they will contain operators for the same spin. In this case, the commutator
increases the number of spins in the product operator by one (forward growth).
The third case involves a spin pair i,j inside the existing coherence. Non-zero
commutators then decrease the number of spins in the product operators by one
(reverse growth). The commutator averaging over all possible combinations of I,,

1+ and I. for spins i, j yields‘”5

[PM,Qij] = 0 i¢ Maj i M (II-67a)
=2PM+1 i¢M,jeMori¢M,jeM (ll-67b)
= (2/3)PM_1 i e M,j e M. (ll-67c)

The selection rule for the growth of the average product operator is AM = il, in
agreement with that for the hopping model. Plugging Eq. "-67 into Eq. "-66, we

can rewrite Eq. "-66 as

2(a): ) 512042 E Z Dii(2PM+l)+ Z Z Dij(§PM-1)]

M M iEMjaM iEMjEM

¥§CM(W{H1PM+1+ W5,_1PM_1) (ll-68)

where W{, H and led are the forward and reverse rate coefficients. The

orthogonal property of the term PM more simplifies the Eq. "-68 to :

ar=aiwt-.cu-t-rwr..cmt (..-69)
The forward rate coefficients, W3}, are defined as
W; = 4 z z: Dij. (II-70)
iedeM

Since the rate of increase of the effective size depends upon the number of spins
outside of the existing spin coherence that are on the periphery of the cluster, we

can rewrite Eq. "-70 as

'.

 

‘E

108

Wj, .—_ 4012mm, (ll-71)
where n,, is the number of spins outside the coherence adjacent to a single spin in
the coherence and n, is the effective number of nuclei at the edge of a coherence.
As shown in Figure "-17, higher dimensional arrangements of spins have more
spins outside a given cluster available to enter the coherence. Therefore, lines

(1-D), squares (2-D), and cubes (3-D) have different forms for n, represented by

n, = 2 (ll-723)
n, = 4(M)1/2 (ll-72b)
n, = 6(M)2/3. (ll-72c)

The parameter n,,, of the order of 1-3, has been used to fit the experimental data,

although we will treat it as fixed (=1) for our 1-D system.

The reverse rate coefficients W}, are obtained by the fact that the norm of all
CM terms is conserved during the evolution time:
E} age) = ‘3} (011,135 + at, 4%). (II-73)
Under the action of the dipolar Hamiltonian 773,, through the Liouville-von Neu-
mann equation, all coefficients CM with odd M are real numbers and the others
are imaginary numbers. Thus, the derivative of the norm of CM can be repre-
sented as
Cild—gf‘+CMi§§‘ = (WL_.ICMIICM-.I + WXI+1|CMHCM+1|)X(-1)M-l- (II-74)
Since the derivative of the norm must be equal zero, the normalization condition
of Eq. "-74 can be given by:
; (W{;_1W54)ICMIICM_1I = o (ll-75)
where W{,_l should be equal to W};,, since the time-dependent CM terms are
variable in the linear equation ( Eq. |l-75). Therefore, the reverse rate coefficient

can be calculated from forward rate coefficient given by Eq "-71.

109

C, is equal to 1 at t=0. The time-dependent CM terms for higher values of
M increase as the preparation time increases. Therefore, the center of mass of
CM moves to higher M values at longer preparation times. Levy and Gleason36

chose to approximate the effective size N by this center of mass:

2 M(C;,CM)
M

N = m . (ll-76)

M

Although the effective size from Eq. "-76 may not correspond exactly to that ob-
tained using the statistical model, it is useful for comparisons with experimental
data. Simulations of MO coherence growth in spin systems of different dimen-
sionalities (1 D, 20 and 30) are shown in Figure "-18. These calculations were
performed with ns obtained from Eq. "-66, nn=1 and D12: 1.0 kHz. The curves
shown are approximately linear, quadratic, and cubic for the one, two and three
dimensional cases respectively, only slightly deviating from these forms at early
times. This program was kindly provided by Dr. K. K. Gleason of the Massachu-
setts Institute of Technology.

3. Experimental
A Multiple-Quantum NMR Studies

The 1H and 19F multiple-quantum experiments were carried out on a Varian
VXR-400$ spectrometer equipped with a 100 watt amplifier (400 and 376 MHz)
and a high power probe with a 5 mm o.d. solenoid coil for 1H and 7 mm o.d.
for 19F. The phase-incremented even-order selective multiple-quantum pulse se-
quence in Fig. "-5 that was used increases the S/N ratio and saves experimental

time compared to the TPPI method, at the expense of information about the

110

 

 

 

 

 

 

 

 

'P
U
‘2

 

3-D
_______ Distribution of M spins
'— in MO coherence
_ _ Permissible distribution
1'0 of M+1 spins (growth)

Permissible distribution
of M1 spins (contraction)

Figure "-16. Schematic diagram of the growth of MO coherence in the systems

of different dimensionalities.

111

 

 

 

 

 

 

160 f - , -
l x'
x l
I
I
120- ,
a I'
a i l ,’
"05 . / I,
.3 sol / x .
on I
/
I
I
/l
..l
2 3 4 5

Preparation time ' 1kHz

Figure "-17. Values of effective size N predicted by the incremental shell model
for spin-1/2 nuclei evenly spaced on a line (1 -D), square (2—D), and cubic (3—D)
lattice. The dotted, dashed, and solid lines represent the 1-D, 2-D, and 3-D MQ

coherence growth curves. Figure is courtesy of Dr. K. K. Gleason.

112

individual line-widths of each order.“0 The 7r/2 pulse lengths are 3.8 us for 1H and
4.6 #8 for 19F, and the basic cycle times are 72 us and 78 #8 respectively. The
t1 value was fixed at 2 [18, and relaxation delays greater than five times T1 were
used. The highest coherence order, nmax, detected is governed by the digital
phase shift increment Acb : A<I>= x/nmax. A 2 ms delay after the mixing time is
used to allow the transverse magnetization to decay. The data are detected with
a 7r/2 pulse, followed by a 100 ps spin locking pulse. A pseudo-10 spectrum
with 64 complex points in the "tl-domain" (phase-incremented) was obtained by
sampling a single complex point in the t2-domain 35 us after the last (spin-lock)
pulse. Unwanted odd order multiple-quantum coherences are eliminated by a 180°
phase shift of the detection pulse at every scan. Because of the periodic nature of
the data, increased digital resolution was obtained by replicating the "FID". The
number of correlated spins N (the effective size) was obtained by least-square
fitting of the orders of coherence (excluding 0 quantum coherence) to a Gaussian
function, according to the widely-employed statistical modelF“""5453-55-56 The MO
intensity distribution should be symmetric, I.1 = I.n theoretically from Eq. "-40.
Since the intensities of the positive MQ orders and the negative MQ orders are
slightly different experimentally, the sum of the positive and negative MQ orders
(In +l.n) was used in the fitting. The sum of all multiple-quantum intensitie [2(lo +

In + l...)] at a given preparation time was normalized to unity.

Since the chemical shift anisotropy of single crystal fluorapatite (hereafter
referred to as FAP) is known (84ppm),‘3‘5 the angles between the c axis of single
crystal FAP ( the a” direction) and the external magnetic field were calculated from
the observed chemical shifts. The PC-based computer program ANTIOPE66 was

used for explicit calculations of MO dynamics of single crystal fluorapatite (FAP)

113

at different orientations. A basic cycle time of 30 us was used. Calculation of spin

dynamics of up to 5 spin-1/2 nuclei is possible with the ANTIOPE program.

B. Sample Preparation and Characterization

Hexamethylbenzene, 06(CH3)6, was obtained from Aldrich and used for the
multiple-quantum experiments. The sample of partially monoclinic hydroxyapatite
(hereafter referred to as HAP-M) was prepared,67 analyzed and provided by Dr.
Bruce Fowler of the National Institute of Standards and the National Institute of
Dental Research. Using the intensity of weak X-ray powder diffraction peaks char-
acteristic of monoclinic hydroxyapatite59 relative to that of a strong peak arising
from the hexagonal form”, along with theoretical calculated intensities for the
_ powdered monoclinic form, his analysis of HAP-M showed that roughly 70 % of
the sample is in the monoclinic form. Its hydroxyl content is therefore assumed
to be highly stoichiometric compared to most hydroxyapatite preparations, espe-
cially precipitated samples. Another sample of hydroxyapatite (hereafter referred
to as HAP-N) was prepared by aqueous precipitation and characterized by many
methods;58'7° a hydroxyl content in HAP-N was determined to be 92% by quanti-
tative 1H MAS-NMR, and 81% by IR.68 The solid solutions of fluorohydroxyapatite
(FOHAP) , Cas(OH)1.xe(PO4)3, with the different fluorine mole fraction (x = 0.24
and 0.41) were synthesized by aqueous precipitation at a boiling temperature.70
Their characterization has been previously reported.”71 A Specimen of single
crystal fluorapatite was kindly loaned by Dr. Bruce Fowler. The color of this sam-
ple is pale yellow, with no obvious inclusions, and the diameter and length are

about 4mm and 7 mm respectively.

114

4. RESULTS

A. 1H Multiple-Quantum NMR Study of Off-Resonance Effects

Hexamethylbenzene was used to check off-resonance effects in the MO NMR
experiments. The half-height linewidth of hexamethylbenzene in the static 1H
spectrum is about 11 kHz. The planar benzene rings of hexamethylbenzene in
the triclinic unit cell form a nearly perfect hexagonal net.72 Since fast reorientation
of each methyl group along its 0;, axis makes the three proton nuclei equivalent,
the CSA of hexamethylbenzene is negligible. The sixfold hopping of the hexam-
ethylbenzene about the Cs axis of the benzene ring lessens the intramolecular
dipolar coupling between ortho-, meta-, and para-methyl groups.73 Thus, breadth
of the peak arises mostly from the intermolecular dipolar coupling.

