Hydrogen bonding in lithium hydrazinium sulfate
by Francis Luther Howell
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Physics
Montana State University
© Copyright by Francis Luther Howell (1969)
Abstract:
The nuclear magnetic resonance spectrum of deuterons in LiN2D5SO4 has been studied in detail
between 78 and 458°K. The spectrum at 78° is related to the deuterons of a static N2D5^+ i o n . The
electric quadrupole coupling constants, the asymmetry parameters, and the orientations of electric field
gradient (efg) tensors relative to the crystal axes were obtained at 78, 193, 298, 338, 368 and
438%deg;K. Above 110°K the lines due to the deuterons of the ND3 group broaden and merge to a
single pair of lines which is still evident at 195°C. The lines corresponding to the deuterons of the ND2
group merge to a single pair between O° and 50°C. These line mergers are related to the motion of the
ND3 and ND2 groups and estimates of the activation energy and jump times of these motions are
made. Because of the symmetry of the crystal the absence of line doubling indicates that ferroelectric
domain reversal may be due to reflection of the structure through the ab plane.
Measurements of the electrical conductivity and the ac dielectric constant were made on both
deuterated and undeuterated samples. The c axis ac dielectric constant was found to be quite large.
Estimates of the activation energy and correlation time for the dielectric relaxation were made.
The spin-lattice relaxation times for the merged ND3 and ND2 lines were measured over the
temperature range from 155°K to 422°K. The results are found to be in good agreement with the
assumption that the relaxation is due to fluctuations in the electric quadrupole coupling energy due to
ND3 and ND2 hindered rotations. In addition, evidence is found to indicate that jumping between ND2
and ND3 groups becomes important at temperatures above 75°C. None of the measurements discussed
indicate that the crystal undergoes a ferroelectric transition below 200°C.
H Y D R O G E N B O N D I N G IN L I T H I U M H Y D R A Z I N I U M S U L F A T E
by
FRANCIS LUTHER HOWELL
A t h e s i s s u b m i t t e d t o t h e Gr a d u a t e F a c u l t y i n p a r t i a l
f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r t h e degr ee
of
DOCTOR OF PHILOSOPHY
i n
P h y s i cs
/4^4y
Head, M a j o i ^ D e j S ^ t m e n t
I /
0
Chairmayr
E x a mi n i n g Commi t t ee
G r a d u a t e Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
December,
1969
iii
ACKNOWLEDGEMENTS
The a u t h o r w i s h e s
to thank
H ea lth"an d the N a tio n a l
assistance
this
Dr .
and f o r
research.
V.
He e x t e n d s
for
encouragement al ways
he i s
grateful
apparatus
for
his
He t h a n k s
help
in
helpful
apparatus
help w ith
the author w ith
thanks
u s e d.
is
discussion,
discussion,
his
for
inform ation
deepest thanks
S h a p , Dan,
to
his
to
th e ir
on t h e
for
the
loan of
the
the loan
o f apparatus
To Fred
assistance with
structure
to
his
w ife,
for
in
Marion,
supplying-
The a u t h o r
and t o
constant
of this
and
o f deuterated
publication.
th e ir
completion
the
e x t e n d e d t o P. LaPere
kindness
p rio r
advisor.
To R. R. K n i s p e l
To D r s . W. C. H a m i l t o n
and A l i c e ,
c o n f i d e n c e w h i c h made t h e
is
used i n
g u i d a n c e and
help w ith
generous
Thanks
extended f o r
h y d r a z i n i urn s u l f a t e
children
thanks
of
financial
the app ar atu s
Robert Parker f o r
the drawings.
F . Ross t h a n k s
for
u s i n g t h e NMR p u l s e a p p a r a t u s .
electronics
offers
p a rticula r
stim ulating
he e x t e n d s
lithium
for
In stitu te s
f o r t h c o m i n g when n e e d e d ..
Blankenburg
for
funds
and c o m p u t e r p r o g r a m s - a n d f o r
experiment.
and f o r
Science Foundation
providing
Hugo S c h m i d t ,
the N a tio n a l
his
h e l p and
program p o s s i b l e .
TABLE OF CONTENTS
Chapt er
Page
LI ST OF TABLES ...................................................................
v
LI ST OF F I G U R E S ..............................................................
vi
ABSTRACT.................................................................................
i x
I
INTRODUCTION ........................................................................
I
II
DEUTERON ELECTRIC FIELD GRADIENT TENSORS AT
7 7 ° K ...........................................................................................
6
Structure
............................................................................
NMR S p e c t r u m .......................................................................
Discussion
..........................................
6
11
16
DEUTERON M O T I O N ...............................................
28
C o n d u c t i v i t y and D i e l e c t r i c C o n s t a n t . . . .
P r e v i o u s NMR S t u d i e s ....................................................
Hi gh T e m p e r a t u r e S p e c t r u m
......................................
Exchange P r o b a b i l i t y ....................................................
D i s c u s s i o n .................................
28
36
41
50
71
EXPERIMENTAL .
. '..............................................................
74
Crystal Preparation
.....................................................
NMR S p e c t r a ........................................................................
C o n d u c t i v i t y and D i e l e c t r i c C o n s t a n t . . . .
R e l a x a t i o n T i m e ...............................................
