• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 Abstract
 Introduction
 Theory
 Experimental equipment and...
 Results and discussion --...
 Results and discussion -- NMR line...
 Results and discussion -- NMR spectrum...
 Results and discussion -- nuclear...
 Appendix A
 References
 Biographical sketch














Title: Nuclear magnetic resonance of 14N in single crystal glycine and of 15N in liquid and solid N2 /
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Permanent Link: http://ufdc.ufl.edu/UF00097507/00001
 Material Information
Title: Nuclear magnetic resonance of 14N in single crystal glycine and of 15N in liquid and solid N2 /
Physical Description: xi, 159 leaves : ill. ; 28 cm.
Language: English
Creator: Ishol, Lyle Milton, 1932-
Publication Date: 1976
Copyright Date: 1976
 Subjects
Subject: Nitrogen   ( lcsh )
Nuclear magnetic resonance   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 157-158.
Additional Physical Form: Also available on World Wide Web
Statement of Responsibility: by Lyle Milton Ishol.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097507
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000173501
oclc - 02987294
notis - AAT9955

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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
    Abstract
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Theory
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
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        Page 15
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        Page 20
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        Page 25
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        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
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        Page 37
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        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    Experimental equipment and procedure
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
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        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
    Results and discussion -- Glycine
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
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        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
    Results and discussion -- NMR line shape of 15N in Nitrogen with O2 impurity
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
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        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
    Results and discussion -- NMR spectrum of 15N in solid 15N-14N
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
    Results and discussion -- nuclear spin-lattice relaxation in liquid and solid 15N2 and 14N2
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
    Appendix A
        Page 156
    References
        Page 157
        Page 158
    Biographical sketch
        Page 159
        Page 160
        Page 161
Full Text











NUCLEAR MAGNETIC RESONANCE OF 1N IN SINGLE CRYSTAL GLYCINE
AND OF 15N IN LIQUID AND SOLID N2












By

LYLE MILTON ISHOL


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













UNIVERSITY OF FLORIDA


1976






































To my wife, Jane













ACKNOWLEDGMENTS


It is not practical to list all those who assisted the author in

the research leading to this dissertation; however a few must be men-

tioned for their especially important contributions.

Most important is Dr. Thomas A. Scott who suggested the problems

and provided essential support, guidance, and assistance throughout all

phases.

Mr. Paul Canepa was especially helpful with electronic and mechani-

cal equipment and also assisted in taking data.

Dr. Atholl Gibson and Dr. James Brookeman must be thanked for'their

frequent and valuable consultations on both theoretical and experimental

aspects.

Professor E.R. Andrew is to be thanked for the glycine crystals

which were grown in his department at the University of Nottingham and

also for the many helpful conversations during his visits to the Univer-

sity of Florida.

Mr. Ralph Warren and the rest of the men in the Physics Shop and

Mr. Pat Coleman always managed to meet last minute requests for shop

support and for liquid helium.

Other students and visiting scientists also shared their expertise

and in some cases provided direct assistance, and finally, the typist,

Adele Koehler, is to be congratulated on turning an illegible rough

handwritten draft into such a fine job.














TABLE OF CONTENTS


ACKNOWLEDGMENTS. . . . . . . . .

LIST OF FIGURES . . . . . . . .

ABSTRACT . . . . . . . . . . .

CHAPTER


I INTRODUCTION . . . . . . . . . . .
14
1.1 High Field NQR of 1N in Single Crystal Glycine .

1.2 NMR in Liquid and Solid Nitrogen. . . . . .

II THEORY . . . . . . . . . . . . .

2.1 General . . . . . . . . . . . .

2.2 Intramolecular Dipolar Coupling . . . . .

2.3 Nuclear Quadrupole Interaction . . . . .

2.4 High Field Nuclear Quadrupole Interaction . . .

2.5 Intramolecular Dipolar Splitting of the NMR Spectrum
of 15N in 15 N14N Hixed Molecules . . . . .

2.6 Nuclear Spin-Lattice Relaxation . . . ....

2.6.1 General. . . . . . . . . . .

2.6.2 The Hubbard Relation . . . . . . .

III EXPERIMENTAL EQUIPMENT ANT PROCEDURE . . . . . .

3.1 Superconductive Magnet System . . . . .

3.1.1 Magnet . . . . . . . . . .

3.1.2 Dewars .. . . . . . . . . .

3.1.3 Sample Probe . . . . . . . . .

3.1.4 Operation of the Superconductive System . .


Page

iii

vii

x


1

1

2

5

5

7

14

21


32

41

41

46

48

48

48

57

58


: : : I







TABLE OF CONTENTS
(Continued)


3.2 Electromagnet System . . . .

3.2.1 Electromagnet . . . .

3.2.2 CryostaL and Si~ple System.

3.3 Spectrometers. . . . . .

3.3.1 Pulse Spectrometer. . .

3.3 2 Continuous Wave Spectrometer

IV RESULTS AND DISCUSSION GLYCINE .

4.1 Gly ine Structure. . . . .

4.2 Electric Field Gradicnc f nsors. .

4.3 Data . . . . . . . .

4.4 Data Analysis. . . . . .

4 .5 Conclusions. . . . . . .

V RESULTS AND DISCUSSION --NMR LINE SHAPE O
WITH 02 IIrPURITY. . . . . . .

5.1 Structure of a Nitrogen. . . .

5.2 Data . . . . . . .

5.3 Data Analysis. . . . . .

5.3.1 Paramagnetic 02 Impurity. .

5.3.2 Ortho-Para Conversion . .

5.3.3 14N Impurity. . . .

5.3.4 Conclusions . . . .

VI RESULTS AND DISCUSSION NTMR SPECTRUM OF
15 -N 4N . . . . . . . .

6.1 Sample . . . . . . .

6.2 Data . . . . . . . .


F. . . . . .tTROGEN































N IN SOLID


Page

. . . . 67

. . . . 67


67

71

71

74

76

76

78

89

98

109


110

110

112

118

118

127

128

133


135

135

135







TABLE OF CONTENTS
(Continued)


Page

6.3 Computer Simulation . . . . . . . .. 140

6.4 Data Analysis . . . . . . . . .. 140

6.4.1 Asymmetric Pake Doublet . . . . . .. 141

6.4.2 Level Crossing. . . . . . . . .. 141

6.5 Conclusions. . . . . . . . ... .... 142

VII RESULTS AND DISCUSSION NUCLEAR SPIN-LATTICE RELAXATION IN
LIQUID AND SOLID 15N2 AND 14N2............... 143
Previous 15 14
7.1 Previous 5N and N, Relaxation Time Studies . . 143

7.2 Data. . . . . . . . .. .. . . . 148

7.3 Crystalline Structure of A-Solid Nitrogen. . . ... 150

7.4 Data Analysis. . . . . . . . .... . 153

7.5 Conclusions .. .. . . . . . . . 154

APPENDIX A . . . . . . . . ... . . . 156

LIST OF REFERENCES. . . . . . . . . .. . . 157

BIOGRAPHICAL SKETCH . . . . . . . . . . 159














LIST OF FIGURES


Figure Pag

1. Pake powder pattern doublet. .... . . . . .... 13

2. Comparison of z-axis projections of rotating vectors and
tensor components. . . . . . * . . . . 24

3. Transformation between the lab axes and crystal axes
during an X rotation ..... . . . . . . . 28

4. Temperature dependence of quadrupolar relaxation time. . 44

5. Homogeneity of the superconductive magnet. . . ... . 49

6. Differences between measured values of magnetic field and
value calculated by exponential decay equation ...... 51

7. Top access dewar system. .... .. . . . ..... 55

8. Re-entrant dewac system. .... . . . ..... 56

9. Sample probe, side view of sample holder . . . ... 59

10. Sample probe, end view of sample holder. . . . . ... 60

11. Gas-flow temperature control system. ... . . . . 61

12. Cryostat used with electromagnet ...... .. . . . 68

13. Sample-gas system. .... .. . . . . . . .. 69

14. Glycine molecule . . . .... . . . . . . 77

15. Typical a-glycine crystal. . . . ... .. . . .... 79

16. Unit cell of a glycine . . . . . . . .... 80

17. A vector rl undergoing a 1800 rotation about the Y axis
would bring it to r2, whereas a reflection in the X,Z plane
2 r381
would bring it to r3, where r2 = -r3 ............ 81

18. Symmetry of rotation patterns for rotation axes in or
normal to the X,Z plane. . . . . .... . . 82

19. C-axis rotation patterns from 14N in a glycine at 50 C . 9J








LIST OF FIGURES
(continued)


Figure Page

20. C-axis rotation patterns from 14N in a glycine at 74 C. 92

21. C-axis rotation patterns from 14N in a glycine at 1480 C. 93

22. C-axis rotation patterns from 1N in a glycine at 1674 C .. 94


23. Rotation patterns from 1N in a glycine at 5 C using axis
normal to (120) plane. . . . . . . . . ... 96

24. Relationship of U, V, W axes to X, Y, Z axes .. . . . 99

25. Unit cell of a N2 . .. . . . . 11
15 15
26. NMR spectrum of N in a N2 with 0.1% 02 added at 625
and 950 G . . . . . . . . . . . . 113
1.5 15
27. NMR spectrum of N in a N2 with 0.1% 02 added at 2000,
3000, and 4000 G . . . . . . . . . . . 114

28. NMR spectrum of 5N in a 15N, wih 0.1% 02 added at 5000,
7000, and 9000 G . . . . . . . . . ... . 115

29. NMR spectrum of 1N in a 1N2 as a function of 02
concentration . . . . . . . . . . . 116

30. Calculated NMR spectrum of protons which are nearest
neighbors of 02 impurity molecules in solid methane ... 120

31. Possible superpositions of Pake doublet and paramagnetic
spectrum which could explain data at 950 and 7000 G. ... 122

32. Calculated ( --- ) and observed ( -- ) NMR spectra of
protons in NH Fe(SO 4)2 . . .. . . . . . 123
14
33. Quadrupcle transition frequencies of N and Zeeman fre-
quency of 15N as a function of magnetic field Ho and angle
9 between Ho and the molecular axis. . . . . . ... 130

34. Qualitative effect on energy levels when the proton Zeeman
transition energy matches a 14N quadrupolar transition
energy in SC(NH2)2 ...... . . . ... . . 131

35. NMR spectra of 1N in a N, enriched to 33% 15N at 5000 and
6000 G . . . . . . . . . . . . . 136
15 15
36. NMR spectra of N in a N, enriched to 33% 5N at 7000 and
9000 G . . . . . . . . . . . 137


viii





LIST OF FIGURES
(continued)


Figure Page
15
37. Calculated NMR spectra of N in a N2 enriched to 33%
15N at 5000 G. . . . . . . . . . . .. 138

15
38. Comparison of NMR spectra of N in a N2 enriched to 33%
15N with computer simulation at 5000 and 7000 G. . . ... 139

15
39. Nitrogen spin-lattice relaxation times TI in N2 (in
seconds) and '4N2 (in milliseconds). . . . . . . 145

40. Correlation times TQ and Tsr in 142 and N2 respectively 147
adT/t)1/2) 14
41. The reduced correlation time ( = T(kT/ Q vs. Tsr 149

42. Crystal structure of B nitrogen. . . . . .... 152














Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



NUCLEAR MAGNETIC RESONANCE OF 14N IN SINGLE CRYSTAL GLYCINE
AND OF i5N IN LIQUID AND SOLID N2

By

Lyle Milton Ishol

June, 1976

Chairman: Thomas A. Scott
Major Department: Physics

Quadrupole perturbed nuclear magnetic resonance (NMR) has been used

to determine the quadrupole coupling constant e2qQ/h, asymmetry para-

meter n, and electric field gradient (EFG) tensor in single crystal
2
glycine. The results at room temperature are e qQ/h = 1.190 kHz, n=0.505,

with the principal axis roughly parallel to the C-N bond direction. As

the temperature increased, e qQ/h decreased. These results are consis-

tent with earlier work, except there is no evidence of a significant

change in the orientation of the EFG tensor at higher temperatures as

reported earlier.
1"5 15
The N~MR spectrum of N in o-Dhase solid 1 N2 with paramagnetic 02

impurity added has been detterained as a function of magnetic field and

02 concentration. A classical Pake doublet was obtained at 650 G as

expected; however, the spectrum became increasingly asymmetric as the

field was increased to a maximum of 9 kG.. Increasing the 02 concentra-

tion broadened the line b;t. did not affect the asymmetry. Various






mechanisms may affect the line shape but none have been found to satis-

factorily explain the asymmetry in all quantitative aspects.

The NMR spectrum of 1N in an a-phase solid containing isotopic
15 15 14 14
molecular species in the ratio 15N : 15N- N: N = 1:4:4 has been obtain-

ed as a function of magnetic field and compared with computer simulations.
15
Besides the Pake doublet due to the N2 molecules, an asymmetric triplet

results from the N- 14N mixed molecule. In the latter case the 14N

nucleus, which experiences a quadrupolar interaction comparable with the

Zeeman Hamiltonian, produces a dipolar interaction at the 1N nucleus

which varies with the magnetic field in accordance with theoretical ex-

pectations. Additional effects occur due to level crossing and to the

asymmetry of the Pake doublet mentioned above.