Figure "-18 shows multiple-quantum spectra of hexamethylbenzene with a
preparation time of 504 [18 obtained with different transmitter carrier frequencies.
The location of the transmitter was varied from the middle of the 1H static lineshape
to 4 kHz off-resonance in intervals of 1kHz. The 2 quantum peak on resonance
(Fig. Il-18a) is more intense than any other peak, but moving the transmitter
from the center of the spectrum diminishes the relative intensity of the 2 quantum
peak, eventually resulting in an inverted peak (Fig. ll-18b to e). The intensities
of the higher-order peaks, shown in Fig. "-18 with the same vertical scale, also

decrease with increasing resonance offset.

Static 1H spectra, normalized to the same peak height, of HAP-M, HAP-N,
FOHAP x=0.24 and FOHAP x=0.41 are shown in Figure "-19. The spectrum of
the low specific surface area (2.1 m2/g) HAP-M sample in Figure lI-19a does not
show evidence of a peak from surface adsorbed water (around 5.6 ppm)“‘9 but

instead only a 2 kHz broad peak, whose width is due to the CSA of the hydroxyl

 

 

C)

b)

 

...JLJLJ

a)

 

 

 

 

 

JUL.)

2 4 6 8 10 12 14 16

n (chquanta)

Figure "-18. 1H PI-MQ—NMR spectra of hexamethylbenzene with different trans-

mitter carrier frequencies. See text.

 

116

 

 

 

 

., J /\\

J/ “L

 

IYVYVITYY'IYTTTITTTTTTIVWYYITIITTrIIITI[TITTrIT1llTllllllTTifilV[III'IT—Y17]

7O 60 50 40 30 20 10 O -20 -40 -60 00111

Figure "-19. 1H 1-dimensional spectra of apatite samples. HAP-M (a), HAP-N
(b), FOHAP x=0.24 (c), and FOHAP x=0.41 (d).

117

group, and homo— and heteronuclear dipolar couplings. The strong signal in HAP-
N around 5.6 ppm (Fig. ll-19b) has been assigned to mobile water at the surface.
The peak of hydroxyl groups in HAP-N is concealed by that of surface adsorbed
water (the specific surface area of HAP-N is 37m"’/g).68 The weak intensity of the
1H MAS-NMR spinning sidebands of the surface absorbed water groups indicates

that homonuclear dipolar couplings among these protons are negligible.69

Figure "-20 shows the 1H multiple-quantum spectrum of the highly stoichio-
metric HAP-M sample for a preparation time of 864 us, as well as the Gaussian fit
corresponding to an effective size of 12.3. The higher orders of multiple-quantum
coherences deviate from a Gaussian distribution,31 being more intense than pre-

dicted.

Figure "-21 displays the intensities of the multiple-quantum peaks for HAP-
M with a preparation time of 864 ps as a function of resonance offset. The
isotropic chemical shift of hydroxyapatite is 0.2 ppm, but the position of highest
intensity of HAP-M (2.8 ppm) is regarded as “on-resonance”, and the transmitter
offset is varied by up to :2 ppm (800 Hz at 9.4T). The change of the transmitter
offset position has only a slight influence on the intensities of the 2 quantum
and 4 quantum peaks, shown in Fig. ll-21a with the same vertical scale. Since
the effective size is measured from the ratios of intensities of multiple-quantum
peaks, we can redraw Figure Il-21a with intensities normalized to the 2-quantum
peak. The resulting intensity profiles, shown in Figure II-21b are not changed
much by varying the transmitter offset by 12 ppm for a preparation time of 864
[18. Figures ll-22a and b have the same parameters as Fig. 21 a and b except for
the preparation time (1872 [18). The intensities (with the same vertical scale) of

multiple-quantum coherences obtained on resonance are always stronger than

118

 

 

 

 

 

 

 

\
\
\

. ,i
.72; .\
s l \
é " . \\
Ii \
c: \
‘ " \

 

n (number of quanta)

Figure "-20. 1H MD NMR spectrum of HAP-M using a preparation time of 864 us.
The dashed line represents a Gaussian fit to the data with effective size N = 12.3.

 

119

 

 

 

 

 

 

1.2
lb)

01.0‘ '

N I

908‘

E

goe'

20.4: .

'7:

502‘

L5 1 I

0.0 2,-ssr-!2;__i

93300

S a);

a. l

a

3

52001

3

g 4

£1001

a I

'5

3 n

o e.f.rtr!vl-———I
o 2 4 6 81012

MQorder

figure “-21. Intensities of the ‘H PI NMR MO orders of HAP-M for various
resonance offsets using a preparation time 01864 #8. a) Intensities on the same
vertical scale. b) Intensities normalized to the 2-0 intensity. On resonance (open
square), 1 ppm off resonance ( open triangle), 2 ppm off resonance (Open circle),

-1 ppm off resonance ( black triangle), and -2 ppm off resonance (black circle).

 

120

 

 

 

 

 

 

 

 

1.2
. b)

O 1.0 ‘ ' A
N i O
O
"' 0.8 1 a
g I
f“ 0.61
E . i
2 041
> .
a ‘ a
5, 0.2 ‘ a
5 .

0.0 - +
SE 300
S a)
2‘
g 3
:5 200‘ o
3 9
> O
'5 ‘ . .
5
,§ 1001 ‘ ‘ a

O

5 , . e
3 ‘ I

o - . - . e . - 2 - . -

O 2 4 6 8 1O 12
MQorder

figure "-22. Intensities of the ‘H Pl NMR MQ order of HAP-M for various
resonance offsets using a preparation time of 1872 us. .a) Intensities with the
same vertical scale. b) Intensities normalized to 2-Q intensity. On resonance
(open square), 1 ppm off resonance (open triangle), 2 ppm off resonance (open
circle), -1 ppm off resonance ( black triangle), and -2 ppm off resonance (black

circle).

121

those obtained off resonance to the upfield side, but are stronger for the 2 and
4 quantum coherences, weaker for the 8 and 10 quantum coherences than those
obtained off resonance to the downfield side. It is difficult to obtain an effective size
fitted by using a Gaussian function from the intensity profile for an off-resonance
transmitter offset at longer preparation times. Thus, the effective sizes obtained
using Gaussian function with different resonance offsets are significantly different
from those obtained on-resonance, due to the distortions of the MO peaks. Such

resonance offset effects complicate the interpretation of MO dynamics.

B. 1H Multiple-Quantum NMR of Hydroxyapatite and Fluorohydroxyapatite
Samples

The normalized intensities of the various orders of MO coherence as a function
of preparation time 1' for the various apatite samples are plotted in Figures ll-23a,
b, c, and d. Figure Il-23a shows that the normalized intensity of the zero-quantum
coherence of HAP-M decreases, and that of the 2-quantum coherence increases
first and then decreases with increasing preparation time; the higher quantum
orders steadily grow within this experimental range of preparation times. The
normalized intensity of the zero-quantum coherence of HAP-N in Fig. ll-23b
shows a different behavior compared to that of HAP-M; it decreases slightly at
earlier preparation times and levels off at longer preparation times. Although the
normalized intensities of the high quantum orders of the HAP-N are less than those
of HAP-M, they present the same behavior with respect to increasing preparation
time as HAP-M. The normalized intensities of the zero-quantum coherence of
FOHAP x=0.24 and x=0.41 shown in Figure ll-23c and d show a slight variation,
and are higher than those of HAP-M and HAP-N.

 

122

 

 

 

 

 

 

 

 

1.0
l o o
0.8 o 0 ° ° 0 o o o
b)
0.6 1
0.4 1
g .
2 0.2
g ‘ ‘ ‘ A ‘ ‘ t l ‘ ‘ ‘
" 0.0 .__._-_,_a_n._fl_i_L_l_L
E
f, 1.0
S a)
2 0.8 1 .
, O
O '0
0.6 ‘ c
. ° ° 0 o o
0.4 ‘
1 A ‘ ‘ ‘ . ‘ ‘ 5 a
0.2 < ‘ a a
D O D O a . I
0.0 ,fl Ll—I—l—l—L.'—._
O 500 1000 1500
Preparationtimems)

figure "-23. The normalized intensities of even-order MQ coherences of apatite
samples as a function of preparation time. a) HAP-M; b) HAP-N; c) FOHAP
x=0.24; d) FOHAP x=0.41. 00 (open circle), 20 (open triangle), 40 (open square)
60 (open circle), and 80 (black square).

Nomalized intensity

1.0

0.8

0.6

0.4 ‘

0.2

0.0
1.0

0.8

0.6

0.4

0.2

0.0

123

 

‘1!
I
I
LI

 

 

 

 

500 1000 1500
Preparation time (us)

124

The effective size obtained from fitting MQ intensities to a Gaussian is useful
for studying MQ dynamics. Figure "-26 shows the experimental effective sizes N
vs. preparation time for HAP-M and theoretically calculated curves for 1-D and
2-D growth using the incremental shell model. The experimental plot is linear in
its early preparation time development, only deviating upward at longer prepara-
tion time (where one expects the weaker inter-chain dipolar couplings to become
more influential). The solid line represents the calculated curve for a one dimen-
sional chains of 1H spins in hydroxyapatite. The value of nn in the incremental
shell model was chosen to be 1, since there is only a single uncorrelated spin
adjacent to each spin at the coherence boundary. The value of no, (related to a
system's dimensionality) was set to 2, representing the total number of adjacent
spins outside of the coherence (the two “end Spins” in a linear chain). The dipolar
coupling D12 between a pair of nuclei is rewritten from Eq l-50 in the Part I as
012 = (72h) (1 — 3cos2 012)/2ri32 (in HZ). (ll-77)
For two neighboring infra-chain protons 344 pm apart (the distance in
hydroxyapatite”) and parallel to the external field, D12 = 2.957 kHz, which is
1/3 of the Pake doublet splitting that would be obtained in a 1H spectrum. We
used in the simulation an average 012, 1.32 kHz, whose value is the root mean
square value of D12 over all powder orientations. Since the value of P2(c03912),
(1 — 3c082 912), is averaged out over all orientations but that of P2(c030)2 is not,
an average D12 is obtained from the square root of an average sz. This follows
the approach used by Levy and Gleason.36 For a 1-D system, a more reasonable
approach is to sum the powder-weighted responses foe each orientation of the
1-D chain. The experimental slope is three times larger than the predicted slope.