74
74
79
86
APPENDIX ".................................
.
91
A p p e n d i x A ............................................................................
A p p e n d i x B ............................................................................
A p p e n d i x C ................................. ......................................
91
98
108
LITERATURE CITED ..............................................................
Ill
III
IV
........................ ....
.
.
V
LI ST OF TABLES
Table
I
11
III
IV
V
VI
Page
C oo rd in a te s o f the C o n s t i t u e n t s o f the U n i t
C e l l o f Li HzS . ........................ .......................................... ....
C o u p l i n g C o n s t a n t s , Asymmet r y P a r a me t e r s and
O r i e n t a t i o n o f P r i n c i p a l Axes o f e f g T e n s o r s
a t 78° and 19 3 ° K ..............................................................
20
N- H . . . X Bond L e n g t h s
24
and Bond A n g l e s . . . .
C o u p l i n g C o n s t a n t s and Asymmet ry P a r a me t e r s
f o r N-D Bonds
o f S e v e r a l C o m p o u n d s ..................
25
C o n d u c t i v i t i e s and C o n d u c t i v i t y A c t i v a t i o n
E n e r g i e s ........................ .....................................
29
Ef g Components
in Crystal
VIII
Efg Components
i n R o t a t e d Sy s t em,
X
XI
.
C o u p l i n g C o n s t a n t s , Asymmet r y P a r a m e t e r s and
O r i e n t a t i o n o f P r i n c i p a l Axes o f e f g T e n s o r s
above 273°K ............................
VII
IX
7
S y s t e m ............
z1
|| a
47
.62
.
.
63
Ef g Components i n R o t a t e d Syst em, z '
18° f r o m
Z 1 Il a
.....................................................................................
64
C a l c u l a t e d Va l u e s f o r t h e T r a n s i t i o n
P robabilities
P-| and P g .................................................
66
I n t e r - s i t e D i s t a n c e s f o r N9D SO. ( A f t e r Ross
and Hami I t o n ( 4 0 ) ) ............................... 4 ..............................
109
vi
LI ST OF FIGURES
Figure
1
Page
S t r u c t u r e o f L i t h i u m H y d r a z i n i u m S u l f a t e Vi ewed
Down t h e c Axi s ................................. .....................................
9
2
D etail o f N-H...N
................................................
9
3
S k e t c h o f t h e H y d r a z i n i u m I o n s Showi ng t h e ND.
and NDg G r o u p i n g
. . . ....................................................
10
T y p i c a l NMR S p e c t r a o f t h e D e u t e r o n s a t L i q u i d
, A i r Temperature
..................................................................
14
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
A n g l e Ab o u t t h e a C r y s t a l l o g r a p h i c A x i s a t
L i q u i d A i r T e m p e r a t u r e ...........................................■. .
17
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
Angle About the b C r y s t a l l o g r a p h i c Axis at
L i q u i d A i r Temperature
............................ . . . . .
18
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
A n g l e Ab o u t t h e c C r y s t a l l o g r a p h i c A x i s a t
L i q u i d A i r Temperature
. . . .
.................................
19
P l o t o f C o u p l i n g C o n s t a n t as a F u n c t i o n
Squar e o f t h e X-H S t r e t c h i n g F r equ enc y
26
4
5
6
7
8
9
.10
11
12
13
14
Log o f E l e c t r i c a l
Bondi ng
Conductivity
o f the
. . . .
vs T e m p e r a t u r e
.
31
Log o f Real and I m a g i n a r y P a r t s o f t h e ac
D i e l e c t r i c C o n s t a n t vs F r e q u e n c y
.............................
32
P l o t o f t h e I m a g i n a r y vs t h e Real P a r t o f t h e
ac D i e l e c t r i c C o n s t a n t
. . ............................................
34
Log o f Real and I m a g i n a r y P a r t s o f t h e ac
D i e l e c t r i c C o n s t a n t vs T e m p e r a t u r e
........................
35
P r o t o n Second Moment vs T e m p e r a t u r e ( A f t e r
C u t h b e r t and P e t c h ^ l ) ) ....................................................
37
P r o t o n Second Moment vs T e m p e r a t u r e ( A f t e r
McCl e m e n t , P i n t a r and P e t c h ( ' ^ ) ) .............................
39
vi i
Figure
15
Page
P r o t o n Sp i n L a t t i c e R e l a x a t i o n Time vs
T e m p e r a t u r e ( A f t e r Mc C l e m e n t 5 P i n t a r and
F e t c h ( 1 2 ) ) . ...............................................
40
NMR Sp e c t r u m o f D e u t e r o n s a t S e l e c t e d
T e m p e r a t u r e s .............................................................
42
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
A n g l e Ab o u t t h e a C r y s t a l l o g r a p h i c A x i s a t 95°C
43
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
A n g l e Ab o u t t h e b C r y s t a l l o g r a p h i c A x i s a t 95°C
44
Q u a d r u p o l e S p l i t t i n g o f D e u t e r o n s vs R o t a t i o n
A n g l e Ab o u t t h e c C r y s t a l l o g r a p h i c A x i s a t 95°C
45
20
Sketch o f Line S e p a ra t io n
52
21
Log Jump P r o b a b i l i t y
vs T e m p e r a t u r e f o r
NDg
.