The nuclear spin-lattice relaxation time T1 of 1N has been measured

in liquid 1N2 for the temperature interval 63 < T < 77 K and in the B-

phase solid for 38 < T < 63 K. The relaxation is attributed to the

spin-rotational interaction in both the liquid and $ solid. Any possible

discontinuity in TI across the triple point is obscured by scatter in the

data. When combined with previous T data for N2, it is concluded that

a rotational diffusion mode? developed for liquids may hold in the 8

solid, and that the -olecules reoriret more slowly in the liquid than in

the solid near the triple point.














CHAPTER I
INTRODUCTION



1.1 High Field NQR of 14N in Single Crystal Glycine


Nitrogen is an important atom in biological material, and nuclear
14
quadrupole resonance of 1N nuclei in biological molecules can provide

information about the structure of the molecules. Problems arise due

to the usually low atomic density and che low resonant frequencies

which combine to give a poor signal-to-noise ratio. Glycine, for ex-
14
ample, has a N nuclear spin density of one nucleus per five C, O,

and N atoms, not counting the five hydrogen atoms. Furthermore, there

are two inequivalent sites, reducing the density of those partici-
14
pating in a resonance by a factor of two. The 14N quadrupole coupling

constant is about 1200 kHz at room temperature and increases only

slightly at 77 K which, combined with the low density, results in a

weak signal that is difficult to observe. To date, the pure quadru-

pole resonance has not been observed to the knowledge of the author.

Three research groups, Andersson, Gourdji, Guibe, and Proctor

[1], Blinc, Mali, Osredkar, Prelesnik, Zupancic, and Ehrenberg [2],

and Edmonds and Speight [3], have reported the quadrupole coupling
2 14
constant e"qQ/h and asymmetry parameter n of N in glycine

obtained from quadruple perturbed NMR. References [l] and [2] also

reported the quadrupole coupling tensor, or equivalently, the direc-

tion cosines of the principal axes of the tensor. Edmonds and Speight








[3] used a powder sample and therefore were unable to obtain the com-

plete tensor.

Andersson et al. [1] used a wide-band Varian spectrometer and

were able to obtain enough data to deduce the complete quadruple

coupling or electric field gradient (EFG) tensor at room temperature.

They reported that the principal axis was roughly parallel to the

axis of the NH3 group. Blinc et al. [2] used pulsed double resonance

methods at a temperature of 1400 C, and reported the principal axis

was about 60' away from the direction obtained by Andersson et al.;

furthermore a much lower quadrupole coupling constant was deduced.

The purpose of the present experiment was two-fold. First, the

laboratory had recently acquired a superconductive magnet capable of

producing up to a 100-kG field and it was decided to employ it in an

investigation of the use of a strong magnetic field to overcome the

weak signal. Next, an attempt would be made to explain the differences

between the data of Andersson et al. [1] and those of Blinc et al. [2].



1.2 NMR in Liquid and Solid Nitrogen


Nitrogen is an important and convenient substance for NMR studies

for a number of reasons. Liquid nitrogen as a cryogenic fluid is

readily available at low cost and requires no special handling other

than that due to its low temperature. Aside from the attractive

cost factor, it is advantageous to have the sample and the cryogenic

fluid both the same, as it facilitates temperature and pressure con-

trol. A temperature region from about 50 K to 77 K in solid and

liquid phases is easily attainable using liquid nitrogen.







More important than the practical aspects mentioned above are

the physical properties. Two nonradioactive isotopes are available,
14 15
the abundant 1N with spin I = 1 and 5N with I = which has only

about 1/3% natural abundance. Three species of diatomic molecules

are therefore available, 1N2 15N- N, and 14N2

Only three nonradioactive elements have spin I = 1, viz., 14N,
7 2
Li, and H. Lithium is a metal and deuterium is such a light atom

that quantum effects are considerably more important than in the
14 14
heavier N. Therefore, N occupies a rather unique position. As

nuclei with spin ! have no quadrupole moment, nitrogen can be chosen

lt 15
with or without quadrupole interactions by selecting 14N2 or N2

respectively, and 15N-14N mixed molecules permit a study of unlike

spin coupling. Three very different spin systems are therefore

available with essentially the same lattice dynamics.

As implied by the name, nuclear spin-lattice relaxation depends

on the coupling between nuclear spins and the lattice. The possi-

bility of selecting either I = 1 or I = nuclei in essentially the

same lattice turns out to be even more convenient in spin-lattice

relaxation studies than the simple fact that the spins are different.
14
It happens that the quadrupolar interaction is dominant in N2 nu-

15
clear relaxation and spin-rotation interaction is dominant in N2

relaxation as 15N2 experiences no quadrupolar interaction. Hence,

two fundamental relaxation processes can be isolated in the same

environment. Furthermore, we will find the particular two processes

are uniquely related in that a change in molecular dynamics due to a

temperature change has opposite effects on the two processes.

It is also interesting to note that S nitrogen, stable from 35 K








to the triple point, 63 K, is a plastic crystal with low molecular

orientational order. Thus rotational and orientational dependent

properties in 6-solid nitrogen have some similarities to the same

properties in liquid nitrogen.

Finally, there are two crystalline forms readily obtained at low

pressures, B nitrogen mentioned earlier, and a nitrogen, a much more

rigid form, stable below 35 K. In a nitrogen, molecules undergo

librational motion about fixed directions in the lattice. The effect

of the librational motion for some purposes is to reduce the magnitude

of the interaction by motional averaging. The resulting line shape

is therefore unchanged except for an overall reduction in width due

to the motional averaging. We therefore can conduct line shape

studies of nuclei in molecules with like spin (I = ) or with unlike

spin (I= l,) in the same crystalline structure.

The purpose of the present work is to study the line shape of 15N

n 15N and in a mixture of 152 and 15-14N samples, and to study the

spin-lattice relaxation times in liquid and 6-solid 15N2 and thereby

fill a gap in existing nitrogen data. In so doing, contributions will

also be made to the understanding of fundamental processes, such as

spin-rotation interaction in solids, which will have applications to

other systems.















CHAPTER II
THEORY




2.1 General



A nucleus with a magnetic moment i in a uniform external magnetic

field H has an energy given by




E = -9*H (2.1.1)
o




The quantum mechanical Hamiltonian is the well-known Zeeman term




7Z = -u-H = -ytIH = -ytiH o (2.1.2)
Z 0 0 oz



->-
where I is the nuclear spin, y is the gyromagnetic ratio, and

H = zH .
o o

The Zeeman energy levels are


Em = = -v hH m ,
mD Z* o


E = -h:, m
m 0


(2.1.3)


where w = -yH .
o o




-6-


Transitions between levels may be induced by application of a

small rotating magnetic field, perpendicular to Ho,



H = H cost, H = H sinwt (2.1.4)
x 1 y 1




as a perturbation of H In practice, an alternating magnetic field

is applied which can be thought of as the sum of two counter-rotating

components. It will be seen later that only one component is effec-

tive in inducing transitions.

The perturbing Hamiltonian is


-iwt eit) (2.1.5)
S= -yh(H I + H I ) = -2YhH (I e- I e )
1 xx yy 1 +-



where the raising and lowering operators I+ = I + iI have been
x y

introduced. These operators have nonvanishing matrix elements only

between states for which m differs by one, so the only transitions

allowed will differ in energy by AE = hi Thus, the frequency of the
0

perturbing alternating magnetic field must be wo in order to induce

transitions. The counter-rotating component -w is 2o( off resonance

and can be ignored for most purposes. The resonant condition, w = 0o,

allows an exchange of energy between the spin system and the alter-

nating field, and this exchange can be observed electronically if the

net exchange is net zero.

The nucleus experiences not only the external field Ho, but also

fields, both electric and magnetic, due to its surroundings. Neigh-

boring nuclei with magnetic moments influence the nucleus of interest

through dipole-dipole coupling. Net electrostatic forces may be







exerted on a nucleus with a nonspherical charge distribution. While

a molecule is rotating, its charges cause a magnetic field at the

sites of its own nuclei, giving rise to spin-rotation interaction.

There are other interactions which are less important in the present

work, and will be disregarded.

An isolated nucleus would resonate at o providing no informa-

tion other than the Zeeman frequency. In practice, effects such as

those mentioned above modify the behavior of the nucleus by adding to

the applied field or by causing other forces on the nucleus. The

nuclei therefore resonate at different frequencies, depending on their

surroundings. The frequency spectrum therefore provides information

about the surroundings and about the interactions taking place.

In order for there to be a net exchange of energy between the

alternating field and the spin system, the spins must have a net ex-

change of energy with a third system, the lattice. This takes place

through the interactions mentioned above as well as through some other

interactions less important here, and the rate of the interaction is

characterized by a spin-lattice relaxation time T V

Much of the discussion below will follow Abragam [4], Andrew [5],

Cohen and Reif [6], or possibly other standard references, and will not

be specifically referenced at each step.



2.2 Intramolecular Dipolar Coupling


Consider one nucleus i at the origin P, and a second nucleus

j at r, with magnetic moments -. and p.., respectively. The magnetic

potential at P due to Pj is








(2.2.1)


4.. = .*V(I/r )
ij J ij


and the magnetic field is


H. = -V
1- ij


(2.2.2)


Performing the indicated operations results in


Hij
H..
13


S - 2 -3
= 3r. .( 'r. .)/r .J ri.
ij J 1] ij 1J


The dipolar Hamiltonian, obtained by letting H = H.. in Eq.
o 1J


(2.2.3)




(2.1.2),


-D =- -- 2 -3
ND -F11.*H. = [ ii.*r 3() r .)(i.r. .]r (2.2.4)
J13 J 1J 11i1


as Hi. interacts with the nucleus i the same way as H .
1j o


If an ex-


ternal field H = zH is now applied, where H >> H.., the total
o o o 1J

Hamiltonian of the two nuclei is


-h y... -
- Y.tH I. + --_3 .
] o jz 3 1 J
1i]


( + T +
3(.'ri..)(T .*r..) I
1 1] 3 1]
2I
r.. J


(2.2.5)


Letting


W -4-
W.. = I.1. -
.1 1 3j


3(I.-r..)( .*r.j.)
2
r..


7/ = -y.fhH I.
3 O 1iZ


(2.2.6)







it is customary to write


W.. =A+B + C+D+E+F
ij


where


A = I. I. (1 3cos20)
1z jz


B = -() (I I. +




C =-(3/2)(I Ij+




D = -(3/2)(Iz I._
1Z J-.


I I.j4)(1 3cos2 )



-iO
+ I. I. )sin0cos0e
jz i+



+ I. I. )sinOcos ei4 = C*
Jz L-


E = -(3/4)(I. I. )sin Oe- 2'



F = -(3/4)(iI s 2i = E*
3- 1-


(2.2.8)


3
The field (2.2.3) H.. is on the order of 1 G. whereas H is 10
1J o
or 10 G, so 1 may be treated as a perturbation of I'. Rescricting

the problem to like spin nuclei, the zero-order energy level is

E = -yhH (m. + m.), and according to first-order perturbation theory,
Z o i j
only those parts of the perturbing Hamiltonian which induce no net

change in M = m. + m. contribute to the energy. Inspection of Eq.

(2.2.8) shows only terms A and B fit this condition, and terms C

through F may be disregarded here. These terms actually permit weak


(2.2.7)




-10-


resonances at 0 and 2o frequencies, which are of no interest in the

present work. The truncated Hamiltonian, good to an excellent approxi-

mation, is therefore


2 2
= -YH (I. + I. ) + (-3cos2)[. I (3)(I I. + I. I.)]
0 Iz ]z 3 iz jz -+ J_ j+
r..
13
(2.2.9)

It is noted that term A can be interpreted as nucleus i experi-

encing the average field due to nucleus j, whereas term B is the

simultaneous 'flipping' of both spins, which can occur when i and j

are like nuclei and therefore in resonance.

There are four states for the case of two spin nuclei, repre-

sented by I++>, 1+->, -+>, 1-->. Term B has no diagonal matrix

elements in this representation; however, linea_- combinations may'-e

used to define a triplet state,



|+> = I++>


0l> = (1//I)(|+-> + i-+>)


I-> = --> (2.2.10)



and a singlet state, 10> = (1/I/)(l+-> |-+>). The singlet state

has no Zeeman energy and isn't coupled to the triplet state by either

/D or Z, and may be ignored.

Energy levels are readily calculated and are



E, = -yH + (2h2/4r3)(1 3cos2e)
1 0




-11-


E = (y2/2r3)( 3cos28)




E_1= yHo + (y2/4r3)(l 3cos28) (2.2.11)




In the remainder of this section only identical spin nuclei in

diatomic molecules will be considered, whose intramolecular separation

is significantly less than the intermolecular nuclear separation.

Thus, the i,j subscripts may be dropped, as the 1/r' factor insures

intramolecular effects dominate.

Two resonant frequencies may be observed at constant H ,


2
w' = E E= yi[H + a(l 3cos 2)]
-1 0 o



l Ew" = E E= yi[H a(l 3cos2e)] (2.2.12)
0 1 o



where a = 3yf/4r3. Or H could be varied and = w' = w" held fixed,

the resonant condition being found at two values of Ho,



H = H* a(3u2 1) (2.2.13)




where H* = w/y and u = cose. H* is the constant central, or ZPeman,

field, and u is the cosine of the angle 6 between Ho and r.

In a polycrystalline sample, any orientation is equally likely,

and for each orientation, there will be two resonant lines, above or

below H* by an amount




-12-


h = H H* = (3u2 1) .(2.2.14)
o



The spectral distribution is just a plot of the density of lines (or

density of orientations) versus h, so the density of orientations must

be known as a function of u.