The dashed line shown in Fig. “-24 is the calculated curve for a two dimensional

 

 

125

 

 

 

 

50
I
40 1
‘ I
2
.. I
g 30
m 4 I
.3
‘87 20 . '
n‘i
‘ I
I
10 ' I
I ..«v
4 I ...v"
. ~G""‘
o ‘ "'1'"-' fl .
0 500 1000 1500 2000
Preparation time (us)

Figure "-24. Effective size N versus MQ preparation time for HAP-M. The solid line
is a theoretical calculation using the incremental shell model for a one-dimensional
spin system with the appropriate dipolar coupling strength and bo adjustable
parameters (see text). The theoretical calculation of a two-dimensional system

is represented as the dashed line.

 

126

 

 

 

 

 

Fe
4 l'
3 cl
L-
g 1
S
2 d
1 f t W t '
1 2 3 4
In (mg)

Figure "-25. In(N) vs. In(r/Tc) for all of the samples of hydroxyapatite and
fluorohydroxyapatite. Solid lines indicate the slopes of data points that are
obtained using least squares fitting. HAP-M (open circle), HAP-N (open triangle),
FOHAP x=0.24 (open square), and FOHAP x=0.41 (black circle).

 

127

system assumed to consist of the Six nearest protons in adjoining columns in the
hydroxyapatite structure. The calculation was performed by using an inter-chain
nearest-neighbor powder-average dipolar coupling D12 of 64 Hz. The value of nn

was set to 6, the number of planar neighbors, and n, was chosen to be 4M"?

The effective size N grows as some power of the preparation time 7' (N 0( 7").
It is convenient to plot In(N) versus In(r/rc) to obtain the growth exponent a as
shown in Figure "-25 , where To is the basic cycle times3 The growth exponent for
effective size vs. preparation time is equal to the slope of this plot. The MO data
for HAP-M and HAP-N show bi-exponential characteristics. The slopes of HAP-M
and HAP-N for short preparation times (shorter than 864 us) are 0.98 (correlation
coefficient = 0.988) and 0.83 (correlation coefficient = 0.998), and those for long
preparation times (longer than 1008 ps)are 1.78 and 1.82 respectively. FOHAP

=0.24 and FOHAP x=0.41 have single exponential characters. The slopes for
FOHAP x=0.24 and FOHAP x=0.41 are 0.54 (correlation coefficient = 0.963) and
0.50 (correlation coefficient = 0.936) respectively.

The decays of the absolute intensity of the sum of all multiple-quantum coher-
ences of the apatite samples are shown in Figure lI-26a as a function of prepa-
ration times. The decay function corresponding to irreversible relaxation can be
assumed to be exponential:

|(1') oc 6Xp(-1'/Td). (ll-78)
The different relaxation times Td of the different MQ coherences can be distin-
guished by taking the logarithm of Eq. "-78. Figure ll-26b shdws a plot of ln(l(r))
vs. the preparation time 1'. Since the Td relaxation time is the inverse of the
slope in Figure II-26b, we have offset the intercept in order to see the slope more

clearly. The Td relaxation times of HAP-M, HAP-N, FOHAP x=0.24, and FOHAP

 

 

128

 

 

 

In(intensity)
A
//‘/H
fl"?

 

 

 

 

 

 

2 - r e .
2500
l a
2000 . ‘
4 I
g 1500 1 2
3 ‘ ' o
1000 1 ‘ o
I
1 ‘) I: g o o
500 . ‘ . : c o
l ‘ a 2
o f r f
0 500 1000 1500
Preparationtlmeots)

figure "-26. Decay of absolute MQ coherence intensity of apatites. HAP-M (open
circle), HAP-N (open triangle), FOHAP x=0.24 (open square), and FOHAP x=0.41
(black circle). a) Intensity vs. Preparation time. b) In(intensity) vs. preparation

time, arbitrarily offset in the vertical direction.

 

129

 

 

 

 

 

 

 

 

 

1.0
a)
A A
1 A 00 .-
2 . T
0.8 “ I : . . g g!
A O 1
a . ‘ ' ‘
.a . 1
8
:C: 0.7 v ! ' I ' f r - ..
E 0.3 l:-
as o
< o
E b) A A 0
0.2 ‘ o a. o
8 O O o O
A 20
. A
0.1 A A A
A A
0.0 ~ . a r f v . . .
0 200 400 600 800 1000
Preparation time (us)

figure "-27. The normalized intensities of 00 and 20 coherences of a single
crystal mineral sample of FAP vs. preparation time at twoorientations with respect
to the external magnetic field. a) 00 at 90° (black circle) and 00 at 62° (black

triangle). b) 20 at 90° (cpen circle) and 20 at 62° (open triangle).

130

 

 

 

 

 

1'
!
i
9.0 ,
g ...r
.5
¢ .
§ 80
§ 0
G
g .
7.0 - . - . - . . , -
0 200 400 600 800 1000

Preparation time (us)

Figure "-28. In(absolute intensity of MO orders) vs. preparation time of single-

crystal fluorapatite at two different orientations. 90° orientation (open circle) and

62° orientation (open triangle).

 

131

x=0.41 are 742, 601, 615, and 610 ps respectively.
C. 19F Multiple-Quantum NMR of a Single Crystal of Fluorapatite

Figures II-27a and b show the intensity of the zero-quantum and two-quantum
coherences of a single crystal FAP at two orientations ( c axis making the angles
62° and 90° with respect to the external magnetic field). Unlike a powder sample
(see fig. "23), an oscillatory behavior is observed for the zero and two quantum
intensities. The frequency of the oscillations of the MO coherences for single-
crystal FAP is proportional to the dipolar coupling. The dipolar coupling infra-chain

of single-crystal FAP at 90° is approximately 2.95 times larger than that at 62°.

The logarithm of the absolute total multiple-quantum intensity for the single-
crystal FAP sample at two orientations vs. preparation time 7' is shown in Figure
"-28. The Td relaxation times of this single-crystal mineral FAP are 694 ps and

645 #8 at 62° and 90° respectively.

5. DISCUSSION

A. Effect of Resonance Offsets in the PI-MO NMR Pulse Sequence on the
Formation of Multiple-Quantum Coherence
Figures "-18, 21, and 22 show the effect of a resonance offset on multiple-
quantum coherences. We can give a possible explanation for this effect. In the
rotating frame, when only the static magnetic field is present, the effective field is
represented by the vector equation
in = 1% — an (II-79)
where Ho is the vector of the main static magnetic field and the magnitude of 63

is the angular frequency of rotation of the unit vector at rf carrier frequency. We

 

‘F

 

132

 

 

 

 

 

figure "-29. The effective magnetic field if,” in the rotating frame in the
presence of an applied rf field 171. The effective magnetic field is the vector

sum of the applied static field 170, the fictitious field ..3/7. and the rf field r‘il.

133

can express the effective field in the rotating frame in the presence of an rf
pulse as the vector sum of fig — iii/7 and H1 (see Figure "29)

1;,” = (H‘o an) +1111 (II-80)
where E is the radiofrequency field. The magnitude of the effective field is rep-
resented by

Heff = [(Ho —w/1)2 + H1211”
l/2

= (1/1) [(7110 - w)2 + (71102] (II-81)

= (1/7) [wa - w)? +wi] ”2
where wo — w is a resonance offset frequency and w = —71H1 is the radiofre-
quency field strength. The Hamiltonian of the radiofrequency along the x direction
in a frame rotating at the carrier frequency can be represented by

11,, = —71H1I,. (ll-82)
A 90° pulse along the x direction flips the initial magnetization (M, = 7Iz) into
the y direction (M, = 7Iy) on-resonance. However, the magnetization of off-
resonance peaks after a 90° pulse is not in the y direction. As mentioned in the
background section, the average dipolar Hamiltonian generated by the 8-pulse
cycle with the proper delays (Pl-MO pulse sequence shown in Fig. "-4) is a driv-
ing force for creating the multiple-quantum coherences. The effectiVeness of the
average Hamiltonian is determined by the preciseness of the 90° pulse length. An

increase of the resonance offset results in an increase of the difference between

1?,” and if; (an imperfect 90° pulse when an off-resonance).

The system evolves under the influence of the internal Hamiltonian alone after
the 90° pulse. The time development of the system in the absence of relaxation
is represented by

100) = exp(-iH.ntt)Iyexp(iH.-mt)- (II-83)

 

134

The average double-quantum Hamiltonian Hyx Is used instead of the internal
Hamilonian Hm in MO NMR experiments. There is no resonance-offset cor-
rection to zero order for the average double-quantum Hamiltonian used in our
experiments,40 but higher order terms quickly contribute to the resonance-offset
effects. Even if the pulses are “hard” 6-pulses that perfectly rotate I, to I, for all
offsets, the existence of the off-resonance term AwI, changes the time develop-
ment of the density operator in Eq. "-83, due to the resonace-offset correction in
some order of the average double-quantum Hamiltonian as time increases.