56
22
Log Jump P r o b a b i l i t y
vs T e m p e r a t u r e f o r
NDg
..
57
23
D e u t e r o n S p i n R e l a x a t i o n Time vs R e c i p r o c a l
T e m p e r a t u r e ............................................................................
60
24
B l o c k Di agr am o f NMR E x p e r i m e n t .............................
77
25
Block
80
26
Equivalent C ir c u it
. .
81
27
C i r c u i t Di agr am o f ac D i e l e c t r i c C o n s t a n t
E x p e r i m e n t ................................................................................
84
28
Dr awi n g o f ac D i e l e c t r i c
85
29
B l o c k Di agr am o f P u l s e d NMR A p p a r a t u s
30
Zeeman L e v e l s o f I = 'I Syst em w i t h Q u a d r u p o l e
P e r t u r b a t i o n .............................................................
95
P o p u l a t i o n s o f N u c l e a r L e v e l s f o r S y mme t r i c
S a t u r a t i o n ................................................................................
I 04
Populations
o f Am = O
I 04
16
17
18
19
31
32
and B r o a d e n i n g
Di agr am o f C o n d u c t i v i t y
of Crystal
Experiment
. . .
. . .
Conductivity
C o n s t a n t Sample Probe
. . . .
o f Nuclear Levels f o r S a t u r a t io n
1 T r a n s i t i o n ...............................................
88
viii
Figure
33
34
-
Page
P o p u l a t i o n s o f N u c l e a r L e v e l s f o r Jumpi ng
Between NDg and NDg S y s t e m ................... .......................
106
Stereoscopic
110
Illu s tra tio n
of Structure
. . . .
ABSTRACT
T h e . n u c l e a r magnetic resonance spectrum o f deuterons
i n LiNgDgSO^ has been s t u d i e d i n d e t a i l bet ween 78 and
4 5 8 ° K.
The s p e c t r u m a t 78° i s r e l a t e d t o t h e d e u t e r o n s o f
a s t a t i c NgDg + i o n .
The. e l e c t r i c q u a d r u p o l e c o u p l i n g
c o n s t a n t s , t h e " a s y m m e t r y p a r a m e t e r s , and t h e o r i e n t a t i o n s
o f e l e c t r i c f i e l d g r a d i e n t ( e f g ) te n sors r e l a t i v e to the
c r y s t a l axes were o b t a i n e d a t 78, 193, 298, 33 8, 368 and
438° K.
Above I l O 0 K t h e l i n e s due t o t h e d e u t e r o n s o f t h e
NDg g r o u p br oaden and merge t o a s i n g l e p a i r o f l i n e s w h i c h
i s s t i l l e v i d e n t a t I 95°C.
The l i n e s c o r r e s p o n d i n g t o t h e
d e u t e r o n s o f t h e NDg gr o u p merge t o a s i n g l e p a i r between
O0 and SO0C.
These l i n e me r ger s a r e r e l a t e d t o t h e m o t i o n
o f t h e NDg and NDg gr oup s and e s t i m a t e s o f t h e a c t i v a t i o n
e n e r g y and j ump t i m e s o f t h e s e m o t i o n s a r e made.
Because
o f t h e sy mmet r y o f t h e c r y s t a l t h e absence o f l i n e d o u b l i n g
i n d i c a t e s t h a t f e r r o e l e c t r i c domai n r e v e r s a l may be due t o
r e f l e c t i o n o f t h e s t r u c t u r e t h r o u g h t h e ab p l a n e .
Measur ement s o f t h e e l e c t r i c a l c o n d u c t i v i t y and t he
ac d i e l e c t r i c c o n s t a n t were made on b o t h d e u t e r a t e d and
u n d e u t e r a t e d samples.
The c a x i s ac d i e l e c t r i c c o n s t a n t
was f o u n d t o be q u i t e l a r g e .
Estimates o f the a c t i v a t i o n
e n e r g y and c o r r e l a t i o n t i m e f o r t h e d i e l e c t r i c r e l a x a t i o n
were made.
The s p i n - l a t t i c e r e l a x a t i o n t i m e s f o r t h e merged NDg
and NDo l i n e s were measured o v e r t h e t e m p e r a t u r e r ange f r o m
I 55°K t o 422° K.
The r e s u l t s a r e f o u n d t o be i n good a g r e e ­
ment w i t h t h e a s s u m p t i o n t h a t t h e r e l a x a t i o n i s due t o
f l u c t u a t i o n s i n t h e e l e c t r i c q u a d r u p o l e c o u p l i n g e n e r g y due
t o NDg and NDg h i n d e r e d r o t a t i o n s .
In a d d i t i o n , evidence
i s f o u n d t o i n d i c a t e t h a t j u m p i n g bet ween NDg and NDg gr oup s
becomes i m p o r t a n t a t t e m p e r a t u r e s above 75° C.
Nane o f t h e
measur ement s d i s c u s s e d i n d i c a t e t h a t t h e c r y s t a l under goes
a f e r r o e l e c t r i c t r a n s i t i o n be l ow 200° C .