A sphere can be defined by all possible positions of the second

nucleus if the first is held fixed, and the density of positions is

constant over the surface of the sphere. Remembering rcosO = ru is

the projection of r along H = zH it is noted that any value of u
o o

defines a circle on the sphere, and a plane containing the circle is

normal to z at a distance ru above the center of the sphere. Another

plane and circle are defined by u + Au, the planes being separated by

rAu. The surface of a sphere between two planes separated by d is

2Trrd, so the surface area between the planes defined by u and u + Au
2
is 2-r Au, independent of u itself. Therefore any value of u is

equally probable, and the spectrum can be considered a plot of the

density of u's versus h, or du/dh.

From (2.2.14)



u ( + I) (2.2.15)




and the spectral distribution, f(h), is



f(h) = ( + 1) (2.2.16)
dh [a


In the case of 1N,'y =:-1Y!, so (2.2.16) becomes




-13-


'ii

I' jl

it
\1.1


/8i;r


/I


L_


tlu!


21a21 h-


Figure 1. Pake powder pattern doublet. The theoretical unbroadened
spectrum (dashed line) is the sum of the two components
(dotted lines) resulting from the transitions shown which are
for the case of a negative gyromagnetic ratio. A typical
spectrum including intermolecular broadening is shown by the
solid line.




-14-


f(h) _- + l)- (2.2.17)



the upper (lower) sign corresponding to the j-> -+ 10>(10> -* I+>)

transition. Front (2.2.14), we note



-21al < h < IaI J-> \0>



-IaI < h < 21j I I0> + |+> (2.2.18)



The two curves overlap in the region -jaj < h < a|l, and result

in a spectrum indicated by the dashed line in Fig. 1. Intermolecular

dipolar coupling causes each line to be broadened, resulting in a

curve such as that shown in Fig. 1 by the solid line. The spectrum

shown in Fig. 1 is known as the Pake doublet after G.E. Pake [7]. It

should be noted that the curve will be reversed if frequency units

vice magnetic field units are used, in addition to a scale factor set

by the gyromagnetic ratio.



2.3 Nuclear Quadrupole Interaction


The charge distribution in a nonspherical nucleus exhibits an

orientational-dependent interaction with the surrounding electric

field due to the external charge distribution. Classically, this may

be written as



E = j p (r)(')d3r (2.3.1)


where p(r) is the nuclear charge density and A(r) is the potential due




-15-


to external charges.

One convenient method of treating the problem is a Taylor series

expansion of i(r) about the origin:



E = 4(0) f pd3r + I f x pd3r+ r+ I f x x Pd3r +
a a, a
(2.3.2)

where x (a = 1,2,3) stands for x, y, z respectively, and where




0, -. etc. (2.3.3)
r=O a r=O



The first term in the expansion is the electrostatic energy of

the nucleus taken as a point charge, and is of no interest as it is

just a constant with no orientational dependence. The second, or

dipole, term vanishes as nuclear theory tells us the wave function has

definite parity so p(r) = p(-r), and since x changes sign in opposite

quadrants, the dipole integral vanishes. For the same reason, all the

even terms vanish.

The terms in the expansion are referred to as monopole, dipole,

quadruple, octopole, hexadecapole, etc., as they are identified with

the type of charge distribution which would cause the corresponding

potential. We have shown the dipole, octopole, and every other higher

term vanishes and the monopole term exists but is of no interest.

This leaves the quadrupole, hexadecapole, and higher terms:


1 3
E = 2 f xx pd3r + hexadecapole and higher (2.3.4)
2 a,B 8




-16-


Now 6(0) is roughly e/r where r is of the order of the radius
ee
of an electron orbit, so the monopole term p(0) f od r is on the

order of Ze2 /r The next nonzero term, 32 /3x 3x f x pd r, is
3 2 2 2 2
on the order of (e/r )(r Ze) = (r /r )(Ze2/r ) where r is on the
e n n e e n
2 2 2 -8
order of a nuclear radius. As r /r e 10 the quadrupole term is
n e
_p
on the order of 10 times the monopole term. Likewise, the hexa-
-8
decapole term is on the order of 10 times the quadrupole term and

so on for succeeding terms. Thus, the quadrupole term is the only

one of interest within the resolution of most spectrometers.

If we replace p with its quantum mechanical equivalent p and
op
define the symmetric tensor Q' f x x p d r, we have the quadrupolar
aa a B op
Hamiltonian to an excellent approximation,



I =1 X 4 Q^ (2.3.5)
a,B



It is convenient to substitute the symmetric, traceless tensor



Q = 3Q 6a QG' (2.3.6)



in Eq. (2.3.5) which then becomes


1 L ,+ Q" Q) (2.3.7)
a, ca a


The potential is due to charges outside the nucleus, and therefore

satisfies Laplace's equation, so 6 is traceless and hence the second
Sin Eq. (2.3.7) vanishes. Here e re selecting any electronic
term in Eq. (2.3.7) vanishes. Here we are neglecting any electronic




-17-


charge distribution which is overlapping the nucleus as it is very

small, especially after subtracting the spherically symmetric part

which is of no interest in orientational studies. We are therefore

left with



= a (2.3.8)




From Eq. (2.3.6) and the definition of Q'~, we have



Qus = f (3x x- 6 s )op d3 r (2.3.9)
up a 6 aS op



Putting in p (r) = o q(r r.), and performing the integration
op k
results in



Q = e L (3x kX 6 r2) (2.3.10)
a k akxk rae
k



where e is the charge of a proton and the sum is over the protons in

the nucleus. The expression in the parentheses could be written



3(x x + x xak) ),
Q = (2.3.11)
oBk 2 Q3 k



We now look at another operator



3(i I + IeT -2
I (op) = -- 6 I- (2.3.12)
as ~2


and assert




-18-


= C (2.3.13)




We notice the two operators are constructed in the same way and

that QaB is a linear combination of the Q ak's. The two operators

obey three conditions: 1) they are symmetric, 2) they are traceless,

and 3) they transform under rotation of coordinate axes in the same

way as second order spherical harmonics. This last condition is equiv-

alent to saying they are second rank tensor operators, or that they

satisfy the same commutation rules with respect to I. These three

conditions are sufficient, using group theory, to prove Eq. (2.3.13)

[8]. A more sophisticated proof calls on the Wigner-Eckart theorem

which shifts the complexity to the proof of the theorem. A simpler,

more direct proof, but also more tedious, comes from direct matrix

multiplication [8].

Remembering Qa. was related to the quadrupole portion of the

nuclear charge distribution, we define the quadrupole moment Q by



eQ = (2.3.14)




which is used in Eq. (2.3.13) to obtain


2 2
eQ = C CI(21 1) (2.3.15)




from which we obtain C = eQ/I(2. -1).

Equation (2.3.8) can be rewritten




-19-


1 eQ 3 -
S= 1 ) (2I + II ) 6 I1
Q 6 1(21 1) a 5 as[ aB


(2.3.16)


We can diagonalize aB and hereafter assume we have done so and

use a,0 without primes to indicate the coordinate system in which as

is diagonal. Thus,


(2.3.17)


u =1 eQ Q (31I2 12)
6 1(21 1) aa a
cx


Labeling the axes is arbitrary, so we define a principal axes

frame by


I zz -> yyl i xxl


(2.3.18)


and in this frame


6 1 eQ ) x(312 2) 2
Q 6 1(21 1) xx -I ) yy (3


I +2 -9 2)
T-) + (3I2 1


(2.3.19)


With the help of Laplace's equation, 4xx + yy + zz = 0,


eQ 2 2 2 )

Q 4 1(21 1) zz z x yy x vy


(2.3.20)


2
Introducing A = e qQ/4I(2I 1), n = (y ')/ 4 Z, eq = zz' and
the raising and lowering oeratrs = (Ix i we obtain
the raising and lowering operators 14. = (I iI ), we obtain
x V


O/ = A 31 I2 (n/2)(I1 + I)
Qz +


(2.3.21)




-20-


If we were to calculate matrix elements for I = 0 or I = using

Eq. (2.3.17), we would find all terms vanish, which means the electric

quadrupole interaction exists only for I > 1. Semiclassically, we

would expect no orientational dependence for I = 0 which has no z com-

ponent, and I = has two states, +- and -, which correspond to a

reversal of spin direction but no change in charge distribution and

again no orientational dependence.

We have introduced the electric quadrupole moment Q, the electric

field gradient (in the direction of maximum gradient) q, the asymmetry
2
parameter n, and a quantity A = e qQ/4I(2I 1). We notice n = 0 when

}xx = yy, which occurs in spherical, cylindrical,or cubic symmetry,

hence the name asymmetry parameter. The maximum value n can have is

unity when x = 0 and y = |zz We also note thac q and Q can

not be determined individually in a nuclear quadrupole resonance ex-

periment as they appear only as a product. The quantity e qQ/h is

called the quadrupole coupling constant, and it is convenient to

include it in the constant A.

The quantization direction of the nuclear quadruple interaction

is set by the z principal axis which is the direction of the maximum

electric field gradient. If other interactions are present, they in

general will have different quantization directions. In the presence

of an external magnetic field, for instance, the Zeeman Hamiltonian

yZ (discussed in Sec. 2.2) is diagonal in a coordinate system with

the z axis parallel to the external field which in general does not

coincide with the z principal axis. Low magnetic field and high

magnetic field cases can be treated by first-order perturbation theory.

The intermediate case with the two interactions of comparable




-21-


magnitude must be solved exactly and results in a much more complex

orientational dependence. The low magnetic field case is of little

interest in the present experiments and will not be discussed. The

high field case will be treated in Sec. 2.4 and the intermediate re-

gion in Sec. 2.5.



2.4 High Field Nuclear Quadrupole Interaction


If a system of identical spins is placed in a magnetic field,

the well-known Zeeman Hamiltonian is



Z = -YH I (2.4.1)
Z oz



from Eq. (2.1.2), the prime being used to distinguish the coordinate

system in which /Z is diagonal.

If nuclear quadrupole interactions are also present, we must add

the quadrupolar Hamiltonian, Eq. (2.3.21):


,2_-2 (1 + 1
= Z + / = -ySHo I + A[3I2- 2 2 ( (2.4.2)
Z Q oz z 2 + j



The unprimed coordinate system refers to the principal axes frame.

Restricting the discussion to the high field approximation where

first-order perturbation theory is valid for we are only inter-

ested in the diagonal matrix elements, . It is convenient at

this point to return to Eq. (2.3.15) which, using xx + + zz = 0,

becomes




-22-


e = e- ( + 2 + I2) (2.4.3)
Q 21(21 1) xx x yy y zz z



Writing the direction cosine of x with respect to x' as a ,

whare x a = 1,2,3 stands for x, y, z, and x', B= 1,2,3 stands for

x',y', z', we have



I a= SI (2.4.4)



Thus,


2 2 2 22 2 2
I = a T + a I + a1 + cross terms (2.4.5)
Ix = allx' a l2 y + 13 z


2 2
and likewise for I and I We are considering only diagonal matrix
y z
elements and cross product terms like

vanish and may be dropped for our purposes.

Putting Eq. (2.4.4) in Eq. (2.4.3) results in



eQ 2 2 2 2 2 2
S= 2 (a I + a I2 + a 1 )
Q = 21(21 I) xx11 x' 12+ y' + 3z'



2 2 2 2 2 22 2 2 2 2
yy 21 x L2 y 23 z zz 31 x' 32 y' 33 z'

(2.4.6)

It is easy to show that


2l,2 7 -m2 2 _
= = = [(I + 1) m2]
y' (2
(2.4.7)




-23-


As the only allowed transitions are those for which Am = +l, we

calculate


= -(2m 1)
x x


7 2

z z


= (2m ) .


The transitions AEQ(m )

evaluated, yielding


=

- 1> are then


AE = eQ(2m 1)
Q(m m 1) 21(21 1)


2 2
al a
12 2
2 2


2
Sa2 a21
131 +yy 2


2
a22 2 1
- -^ -I-a


2 2
Sz( a31 a32
zz 2 2


r a33


eQ(2m 1) 3 2 3 2 3 2
S21(21- 1) 2xxa13 + 2yya23 + 2zza33) (2.4.9)



From Eq. (2.4.8) we see that EQ( + 1 -m) = -EQ(m m 1) and from


Sec. 2.1, AE = AE(- = -YhHo.
Z (i -L m. 1) Z(-M + 1 --m)


Hence the ob-


served transitions are in pairs,



AE+ = AE AE (2.4.10)




where the upper sign signifies the (m m 1) transition and the lower

sign the (-m + 1 -m) transition.

We are not concerned with the Zeeman term which depends only on


(2.4.8)





-24-


Projection
on z axis
+ or -.


Projection
on z axis
always +.


a. Vector


b. Tensor


/
/


3600 9-


c. Projections of vector ( -- ) and tensor
component ( --- ) on z axis.




Figure 2. Comparison of z-axis projections of rotating vectors and
tensor components. The period of the tensor component is
that -of the vector.