The extent to which resonance-offset effects influence the MO spectrum is
sample dependent. The 1H CSA of hexamethylbenzene is nearly zero due to the
fast reorientation of the methyl groups, and that of hydroxyapatite is estimated to
be roughly 14 ppm.74 The placement of the transmitter carrier frequency away from
the isotropic chemical shift for hexamethylbenzene shows a severe destmctive
interference of the multiple-quantum coherences for a preparation time 01504 ps.
The 2-quantum order seems to be more influenced by resonance offset effects
than are the higher orders. As resonance offsets increase, the intensities of the 2-
Q coherence become weaker and more out-of-phase. To illustrate the magnitude
of off-resonance error in the 1H MO spectrum of powdered HAP-M, the MO spectra
of HAP-M was recorded with the transmitter positioned at 2ppm (800Hz) upfield,
and in a separate experiment 2 ppm downfield, of the highest position in the
pattern. For a preparation time of 862 [18, the MO spectra show no distortion, but
for a preparation time of 1872 ps the spectra were distorted due to the cumulative

effect of off-resonance terms.
B. Dimensionality Effects in the Multiple-Quantum NMR of Hydroxyapatite

Different samples show different growth curves of MO coherences with in-

 

135

creasing preparation time. However, it has been demonstrated by Gleason et. al.
that when the preparation time is scaled by the square root of the second moments
of the individual samples, samples having the same dimensionality show universal
MO growth curves.36 Therefore, the growth of MO coherences is related to the
dimensionality.36 The study of dimensionality using MO NMR has been performed
with two- and three-dimensional spin systems. Hydroxyapatite is a good model
for studying one-dimensional spin systems since it has linear chains of uniformly-
spaced proton spins, fairly widely separated from each other. The growth of MO
coherences for HAP-M at preparation times up to 864 ps in Fig. "-24 is linear
as is the calculated one-dimensional growth in the incremental shell model. In
qualitative agreement with the calculation for two dimensional growth, an upward
curvature in the experimental curve at longer preparation times may reflect growth
to other columns of spins. Lacelle53 has discussed the possible significance of
the different growth exponents observed for various spin systems. The growth
exponents of 1- and 2-dimensional spin systems obtained from the incremental
shell model prediction after the initial induction time are 1.00 and 2.05 respec-
tively. The bi-exponential character of the curve for HAP-M shown in Figure "-27
suggests a growth of MO coherences with a change in the dimensionality. The
growth exponents of HAP-M (0.98 for preparation times shorter than 864 ps and
1.78 for times longer than 1004 ps), which are close to those predicted by the
incremental shell model for 1- and 2-dimensional growth, show a 1-dimensional
character at short preparation times arising from intra-chain dipolar couplings, and
a higher slope at longer preparation times arising from the more 2-dimensional
spin system corresponding to inter-chain dipolar couplings. Of course, the repre-

sentation of the inter-chain growth as 2—dimensional is inexact, since it has some

 

136

three-dimensional character. Further more, when the intensities of the orders of
HAP-M are fit to an exponential, the fit is somewhat improved over a Gausian
fit, and the parameter characterizing the exponential fit is observed to increase
linearly at all preparation time. Thus, the effects of different dimensionalities will
require further study, possibly by using powdered fluorapatite samples.

C. One-Dimensional Cluster Model

Clustered samples give rise to a strong intensity of the zero quantum peak
compared to non-clustered samples.75 From figure "-23, the strong normalized
zero quantum intensities of HAP-N, FOHAP x=0.24 and FOHAP x=0.41 compared
to that of HAP-M qualitatively demonstrate the existence of vacancies and/or
fluorine ion substitutions that make clusters in a one-dimensional chain. The
growth exponent of an infinite 1-dimensional chain theoretically predicted by
. the incremental shell model is 1.00. The growth exponent of HAP-N at early
preparation times (0.83) clearly shows evidence for a hydroxyl deficiency. The
lesser slopes of FOHAP with increasing mole fraction of fluoride ion in fig. "-25
indicate that the FOHAP samples having higher mole fractions of fluorine contain
higher defect densities. Thus, MO NMR shows evidence of the interruption of
hydroxyl groups in apatite samples.

figure "-30 shows hydroxyl groups in an apatite chain segmented by vacan-
cies and/or fluorine ions. The run number of a group of hydroxyls is defined as
the number of contiguous hydroxyl groups between two “defects” on either side,
which can be either vacancies or fluoride ions. Since MO coherences are created
by the homonuclear dipolar interaction, the existence of vacancies and/or fluoride
ions between hydroxyl groups in an apatite chain hinders the propagation of MO

coherences. In order to compare experimental MO spectra with calculated MQ

137

spectra for given defect densities by the statistical model, we assume the follow-
ing:
1. defects are randomly distributed (ideal solid solution);
2. MO coherence growth occurs only along the linear chains;
3. MO coherence do not grow across vacancies or fluorine atom substitutions;
4. MO results for stoichiometric hydroxyapatite (HAP-M) yield effective size for
each preparation time 7'.
Since the sum of the intensity of all MO orders in the absence of decay of MO
coherences is the same regardless of preparation times, it can be normalized to
one. The intensities of MO orders, In(r), can be represented using the normalized
factor74 from Eq. "-39 as

In(r) = CZn/4N (ll-84)
where C is equal to 1 for a non-selective experiment and 2 for an even-selective
experiment. Since the intensity of each order of MO coherence depends on the run
number and the mole fraction of spins present in a given run number, the intensity
of each order of coherence for a defect-containing sample can be calculated by
summing over the contribution from each run number combined with a Gaussian

distribution, as given by the following equation

6 _
mm) = 2 ”(13:23) (4M) 1+ for N, s 6
r=2
27 x,(N,1r)_l/2e$p(—n2/N,) for N, > 7. (II-85)

where x, is the mole fraction of spins in a run number, and N, is set equal to the
run number for run lengths up to the effective size. The value of N, is set equal
to the run number when the run number is less than effective size; otherwise, N,

is set equal to the effective size.

 

 

 

 

 

 

 

138
C
. . 2 OH O H
. | I
x x :F or Defect H _ O
I _ F 436pm F
. Run number=1 :l 2129'“
_ H _ H
x | |
. O O
. I II
. Run number=5 O
' III
e — 3 203m l?)
x F
x F
. III

figure "-30. Arrangement of hydroxyl groups separated by vacancies and/or
fluoride ion substitutions in apatite samples. The distance between hydroxyl
groups and fluoride ions has been obtained by NMR studies.75'77 The substitution
of fluoride ions gives three configurations (l-Ill) along the crystallographic c axis

in FOHAP. The dotted lines denote hydrogen bonds.

139

D. Estimation of Defect Densities Using the 1-D Cluster Model.

Even if the MO intensities of experimental data obey a Gaussian distribution
perfectly, the theoretical MO intensities of defect-containing samples obtained
from Eq. "-85 may not be fitted very well by a Gaussian distribution due to the
summing of contributions by different run numbers. Therefore, the comparison
between theoretical and experimental 4Q/20 intensities for a given preparation
time is more relevant than a comparison of effective sizes. Experimentally, since
MQ intensities do not correspond to a Gaussian distribution exactly due to strong
intensities of the higher order peaks, we recalculated the effective size of HAP-M
using the 40/20 intensity at each preparation time. Since the contribution of MO
intensities from a 2-D spin system complicates the calculation of defect density
at longer preparation times, we use MO data from the linear portion of the curve
corresponding to 1-dimensional growth in Fig. "-24. The mole fraction of the
run numbers in a 1dimensional chain can be calculated by percolation theory.78
Figure "—31 shows run number distributions for various defect densities. The
center of mass of a given run number distribution moves to a larger run number

value as the defect ratio decreases.

The result of calculations based on this 1-D cluster model and experimental
data is shown in figure "-32. The calculation program is given in Appendix 1C. The
HAP-N from MD data are closer to the curve calculated for an 8% defect density
(that obtained from 1H MAS-NMR) than 0 that calculated for a 19% defect density
(that obtained from IR). The experimental curves for the FOHAP samples are
lower than the calculated curves. There are several possible explanations for the
deviation between the calculated and experimental curves for FOHAP samples.

First, hydroxyl vacancies exist in the one dimensional chains of precipitated

 

a?

 

140

apatites (compare the difference between HAP-M and HAP-N). These vacancies
in the hydroxyl chains can create additional defects in addition to those created by
fluoride substitutions, and could explain the discrepancy between the calculated
and experimental curves of FOHAP. The experimental curve of the FOHAP x=0.24
sample is close to the theoretical curve for a 41% defect density corresponding
to a 22% hydroxyl deficiency. Secondly, if fluoride ions substitute randomly for
hydroxyl groups, the molar ratio of the configuration III to configurations l or II
shown in Figure "-30 for FOHAP x=0.24 is 0.32. Yesinowski et. al. have
shown that the peak arising from the configuration III is not detected with 1H
MAS-NMR.69 If configurations l and II are more preferable than configuration
III, the decrease of the number of fluorine ions in a row increases the mole
fraction of small run number. Therefore, non-random substitutions of fluoride
ions in FOHAP can result in the lower experimental intensity compared to the
calculated one. Thirdly, the eight-pulse MQ sequence for solids shown in Fig.
"-6 ideally eliminates heteronuclear dipolar couplings and J couplings of 1H and
19F.75 In these experiments, we have given applied the MO pulse sequence at
the proton frequency. The effectiveness of the decoupling is determined by the
basic cycle time, the time scale of mutual spin flips of the irradiated spin system
T2” and the strength of the heteronuclear dipolar coupling between 1H and 19F. A
heteronuclear dipolar coupling between two different spin 1/2 nuclei is represented
by
D15 = (7n,h)(1 — 3cos2 912)/3r?2- (ll-86)

The 1H-‘9F dipole coupling in configuration I parallel to the external magnetic field
is equal to 7.27 kHz. The isotropic chemical shift of hydroxyl groups not adjacent

to a fluoride ion is 0.2 ppm and those of configurations l and II are 1.2 ppm and

 

141

 

 

 

 

 

Run number

figure "-31. Run number distributions for various defect densities in .a one-
dimensicnal chain. The mole fraction of spins in a given run number is calculated
by percolation theory"; 8% (open circle), 24% (open square), and 41%( cpen

triangle).