CHAPTER I
An e x p l a n a t i o n
certain
INTRODUCTION
o f the f e r r o e l e c t r i c
h y d r o g e n bonded c r y s t a l s *
behavior of
derives
t h e e n v i r o n m e n t and m o t i o n w i t h i n
the
from un derstanding
crystal
o f the protons
forming
t h e h y d r o g e n bonds.
X - r a y and n e u t r o n
studies
a r e commonl y used t o
locate
in
the u n i t
information
d ie le c tric
cell
o f the
crystal,
the p o s i t i o n s
Conductivity
c o n s t a n t measur ement s a l s o y i e l d
about the motion o f the c o n s t i t u e n t s
inform ation
usefulness
on m o t i o n i s
in
and
inform ation
o f the c r y s t a l .
o f neutron studies
lim ited
o f atoms
and may be used t o o b t a i n
on t h e m o t i o n o f t h e a t o ms .
However, the
d iffraction
to obtain
the f o l l o w i n g
respect.
S i n c e ex c hange bet ween s i t e s ,
such as h i n d e r e d r o t a t i o n
a group, probably
in
time spent
the t o t a l
"jumping"
time,
the s t a t i o n a r y
neutron
takes
such a f a s h i o n
bet ween s i t e s
is
location
o f the
studies
group.
give
that
a very small
the neutron d i f f r a c t i o n
d iffra ctio n
vibra tion al
place
results
the
part of
w ill
re flect
Because o f t h i s ,
good i n d i c a t i o n
of
.
of
or t o r s i o n a l , m o t i o n s , f o r which time spent o u t
o f the e q u i l i b r i u m
site
is
large,
but
little
inform ation
* D i s o u s s i on o f . hy dr ogen bonded f e r r o e l e c t r i c s may be
found i n r e f e r e n c e I .
For a d e t a i l e d d i s c u s s i o n o f KHgPO^,
a r e p r e s e n t a t i v e o f t h i s c l a s s w h i c h has been s t u d i e d i n
d e t a i l , see r e f e r e n c e s 2, 3, 4, 5, 6 , and 7.
" 2about motions
such as ar e o f p r i m a r y
Conductivity
d iffic u lt.
sm all,
cm o r
less,
can i n t r o d u c e
to
bulk
properties
distinguish
this
re la tive ly
large
the v a rio u s
errors.
At be st,
c o n s t a n t deal
maki ng i t
atomic e f f e c t s
d iffic u lt
unambiguously.
An e x p e r i m e n t wh i c h has p r o v e n p a r t i c u l a r l y
information
nuclei
the o b s e r v a t i o n
is
resonance
wi de l i n e ,
of
o f primary
the n u c l e a r magnetic
inte rest
in
solids
and p r o t o n w i d e - l i n e
-measurement o f l i n e
time.
width
is
equivalence
it
is
generally
re la tive
are
possible
The
a single
lim ited
to
relaxa­
location
local
the proton
Because o f t h e i r
* D i s c u s s i on o f t h i s
r e f e r e n c e s 8 and 9.
is
much can be i n f e r r e d
and t h e i r
possible with
moments.
the p r o t o n .
NMR s t u d i e s
H o w e v e r , no measure o f t h e
environment
is
and o f t h e s p i n - l a t t i c e
From such s t u d i e s
motion o f the p r oto n s
ele ctric
in
a b o u t t h e e n v i r o n m e n t and m o t i o n o f
NMR s p e c t r u m o f t h e p r o t o n
crystal.
useful
(NMR) s p e c t r u m . *
The n u c l e u s
tion
ar e
and c r y s t a l
and d i e l e c t r i c
o f the c r y s t a l ,
obtaining
work.
commonly e n c o u n t e r e d ar e
and i m p u r i t i e s
measur ement s o f c o n d u c t i v i t y
with
in
o f h y d r og en bonded c r y s t a l s
The c o n d u c t i v i t i e s
~1
defects
studies
interest
about
in the
ele ctrical
since
it
has no
a p p r o x i m a t e c h e mi c a l
to s u b s t i t u t e
deuterons
type o f experiment
is
for
included
in
-3protons.
This
can be done by c r y s t a l I i z i n g
water s o l u t i o n
The d e u t e r o n
in
f r o m a heavy
t h e case o f most w a t e r s o l u b l e
has an e l e c t r i c
with
the e l e c t r i c
fie ld
This
interaction
that
the d e te r m in a t io n
site
is
q u a d r u p o l e moment and i n t e r a c t s :
gradient
modifies
compounds.
its
(efg)
at
its
NMR s p e c t r u m i n
o f the efg
location.
such a f a s h i o n
te n s o r at the deuteron
made p o s s i b l e .
Lithium
h y d r a z i n i urn s u l f a t e
c r y s t a l , ( 1 0 ) and n e u t r o n
structure
diffra ctio n
have e s t a b l i s h e d
gr o u p a r e a l l
resonance
involved
studies
have been c a r r i e d
(LiHzS)
in
that
o f the p roto n s
jumping o f the p r oto n s
a ferroelectric
s t u d i e s ( I I ) o f t he
the pr ot on s
hydrogen
o u t . (12,13)
is
bonds.
o f the hydrazine
Nucl ea r magnetic
and Li ^ n u c l e i
i n Li HzS
Based on t h e s e s t u d i e s ,
bet ween bond s i t e s
and o t h e r m o t i o n s
have been h y p o t h e s i z e d .