-25-


H and the nuclear species. It is sufficient to measure the difference
o

in energies between the two transitions, which is 2AE In frequency

units,



3eQ(2m I) 2 2 2
2Av = (e(m-1) a + a 2 + a (2.4.11)
2v =21(21 1)h xx 13 +yy23 + azz 33



This is a convenient point at which to note the physical inter-

pretation of the frequency separation 2Av in terms of the orientation

of the external field H with respect to the principal axes x, y, z.
o

We recall a13 is the direction cosine of the x principal axis with

respect to the direction of H (which is parallel to z') and likewise
o
2
for a23 and a33. Thus, x a3 is the magnitude of the x component of

the electric field gradient tensor times the square of the cosine of

the angle between the x component and Ho, and so on for the other

terms. We note xx and Ly have the same sign, which is opposite to

the sign of Qzz

The observed frequency shift Av is proportional to the sum of

the three 'projections', remembering the z component is opposite in

sign to the others.

We also note heir the significance of the square of the cosine.

If the EFG components were simple vectors which have a direction and

sense (or sign), we would expect any effects to reverse signs every

1800 as shown in Fig. 2. Noting in Fig. 2(c) that the tensor 'pro-

jection' goes through two complete cycles in a 3600 rotation, we

expect a 20 to appear which will come from the cosine squared factors.

In general, the orientations of the principal axes are unknown

as are the magnitudes of the 's. We therefore express the EFG'
c x





-26-


tensor in the lab frame. Using a B, as before and the fact that aB

is diagonal (a = 6 a ) we have



4x'x' = (a- )aWyya6 = a.y6 Y66a6
a y,6 Y,6



= C a6 66a6 (2.4.12)



and therefore


2 2 2
z'z' = a63663 = a13 xx + a23 yy +33zz (2.4.13)




Finally, Eq. (2.4.11) becomes



A = 3eQ(2m 1i) = K (2.4.14)
2 21(21 1)h z'z' Kz'z



with the definition of K obvious.

We are interested in finding e qQ/h (or A = e2qQ/4I(2I 1)), n,

and the orientation of the principal axes. By measuring 2Av at

enough different orientations and using the transformation Eq. (2.4.13),

the desired quantities can be worked out. Equation (2.4.13) is not

very convenient, however, a- the ai3, a23, a33 terms are different

for each orientation. We therefore pick a crystal frame X, Y, Z which

is known and may contain some of the crystal axes. The transformation

from the lab x', y', z' frame to the X, Y, Z frame is known, and the

transformation from the X, Y, Z frame to the x, y, z principal axis

frame will be constant as both frames are fixed in the crystal.




-27-


The procedure is to find the EFG tensor in the crystal frame by

experimental methods, then diagonalize the tensor. Diagonalization

will yield the EFG tensor in the principal axis frame as well as its

orientation with respect to the crystal axes. Volkoff, Petch and

Smellie [9] outlined the procedure in 1952 and it has become a rather

standard method since then. The Volkoff method more recently was

shown to have possible ambiguities [10]. Furthermore it requires

more data than is necessary in many cases as discovered in the present

work (discussed in Chap. 4) and as recently reported by El Saffar

[11]. Nevertheless, the Volkoff method provides a very convenient

starting point and it is easy to modify it to reduce the quantity of

data required.

The tensor


yP'
'Xx XY 'Xz


XY YY :YZ (2.4.15)


xz Yz zz



is symmetric as displayed and has only 5 independent elements using

the vanishing trace property. Diagonalization gives



xx 0 0 -eq(l-n)/2 0 0


0 yy 0 = 0 -eq(l+n)/2 0 (2.4.16)


0 0 (zz0 0 eq
z.Z1


and the direction cosines of the principal axes. We recall the




-28-


H = H z
0 0


-y ,,X



//






z


X = y'
Y z'coseX + x'sinOX
Z =- -z'sinO + x'cose6




Figure 3. Transformation between the lab axes and crystal axes during at
X rotation. Cyclic permutations of X, Y, Z result in the
transformations for Z and Y rotations.




-29-


elements all contain the factor K = 3eQ(2m 1)/21(21 l)h so only
2
e qQ is actually determined, rather than eq and eQ.

We start experimentally by selecting crystal axes X, Y, Z and

aligning the crystal in a magnetic field H = z'H with X, Y, Z paral-
C o
lel to y', z', x' initially. The crystal is then rotated about the

X = y' axis through an angle 0X and the frequency separation 2AvX

observed and plotted versus 0X, the plot being termed the X rotation

pattern. Figure 3 shows the relation between the crystal and lab

frames for an X rotation.

Using the transformation equations in Fig. 3, we obtain


2
SZ' = (coseX a sine ) a2
z'z' ,z Xz
z z



= (YY + ZZ) + !(4YY ZZ)Cos2x X YZsin2 X (2.4.17)




which is put in Eq. (2.4.14), yielding



2AX = AX + BX 28X + C sin2eX




= AX + (B + C) cos2( 6 (2.4.18)



where


AX = -K( + K = -K


BX= K($ =Z)




-30-


CX = -K yz



Cx
X
tan2x = Bx
X B x


(2.4.19)


Thus the X rotation yields the diagonal elements and i Z. Similar

equations may be obtained by cyclic permutation, so XZ and
be obtained from Y and Z rotations, respectively.

Diagonalization of the EFG tensor requires solving the cubic

equation


3
y ay- b = 0


(2.4.20)


where


2 2 2 + 2
a = K( + + YY YYZZ ZZ4X
XY YZ XZ XX YY YY ZZ ZZXX



3 2 2 2
b = K (4 + 24 4 4 4 4 tp ( 4 4 ).(2.4.21)
S XXYYZZ XYYZ XZ -fXYYZ iYY XZ ZZ X



We find


Yn 2(a/3)cos(a 2 rn/3), n=1,2,3


(2.4.22)


cos3a = (lb /2)(3/a)3/2


0 < a < 1w/


Y3 = Kpzz




-31.-


These choices insure



Y3 = 131' ,3 y IY2 Ivll (2.4.23)




and finally we obtain



S= (Y1 Y2)/3 = 3 tana (2.4.24)



and



le2q/h) = eQ zz/hI = [21(2I 1)/3(2m 1)].IK I zz (2.4.25)




The diagonalization process yields the direction cosines of the

x, y, z principal axes with respect to the X, Y, Z crystal axes:



n n n 1,
Din D2n D 2 + 2 2 ) (2.4.26)
In 2n 3n (D + D + D )
in 2n 3n



where XA' P1' V1 are the direction cosines of x with respect to X, Y, Z

and likewise for n = 2 and 3.

The relative signs of A v for any n are fixed, but all
n n n
three signs may be reversed without reversing the signs for the other

values of n. Thus, a right- or left-handed coordinate system may be

chosen.

Volkoff notes that only five readings are required in the most

general case, such as 86 = ey = 8 = 450 and any two of X = 8 =

86 = 00. This would require a very accurate knowledge of the




-32-


orientations and in the case of inequivalent sites, identification of

the transitions might be difficult.

Additional data increase accuracy and may be helpful in detect-

ing orientation errors and in identifying the transitions. Some of

these effects as well as the single rotation method will be discussed

in Chap. 4.



2.5 Intramolecular Dipolar Splitting of the NMR Spectrum of 1N in

N- 14N Mixed Molecules
15 14
We next consider the case of N- 14 mixed molecules with nuclear

spins I = and I = 1 for the two isotopes, respectively. The total

Hamiltonian is taken as



V = !(15) + 71(14) + 74(14) + 1D(15-14) (2.5.1)



15 14
The region where the 1N Zeeman and 1N quadrupolar energies are of
15 14
the same order will be considered. The N- 14 intramolecular di-

polar energy is much less and may be treated by first-order pertur-

bation theory. We note at this point that the asymmetry parameter n

is zero for N2.

We proceed in the following manner. The NQ (14) term in zero

field gives rise to two energy levels corresponding to m = 0 and

m = 1 with state functions 1i>, 10>, 1-1>. Adding an external mag-

netic field H removes the degeneracy and the state functions become

linear combinations of 11>, 0j>, 1-1>:


n>> = Uin l>> + an > + 3nl- >
-n ln 2n3ni-


(2.5.2)




-33-


Working in the principal axes (PA) frame, we calculate the a 's
mn
using l (14) and Z' (14). Next, '1 (15) (with state functions I+-_>,

1-> in a frame where H = zH ) is expressed in the PA frame, and
o o
the state functions are linear combinations of I+>, I->:



C+> = 8l\+-> + S2-> (2.5.3)



To first order, the total state functions are taken as products

of the exact uncoupled state functions,



I n> = In I+> (2.5.4)




_VD(15-14) is treated as a perturbation causing a first-order
14
shift which is dependent on the state of the 14N nuclei.

From Eqs. (2.3.21) and (2.1.2) we have



Q,Z(l4) = 7Q(14) + Iz(14)



2 -2 ( -2
A(3I I ) yi-H (2.5.5)
z o



where x, y, z is the PA frame.

As the molecule is axi.illy symmetric (and therefore n = 0), we

may arbitrarily select the orientation of x and y and therefore choose

x in the plane of H and z. Ths., H = zH ccsO + xH sine where 6 is
o 0o o o
the angle between H and z, and
0





-34-


S 3- 2
N4 (14) = A(3I1 I
iQ,Z z


yh(I H cosa + I H sin9)
z o x o


= A(31 I ) yhl H cos6
z z o


"hH sinG
0
o
2


(I + I_). (2.5.6)
+r -


Introducing


2yhH
L = 2
e qQ


E sin8
T u = cos, v =--
2A'
r/2


we obtain


QZ (14)


= 2A(312 -2 )/2
z


- Lul Lv(I + I )/
z + -


Matrix elements are calculated and displayed in marrix form


Lu

= 2A -Lv

0


-Lv 0

-1 -Lv

-Lv + + Lu


The secular deteniinant



I/ Ej = o



results in the cubic equation



T3 T(3/4 + L2) 1/4 (1/2)L 2(3u -


(2.5.9)


(2.5.10)


1) = 0 (2.5.11)


(2.5.7)


(2.5.8)


2 ]




-35-


where we have used T = E/2A from (2.5.7). The solutions of a cubic

equation



3
T -- a1T a = 0 (2.5.12)
1 o


are



T_i = 2(a /3) cos(4/3)



1
TO = 2(a1/3) cos0(/3 + 120)




T1 = 2(a /3) 2cos(4/3 + 2400) (2.5.13)


1.I

where cos4 = (a0/2)(a1/3) / Parker [12] has tabulated T 's for
n

various angles, external fields, and spins.

The assignment of subscripts to the reduced energies T is made
n

by examining the behavior of T in the limit of small H where -1, 0, 1
n o

are good quantum numbers. In the region of interest, linear combina-

tions must be used as expressed in Eq. (2.5.2) which can be written as

a column vector,



r
SIn


n> = an (2.5.14)


"-in


In our notation, |i',' > = E > becomes
n n 'n







Lu -Lv


-Lv -1 -Lv


0 -Lv + Lu


which can be solved

Introducing


2 2 2
for a 's using a + a + =
mn In On -In


Cin = (. Tn Lu), C = Lv, Cin



2 2 2 2 2 2 ,
C = 2C + C C2 + C C )
n In On in n -In On


= (- Tn + Lu),


(2.5.16)


the solutions are



an = C nC ]n/C, a On = Cln CI /C, 0. = CO C /C (2.5.17)
In On -In n On -In In n -In On In n


The complexity of these rather simple appearing equations can
2
only be appreciated by expressing them in terms of H e qQ, and 8;

however, the substitutions in (2.5.7) and (2.5.16) greatly simplify

the calculations.
14
Up to this point only 1N has been considered. From here on we

must add a 14 or 15 label to avoid confusion. The Zeeman Hamiltonian


z (15) expressed in the PA frame has a form similar to Z (16) in Eq.

(2.5.6):



y(15)tH sine
o z 4
(2.5(15).1)co (15) -
(2.5.18)


-36-


an I


aOn


^In


= Tn


aln


"On


a-ln


(2.5.15)




-37-


In the frame with H parallel to the z axis, the exact states

are l4->, I|->, but here H forms an angle e with the z axis and

linear combinations of the exact state functions must be used:



i+> = 81 > + 82 ->




61
(2.5.19)
2+




in column vector form.

Letting B = ()Y(15)hH cos6, C = ()Y(15)tH sin6, Eq. (2.5.18)

becomes


W(15) = -2BI(15) C[I(15) + I (15)]



-B -C
(2.5.20)
-C C



in matrix form, and


-B -C 81. 1

= +y(15)AH (2.5.21)
-C C I 82 B2



2 2
This equation together with the normalization condition, 61 + 2_+ = i,

can be used to obtain




-38-


8
+ = cos- ,
1+ 2= -si




1- s2


8
0
cos-
OS


8
sirn-
2


.8
S2 = sin-



8
82= co
2- 2


2
--sin

2


We can now form the product functions | n > Eq. (2.5.4).

We next consider the perturbation term,




()'%2 5 (1
lD(15-14) 3 5)I(14) 2
r r
(2.5.24)


which comes from Eq. (2.2.5). In the PA frame, r/r = z, so



7/ (15-14) = D{(1/2)[1 (15)I_(14) + I_(i5)I+(i')] 21 (15)1 (14)}

(2.5.25)


where D = y(15)y(1L)T2/r3. We recognize the B and A terms, respec-

tively, of Eqs. (2.2.8) with 6 = 0, and note all the other terms vanish

due to a sinO factor. We recall the angle 0 in Sec. 2.3 was between
-*-
r and the z axis which is taken to be zero here.