142

 

 

 

 

 

 

0.4

0.3 ‘ °
2
.6 I
5 0.2 « I
.5 t
3

0.1 «

0.0 . . . . . ,

200 400 600 800

Preparation time (us)

figure "-32. 1-D cluster model of MO growth for various defect densities in
apatite, compared to experimental data. Lines for the various defect densities are
calculated using Eq. "-84. From top to bottom, lines correspond to 8%, 19%,
24%, and 41% defect densities. HAP-M (open circle), HAP-N (open square),

FOHAP x=0.24 (open triangle), and FOHAP x=0.41 (crosses).

143

 

 

 

 

 

 

0.4
0.3 r f
2:
'75 1
C
2
.5 a
CS, 0.2
v 1 ------ "'"
0.1 - ....... ‘5 ....... A
a" ........ .— -------
‘ A .... 1'- ---- '
firzzzzrsisz“‘ I I
0.0 . r - r ' .
200 400 600 800
Preparation time (us)

figure "-33. Comparison between experimental 40/20 intensities and calculated
40120 intensities for FOHAP samples after excluding the two hydroxyl groups
adjacent to fluoride ion. From top to bottom, lines correspond to 24% and 41%.

FOHAP x=0.24 (open triangle) and FOHAP x=0.41 (black square).

144

1.5 ppm respectively.69 However, the proton adjacent to a fluorine atom will be
split into a Pake doublet. This splitting produces a resonance for these protons
of typically several kHz, sufficient to prevent them from effectively participating
in normal MO coherence growth. Thus, we assume that the protons adjacent
to a fluorine atoms do not participate in MO coherences. Figure "-33 shows
a recalculation of the 4Q/20 intensities for 24 % and 41 % defect densities
using Eq. "-85, when terminal OHu-F groups in a run are excluded. The
experimental 40/2Q intensities of FOHAP x=0.24 and FOHAP x=0.41 in Fig.
"-33 are closer to the calculated 4Q/20 intensities than those in Fig. "-32,
suggesting that this resonance offset effect may be at least partially responsible
for the initial discrepancy. Finally, a different coherence in the decay time of
different run numbers of spins and/or the different MO order might result in a
. difference between calculated and experimental 4Q/20 intensities. However, the
logarithmic plots in Figure Il-26b do not show two different slopes for a the decay
time within experimental limits, which means that different MO coherences do not

have different decay times.
D. 19F Multiple-Quantum NMR Dynamics of Single Crystal Fluorapatite.

The oscillatory behavior of MO dynamics in a liquid crystal has been previously
shown experimentally and theoretically for finite spin systems.5‘f56 Munowitz has
theoretically predicted such oscillations for small oriented linear arrays of uniformly
spaced spins under the Hamiltonian 71,, where

Hz: = 1/3 ED141211“ + Iszzlc)- ("'87)
The theoretically predicted oscillations are periodic for two spins but are damped
for larger arrays (very small oscillations for arrays of 6 spins).5o The oscillatory

frequency of the intensities of 1 Q and 20 coherences for a 2 spin system under

 

 

145

 

 

 

 

 

 

 

 

 

 

 

1.0 ‘17 1’6
. 0e b) °° 0°
c o co
. e
0.8 J °° o c
‘I‘ O O
0.6 ‘ 0° .‘ ‘9‘ c o ‘A
l o 2 2
4 -l ‘ 0 0° ‘ A 0°
0. ‘ .. Cocoa ‘ ‘ f“
g: ‘ A A ‘
g 0.2 1 ‘ A A
E 1 ‘ ‘A A.
- A
..A - f 2 , - T , ‘Ap‘ - .
E 0.0 _
a 1.0 "In; ‘In‘ :6— _
g do a) ‘. ‘A °° L"
2 , o 9 A °
0.8 o I ‘ O
f O I ‘ O
4 0 ‘ O
0.6 ‘ 0. 3°
‘0 9‘
0.4 1 A . ° .
l I O O ‘
A o
0.2 ‘ A a, o ‘.
‘ A. 0 Do ‘A
0.0 «ll-.9- ' v V ovowo v r - f ‘L—l
O 200 400 600 800 1000 1200
Preparation time (us)

figure "-34. Simulated MQ dynamics of 2, 3, 4, and 5 spin systems under 11,,
for an oriented linear chain using ANTIOPE66 (see text). a) 2 spin system; b) 3
Spin system; c) 4 spin system; d) 5 spin system. 00 (open circle) and 20 (open

triangle).

 

Normalized intensity

146

 

 

 

 

 

 

 

 

1.0 o
< 0o d)
0.8 ‘
O . ‘ O
0.6 ‘ o “ ‘5‘ ¢O°°°O°° 0°
1 1 6° °°22°
e 0 ‘ 5“ ‘A
0.43 ‘ 0 ° ‘AAA“ A
‘ 4 °oe¢ A
02. ‘ ‘f
l “
0.0 a e . , .. e -
1 0 f
1 on C) ‘A‘O“
A
0.8 ‘ o ‘A
9 A‘I A
J ‘ ‘A
0.6 ec‘A . 00.00 ‘I
‘ Ag 0 5‘“. e
0.4 1 9 ° 0 °
. 00000 O
. . e
0.2 ‘ oo o
f “ 000.0
0.0 ‘FL v T r I V v v I v 1 f
O 200 400 600 800 1000 1200

Preparation time (us)

 

147

71,, is (1/3)D12.

Figure "-34 shows the simulated MO dynamics for oriented chains of 2, 3,
4, and 5 spin systems under the Hamiltonian 71,, as a function of preparation
times using ANTIOPE.66 The distance between nuclei is 344 pm; the same as
that in fluorapatite, and a 62° orientation with respect to the external magnetic
field was chosen for simulations. The MO dynamics of the 2 spin system ( Fig.
lI-34a) are both oscillatory and periodic. The dipolar interaction between two
spins (d12 = 344pm) is 442.6 Hz, and the periodicity of the MO dynamics for two
spin system is 1129.6 ps [1/(2'442.6 Hz)]. An increase in the size of the spin
system results in a apparent disappearance of periodicity in the MO dynamics.
This is likely due to the destructive interference of the many frequencies present
when there are a larger number of unequal dipolar couplings present in a spin
system: the actual periodicity may occur at time intervals too long to be revealed
by the simulation. The MO dynamics of a 5 spin system ( Fig. ll-34d) shows the

oscillatory behavior, but no apparent periodicity.

The theoretical calculation of MO dynamics under even different Hamiltonians
(71,, and H,,) show the periodicity and oscillatory and periodic behavior for
both 2 and 3 spin systems. The oscillatory behavior of MO dynamics in single
crystal fluorapatite appears to be damped and periodic. We have no independent
measurements of the defect density in this particular crystal of mineral fluorapatite,
although mineral fluorapatite generally contain defects caused by substitutions
such as hydroxyl and chloride ions. We can attempt to model the observed MO
dynamics of this defect-containing sample somewhat along the lines of the 1-D
cluster model previously described. However, the intensities of the various orders

will be obtained in a different manner. For each run number, we will use the

 

148

intensities of the individual orders of MO coherence calculated from the ANTIOPE
simulation. The overall intensities of a given order n at preparation time 7' i given
by following expression
Jinn) = 22 err(n,r) (II-88)

where r refers to the spin system orf-size r. The obvious limitation of this approach
is that run lengths < 5, which are expected to yield high-damped oscillation, are
ignored, since no calculations are available. Figure "-35 shows the mole fraction
of spins in various run numbers from 1 to 5 and the sum of the mole fraction
with different defect densities obtained by using percolation theory.78 For defect
densities larger than 40%, the sum of the mole fraction of spins in runs 3 5 is
about 0.8. Therefore, the contribution of runs having more than 5 spins to the
MO dynamics for defect densities larger than 40% is small. However, for smaller
defect densities, the larger runs will dominate MQ dynamics. Nevertheless, since
these runs will exhibit more damped behavior, we can hope to reproduce the

essential features of the experimentally-observed oscillations.

Theoretical intensities of MO coherences for an oriented spin system are
calculated by using Eq. "-88. Figure lI-36a and b show a comparison between the
experimental 20 intensities of single crystal FAP, and calculated 2Q intensities for
assumed 30% and 50% defect densities at two different orientations (62° and 90°
with respect to the external magnetic field) as a function of preparation time. The
calculated 20 intensifies for both 30% and 50% defect densities at the two different
orientations show similar oscillatory behaviors. Although the defect density of
single crystal mineral fluorapatite cannot be reliably estimated, the agreement
does not appear to improve as the defect density decreases from 50% to 30%,

especially in the positions of the maxima and minima in Figure Il-36a. A defect

 

149

density less than 30%, which is quite reasonable, would appear to yield even
better agreement if the same trend occurs. 19F MO NMR spectra of single crystal
FAP (not shown here) at two orientations (62° and 90°) display 60 peaks at
shorter than 1ms. This indicates that a considerable mole fraction of larger than
6 spin systems is present in mineral single crystal FAP. Thus, since the sum of
the mole fraction of defect densities larger than 50% is greater than 90%, we
can presume that the defect density is less than 50%. The contribution of larger
than 6 spin systems to the MO dynamics hinders the reliable measurement of the
defect density from comparisons between calculated and experimental plots. In
spite of the existence of a variety of spin system in single crystal fluorapatite, the
oscillatory behavior of MO intensities of single crystal FAP in fig. "-27 indicates

that the fluorine atoms in chains are interrupted by defects.