Although.the
questions
studies
m e n t i o n e d above answer many
about the s t r u c t u r e
undergo c e r t a i n
motions,
several
unanswered.
S p e cifica lly,
protons?
there
Is
decomposition
structural
Is t h e r e
the
temperature
attempt to
that
w h a t a r e t he m o t i o n s
tra nsition
o f the c r y s t a l ?
changes o c c u r i n
the c r y s t a l
t he o r d e r i n g
changes p o l a r i z a t i o n ?
answer t h e s e q u e s t i o n s
the p r o to n s ,
important questions
a fe rro e le ctric
any change i n
crystal
and i n d i c a t e
o f t he
bel ow t h e
What ,
as i t
r e ma i n
is
if
any,
heat ed?
o f t h e p r o t o n s when
T h i s wor k d e a l s w i t h
the
by a s t u d y o f t h e n u c l e a r
"4magnetic
resonance spectrum o f
crysta l,
L i NgDgSO^,
conductivity
deuterated
deuterons
in
and by measur ement s o f
and ac d i e l e c t r i c
the d e u te r a te d
the e l e c t r i c a l
co n s ta n t o f the
normal
crystals.
The s p e c t r u m o f t h e d e u t e r o n s was s t u d i e d
temperature
a t w h i c h j u m p i n g among s i t e s
and t h e r e s u l t s
are d i s c u s s e d
in
spectrum a t h i g h e r tem pe ratu re
and t h e t h i r d
chapter is
motion.
this
o f the
d e u t e r o n s was c a r r i e d
On t h e
A complete
the deuteron
o f temperature
basis
reflects
through
line
bel ow t h e
becomes i m p o r t a n t *
Chapter I I .
devoted to
reveal
addition,
and
The NMR
in te r-site
t h e measur ement s wh i c h
s t u d y o f t h e NMR s p e c t r u m
out to
above I BO0 C.
In
w i d t h was measured as a f u n c t i o n
the c r i t i c a l
temperature
o f t h e s e measur ement s a c a l c u l a t i o n
ex change p r o b a b i l i t y
f o r the motion causing the
change was d o n e .
was hoped t h a t
It
exchange
t he e l e c t r i c
regions.
o f the
lin e width
fie ld
g r a d i e n t measur ed a t e l e v a t e d t e m p e r a t u r e s w o u l d r e f l e c t
the
structural
as a r e s u l t
change s u g g e s t e d by C u t h b e r t and P e t c h ^ ^ )
7
o f t h e i r o b s e r v a t i o n o f a change i n t h e Li
NMR
spectrum w i t h
temperature.
The d i e l e c t r i c
o f the d i e l e c t r i c
b e h a v i o r was s t u d i e d
anomal y a t a phase t r a n s i t i o n
what t y p e o f f e r r o e l e c t r i c
Finally
because t h e n a t u r e
tra nsition
the s p i n - l a t t i c e
indicates
occurs.
relaxation
t i m e was measured
-5for
deuterons
o f t h e gr oup s wh i c h under go m o t i o n .
the efg data di scus sed
mechani sm was r e l a t e d
in
Chapter I I I ,
to the
t o ex change bet ween s i t e s .
the motions
obtained
the
fluctuations
by t h e d i f f e r e n t
relaxation
in
The a c t i v a t i o n
Usi ng
t h e e f g due
energies
for
measur ement s were
compar ed.
R elatively
mi ne t h e v a l u e s
in
f ew s t u d i e s
to these
significance,
inform ation
energies
while.
it
bonds .
Because o f t h e
bonds f o u n d i n
about the efg te n so rs
numerous
compounds o f b i o l o g i c a l
and t h e a c t i v a t i o n
bet ween bonds o f t h i s
Summari zed i n
a table
in
Chapter
on some o f t h e n i t r o g e n
been s t u d i e d .
o f deuterons
w o u l d seem t h a t s i m p l y t h e a c c u m u l a t i o n
f o r motion
inform ation
out to d e t e r ­
o f the efg t e n s o r s a t the s i t e s
N-D...N or N -D ...0
analogies
have been c a r r i e d
type
11 i s
is worth
pertinent
compounds w h i c h have
of
CHAPTER I I
DEUTERON ELECTRIC FIELD GRADIENT TENSORS AT 77°K
STRUCTURE. •
The s t r u c t u r e
of
lithium
( L i N2H5SO4 ) has been s t u d i e d
and n e u t r o n ( l l )
coordinates
h y d r a z i ni um s u l f a t e
by means o f x - r a y M 4 , 1 5 , 1 6 )
d iffra ctio n .
Table
o f the c o n s t i t u e n t s
I compares t h e
o f Li HzS f o u n d by t h e s e
experiments.
Li HzS i s
o r t h o r h o m b i c o f space gr oup Pbn2-| , 0 5 )
fo u r molecular u n its
Balasubramanian(IT)
dimensions
a=9 . 9 3 ,
found f o r the
0 *
c = 5 . 1 8 A.
cell
The l i t h i u m
centers
in
Fig.