We are interested only in the first-order energy shifts due to

D!(15-14), given by the diagonal matrix elements .
D3. t l


(2.5.22)


i > =


(2.5.23)




-39-


Expressing Eq. (2.5.20) in matrix form,


0 1 0 c -
0 0 0
D 001
E 00 1 0
0 0 0


1
+ I
v2T


0 0 0' 1 0 0 0
1 0 0 0 0
0 1 0 0 0 0 0 -1 0 -1


(2.5.26)


Diagonal elements can be calculated


(aln,a0n, -1n)


01 0 an

0 01 a-
On

'OOOJ -ln


0 0 01 1
(B +B +)
1- 2- i oj 0^
$I-J


= (alna0n + a0na_-n) (l 21)



and so on for the other terms. The result is


AEn = D (a0nan + a0na_-)(BI + ) (ain
n = On In On -In 1a2) 2n


D (na + a a ) (I,
On In On -In r I


2 2
-i.n' 1+


2 1 O
- a -)cos9
-In 0


(2.5.28)


-- -AE
n.


The 15N + transition will occur, to zero order, at

hwi = YTiH which is of no interest and will be shifted by AE+ AE =
c o n+ n-
2AEn+. We therefore calculate only



2 AE,+ D '2(aoa + C a 0 )sine 2(a a2 )cos6 (2.5.29)
n+ L On In Cn -In -In


(2.5.27)


HD(15-14) =


-2]




-40-


It is useful to examine the 6 dependence of 2AEn+. We note from

Eqs. (2.5.11) and (2.5.12) that a and al have the same value at 0

and 180 6, and thus 4 and subsequently T_1, T, T1 also have the

same value. Therefore, from Eqs. (2.5.16), C n(0) = CIn(180 e),

COn (6) = C n(180 e) and C (e) = C (180 6). Using these equations,
On On n n
we find from Eqs. (2.5.17) that a ln() = a _n(180 6) and

aOn(6) = a0n(180 6). We now inspect Eq. (2.5.29) and note (a0 aln+

a0na- n) and sine have the same value at 6 as at 180 6, while both

In -In
value or sign. Thus 2AEn+ needs to be calculated for the first quad-

rant only, as it merely repeats the same values in the second quadrant.

As the sample was polycrystalline, we again must work out the

powder pattern spectrum. In Sec. 2.2 it was shown that any value of

cosO was equally probable so the unbroadened spectrum was the 'density'

of u(= cosO) versus h, the dipole contribution to the field, or

du/dh. It was easy to express u in terms of h, and thereby obtain an

analytic expression.

In principle the same procedure would work here, but it is not

possible to express u in terms ofv (equivalent to h in Sec. 2.2). It

is possible in both cases, however, to numerically calculate v (or h)

for various 6's and weigh the results by a factor depending on the

probability of each 8 occurring.

We center a sphere on the origin and note that a given 6. and Ae
1
define an annular area on the surface of the sphere given by



A.; = (lergth)(width) = (27rsin0.)(rAe)


= (constant) sinO. (2.5.30)




-41-


in the limit of srall Ae. The intensity of the signal at frequency v

due to molecules with orientation 9. is proportional to the number of

molecules with that orientation, or sinE.. We therefore plot sini

versus v to obtain the unbroadened spectrum, or convolute with a

Gaussian function to obtain the intermolecular broadened function.

A computer is used to perform the calculations, taking values of

0. between 0 and 900 with a sufficiently small Ae to adequately repre-

sent the smooth curve.



2.6 Nuclear Spin-Lattice Relaxation


2.6.1 General


The resonant condition is detected by noting the energy absorbed

in driving nuclear spins to a higher energy state or by noting th&

energy given off when they return to a lower energy state. The radio

frequency field H1, discussed in Sec. 2.1, is equally likely to induce

transitions either way which would result in zero net energy exchange

if there were the same number of spins in the two states involved.

In practice, the population of the lower energy state is slightly

favored according to the Boltzmann distribution if the system is in

thermal equilibrium. Therefore, a resonant rf signal will give up a

net amount of energy to the system as long as there are more spins in

the lower energy state.

It can be seen that the rf signal will tend to destroy the popula-

tion difference by inducing transitions from the more highly populated

state at a faster rate. The experimental fact that the system returns

to the equilibrium distribution, perhaps at a rate fast enough to over-

come the tendency of the rf signal to destroy the equilibrium condition,




-42-


is evidence that there must be some other mechanism which couples with

the spins.

We find there are several mechanisms by which the spin system can

give up energy to the lattice and thereby return to equilibrium. The

rate at which this occurs is termed the relaxation rate and its recipro-

cal is the spin-lattice relaxation time T1.

Taking the simple condition of a spin system in a magnetic field

H where H is parallel to the z axis. we see the population difference
o o

results in a net magnetization M parallel to H which has some equili-
z o
brium value M The magnetization M approaches M at a rate propor-

tional to the difference M M resulting in the exponential equation
o Z


M (t) = M (1 e-t/l) (2.6.1)
Z 0

where t is measured from the time at which the spin order was completely

destroyed (M (0) = 0).
z
The spin system exchanges energy with the lattice in several ways.

For instance, in Sec. 2.2 it was noted that the magnetic moment of one

nucleus produces a magnetic field at the sites of neighboring nuclei.

If the nucleus is undergoing some type of motion within the lattice, it

will produce a changing magnetic field at other sites. We might imagine

this varying field as a sum of its Fourier components. If some of the

components are of the appropriate frequency, they can induce transitions

in neighboring spins, allowing the nonequilibrium spin system to return

to equilibrium. This relaxation towards equilibrium through a coupling

of the spins with the lattice through the field produced by nearby

nuclear dipoles is appropriately called dipolar relaxation.

A nucleus having a nuclear quadruple moment in an electric field




-43-


gradient which is time dependent due to motion within the lattice exper-

iences nuclear quadrupole relaxation. Considering the magnetic moment

of a molecule as a whole, a time-dependent field is produced at the

nuclear sites within the molecule itself due to the rotational motion

of the molecule which produces spin-rotational relaxation.

There are many other relaxation processes which are of no interest

to the present experiment. It might be that several processes simultane-

ously contribute to the relaxation and the observed relaxation rate is

the sum of the rates due to individual processes,


1/TI = 1/TID + I/T1Q + 1/Tlsr + ... (2.6.2)

where TD TQ, Tsr are the dipolar, quadrupolar, and spin-rotation

relaxation times, respectively. It may be relatively difficult in such

a case to determine the relaxation times individually.

The molecular motion responsible for producing the fluctuating

electric or magnetic fields at the site of a nucleus must have frequency

components in the spectral density at resonance and must be significant

enough to induce transitions. In the case of dipolar and quadrupolar

relaxation, a reorientation on the order of one radian at the appropri-

ate rate is required, whereas spin-rotation relaxation involves only a

change in angular momentum. In order to relate this molecular motion

to the temperature of the sample, we use a quantity called the correla-

tion time which is something like the time required by the molecule to

undergo the appropriate change in orientation or angular momentum. The

quadrupolar correlation time %: is given []3] by the integral of the
QQ
autocorrelation function of P2(cos6) where 6 specifies the orientation

of the molecule, and the spin-rotational correlation time sr is given

[13] by the integral of the time: autocorrelation function of molecular




-44-


Low T
Long TQ
W0Q >> 1
oQ


High T
Short TQ
STQ << 1
oQ


High w 0



Low w \


Temperature dependence of quadrupolar relaxation time.
(a) Spectral density of correlation time versus frequency.
(b) Qualitative dependence of T1Q versus temperature. Fre-
quency independence of TIQ at high temperature is a result of
the flatness of the high temperature curve in the upper figure


Figure 4.




-45-


angular momentum.

As a sample cools, collisions are more frequent due to increased

density. A molecule therefore requires more time to change orientation

through a large angle as the random collisions cancel each other out to

a greater extent. The correlation time T therefore increases with

decreasing temperature. On the other hand, a single collision can even

completely reverse the angular momentum of a molecule, and therefore,

increasing the frequency of collisions reduces the spin-rotational cor-

relation time T
sr

The reciprocal of the correlation time is something like the fre-

quency of the changing electric or magnetic field. If this frequency is

very high so o r < 1 where w is the Larmor frequency of the nucleus,
o o

there will be frequency components over a very wide range, resulting in

only a small proportion near w The mechanism will be fairly ineffec-

tive under these conditions. On the other hand, if the correlation time

is very long, it corresponds to a very low frequency, and rT >> 1.

Most of the frequency components will be very low, and again the mechan-

ism will be fairly ineffective. Between the two extremes, where m r T 1,

the frequency is on the order of the Larmor frequency and the relaxation

mechanism attains maximum effectiveness and T is a minimum.

Figure 4 illustrates the case of quadrupolar relaxation. Figure

4(a) shows the spectral density of the changing electric field gradient

at both high and low temperatures. A maximum at w would occur at some

intermediate temperature. We note that if m is changed by changing

the external field, for instance, there will be little effect in the

high temperature region, but in the low temperature region, reducing uo

will increase the effectiveness and shorten T,.
.L




-46-


Converted to T1 dependence on temperature, the plot would be some-

thing like Fig. 4(b).


2.6.2 The Hubbard Relation


We now restrict the discussion to the high temperature range where

wT r<< 1 and where TiQ is independent of frequency o In this region
o Q 1Q o
the spin-rotational component of the relaxation time is given by [14]


S = (4/3)(I kT )(2nC) (2.6.3)
o sr

where I is the molecular moment of inertia, T is the temperature, and
o
C is the spin-rotational constant.

The quadrupolar relaxation time is given by [4]

-1 2 2
T = (3/8)(e qQ/-)2 (2.6.4)
1Q Q

We now look at the relation between Tr and Q. If sr is long
sr Q sr

enough to allow the molecule to undergo large angle reorientation

between collisions, we have Q < T Furthermore, changing TT does

not directly have much effect on T On the other hand, we may increase

the frequency of collisions and thereby reduce T to the point where
sr

many random collisions are required before the molecule reorients

through a large angle. In this region T Q> r and a change in T
Q sr sr

will be accompanied by an opposite change in Q. This complementary

effect has been shown to obey the condition

T r = /6kT (2.6.5)
Qsr o

by Hubbard [14] using a rotational diffusion model. The Hubbard rela-

tion can be written


rT* = 1/6
Q sr


(2.6.6)




-47-


where reduced correlation times are defined by T* = T(kT/I )0. The

reduced correlation times are the correlation times expressed in units

of time required for a classical freely rotating spherical molecule to

rotate through one radian. The Hubbard relation was based on spherical

molecules; however Kluk and Powles [15] have shown it holds for linear

molecules as well.













CHAPTER III
EXPERIMENTAL EQUIPMENT AND PROCEDURE



3.1 Superconductive Magnet System


3.1.1 Magnet


The heart of the system is an RCA Superconductive Magnet Type

SM-2841 capable of producing a maximum field strength of 100 kG at a

current level of 91 A. The solenoid itself is about 7 in. long and 7

in. in diameter with a bore of just over two inches and is wound of

Nb Sn superconductive ribbon. Overall dimensions, including the form

(housing), but excluding mounting studs, are 10 in. long, 7.2 in. out-

side diameter, and a 2.03 in. bore.

The magnet is equipped with a magneto-resistive probe for field

strength measurement, which also turns out to be useful as a thermometer

during cool-down. With 50 mA dc through the magneto-resistive probe the

voltage varied from about 700 mV at room temperature to 84 mV at liquid

nitrogen temperature and 2.84 mV at liquid helium temperature at zero

field. Thus it was easy to determine when the magnet had reached liquid

nitrogen temperature during precooling and when liquid helium began

collecting during helium transfer.

At liquid helium temperature, again with 50 mA dc, the voltage

across the magneto-resistive probe varied from 2.84 mV at zero field to

about 20 mV at 100 kG. Once calibrated, it was accurate to better than

20 G.


-48-





-49-


Ffl T- --FT----- -


- xxx


X >0


/
o






/ ./
<<




\ \









/
00 x









X X9 x!0
\ \
N.XX 0


x xx










-X X eOK 0 *X .

^ ,- - 7



---0-o'^


U)
-)-




c( .:




CO
aw a


r- 0
ca




4-J
0




ca -a




e,
ca ,















0 a 0
cfl *H






UNU









1 H 0 ,
S >^ i
*HQ
9 u
UJ















o4 .,4









- ..
OU
i: a:
0 V4 0
a)^ Q) ^

^0 (a4




















60
4JJ 4J


S U 1>
.n1 a)<









H 34-14











*-H
tU ll

C >

(U*3
(NU)-
0 f


ssneD


-.~-Jr-----rud~-u




-50-


The magnet is equipped with a superconductive switch across the

terminals for semipersistent operation. A 300-mA current through a

resistance wire in the switch keeps the switch in its normal state.

Shutting off the heater current allows the switch to cool and become

superconductive. The superconductive path is then complete except for

the resistance of the copper current terminals and contact resistance

where the ends of the coil and the ends of the switch are soldered to

the current terminals.

On one occasion the switch heater wire was accidentally burned out,

making it impossible to charge the solenoid, so the epoxy-potted switch

had to be disassembled and repaired. Unfortunately, this happened

before the first resonance was obtained so behavior of the magnet with

the original switch as delivered was not determined.

