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 SpinLattice 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 OrthoPara 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 SPINLATTICE 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 ASolid 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 zaxis 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. Reentrant dewac system. .... . . . ..... 56
9. Sample probe, side view of sample holder . . . ... 59
10. Sample probe, end view of sample holder. . . . . ... 60
11. Gasflow temperature control system. ... . . . . 61
12. Cryostat used with electromagnet ...... .. . . . 68
13. Samplegas system. .... .. . . . . . . .. 69
14. Glycine molecule . . . .... . . . . . . 77
15. Typical aglycine 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. Caxis rotation patterns from 14N in a glycine at 50 C . 9J
LIST OF FIGURES
(continued)
Figure Page
20. Caxis rotation patterns from 14N in a glycine at 74 C. 92
21. Caxis rotation patterns from 14N in a glycine at 1480 C. 93
22. Caxis 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 spinlattice 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 CN 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 oDhase 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 aphase 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 spinlattice 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
spinrotational 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 signaltonoise 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 wideband 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 twofold. First, the
laboratory had recently acquired a superconductive magnet capable of
producing up to a 100kG 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 15N14N 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 spinlattice 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 spinlattice
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 spinrotation 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 6solid 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 1514N samples, and to study the
spinlattice relaxation times in liquid and 6solid 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
spinrotation 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 wellknown Zeeman term
7Z = uH = 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 counterrotating
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 counterrotating 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 dipoledipole 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 spinrotation 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 spinlattice 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 zeroorder energy level is
E = yhH (m. + m.), and according to firstorder 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 2r 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 < al, 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
orientationaldependent 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 WignerEckart 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 firstorder 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 wellknown 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
firstorder 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
 ^ Ia
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 zaxis 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(m1) 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(ln)/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 lefthanded 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(1514) (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 firstorder 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>, 11>. Adding an external mag
netic field H removes the degeneracy and the state functions become
linear combinations of 11>, 0j>, 11>:
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(1514) is treated as a perturbation causing a firstorder
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 ) yiH (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
aln
(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(1514) 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/ (1514) = 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 firstorder energy shifts due to
D!(1514), 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^
$IJ
= (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(1514) =
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 SpinLattice 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 spinlattice 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 et/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 timedependent field is produced at the
nuclear sites within the molecule itself due to the rotational motion
of the molecule which produces spinrotational 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 spinrotation
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 spinrotation 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 spinrotational 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 spinrotational 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 spinrotational 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 spinrotational 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
SM2841 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 magnetoresistive probe for field
strength measurement, which also turns out to be useful as a thermometer
during cooldown. With 50 mA dc through the magnetoresistive 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 magnetoresistive 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
0o'^
U)
)
c( .:
CO
aw a
r 0
ca
4J
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 3414
*H
tU ll
C >
(U*3
(NU)
0 f
ssneD
.~Jrrud~u
50
The magnet is equipped with a superconductive switch across the
terminals for semipersistent operation. A 300mA 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 epoxypotted 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 linewidth data points are in
gauss, and were determined by peaktopeak 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 pointtopoint 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 3/4in. DSV which was advertised tc have a homogeneity of 5 G. No
such region was found. IK appears the homogeneity was closer to 79 G
over the 3/4in. DSV.
The peaktopeak 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 0.9in. 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 0.55in. 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 11.5h 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) (31.3)
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 104 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. Reentrant 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 iitrogen
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 builtin 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 reentrant 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 standoff distance from room temperature.
A gasflow temperature control system to be described later used
in conjunction with the reentrant 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 trialanderror 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
Oring 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. Gasflow temperature control system.
62
support. The crystal was mounted inside a cylindrical cagelike 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 reentrant tail of the dewar,
leaving a small space above the nylon head. Hot (or cold) gas is passed
up to this space through a 3/16in. 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
1/4in. 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 gasflow 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 250 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 250 resistance in che liquid nitrogen supply, with addi
tional control achieved by means of a resistancewire 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 cooldown 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
8183 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 cooldown, 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 45min cooldown from liquid
nitrogen temperature.
Onceaday 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 CFC100 power supply was used to charge the solenoid. A
40turn wirewound 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 40turn 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
magnetoresistive 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 20 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 deenergize, 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 V40123B 12inch electromagnet with a 3.5in. 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 V2100B
magnet power supply and a Varian VFR2100 Fieldial field regulator were
used to operate the magnet in the fieldregulation 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 KelF 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. Samplegas 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 exchangegas
pressure outside the copper can to Jary thermal contact with the cryo
genic fluid.
Figure 13 is a schematic of the samplegas 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 samplecell temperature would be maintained at about 6567 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 FabriTek 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 NQRperturbed 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 NQRperturbed 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 FabriTek Instrument Recorder and
T1 was then calculated by th PDP 8/E computer using a FOCAL program. A
series of 90c90 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 pairwise 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, t90r180 following complete saturation,
was also used. In this method, a Princeton Applied Research TDH9
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
5kHz sweep at 270 kHz turned oit to be nonlirear, and therefore
75
magnetic field sweep was normally used.
A Princeton Applied Research Model HR8 LockIn 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 FabriTek 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 XY 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 aglycine 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 aglycine ,: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 Yaxis
rotation and the X,Z plane reflection are equivalent. Hence
the EFG tensors from sites A and B are related by a 1800
Yaxis 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 Yaxis 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 Yaxis
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 axis 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 offdiagonal ele
ment comes from each rotation, and offdiagonal 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 Vaxis 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 Vaxis
rotation, the following tersor 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 offdiagonal
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 Zaxis 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 caxis 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