6. Conclusions

We have shown that multiple-quantum dynamics in an essentially one-
dimensional distribution of spins leads to an initial linear dependence of effective
size N upon preparation time, in qualitative agreement with predictions of the in-
cremental shell model.36 Further work is needed to determine whether the lack of
good quantitative agreement in this initial region arises from experimental consid-
erations, or instead from limitations of the theoretical model, and whether further
refinements of the model may improve its predictive accuracy. The MO growth
exponents also distinguish the different dimensionalities of spin systems and the
presence of clustering. The contribution of the two different dimensionalities to the
growth of the MO coherences is visually illustrated by the MO growth exponent

plot. Two growth exponents of HAP-M agree well reasonably with those of the

 

150

 

 

 

 

 

‘ I
0.8 ‘ .
he;
5 0.6 ' I g
8 1
g 0
§ 0.4 .
I c
‘ A A
a
0.2 - 9 a a
. O 8 . : I I:
00 LL 9 - . . t =.
0.0 0.2 0.4 0.6 0.8

Defect density

 

figure "-35. Mole fraction of spins in various run numbers and sum of the mole
fraction with different defect densities using percolation theory." Run number =
1 (open circle), run number =- 2 (open triangle), run number = 3 (open square),
run number a 4 (cross in open square), run number a 5 (cross), and sum of 1

- 5 spins (black circle).

151

 

 

 

 

1.0
Cl
4 on b) $0
0.8 ‘ Cl DUO 13
1 A D D
, DAAD U UAAA
0.6 D M
AA 0 A
0.4 1 El AAA GAGA 83 26 A
, A O 4‘ O @O O a
a « is
g l
'S 00 - r . . r . - ' -
B .
-% 1.0
g a) D b
5 0.8 1 D an

a We a
0.6 . A AA
DA
A AARm
DA 0 O A
0.4 mo 0
0.2 if? 0 0 0
o O
0.0 3 ' ' ' r r v r t v , -
0 200 400 600 800 1000 1200
Preparationtlfneols)

j

 

 

 

figure "-36. Comparison between experimental 20 intensities (open circle) of
single crystal FAP and calculated 20 intensifies for two assumed defect densities
[30% (open circle) and 50% (cpen square)] at two different orientations. a) 62°
orientation with respect to the external magnetic field; b) 90° orientation with

respect to the external magnetic field.

 

 

152

incremental shell model. Linear behavior of the effective sizes vs. preparation
time has previously been observed generally only for finite clusters, but the slope
at longer preparation times in these cases is observed to level off,57 in contrast to
our results. It is interesting to note that the hopping model also predicts a linear
dependence for clusters of 21 spins, but it cannot treat infinite spin systems, and

has not predicted MO dynamics for different dimensionalities.

It is feasible to model the one-dimensional chains due to their simple configu-
ration. 1H MQ—NMR is sensitive to defects in one-dimensional apatite chains, and
the defect density can be estimated by using the one-dimensional cluster model.
The comparison of 4Q/2Q intensities between HAP-M and HAP-N for given prepa-
ration times directly shows the presence of vacancies in the non-stoichiometric
hydroxyapatite sample. The defect density of HAP-N obtained using MQ NMR
is in good agreement with the 8% defect density obtained from 1H MAS-NMR;68
however, the calculated 4Q/ZO intensities based on the defect densities of the
FOHAP samples deviate considerably from MQ experimental 40120 intensities.
This discrepancy may arise from three possible sources; additional defects in the
hydroxyl chains other than fluorine atoms (i. e. vacancies), non-random fluorine
substitutions, and/or off-resonance effects caused by dipolar coupling of the hy-
droxyl protons to 19F nuclei. Partially deuterated hydroxyapatite could be used
to test the 1-D cluster model wi’d'l knowledge of the hydroxyl content, since a
deuterium randomly substitutes for a proton and the dipolar interaction between
a deuterium and a proton is small. The first explanation could be tested by a
detailed chemical analysis of the hydroxyl content of the crystals, which is quite

difficult to perform.

Since the dipolar interaction in single crystal samples can be adjusted by

“..-

 

 

153

field, we hope that 19F MO NMR of single crystal fluorapatite could be used
to separate the MO dynamics of different dimensionalities. For example, if the
orientation of intra-chain fluoride ions is fixed at 54.7° (magic angle), the non-zero
dipolar coupling of inter-chain fluoride ions would create MQ coherence in a 2-D
or fashion. Our 19F MQ-NMR results for a single crystal of mineral FAP show an
oscillatory behavior of the MO peaks that does not permit measurement of the
effective size at each preparation time. A less deficient synthetic single crystal
sample might give information about dimensionality in more detail. Since the
orientation of individual crystallites can be selected by relating them to a position
in the CSA powder pattern, MQ-NMR can be performed on a specific orientation
by selectively saturating all but a single frequency in the pattern.79 Selecting the
orientation in a powdered sample thus also permits the relative magnitude of the

. intra- and inter-chain dipolar couplings to be “tuned” at will.

 

l! i”: “m5 11'- 55‘...” .\ VA .

154

7. References

1. W. A. Anderson, Phys. Rev. 104, 850 (1956)

2. J. I. Kaplan and S. Meiboom, Phys. Rev. 109, 499 (1957)

3. S. Yatsive, Phys. Rev. 113, 1522 (1959).

4. W. A. Anderson, R. Freeman, and C. A. Reilly, J. Chem. Phys. 39, 1518
(1963).

5. K. A. McLauchlan and D. H. Whiffen, Proc. Chem. 800., 144 (1962).

6. A. D. Cohen and D. H. Whiffen, Mal. Physics, 7, 449 (1964).

7. J. I. Musher, J. Chem. Phys. 40, 983 (1964).

8. M. L. Martin, G. J. Martin, and R. Couffignol, J. Chem. Phys. 49, 1985 (1968).

9. K. M. Worvill, J. Magn. Reson. 18, 217 (1975).

10. P. Bucci, M. Martinelli, and S. Santucci, J. Chem. Phys. 52, 4041 (1970).

11. P. Bucci, M. Martinelli, and S. Santucci, J. Chem. Phys. 53, 4524 (1970).

12. G. Lindblom, H. Wennerstroem, and B. Lindman, J. Mag. Reson. 23, 177
(1976)

13. R. B. Creel, E. D. von Meerwall, and R. G. Barnes, Chem Phys. Lett. 49,
501 (1977).

14. D. Dubber, K. Deorr, H. Ackmann, F. Fujara, H. Grupp, M. Grupp, P. Heitjans,
A. Keoblein, and H. J. Steockmann, Z. Physik A, 282, 243 (1977).

15. R. R. Ernst and W. A. Anderson, Rev. Sci. Instr. 37, 93 (1966).

16. H. Hatanaka, T. Terao, and T. Hashi, J. Phys. Soc. Jpn. 39, 835 (1975).

17. W. P. Aue, E. Bartholdi, and R. R. Ernst, J. Chem. Phys. 64, 2229 (1976)

18. G. Bodenhausen, Prog. NMR Spectrosc. 14, 137 (1981).

19. U. Piantini, O. W. Sorenson, and R. R. Ernst, J. Am. Chem. Soc. 104,
6800 (1982).

 

 

155

20. A. Bax, R. Freeman, and S. P. Kempsell, J. Am. Chem. Soc. 102, 4849
(1980).

21. A. Bax, R. Freeman, and S. P. Kempsell, J. Magn. Reson. 41, 349 (1980).

22. A. Bax, R. Freeman, T. A. Frenkiel, and M. H. Levitt, J. Magn. Reson. 43,
478 (1981)

 

23. A. Bax, R. Freeman, and T. A. Frenkiel, J. Am. Chem. Soc. 103, 2102
(1981).

24. L. Braunschweiller, G. Bodenhausen, and R. R. Ernst, Mal. Phys. 48, 535
(1983).

 

25. M. H. Levitt and R. R. Ernst, Chem. Phys. Left. 100, 109 (1983).

26. W. S. Warren, D. P. Weitekamp, and A. Pines, J. Chem. Phys. 73, 2084
(1080).

27. W. S. Warren and A Pines, J. Chem. Phys. 74, 2808 (1981).

28. S. Sinton, D. B. Zax, J. B. Murdoch, and A. Pines, Mal. Phys. 53, 333 (1984).

29. W. S. Warren, S. Sinton, D. P. Weitekamp, and A. Pines, Phys. Rev. Left.
43, 1792 (1979).

30. Y. S. Yen and A. Pines, J. Chem. Phys. 66, 5624 (1983).

31. J. Baum, M. Munowitz, A. N. Garroway, and A. Pines, J. Chem. Phys. 83,
2015 (1985).

32. A. N. Garroway, J. Baum, M. Munowitz, A. Pines, J. Magn. Reson. 60,
337 (1984).

33. J. Baum, K. K. Gleason, A. Pine, A. N. Garroway, and J. A. Reimer, Phys.
Rev. Left. 56, 1377 (1986).

34. K. K. Gleason, M. A. Petn'ch, and J. A. Reimer, Phys. Rev. 8.36, 3259

(1987).