The magnet was advertised to have a homogeneity of 5 G over a 3/4-

in. diameter spherical volume (DSV). In order to test the field char-

acteristics and to have a convenient nitrogen reference signal, a

liquid ammonia vial was prepared which had inside dimensions of about

7/16 in. long and 3/16 in. diameter. The magnet was mounted with the

bore (and hence H ) vertical and the ammonia sample was mounted horizon-
o

tally.

Variation of the field and line width with position of the saTple

along the axis of the solenoid are shown in Fig. 5. The abscissa is in

inches above an arbitrary reference point on the axis near the center

of the solenoid. The ordinates of the line-width data points are in

gauss, and were determined by peak-to-peak measurements of the deriva-

tive signal obtained by a cw spectrometer. The ordinates of the mag-

netic field strength H are in gauss above an arbitrary reference of
0




-51-


1 -b


0-



-1


-2


-3














m





1


o ^


-1













Figure 6.


5 10
Time (h)



Differences between measured values of magnetic field and
value calculated by exponential decay equation. (a) Decay to
zero field assumed. Smooth curves are drawn through data
points at 0.50, 0.55, and 0.60 inches from Fig. 5. (b) Decay
to intermediate value.




-52-


about 89 kG, and represent the zero crossover of a sometimes quite wide

derivative line.

The data points had to be corrected for a varying field drift rate,

the spectrometer frequency sweep rate, direction of sweep, and the

spectrometer time constant used.

As the field measurements are an average over the sample volume,

they cannot be taken directly as a point-to-point variation along the

axis of the bore. Considering the sample size, however, it should be

possible to move it about 0.45 in. and still keep it entirely within

the 3/4-in. DSV which was advertised tc have a homogeneity of 5 G. No

such region was found. IK appears the homogeneity was closer to 7-9 G

over the 3/4-in. DSV.

The peak-to-peak amplitude of the Cerivative signal was roughly

inversely proportional to the line width over most of the region,/as

expected. However, the maximum amn'litude occurred at 0.9 in. on the

reference scale and unexpectedly decreased rapidly as the sample was

raised until the signal was hardly discernible above 1.05 in.

The 0.9-in. position was chosen for the center of the field as the

best combination of linewidth and signal amplitude.

The field typically decayed at a rate of 0.8 to 0.9 G/min. in the

semipersistent mode. Values of the field recorded with the sample at

the 0.55-in. position were first crudely fitted to the exponential decay

equation

H(t) = H(0)exp(-t/T) (3.1.1)

by a least squares method. Differences between the data points and the

values calculated according to Eq. (3.1.1), H.(t) H(t), are plotted

in Fig. 6(a) as solid circles.




-53-


As the value of the field depends on both drift rate and variation

of the field with position of the sample, an error in the assumed drift

rate causes an erroneous determination of field variation with position.

This effect is apparent in Fig. 6(a) where data points at positions

other than 0.55 in. are shown as open circles which are offset from a

smooth curve through the solid circles.

Clearly Eq. (3.1.1) is not adequate over the 11.5-h duration of

the experiment. Hence, the equation


H(t) = H(-) + [H(0) H(m)]exp(-t/T) (3.1.2)


was used. A plot of H.(t) H(t) versus time is shown in Fig. 6(b)
1
where it is seen the fit is far better and entirely adequate.

Equation (3.1.1) was


H(t) = 89,411 exp(-t/1755.5) (3-1.3)

and Eq. (3.1.2) was


H(t) = 81,774 + 7639.1 exp(-t/144.6) (3.1.4)

where t is in hours and H in gauss.

Some variation in the values (f H(@) and T occurred on succeeding

runs, but generally H(-) could be considered constant during any given

run. On occasion the drift rate wao nearly double the typical 0.8 or

0.9 G/min value. During one run, the druft rate increased with time as

though H(-) were decreasing.

Maintaining full current in the leads appeared to have no effect

on the drift rate, but the magnet was not operated for extended periods

with full current in the leads due to increased liquid helium consump-

tion.

In some runs, the field was deliberately increased beyond the




-54-


desired value, then decreased. The drift rate was affected only the

first few minutes before returning to a behavior similar to that de-

scribed by Eq. (3.1.2).

It appears that Eq. (3.1.2) is normally quite accurate over opera-

ting periods of up to 23 h used thus far. At some time the value of

H(-) must decrease due to the small resistance in the solenoid circuit,

but no long term study of this effect was undertaken.

Several field determinations during each run using the ammonia

sample were adequate for drift determination and would probably still

be necessary even if the long term behavior were known.

When the magnet was first installed, it frequently quenched for

reasons which were not initially understood. As experience was gained,

some causes were identified and eliminated. One cause of quenching

which could not be eliminated is thought to be connected with the're-

paired superconductive switch. The current through the switch is the

difference in the solenoid current and the current in the leads to the

power supply. In the semipersistent mode the switch current therefore

increases as the power supply current is reduced. The magnet quenches

fairly consistently if the switch current becomes greater than about

50 A. This can be explained in terms of resistance in the switch or
-4
switch solder connections. A resistance of 10-4 and 50 or 60 A pro-

duces about as much heat as the 300 m4 in the heater wire used to keep

the switch normal. Thus the I'R heat associated with a small resistance

at the solder joints may be enough to drive the superconductive switch

normal even though it is in contact with large copper heat sinks.

The result was that the power supply had to be left engaged carry-

ing about 30 A to keep the current in the sui.erconductive switch below

50 A. This of course increased the helium consumption and possibly




-55-


Helium capillary
valve


Capillary
for
helium flow


flange


_- Sample access


S Vacuum space


SLiquid helium




- Liquid nitrogen











Flange for suspending
magnet


Magnet


Sample location



Heaters


Figure 7. Top access dewar system.








Mounting flange








,Vacuum space



. Liquid helium


Liquid nitrogen













.Magnet



-Sample location


Radiation shield


Figure 8. Re-entrant dewar system.


-56-


IL'




-57-


contributed to electronic noise. It also made the system susceptible

to power failure which would not be conducive to long term operation.


3.1.2 Dewars


Two interchangeable sets of dewars were obtained for use with the

superconductive magnet. The first set, manufactured by Sulfrian

Cryogenics, Inc., consisted of separate nitrogen and helium dewars with

top access. It was complemented by a Janis Research Company Super Vari-

Temp insert assembly which supported the magnet, provided access to the

working space, and provided sample temperature control. The system is

shown schematically in Fig. 7.

The other system consisted of a Cryofab unit construction i-itrogen

and helium dewar with a common vacuum space and equipped with a re-

entrant access tail. It is schematically shown in Fig. 8.

The first system has the advantage of low temperatures operation

and built-in temperature control. Liquid heliiun flow through the capil-

lary in conjunction with either electrical heaters in the bottom of the

insert or a heater added to the sample holder permits operation from 1.5

to 300 K. The obvious disadvantage Ls the distance from the top access

flange to the working center of the magnet, about 50 in.

Lack of adequate vertical clearance in the laboratory made it

necessary to either build the sample probe in sections cr to shut the

system down and lower it through the floor in order tc insert the

sample. Structural steel in the concrete floor prevented operation in

the lowered position.

The re-entrant dewar had the significant advantage of easy sample

access. The center of the magnetic field was only about 11 in. up from




-58-


the outside bottom of the dewar and thus the sample probe could be in-

serted or removed while the magnet remained at high field. Temperature

control was less convenient, partly because it was not built in and

partly because of the short stand-off distance from room temperature.

A gas-flow temperature control system to be described later used

in conjunction with the re-entrant dewar permitted operation over a 90

to 44 K sample temperature range. The desired sample temperature could

be attained and stabilized fairly rapidly but with increasing diffi-

culty near the extremes of the range.

One problem required constant attention during operation. The

helium space in the dewars was connected to the helium recovery system

through the current lead feed through and also through a larger port.

Severe thermal oscillations occurred occasionally which could be stopped

by a trial-and-error adjustment of valves in the recovery lines. In

general, the main valve in the recovery line from the large port in the

dewar could not be opened very wide without inducing these oscillations.

The dewars were suspended on a large steel plate which could be

raised and lowered on a vertical track by means of a cable and winch

arrangement. The dewars alone or the whole assembly could be lowered

either co gain access to the magnet or to obtain adequate overhead

clearance for sample probe installation in the top access dewars.


3.1.3 Sample Probe


A sample probe, shown in Figs. 9 and 10, was constructed to posi-

tion and orient the sample as well as to provide temperature control

and rf connections.

The outer conductor of the rf coaxial cable doubled as structural




-59-


Hot/cold gas


Nylon head


Sample
Thermocouple

RF coil -
Holes for gas
flow and
rf leads




Thermocouple
leads








Vacuum insulated
hot/cold gas
return and
thermocouple lead
feedthru


Worm gear

Rotating sample holder


Support and rf conductors












Teflon spacers and
gas barriers





SInsulation




..-Bottom flange
O-ring to seal against
Sdewar bottom



RF connector


HIot/cold gas exhaust


Sample probe, side view of sample holder.


Figure 9.





-60-


Hot/cold gas








Sample


Vacuum jacket -
























Hot/cold gas in


Antibacklash spring




Rotating sample holder

Worm

Worm gear


-Reference collar








Worm shaft




















Distance piece for
thermal isolation


-Index mark (fixed)
-Scale (rotating)


Figure 10. Sample probe, end view of sample holder.




-61-


120 VAC

To sample A Variac
probe

[t :


Hot air Dry air Pressure
exane e at4 Regulator
exchanger Dessicator


Hot air supply


To sample
probe


Heater


Vacuum insulated
transfer tubes


VAC


N2 gas


Liquid N2


Cold N9 gas supply


Figure 11. Gas-flow temperature control system.




-62-


support. The crystal was mounted inside a cylindrical cage-like holder

which had worm gear teeth cut around its center. The angular rotation

of the crystal could be controlled to within 2 minutes of arc relative

to the initial orientation, and the initial orientation could usually

be accurately determined from the data. Slack in leads from the coil

to the top of the solid rf coax permitted a full 1800 crystal rotation.

The entire unit fit snugly in the re-entrant tail of the dewar,

leaving a small space above the nylon head. Hot (or cold) gas is passed

up to this space through a 3/16-in. stainless steel tube which was

vacuum jacketed up to the top spacer. The gas flows back through and

around the nylon head to the space below, where it exhausts through a

1/4-in. vacuum jacketed tube.

A uniform temperature is maintained in the space above the top

Teflon spacer by a combination of gas flow rate and insulation. Near

room temperature there is only a small temperature gradient between the

sample area and ambient temperature outside the dewar, providing little

heat loss. At higher temperatures, a much higher hot gas flow rate is

used which offsets the greater heat loss through the in:;ulation withcut

significantly cooling the gas. A similar situation applies at low

temperatures.

A thermocouple close to the sample was used to measure temperature

and was adequate in view of the weak temperature dependence encountered.

The gas-flow systems for operation both above and below room tem-

perature are shown in Fig. 11. For high temperature operation, air was

passed through a pressure regulator, desiccator, and electrical heat

exchanger. Manual adjustment of inlet air pressure and heat exchanger

voltage provided better than 1 K stability with reasonably infrequent:




-63-


adjustments.

The low temperature system operated on cold nitrogen gas boiled

off by passing current through a 25-0 resistance immersed in liquid

nitrogen. Stability was enhanced by passing the gas through copper

tubing wound around a copper block which provided thermal inertia. Tem-

perature variation was predominantly provided by varying the current

through the 25-0 resistance in che liquid nitrogen supply, with addi-

tional control achieved by means of a resistance-wire heater on the

copper block.

The systems were operated over a range from 90 to 440 K. Higher

temperatures would have been readily obtainable with no modification to

the system, the limit being the ability of the sample probe to withstand

the heat. Initially, an rf coaxial cable was inserted in the probe all

the way up to just below the sample holder, but plastic spacers inside

the cable melted at 1670C. The nylon worm gear assembly also deformed

and seized at high temperatures. These problems were corrected by use

of rigid rf conductors as shown in Fig. 9 and by slightly increasing

clearances between moving parts.

Somewhat lower temperatures should be possible with some modifica-

tion to the system. The inlet gas tube could be passed through the

liquid nitrogen to insure it is as cold as possible prior to entering

the vacuum transfer tubes. Also, pumping on the exhaust tube to in-

crease gas flow rather than passing current through resistors in liquid

nitrogen would have the additional effect cf depressing the boiling

point.




-64-


3.1.4 Operation of the Superconductive System


It was generally best to start cool-down about three days before

the actual run. Simply maintaining liquid nitrogen in the liquid nitro-

gen dewar would cause the magnet temperature to lower to an estimated

81-83 K in that time period and an extra day would result in lowering

the temperature 2 to 3 degrees.

It was also possible to precool rapidly by introduction of liquid

nitrogen into the helium dewar in direct contact with the magnet, but

this method required back transfer which was more trouble than keeping

the outer jacket full of liquid nitrogen.

Overhead clearance limited the length of the helium transfer tube

so when fully inserted it would reach roughly one third of the way dcwn

from the top. Thus, a vacuum jacketed extension was required, but it

was also desirable to be able to pull the extension up out of the

liquid once liquid helium was in the dewar. This was especially impor-

tant to avoid passing warm gas through the liquid when starting a

replenishment transfer. Therefore, a vacuum jacketed extension was

made which extended through the top flange of the dewar where it was

terminated with a ball valve.