I,
d i m e n s i o n s a = 8 . 99,
these t e t r a h e d r a
channels
These c h a n n e l s
nitrogen
report
b=9.94,
As i s
and
a t the
indicated
have common a p i c e s
running p a r a lle l
contain
cell
B r o w n ( T'^) has
atoms a r e l o c a t e d
o f oxygen a t o ms .
a r e h y d r o g e n bonded t o atoms
and t o
Padmanabhan and
b = 8 . 969 , and c = 5 . 178 A.
and s u l f u r
with
c e ll.
( h e r e a f t e r d e n o t e d by PB)
of tetrahedra
a lattice -w ork
axis.
per u n i t
^ith
and f o r m
to the c
t h e h y d r a z i ni um i o n s wh i c h
in
atoms a d j a c e n t
the s u r ro u n d in g l a t t i c e
in
the c d i r e c t i o n .
The
*The exchange i n t h e a s s i g n m e n t o f t h e a and b c r y s t a l
axes by PB w i l l n o t be f o l l o w e d i n t h i s t h e s i s , r a t h e r t he
a s s i g n m e n t made by Brown w i l l be o b s e r v e d .
-7TABLE I
At omi c Parameters
Standard D e v ia t io n
S
X
y
Z
0 ( 1)
X
y
Z
0( 2)
X
y
Z
0 ( 3)
X
y
Z
0(4)
X
y
Z
N (1 )
X
y
Z
N(2)
X
y
Z
Li
X
y
Z
H(I)
X
y
Z
O b t a i n e d by X - r a y and N e u t r o n S t u d i e s
I s T h a t Quot ed by P & B f o r T h e i r Data
, r\
Brown M bI
Van den U 6)
Hende & Padmanabhan & H U S t a n d a r d ( "11)
Boutin
Balasubramanian
Deviation
0.1286
0.1579
0.250
0.1290
0.1589
0.250
0.126
0.159
0.250
0.001
0.001
0.002
0.106
0. 191
0.519
0.1068
0.1870
0.5434
0.106
0.193
0.533
0.002
0.001
0. 0 0 2
0.154
0.010
0.220
0. 1511
0.0132
0.2296
0.153
0.012
0.229
0.002
0.001
0.001
0.259
0.230
0.149
0.2558
0.2335
0.1626
0.259
0.230
0.151
0.001
0.001
0.002
0.497
0.302
,0 . 5 9 6
0.4904
0.3048
0.6016
0.498
0.304
0. 600
0.001
0.001
0. 0 0 2
0.418
0.002
0.746
0.4193
0.0285
0.7405
0.414
0. 024 ■
0.743
0.001
0.001
0. 0 02
0.217
0.440
0.742
0.2137
0.4460
0.7346
0.213
0.440
0. 739
0.001
0.001
0. 0 0 2
0.437
0.332
0.253
0.433
0.339
0. 271
0.422
0. 336
0.256
0.003
0.003
0.003
0. 381
0.118
0.796
0.004
0.003
0.003
.
-8TABLE I
Brown(I 5 )
H (2 )
x
V
Z
H( 3)
x
Y
Z
H( 4)
x
Y
Z
H( 5 )
x
Y
Z
(continued)
Van den (TG)
Hende & Padmanabhan & 1 ) S t a n d a r d ( T l )
Boutin
Bal asubramanian
Deviation
0.467
0.036
0. 571
0.004
0.004
0.004
0.189
0.345
0.677
0.003
0. 0 0 3
0.003
0.247
0.428
0.924
0. 0 05
0. 0 0 4
0.004
0.292
0.469
0.601
0.004
0. 0 0 2
0.003
Fig. I
A Vi ew o f Li HzS P r o j e c t e d
( A f t e r Brown
Down t h e c A x i s .
H' *---nhI
.H3N--- N0xZ
-N H j
of"'
% ,N — N
^
\ ‘0
\
H'.*----nh3
o' /
H
* N 3N -------N ^ h
V
Irik----- NMi
O-'
,
/
•'i'--- Nc-
H
H T -------- N H 1
0V
H1N ------N-.
(O)
Fig. 2
Two A l t e r n a t e A r r a n g e me n t s P o s s i b l e f o r
N- H. . . N- H. . . Chai n i f t h e Hydrogen Bonds A r e Or der ed
( A f t e r Brown O ^ ) )
i
gr oup
gr oup
F i g . 3.
S k e t c h Showi ng t h e Gener al A r r a n g e m e n t o f t h e
NDg and NDg Groups and t h e B o n d i n g Between H y d r a z i n e Gr oups .
-11proton
in
a h y d r o g e n bond i s
t h e two atoms w i t h
the
proton
other.
is
which
it
found at a p o s i t i o n
f o r ms
t h e bond.
between
In general
n e a r e r t o one o f t h e bonded atoms t h a n t h e
The bonded atom w h i c h
denoted the do nor,
is
c lo s e s t to
and t h e f u r t h e r
the proton
bonded atom i s
is
called
the a c c e p t o r .
I n Li HzS, t h e d o n o r t o p r o t o n d i s t a n c e s a r e
O
a b o u t I A, and t h e a c c e p t o r t o p r o t o n d i s t a n c e s ar e
O
r o u g h l y 2 A.