156

35. M. A. Petrich, K. K. Gleason, and J. A. Reimer, Phys. Rev. 8.36, 9722
(1987).

36. D. H. Levy and K. K. Gleason, J. Phys. Chem. 96, 1992 (1992).

37. G. Drobny, A. Pines, S. Sinton, D. P. Weitekamp, and D. Wemmer, Faraday
Symp. Chem. Soc. 13, 49 (1979).

38. R. Ryoo, S.-B. Liu, L. C. de Menorval, K. Takegoshi, B. Chmelka, M.
Trecoske, and A. Pines, J. Phys. Chem. 91, 6575 (1987).

39. B. Chmelka, J. G. Pearson, S.-B. Liu, R. Ryoo, L. C. de Menorval, and
A. Pines, J. Phys. Chem. 95, 303 (1991).

40. D. N. Shykind, J. Baum, S.-B. Liu, A. Pines, and A. N. Garroway, J. Magn.

Reson.
76, 149 (1988).

41. S. Sinton and A. Pines, Chem. Phys. Left. 76, 263 (1980).

42. M. Munowitz and A. Pines, Science 233, 525 (1986).

43. W. V. Gerasimowicz, A. N. Garroway, and J. B. Miller, J. Am. Chem. Soc.
1 12, 3726 (1990).

44. B. E. Scruggs and K. K. Gleason, J. Magn. Reson. 99, 149 (1992).

45. B. E. Scruggs and K. K. Gleason, Macromolecules, 25(7), 1864 (1992).

46. M. Gold man, “Quantum Description of High-Resolution NMR in Liquids”
(Oxford University Press, New York, 1988).

47. C. P. Slichter, “Principles of Magnetic Resonance”, 3rd ed. (Springer-Verlag,
New-York, 1989).

48. R. R. Ernest, G. Bodenhausen and A. Wokaun, “Principles of Nuclear
Magnetic Resonance in One and Two Dimensions” (Oxford University Press,

New York, 1987).

 

1,,

 

49.
50.
51 .
52.
53.
54.
55.
56.

57.
58.
59.
60.

61.
62.

63.

65.
66.
67.
68.

157

D. P. Weitekamp, Adv. Magn. Reson. 11, 111 (1983).

M. Munowitz, Mol. Phys. 71, 959 (1990).

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

M. Munowitz and A. Pines, Adv. Chem. Phys. 66, 1 (1987).

S. Lacelle, Advances in Magnetic and Optical Resonance 16, 173 (1991 ).
A. Wokaun and R. R. Ernst, Mol. Phys. 36, 317 (1978).

J. Baum and A. Pines, J. Am. Chem. Soc. 108, 7447 (1986).

J. B. Murdoch, W. S. Warren, D. P. Weitekamp and A. Pines, J. Mag. Reson.

60, 205 (1984).
M. Munowitz, A. Pines, and M. Mehring, J. Chem. Phys. 86, 3172 (1987).

M. I. Kay, R. A. Young and A. S. Posner, Nature (London), 204, 1050 (1964).

J. C. Elliott, P. E. Mackie, and R. A. Young, Science, 180, 1055 (1980).
A. Abragam, "Principles of Nuclear Magnetism" Oxford Univ. Press,
Oxford (1961).

M. Munowitz, “Coherence and NMR’ Wiley, New York (1988).

J. von Neumann, “Mathematical Foundations of Quantum Mechanics”,
Princeton University Press. Princeton, NJ (1955).

U. Haeberlen, "High Resolution NMR in Solids: Selective Averaging", Adv.
Magn. Reson., Supplement 1, . Academic Press, New York (1976).

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

J.L. Carolan, Chem. Phys. Left. 12 (1971) 389.

F. S. de Bouregas and J. S. Waugh, J. Magn. Reson. 96, 280 (1992)
BC. Fowler, Inorg. Chem. 13 (1974) 207.

J. Arends, J. Christofferson, M. R. Christofferson, H. Eckert, B. O. Fowler,

J. C. Heughebaert, G. H. Nancollas, J. P. Yesinowski and S. J. Zawacki,

“"3

 

 

158

J. Cryst. Growth 84, 515 (1987).

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

70. EC. Moreno, M. Kresak, and RT Zaradnik, Caries Res. Supp/.1 142 (1977)

71. JP Yesinowski and M.J. Mobley, J. Am. Chem. Soc.105 6191 (1983);
JP Yesinowski, R. Wolfgang, and M.J.. Mobley, in Adsorption on and
Surface Chemistry of Hydroxyapatite, Mlsra, D. N., Ed., Plenum: New York,
pp151-175.

72. R. W. C. Wyckoff, Crystal Structures (lnterscienoe New York), 6, part 1,386
(1963).

73. E. R. Andrew, J. Chem. Phys. 18, 607 (1950).

74. L. B. Schreiber and R. W. Vaughan, Chem. Phys. Left., 28, 586 (1974).

75. K. K. Gleason, Concepts in Magn. Reson. Appeared.

. 76. W. Van der Lugt, D. l. M. Knottnerus, and W. G. Perdok, Acta Crysallogr.,
Sect. 8: Crystallogr. Cryst. Chem. B27, 1509 (1971).

77. A. M. Vakhrameev, S. P. Gaguda, and R. G. Knubovets, J. Stmct. Chem.
(Engl. Transl.), 19, 256 (1978).

78. D. Stauffer and A. Aharony, “Introduction to Percolation Theory” Taylor &
Francis, London, Washington, DC (1992).

79. L. B. Moran and J. P. Yesinowski, Chem. Phys. Left., To be submitted.

-D‘i“;; hffi.m1
I

 

 

159

APPENDIX A

Pulse program for even-order selective phase-incremented or TPPI multiple-
quantum pulse sequence for solids for Varian VXR spectrometer. This program

is courtesy of T. Barbara of Varian Associate.

/*Pulse sequence: G_INMQ_LOCK*/
[*Pseudo l—D even order MQ generation with a detection using spin locking sequence.
Phase cycle to remove odd order quanta by alternating excitation phase between x and
-x and adding memory.*/

/* VARIABLES:

del: delay between 8 pulses ps

delp: delay between 8 pulses, 2*del-l-pw ps

mloop: the number of loop (basic cycle time*mloop = preparation time)

shift: the angle phase incrementation (determine the maximum detectable MQ order)
tlinc: the time to increment the phase

d3: the time between mixing and detection period to eliminate the remaining transverse
magnetization

d4: the evolution time*/

#include <standard.h>

static int table 1[4] =
static int tab182[4] =
static. int table3[4] =
static int table4[4] =
static int tableS[4] =
static int tab1e6[4] =
static int table7[4] =

9 9

1

9 9

C

9 9

{0220
{0022
{2002
{1,1,3.3
{3311
{0202
{1313

O

};
};
. , };
}
i

9 9 9

9 9 9

pulse sequenceO

{

double del, delp, ddel, ddelp, mloop, shift, d3, d4, tlinc:
char trig[MAXSTR];

extem double getvalO;

del = getva1(“del”);
delp = gctva1(“delp”);

 

 

160

mloop = getval(“mloop”);

shift = getval(“shift”);
d3 = getval(“d3”);
d4 = getval(“d4”);
tlinc = getval(“tlinc”);
getstr(“trig”,trig);

ddel = del - rofl - rof2;
ddelp = del - rofl - rof2;

settable(tl, 4, tablel);
settable(t2, 4, table2);
settable(t3, 4, table3);
settable(t4, 4, table4);
settable(tS, 4, tableS);
settable(t6, 4, tableé);
settable(t7, 4, table7);

setreceiver(t7);
stepsize(shift,TODEV);
initval(mloop.v 1);

initva1( np/2.0,v7);
initdelay(tlinc,DELAY5);
assign(zero,v 10);

loop(v7,v9);
if(trig[0 =’y’)
{

xgate(l .0);
}

xmtrphase(v10);
delay(d1); delay(rofl);
incr(v10);

rcvroffO;
starthardloop(vl);
delay(ddel);
rgpulse(pw,tl,rofl,rof2); f" x -x -x x */
delay(ddelp);
rgpulse(pw,tl ,rofl ,rot‘Z);
delay(ddel);
rgpulse(pw,tl,rofl ,rof2);
delay(ddelp);

 

161

rgpulse(pw,tl,rof1,rof2);

delay(ddel);

rgpulse(pw,t3,rof1,rof2); /* -x x x -x */
delay(ddelp);

rgpulse(pw,13,rofl ,rof2);

delay(ddel);

rgpulse(pw,t3,rof1,rof2);

delay(ddelp);

rgpulse(pw.t3.rofl.rof2);
endharleOPO;

xmuphase(zero); /* reset small angle shift to zero */
delay(d4);

incdelay(v10,DELAY5); /* generate t1 delay If tlinc=0, TPPI*/
/*otherwise, phase incremented method*/
stanhardloop(v 1);
delay(ddel);
rgpulsc(pw,t4,rofl,rof2); f“ y y -y -y */
delay(ddelp);
rgpulse(pw,t4,rofl ,ron);
delay(ddel);
rgpulse(pw,t4,rof1,rof2);
delaflddelp);
rgpulse(pw,t4,rof1,rof2);
delay(ddel);
rgpulse(pw,t5,rof1,rot?); /* -y -y y y */
delay(ddelp);
rgpulse(pw,t5,rof1,roa);
delay(ddel);
rgpulse(pw,15 ,rof l ,rof2);
delay(ddelp);
rgpulse(pw,t5,rof1,rof2);
endhardloopO;
rcvron();
delay(d3); /* delay to allow decay of transverse coherences */

rgpulse(pw, t6, rofl, rof2); I“ x -x x -x */
rgpulse(p1, t7, rofl, rof2); /* y -y y -y spin loclcing*/
delay(alfa);

acquirc(2.0,l/sw);

endloop(v9);