This extension could be lowered for insertion of the transfer tube

and for initial transfer when it was desirable to have the cold gas

pass up around the magnet from below. For replenishment, it could be

raised part wwy, and between transfers it could be raised well above

the liquid helium to reduce the heat path.

During cool-down, a pressure in the supply dewar of about 5 in.

H2,0 above recovery line pressure wMuld be the maximum the recovery

system would take, but once liquid helium started collecting, 40 in.




-65-


H20 pressure or more could be used. Only a few minutes would be re-

quired to collect about 25 liters after a 45-min cool-down from liquid

nitrogen temperature.

Once-a-day replenishment was generally adequate. Helium consump-

tion was about 2 liters/h during the time the field was being either

increased or decreased and was as low as 0.3 liter/h with the system

shut down. Vibrations fed back from the recovery compressor, thermal

oscillations, changes in recovery line pressure, current in the electri-

cal leads, and possibly other factors affected the helium consumption

which usually averaged about 20 liters/day including that lost while

cooling the warm transfer tubes during replenishment transfers.

Moveable level resistors in both the experimental and supply dewars

made it very easy to monitor liquid helium levels. The rather simple
-z
level indicators exhibited only a small needle deflection as the resis-

tor passed from one side of the surface to the other so needle movement

while dipping the resistor in and out of liquid helium was a much more

positive indication than absolute needle position.

A Magnion CFC-100 power supply was used to charge the solenoid. A

40-turn wire-wound potentiometer, motor driven at a selected rate, con-

trolled the rate of current increase. A maximum rate of 2.0 A/min was

allowed, which required 45.5 mini to bring ti:e magnet up to 100 kG. It

was discovered that the 2ore of che 40- turn potenticmeter had a high

coefficient of thermal expansion qrnd ,o,,ld sometimes shrink so much

that the wire would loosen and cause intermittent contact with the wiper

which would result in tha magnet quenching. The 40-turn potentiometer

had to be replaced with one having only 10 turns. With this potentio-

meter the fastest charging rate without e:::ceeding 2.0 A/min was




-66-


1.6 A/min which increased charging time by 25%.

Prior to placing the magnet in the semipersistent mode, the

magneto-resistive voltage and power supply dial reading were recorded.

The power supply current could then be reduced to about 30 A with the

magnet in the semipersistent mode. At the end of the run, the magneto-

resistive voltage was again noted which would be somewhat lower than

the initial reading. The power supply current would be brought up to

a proportionately lower reading prior to opening the superconductive

switch. The power supply current and the magnet current would be close-

ly matched in this condition, any difference between them being through

the superconductive switch. When the switch is opened by driving it to

the normal state, the current in it causes a voltage to develop across

it, normally less than 25 mV. In case of too great a mismatch of

currents, the voltage would be high enough to cause the power supply to

quench which would result in the magnet discharging through a 2-0 shunt

across the terminals. Under these conditions the magnet would be de-

energized in a few minutes with little loss of liquid helium.

As mentioned already, various causes would result in quenching the

magnet, and it would very rapidly de-energize, generally boiling off

all the liquid helium in the dewar but not warming up appreciably. The

recovery system was not capable of handling such a large flow rate, so

most of the helium would be lost to the atmosphere through a safety

valve. A helium gas bag is needed to prevent this loss. It appears

that the primary remaining cause of quenching is excessive current in

the superconductive switch which can be avoided by leaving the power

supply on and carrying part of ths current.




-67-


3.2 Electromrgnaet System


3.2.1 Electromagnet


A Varian V-4012-3B 12-inch electromagnet with a 3.5-in. gap was

used for the 1N line shape and relaxation studies. The magnet was capa-

ble of providing magnetic fields of up to 9000 G. A Varian V--2100B

magnet power supply and a Varian V-FR-2100 Fieldial field regulator were

used to operate the magnet in the field-regulation mode.

Drift rates of up to about 1 G/h were encountered whereas a change

of as little as 0.03 G would sometimes cause significant inaccuracies

in the data. It was sometimes possible to correct for the drift rate

once it was known; however the drift rate varied and even reversed direc-

tion throughout the day, especially when the laboratory temperature was

fluctuating due to intermittent operation of air conditioning.


3.2.2 Cryostat and Sample System


A schematic diagram of the cryostat is shown in Fig. 12. The

sample cell was made of Kel-F and had a volume of about 1.8 cm The

space around it contained helium exchange gas for thermal contact with

the surrounding copper can which contained the heater and thermometer.

Exchange gas in the brass can was used to control thermal contact with

the cryogenic fluid outside the brass can. The sample gas down pipe

doubled as the center rf conductor, and the sample assembly was sup-

ported by a double walled cabe which also served as the outer rf con-

ductor. The space between the walls was evacuated to thermally isolate

the sample gas down pipe from the surrounding cryogenic fluid. The

sample gas down pipe was electrically insulated from the external




-68-


Outer rf conductor


Sample gas down pipe
and center rf conductor


Brass can




GaAs thermometer

Heater wire --RF coil
Sample cell
Copper can -4i 1 ....


Figure 12. Cryostat used with electrormagnet.





-69-


Pressure/vacuum
gage

E I- /






Oxisorb


I L
High 02 cylinder Sample
pressure cell
cylinder
Cryopump Cryopump


Figure 13. Sample-gas system.




-70-


plumbing by an arrangement not shown on the drawing.

Temperature control was provided for by a GaAs thermometer and

resistance heater wire installed on the copper can surrounding the

sample cell containing exchange gas, and by varying the exchange-gas

pressure outside the copper can to Jary thermal contact with the cryo-

genic fluid.

Figure 13 is a schematic of the sample-gas system. The sample gas

was normally stored in the high pressure cylinder. Prior to releasing

the sample to the rest of the system, the pressure was reduced to approxi

mately one atmosphere by immersing the cylinder in liquid nitrogen. The

gas was then moved around by controlling the temperature of the sample

cell, the high pressure cylinder, and the two cryopumps. Ideally,

thermal isolation of the sample cell and down pipe would have permitted

operation with liquid helium as a cryogenic fluid over a wide tempera-

ture range. In practice, if helium was used during the sample conden-

sation stage, the down pipe would freeze up and block before the sample

was condensed in the sample cell. Part of the problem was due to an

uninsulated tube which passed up through the cryogenic fluid connecting

the copper sample cell to room temperature. This provided some thermal

contact even when the exchange gas was removed from the outside of the

copper can. Rather than modifying the cryostat, pumped liquid and solid

nitrogen were used while condensing the sample and for operation above

50 K.

The sample-cell temperature would be maintained at about 65-67 K

while the remainder of the sample system was gradually warmed, thus

condensing the sample in the sample cell as it boiled off from other

parts of the sample system. The pumped liquid or solid nitrogen could




-71-


then be used as the _ryogenic fluid (or solid below 63 K) or it could

be replaced by liquid helium for lower temperature operation.



3.3 Spectrometers


3.3.1 Pulse specLromne':er


Various pulse spectrometer combinations were used depending on

equipment availability and the type of measurements to be made.

It was normally not possible to observe the free induction decay
14
(FID) of 1N in a glycine directly without signal averaging. Thus it

was necessary to adjust the frequency by small increments and signal

average after each adjustment to first detect the signal and then to

determine the frequency, a very tedious process. Under these coLditions,

the Fast Fourier Transforr (FFT) technique was employed. The Fourier

transform of the FID is the frequency spectrum or absorption line with

respect to the oscillator frequency as the reference. Thus, if the

oscillator frequency is displaced below the central frequency, the peak

of the absorption line will be offset above the origin, and the amount

of offset is added to the oscillator frequency to determine the central

frequency. Hence it is only necessary to set the oscillator frequency

close to the frequency oF the signal.

Complex Fourier transformation capability would provide information

as to whether the offset is above or below the oscillator frequency.

Equipment used in the present work did not have complex capability, and

only the frequency differe.ncs could be fond. The ambiguity as to

w. hccher the offset is above or below the oscillator frequency had et be

i:etsolved by othar mi.eans. The usual method was to change the oscillator




-72-


frequency and note whether the offset increased or decreased.

It is not only unnecessary to set the oscillator frequency to the

central frequencyof the line, it is undesirable without complex Fourier

transform capability. If the oscillator frequency is within the fre-

quency spectrum of the line, pacts of the spectrum equal distances above

and below the oscillator frequency will be added, thus distorting the

frequency spectrum.

The spectrometer used in conjunction with the FFT equipment had

frequency stabilization provided by a General Radio Model 1061 Frequency

Synthesizer. It could be set digitally in 10 Hz increments and incor-

porated a search mode by which any digit position could be replaced by

a continuously variable value. For example, replacing I the 10 Hz digital

control permitted a continuous variation over a range from -20 Hz to

100 Hz below or above the frequency set on the other controls.

An F and H Instruments Pulse Program Generator PPG 45 permitted

considerable flexibility in choice of pulse sequence, width, separation,

repetition rate, etc.

A locally made Quadrature Phase Sensitive Detector was especially

convenient when it was desired to monitor the tuning while recording

data. When the frequency and phase are properly adjusted, the FID ampli-

tude is a maximum while the quadrature output is flat, and a drift off

resonance is easier to detect relative to a flat response. This of

course was not used when the oscillator was deliberately set off reso-

nance.

Each FID was recorded by a Bionation Model 801 Transient Recorder

and transferred to a Fabri-Tek Instruments, Inc., Model 1072 Instrument

Computer for accumulation.




-73-


After accumulation of an adequate number of signals, a Digital

Equipment Corporation PDP 8/E computer was used to perform the FFT.
14
While observing the NQR-perturbed 1N Zeeman resonance in glycine,
14
the frequency of the unperturbed 14N Zeeman line was periodically deter-

mined by means of a liquid ammonia sample. From these data and the

calculated drift rate of the superconductive magnet, the unperturbed

frequency was calculated for each time at which data were recorded.

After performing the FFT, the displacement of the line center from

the origin was measured in frequency units and the correction was

applied to the oscillator frequency, resulting in the actual frequency

of the NQR-perturbed line. Finally the difference between this fre-
14
quency and the frequency of the unperturbed 14N Zeeman line was deter-

mined which was the frequency shift of interest.
15
When using the same spectrometer co measure T1 of N in liquid and

$-solid 152, the amplitude of the FID was sampled over a short section

near the origin. Successive amplitudes were recorded in successive

channels of the Biomation Transient Recorder. When enough data were

recorded, they were transferred to the Fabri-Tek Instrument Recorder and

T1 was then calculated by th- PDP 8/E computer using a FOCAL program. A

series of 90-c-90 pulse sequences were used, separated by at least 6T1

in order to allow the spin system to reach equilibrium before the next

sequence. Thus, the first pulse would yield the amplitude at equilib-

rium and destroy the z component of the magnetization, and the second

pulse would yield the recovery amplitude after time t. The difference

between the two amplitudes decreases exponentially with increasing t,

and the pair-wise differencesversus t were used to calculate T1.

This method of measuring T1 had the advantage of requiring only a




-74-


single sampling point on each FID. It was not necessary to determine

the base line which would be required to get the correct amplitude, as

only differences in amplitudes were required. It had the disadvantage

of requiring a long wait between pairs of pulses as a new determination

of the equilibrium magnetization was required each time. The long wait

also subjected the data to more errors due to magnetic field drift and

instabilities of the electronics.

Another pulse sequence, t-90-r-180 following complete saturation,

was also used. In this method, a Princeton Applied Research TDH-9

Waveform Eductor was used to record the echo following a 1800 pulse.

The echo was then displayed on an oscilloscope where the echo amplitude

could be determined by comparison with a scale on the screen. The base

line on both sides of the echo was visible so the absolute amplitude

could be determined.

This method was much faster, as it was not necessary to wait be-

tween pulse sequences. However, it depended on visual estimation of the

pulse height rather than an electronically determined amplitude averaged

over a short section.


3.3.2 Continuous Wave Spectrometer


A continuous wave (cw) spectrometer was used to determine the 15N

15 15 15 14 14
NMR line shape of solid N2 and of a I: N- N: 2 = 1:4:4 mixture.

A locally constructed Robinson oscillator could be operated over

the entire range of frequencies (270 kHz to 4 iMfz) with the appropriate

rf coil and capacitance in the sample circuit. It could be frequency

swept, the sweep being controlled by vciage variable capacitors. A

5-kHz sweep at 270 kHz turned oit to be nonlirear, and therefore




-75-


magnetic field sweep was normally used.

A Princeton Applied Research Model HR-8 Lock-In Amplifier was used

to detect the signal and a Hewlett Packard Model 201C Audio Amplifier

and a Ling Electronics Model TP100 Power Amplifier provided field modu-

lation.

The derivative line shape was recorded by the Fabri-Tek Instrument

Computer which was also used to integrate the line shape to obtain the

absorption line shape. Both derivative and integrated line shapes were

recorded on paper by an X-Y recorder.














CHAPTER IV
RESULTS AND DISCUSSION GLYCINE



4.1 Glycine Structure


3+- -
Glycine, NH2CH2COOK (or NH1 CH200 ), is the simplest of the amino

acids. It crystallizes in one of three forms, a, 8, or y, depending on

the preparation conditions. Crystals grown from an aqueous solution

form a glycine, the most common and stable. Addition of ethyl alcohol

to the solution results in 8 glycine, and y glycine is obtained by addi-

tion of acetic acid or ammonium hydroxide to make the solution acidic

or basic.