The n i t r o g e n i s t h e d o n o r i n each N - H ...0
bond.
part
is
The n i t r o g e n
in
three
N - H . . .0
bonds f o r m
o f the
crystal.
sites
in
NHg
bonds.
end o f t h e
NgHg
The n i t r o g e n
N - H . . .0
sites
3,
N-H...N-H...
gr oup t a k e s
at the
Following
site
a t the
bond and i s
4 and 5 i s
neutron d i f f r a c t i o n
o f the s i t e
pictured
the
chains along the
notation
t h e s e bonds a r e r e f e r r e d
o th e r proton
in
Fig.
They f u r t h e r
NH2
the
bonded t o one oxygen and two o t h e r n i t r o g e n s .
la tte r
all
at
gr oup s
indicated
as s i t e
N 2 Hg
end o f t h e
d e n o t e d as s i t e
in
I.
Fig.
These
c direction
the proton
2 wher eas t h e
group i s
The l a b e l i n g
I.
end
in a
of
From t h e i r
s t u d y , PB have c o n c l u d e d t h a t most o r
2 protons
are ord er ed
2a as opposed t o
conclude
takes
NH2
to
o f PB,
NHg
place
that
in the arrangement
the arrangement in
no h i n d e r e d
rotation
of
Fig.
NHg
or
a t room t e m p e r a t u r e .
NMR SPECTRUM.
The
NMR
spectrum o f deuterons
in
L i HzS was s t u d i e d a t
2 b.
-12temperatures
f r o m 78°K t o 4 6 3 ° K.
78°K does n o t r e f l e c t
gr o u p and w i l l
any m o t i o n w i t h i n
be d i s c u s s e d
at hig her temperatures
discussed
in
in
in
this
re flects
Chapter I I I
measur ement s w h i c h
.The s p e c t r u m o b s e r v e d a t
w ithin
the
hydrazine
chapter.
The s p e c t r u m
m o t i o n and w i l l
be
the context o f other
have t o do w i t h
motion o f the deuterons
the c r y s t a l .
In f i r s t
o rd e r the
interaction
q u a d r u p o l e moment o f
the
gradient
site
(efg)
a t the
d e u t e r o n NMR l i n e
into
below the p o s i t i o n
o f the deuteron
two l i n e s
in
spaced e q u a l l y
re la tive
H0 , t h e s p e c t r u m c o n s i s t s
However, t h e r e
are o n l y f i v e
with
three
if
the
above and
Since t h e r e
t o t h e a p p l i e d dc
magnetic f i e l d
one o f t h e f o u r
splits
fie ld
a unit c e l l , for a rb itra ry
o f the c r y s t a l
non-equivalent.
nuclear
and t h e e l e c t r i c
o f the unperturbed l i n e .
are tw en ty de ute rons
orie nta tion
deuteron
bet ween t h e
For e x a m p l e ,
hydrazine
sites
of fo rty
lines.
which are c h e m i c a l l y
the deuteron a t
gr o u p s o f a u n i t
site
cell
I o f any
interacts
an e f g p o s s e s s i n g t h e same component s as each o f t h e
other s ite
I
deuterons
o f the c e l l .
The f o u r s i t e
I*
*Appendix A c on tains a d is c u s s io n o f the nuclear
q u a d r u p o l e i n t e r a c t i o n and t h e t e c h n i q u e o f n u c l e a r
m a g n e t i c r e s o n a n c e s p e c t r o s c o p y empl oyed i n t h e d e t e r m i n a ­
t i o n o f the efg t e n s o r c a r r i e d out in t h i s s t u d y .
-13deuterons
d iffe r
coordinate
sys t ems
are d i a g o n a l .
the c r y s t a l
is
only
in
the d i r e c t i o n
i n which
A result
sy mmet r y
the ir
of this
(see F i g .
o f t h e axes o f t h e
respective
chemical
I)
is
equivalence
that
if
made p e r p e n d i c u l a r t o one o f t h e c r y s t a l
ten
pairs
crystal
of lines
axis
only
are o b s e r v e d .
five
pairs
S p e c t r a were o b t a i n e d
crystal
77° K,
axes.
Fig.
First
the p o s i t i o n
the
o f the s p e c t r a l
about the th r e e
spectra
following
lines
was ,done a t f i x e d
the
individual
This
is
lines
with
frequency.
spectral
shown c l e a r l y
which
in
These p l o t s
curve o v e r l a y s .
exclude
points
angle,
were n o t a l l
were used i n
the curve
having
a least
the experiment
orientations
distinguishable.
fiv e pairs
positions
ten p a irs
were t h e n p l o t t e d
rela tive
by u s i n g a s e t o f s i n e
r o ugh f i t
error.
squares
of
a r e merged
<j>, o f t he c r y s t a l
large
to the
from the
4b w h e r e , o f t h e
were r o u g h f i t
By u s i n g t h i s
data p o i n t s
calculate
Fig.
The l i n e
a g a i n s t the o r i e n t a t i o n
t o H0 .
gauss s i n c e
s h o u l d be d i s c e r n i b l e ,
the o th er f i v e .
fashion.