I

 

FL"
f5

 

 

162

APPENDIX B
Selection rule of 1-spin/2-quantum average Hamiltonian Hy, in hopping model

If the spin angular momentum operator 1,, is expressed for a spin with index
3,- by a ket jejaj). the Liouville state that is formed is a product of K single-spin

operators that can be written by the abbreviated notation

|Knp)=lslal ............ 850k) ' ‘ (A-1)
where
n = Z or". (A'2)
j=l,K

The scalar product between two Liouville state vectors is orthornormal and is de-
fined as
(AlB) = Tr{A+B} (A-3)
where A+ is the Hermitian adjoint of A. Since the superoperator 7:1 is defined by
710.4) = |[’H, .41), (A-4)
matrix elements involving the superoperator are given by
(AWE) -_- (Al [71,19]) -_- Tr{A+[H,B]}. (A-5)
The selection rules in the Liouville space from the Hamiltonian in Eq. lI-47 are
calculated using Eq. A-5 and the commutation relations
[I,,-, I,,] = A6,,- (A-6a)
and
[I+i1I—j1 = 21260 (A-6b)
Since the selection rule of the average dipolar Hamiltonian from the pulse se-
quence in Fig. ll-4 can be evaluated using the numerator of Eq. Il-47, we can
rewrite Eq. A-5 as
(KnplfiDIK’n’p’) = (1(an [120,K’n'p’]). (A-7)

After K ' = 1 and n' = 0 (11,) is chosen as an initial condition, the square bracket

 

 

 

 

163

part of Eq. A-7 can be written using only the spin operators
[[+1I+2 + Lil—2, [12] = [[+1I+2,1'1z] + [1—11-2, I121 (A-8)
using a commutator algebra
[A, B + C] = (A, B] + [A, C]. (A-9)
Since 1+2 and 1.2 commute with [1,, we can rewrite Eq. A-8 using the commuta-
tion relations in Eq. A-6 as

= I+2[I+1.1121+I—2lI—1ll'12] = -I+2I+1-I—2I—1 (Pl-10)

[.4. BC] = [A, B]C + B[A, C]. (A-11)

The values of K and n of the first term in Eq. A-10 are 2 and 2, and those of the

second term in Eq. A-10 are 2 and -2 respectively. Therefore, the solution of Eq.
A-7 is .

= (Knpl - 1+2I+1 - 1.2111) = (K711018138)+<Knp|2l-2,p'>- (A42)

' Since the scalar product of the Liouville state is orthonormal, Eq. A-12 is non-

zero at K-2, n=2 and K-2, n--2.

When It" a 2 and n' =2 ( from the 1,21“ term), [HD,K'n'p'] can be also
written using the spin operators
[1+1I+2 + 1.11-2,I+2I+1] = [1+1I+2, 1+2I+11 + [I—tI—z, 1+2I+1] (A-13)
Since the first term [I+II+2,I+2I+1] commutes, we solve for the second term
[I_1I_2,I+2I+1] using Eq. A-11
= [+211—11—2J+11 1' 11-11—2. I+211+1 = -I+1I—1122 + 1121—21” (A44)
Therefore, only when K=3 and n=0 is (Knpl’lez, 2, p) non-zero. The detailed
calculation of the selection rule for K ' = 2 and n' =- -2 is omitted since it is similar
to the calculation of the selection rule for It" = 2 and n' = 2. Only when K = 3

and n = 0 is (Knpl’lefz,—2, p) non-zero too.

 

 

 

164

We will calculate the general solution of the selection rule using Eq. A-7. The

square bracket in Eq. A-7 can be represented by

[Hal K 'n'P'] = [.2 (1+il'+j + I_,-I_,-), K'nlp'] (A'1 5)
1)]

We can rewrite Eq. A-15 using Eq. A-9 and A-11 as

= 2 1+,- [I+j,K'n'p'] + Z [LI-itK'nIP']I-l-I

i>j t'>j

+ XI..- [I_,-,K'n'p’] + 2 [I_,,K’n'p']I_,. (A-16)

The solutions of): [1+], K 'n'p'] and :2: [I+,-, K 'n'p'] do not change the value of
K ' but change 12' tr]; n' + 1 due to the commutation relations (Eq. A-6a and A-6b).
The lowering operators in the square brackets of Eq. (A-16) do not change the
value of K' but decrease n' to n' — 1. Since the operators outside the square
brackets in Eq. A-16 increase K ' to K ' + 1, and change n' to n' :t 1 according to
(Knplflle'n'p'> is non-zero when K is equal to K ' + 1 and n is equal to n' j: 2.

When we solve the Eq. A-7 as(KnijD|K'n'p') = ([Knp,HD]jK'n'p'), the
non-zero condition of (Knp|Hp|K'n'p'> is that K is equal to K ' — 1 and n is

equal to n' :l: 2.

Therefore, the selection rule of the average Hamiltonian 71,, in the Liouville

spaceisK=K'i1andn=n'i2.

165

APPENDIX C
C programs for calculation of the ratio of 40/20 intensities for a 1-D chain
with different defect densities.

These programs calculate the ratio of the 4Q/2Q intensities of defect containing
1D chains for various preparation times using the statistical model, combined with
percolation theory expressions that provide the mole fraction of spins in a run

number.

/* Calculation of 4Q/2Q intensity using the statistical model and the percolation theory*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
double x,y;
double exv().sqfl().exr>2().pow0;
mainO
{
int i,j,n,l,m,p;
float k,s,h,sum2,pp,sump2,sum2p,sum4,sump4,sum4p,rat2,rat4;
char filename[20];
FILE *fp;

printf("Enter filename: \n");
scanf("%s",filenamc);

fp=fopen(filename."wr");

pfintt‘C'What is the ratio of the defect’?\n");
fprintf(fp,"What is the ratio of the defect. ");

scanf("%f",&k);
fprintf(fp,"%f\n", k);

printf("What is the maximum run number. ");
fprintf(fp,"What is the maximum run number?\n");

scanf("%d",&l);

 

166

fprintf(fp,"%d\n", 1);

printf("What is the efi'ective size?\n");
fprintf(fp,"What is the effective size?\n");

scanf("%f",&h);
fprintf(fp,"%f\n". h);

s=1.0-k;

sum2=0.0;
sum4=0.0;
sum2p=0.0;
sum4p=0.0;

[*The first part of Eq. 11-82 for 2-Q intensity. The decimal number (efiective size from
Gaussian function) cannot be used to calculate a factorial so that the round up values are
used.*/

if(h<=6.0)
{ for(i=2;i<=l;i++)
{ x=(doub1e)i;
if(x<=h)
{s}um2+=x*pow(k,2.0)*pow(s,x)*fac(2*i)/(fac(i-2)*fac(i+2))/pow(4.0,x);

else
{
P=round(h);
PP=(float)p;
sum2 +=x*pow(k,2.0)*pow(s,x)*fac(2*p)/(fac(p-2)*fac(p+2))/pow(4.0,pp);
1
1

[*The first part of Eq. II-82 for 4-Q intensity. The decimal number (efi‘ective size from
Gaussian function) cannot be used to calculate a factorial so that the round up values are
used.*/

for(ifl;i<=l;i++)

{

 

167

x=(double)i;
if(x<=h)
{
sum4 +=x*pow(k,2.0)*pow(s,x)*fac(2*i)/(fac(i-4)*fac(i+4))/pow(4.0,x);
1

else

1
p=round(h);
pp=(float)p;
sum4 +=x*pow(k,2.0)*pow(s,x)*fac(2*p)/(fac(p-4)*fac(p+4))/pow(4.0,pp);
}
}
1

/*The second part of Eq. II-82 for 2Q and 4Q intensity. */

else
I
for(i=2;i<=6;i++)
{
x=(double)i;
sum2p +=x*pow(k,2.0)*pow(s,x)*fac(2*i)/(fac(i-2)*fac(i+2))/pow(4.0,x);
}

for(i=4;i<=6;i-H-)
{

x=(double)i;
sum4p +=x*pow(k,2.0)*pow(s,x)*fac(2*i)/(fac(i-4)*fac(i+4))/pow(4.0,x);
} .

for(i=7;i<=1;i-H-)
{
x=(double)i;
if (x<=h)
{
sum2p +=x*pow(k,2.0)*pow(s,x)l(sqrt(x)*1.77)*exp(-4.0/x);
sum4p +=x*pow(k,2.0)*pow(s,x)/(sqrt(x)* 1 .77)*exp(-16.0/x);

1

else

1
sum2p +=x*pow(k,2.0)*pow(s,x)/(sqrt(h)* 1 .77)*exp(-4.0/h);

 

168

sum4p +=x*pow(k,2.0)*p0W(8.x)/(Sq1't(h)*1-77)*6XP(' 16W“);
}

1
}

printf("The ratio of each multiple quantum signal\n");
fprintf(fp,"The ratio of each multiple quantum signal\n");

sump2=(sum2+sum2p)/(sum2+sum2p);
sump4=(sum4—I-sum4p)/(sum2+sum2p);

printf("2quantum => %f, 4quantum => %f\n",sump2,sump4);
fprintf(fp,"2quantum ==> %f, 4quantum => %t\n",sump2,sump4);

fclose(fp);
}

[*Calculation of factorial’V

fac(m)
int m;
{
int i, fac;
fac=l;
for(i=l;i<=m;i-H~)
fac=fac*i;
rettn'n(fac);
}

/* Calculation of round up*/

round(x) '
float x;
1
int ix;
float fx, gx;
8X=X;
ix=(int)gX;
fx=(float)ix;
gx-=fx;
if(gx>=05) ix+=1;
returnfiX):
I

 

“‘illillllillili“