Structures of the a, 8, and y forms were reported by Albrecht and

Corey [16], litaka [17], and litaka [18], respectively, and Marsh [19]

reported a refinement of the structure of a glycine.

Crystals studied in the present work were of the a glycine form,

grown, cut, and mounted at the University of Nottingham. Reference [16]

has some excellent drawings of a glycine, though reference [19] probably

contains more accurate data.

The molecule is shown in Fig. 14. Two enantiomorphous forms of the

molecule occur. The second form would be obtained by reversing the sign

of all the numbers in parentheses in Fig. 14, i.e., putting atoms shown

above the 0 0 CI plane below and vice versa. Only one form occurs

in ( and y glycine, whereas both are found in a glycine.

The two forms described above would be mirror images of each.other


-76-




-77-


(0.0)


1.255


i(0.006)
1.523


HV(0.561)
P

H (-0.859)


CI 0i
(0.0)


N(0.437)


0 (0.0)


.122)


HIII
SIII


H (0.467)













HI


01 (front) C

OI[ (behind)


Figure 14.


Glycine molecule. (a) Viewed normal to the 01, O11, CIi
plane. Bond distances are in A. Numbers in parentheses are
distances in A above (+) or below (-) the plane. (b) View
parallel to the 01, 0I, CII plane showing the molecule is
rather flat except for the protons.





-78-


in a reflection plane parallel to the O, O C plane. A point in-

version of one form would also result in the other form, in which case

any line, such as the lines representing bonds in Fig. 14, would be

antiparallel to the corresponding line in the other form.

The a-glycine crystal is monoclinic with four molecules per unit

cell. Unit cell dimensions are [19]

0
a = 5.1020 0.0008 A

b = 11.9709 + 0.0017 A
0
c = 5.4575 t 0.0015 A

B = 1110 42.3' 1.0'

A typical crystal is shown in Fig. 15, together with a set of

crystal axes E = X, Y, Z, which will be used in the discussion. Figure

15 is similar to Fig. 1 in reference [1] for purposes of consistency.

Figure 16 shows the arrangement of molecules in a glycine. The

molecules are identified by Al, A2, B1, 2 corresponding to D, C, A, B,

respectively, in reference [16]. The N atom in B1 is bound to 0 in

B2 and to 0 in an adjacent B2 molecule, binding B1 and B2 layers in a

double layer. The double B layer is loosely bound to the double A layer

by van der Waals forces, which is the reason for the (010) cleavage

plane.


4.2 Electric Field Gradient Tensors


There are 4 inequivalent N sites; however, B, and B2 are related

by point symmetry. Components of the EFG tensors at the N site in B1

and B2 are therefore antiparallel, and being a second rank tensor,

parallel and antiparallel cannot be distinguished. Thus, EFG tensors

are equivalent at the two sites. Considering the A molecules, there




-79-


- b,Y


S= 1110 42.3'


Figure 15.


Typical a-glycine ,:cyC:al. The {011} and {120} Faces appear
most frequently although {210} faces also appear. The {010}
faces are readily obtained by cleavage.


i010




























A2 B1 B2 A1
X________^i^


01 C i.


N




s- c> A



^9R



-^-


Figure 16.


Unit cell of a glycine. Axes are defined in Fig. 15. Mole-
cules A1 and B, are of the enantiomorphic form shown in Fig.
14, and are related by point inversion symmetry to A2 and B2.
Molecules B1 and B2 form a double layer which is loosely
bound to the AI, A2 layer.


-80-




-81-


Figure 17.


I
3 |





\1


A vector r1 undergoing a 180' rotation about the Y axis would
bring it to r whereas a reflection in the X,Z plane would
bring it to rj, where r2 = -r3. A tensor component has no
positive or negative sense, and therefore the 1800 Y-axis
rotation and the X,Z plane reflection are equivalent. Hence
the EFG tensors from sites A and B are related by a 1800
Y-axis rotation or, equivalently, an X,Z reflection plane.




-82-


Rotation axis


Figure 18.


Symmetry of rotation patterns for rotation axes in or normal
to the X,Z plane. Let A' and B' be projections of A and B,
respectively, in the rotation plane for any axis in the X,Z
plane. The projections are symmetric about the Y axis and
about the XZ plane (some detail omitted below the X,Z plane
to reduce clutter). For a Y-axis rotation, rotation patterns
frcm sites A and B merge as projections in the rotation plane
are parallel.




-83-


are two inequivalent EFG tensors, one for site A and one for site B.

Reference to Fig. 16 shows that a 1800 rotation of the A layers

about an axis parallel to the Y axis would make the A layer equivalent

to the B layer. The two EFG tensors are therefore related by such a

180 rotation, or equivalently by a reflection in the X, Z plane, as

shown in Fig. 17.

It is seen from Fig. 18 that the rotation patterns obtained from

sites A and B are symmetric for any axis in the X, Z plane when the ini-

tial orientation is chosen with H in the X, Z plane or normal to it.

Site A and a clockwise rotation results in the same pattern as site B

and a counterclockwise rotation. Furthermore, this requires the two

patterns to coincide when H is in the plane or normal to it. A rota-

tion about the Y axis would result in a single curve, as the projections

of both EFG tensors on the rotation plane would coincide.

These considerations affect the choice of rotation axes. A Y-axis

rotation would result in a single curve for (2Av)y, taking no advantage

of the fact that there are two inequivalent sites, other than alignment

information. A single curve would confirm that the rotation a-xis was

parallel to the Y axis, and two curves would mean the axis was not quite

parallel to Y.

A rotation axis in the X, Z plane would result in two curves, sym-

metric about the 8 = 0 or 9 = 90' orientations. Thus, the 6 0 orien-

tation could be accurately determined, and if the curves were not sym-

metric in amplitude or offset, it would be due to the rotation axis

being somewhat out of the X, Z plane. Again, the symmetry of the two

sites would provide only orientation information, and cbree such rota-

tion patterns would be needed to solve the problem completely.




-84-


Using the convention that upper signs go with site A and lower with

site B where the signs differ, X, Y, and Z rotations would result in the

following rotation patterns:


(2A)x = Ax

= A


(f =S
(2Av) = A
Y I


+ (B + C ) cosO2( 6X

+ Bxcos26X + C sin29X
X xY- x x


+


(B2 + C ) os2(2 -
Y Y Y


= A + B cos2y + C sin2y


(2Av) = Az

Az


+(Bz + cz) cos2(e 6Z)

+ B cos26 + C sin2e6
z z- z z


where tan25 = C/B.

Any of the six curves will furnish the diagonal elements of the

EFG tensors, which are the same for both sites. One off-diagonal ele-

ment comes from each rotation, and off-diagonal elements from the two

sites have the same magnitude, differing in sign only in two cases. The

complete tensors, in terms of coefficients in equation (4.2.1), are

-2AX TCz -Cy

CK C -2A, TC (4.2.2)


-Cy +Cx -2Azi

where other choices could have been made for the diagonal elements.

Here K = 3eQ/2L, as in (2.4.11) and will be suppressed for simplicity

in the following.

The symmetry of the EFG sensors at sites A and B is displayed in

(4.2.2), where the only differences in the elements are y = and
XY XY


(4.2.1)




-85-


A B
S= -Y. For a general choice of crystal axes, the elements of the
YZ YZ*
A tensor would not have the same magnitude as those of the B tensor,

unless by coincidence.
U
Suppose a E = U, V, W frame is chosen, and a V-axis rotation is

performed. The rotation patterns for sites A and B would be

(2Av) = AV + B cos29V + Cvsiln26


(2Av)) = A + Bvcos26 + C'sin2e (4.2.3)

for sites A and B respectively, the prime indicating site B. The cor-

responding tensors would be


A BV V -CV
UV

-UV -2AV 4 VW
S- (4.2.4)


-CVv VV + BV

-C' A' + B'
V VW V

the upper and lower values belonging to the site A and site B tensors,

respectively, as indicated by the primes. Here i's have been used to

avoid confusion in distinguishing between the E and U frames. There

are four unknowns, compared to only two which would result in tensor

(4.2.2) if a single Y rotation were performed. If the tensor (4.2.4) is

transformed to the Z frame, it must have the symmetry of (4.2.2) which

permits solving for the four unknowns.

The transformation is

S= = a lj (4.2.5)
k L i ii kJ 1jij

where k, 1 stand for X, Y, Z, and i ,j U, V, W, and aki is the direc-

tion cosine of the k (= X, Y, or Z) axis with respect to the i (=U, V, or
-1
W) axis. Note aj = alj
jl 13




-86-


Using primes to indicate elements of the B tensor in (4.2.2) we

have k + k1 = 0, where the upper sign holds for kl = XY or YZ, and
kl kl
the lower for kl = XX, YY, ZZ, and XZ. This results in six equations:


X = a xiaxj(ij ) = 0 (4.2.6)
i,j


Y- = aiaj (ij j) = 0 (4.2.7)
1,j


Z Z = a aj (i ') = 0 (4.2.8)
1,J


XZ- Z =J. aXaZj(ij 'j) = 0 (4.2.9)




1,J
4 + Y' ax.iayj( J + 4) = 0 (4.2.10)



z 4. = aYivZj(ij+ ij) = 0 (4.2.11)
1,j

Equations (4.2.10) and (4.2.11) can be solved for (UV+ 'UV ) and

(VW + ,' ) which are the only unknowns appearing. Any two of the first
VW VIM
four equations can be solved for (,UV and (, "), the only

unknowns in these equations. Finally, (i.U + iUV) arnd (UV IW) can

be solved for an and J'V' and likewise for V and I' using the other

two equations.

Thus, a single rotation actually overspecifies the solution, as

only four of the six equations are required. The only disadvantage is

that less information about ulignn.ent accuracy is provided. Misalign-

ment will cause the solution to depend on which four of the six equa-

tions are chosen.

As an example, calling Andersscn's [1] third rotation a V-axis

rotation, the following ter-sor results:




-87-


-322.4 %UV -199.3
1307.3 1',V -997.0


UV 1540.0 VW (.2.
(4.2.12)
iUV -400.0 $'


-199.3 VW -1217.6
-997.0 -907.3

Here the U, V, W axes have been chosen such that the direction cosines

of X, Y, Z with respect to U, V, W are

.7839 -.5313 .3212

(a..) = .6208 .6709 -.4056 .(4.2.13)

0 .5173 .8558

The equations (4.2.6) through (4.2.9) can be used with values from

(4.2.12) and (4.2.13) to solve for (UV $' ) and (.W i ). One of

the first three equations is a linear combination of the other tUo, and

therefore any two of the first three give consistent results. The

values of i i,' V and i' are listed in Table 1 together with

the equations used in obtaining the values.

Table 1. Values of the unknown EFG tensor elements calculated
using the equations listed together with Eqs. (4.2.10) and (4.2.11).


Solution Equations
number used UV UV VW .W

I (4.1.6, 7, 8) -247 -281 -615 -286
II (4.1.6, 9) -258 -269 -587 -314
III (4.1.7, 9) 107 -634 -74 -827
IV (4.1.8, 9) -278 -249 -615 -286



The values in the table were calculated by transforming frci the

U, V, W frame to the X, Y, Z frame, using only the syr.metry properties

of the tensor in the latter frame. The actual transformations ca:n now

be performed using Table 1 values in tensors(4.2.12) and (4.2.13).




-88-


The following four

of values in Table 1:

I 426

T837

-814
-840






S827


-827




-53
III
906

5 34

-827


tensors result, corresponding to the four sets


T837

598

548




+827

573
623

+554




-534

598


F859


-814
-840

548

-1024




-827


554

-999
-1049




-827

859

-545
-1504


(4.2.14)









(4.2.15)










(4.2.16)


452
IV 4 +843 -827
400

843 72 538 (4.2.17)
624

-827 538 -1024

It is noted that if averages are taken where the A and B site

elements differ, all the diagonal elements and the X, Z off-diagonal

elements are consistent among all four sensors. Furthermore, it seems

clear that (4.2.16) should be disregarded. The spread in values in

(4.2.16) arises from division by the difference of two nearly equal

numbers. Tensor (4.2.14) is the most consistent and the X, Y and Y, Z




-89-


elements fall between corresponding values in (4.2.15) and (4.2.17).

Therefore, the complete tensor would be

426 +837 -827

-837 598 548 (4.2.18)

-827 548 -1024

using only the third rotation reported by Andersson [1].

According to (4.2.18) a Z-axis rotation should give


(2Av) = 512 + 84icos2(e f 47.90) (4.2.19)


whereas Andersson [1] reported


(2Av), = 395 + 870cos2(9 + 390) .(4.2.20)
L. ,Z

The differences are probably due to uncertainty in orientation of

the axes which Andersson et al. [1] suggested may have accounted for

the discrepancies they observed.

Using two or three rotation patterns to solve the problem would

result in even greater overspecification, and make the problem of de-

ciding which equations to use even more difficult.


4.3 Data


In the present experiment, two rotation patterns were obtained at

about 50 C and one pattern each at 740 C, 1480 C, and 167 C.

The c axis was chosen for the first series for several reasons.

First, cleavage along the (010) and (010) planes results in a roughly

hexagonal crystal, facilitating shaping and aligning the crystal. Next,

Andersson et al. [1] had performed a c-axis rotation, so there was a

reference to use in initially obtaining a nitrogen resonance in glycine,

and a direct comparison could be made with that work. Finally, it was




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