Fo r many c r y s t a l
lines
obtained at
re la tive
c e n t e r o f t h e s p e c t r u m was measur ed d i r e c t l y
r e c o r d e r c h a r t and e x p r e s s e d i n
to a
are observed.
typical
in
H0
a x e s , a t most
a t 2^° i n t e r v a l s
The d a t a was a n a l y z e d
and o f
the f i e l d
Wi t h H0 p a r a l l e l
of lines
4 displays
efg tensors
it
was p o s s i b l e
The r e m a i n i n g
c o mp u t e r p r o g r a m t o
to
|j
Fig.
to
4- a .
77°K S p e c t r u m ,
Hq wi t h c I t o H0 .
Crystal
Oriented
I O0 f r om
Fig.
to
4-b.
77°K S p e c t r u m ,
H0 w i t h c J_ t o H0 .
Crystal
Oriented
27° f rom
a Il
2 av = A-j + B-| cose + Ci . s i n e
which be s t f i t
the data.
Figs.
t h e qu adr u. p ol e s p l i t t i n g
curves
for
rotations
about the three
I n t h e case t h a t
symmetric,
principal
(I
a rotation
axis
4,
5,
for
and 6 a r e p l o t s
the f i v e
crystal
deuteron
sites
ax e s .
the efg t e n s o r is
a b o u t an a x i s
of
c y lin d r ic a l Iy
p e r p e n d i c u l a r t o t he
o f t h e e f g t e n s o r makes 2 av p r o p o r t i o n a l
to
+ 3 cos 2e) where e measur es t h e a n g l e bet ween t h e
principal
axis
and t h e m a g n e t i c f i e l d
bet ween a maximum p o s i t i v e
negative
value
at
0=90.
the p o s i t i v e
value
an a r b i t r a r y
axis
to
(2. 1)
the p r i n c i p a l
parallel
plots
value
for
Hq .
8=0 , and a maximum
The n e g a t i v e v a l u e
in a b s o lu te magnitude.
includes
axis,
to the p r i n c i p a l
o f 2Av ( F i g s .
5,
n e g a t i v e maximum v a l u e s
the o r i e n t a t i o n
although
it
axis.
6 , and 7)
2Av t h e n r a nge s
does n o t
is
just
A rotation
include
it
can be seen t h a t
the
ar e a p p r o x i m a t e l y t h e same
nearly
cylindrical
symmet r y o f t h e
values
o f t h e component s o f t h e e f g t e n s o r r e l a t i v e
axes a r e o b t a i n e d
axis.
This
indicates
t he
bonds s t u d i e d .
f r o m t h e A , B and C o f
The e f g t e n s o r was t h e n d i a g o n a l i z e d
constant,
Hq
By e x a m i n a t i o n o f t h e
rotation
coupling
about
H0 p e r p e n d i c u l a r
independent o f the
crystal
half
and v a l u e s
t h e asymmet r y p a r a m e t e r ,
The
Eg.
to the
2.1.
o f the
and t h e a n g l e s
-16relat'ing
p rin cipal
were o b t a i n e d .
axes o f t h e e f g
This
data
The e s t i m a t e d e r r o r s
the follo w in g
is
axis
is
II.
included
i n Table
11 a r e
about d i f f e r e n t
parallel
to
axes
that
to
than h 0 .
better
but with
in itia l
I 0 and e s t i m a t e d
to
pa rallel
the
alignment o f
the
values
presumably
Such
the c r y s t a l
checks
was a c c u r a t e
Error
than 0 . 1 ° . '
curves.
the
rotation
The d a t a p o i n t s
To t e s t
used i n t h e
the e f f e c t
t e n s o r c omponent s
in
d a t a was f e d i n t o
included
least
in
-
squares
of errors
the p r i n c i p a l
on
axis-
the d i a g o n a l i z a t i o n
o f e r r o r were based on t h e r e s u l t i n g
.
efg
to
angle is
the o u tp u t.
The ' d e u t e r o n
structure.
axis
in
p r o g r a m and t h e e s t i m a t e s
DISCUSSION.
rotations
0.1°.
o f the efg
in
axis.
a n g l e was r e a d d i r e c t l y
system a s e t o f v a r i e d
varia tion
so t h a t
The r o t a t i o n
6 and 7 a r e t h o s e f i n a l l y
the
obtained during
chec k t h e a l i g n m e n t .
Figs.
of
is
a crystal
same c r y s t a l
to
calculation
to
set
based on
t h e s p e c t r o m e t e r he a d ,
believed
5,
be l e s s
One s o u r c e o f e r r o r
the spectra
Hq was used t o
indicated
in
not q u ite
The a g r e e me n t bet ween
crystal
in Table
considerations.
rotation
the
given
misalignment o f the c r y s t a l
the
ten sor to
results
(See A p p e n d i x C)
are r e l a t e d
to
t h e Li HzS
2 Av ( kHz)
Fig.
5
O
O
O -
200
-200
■
O
O
Quadrupole S p l i t t i n g P a tte r n s f o r R o ta tio n about the
a C r y s t a l l o g r a p h i c Axis at L i q u i d A i r Temperature.
2 Av(kHz)
Fig.
6
o
o
o
200
- 200
o
o
Q u a d r u p o l e S p l i t t i n g P a t t e r n s f o r R o t a t i o n a b o u t t he
b C r y s ta llo g r a p h ic Axis at Liq uid A i r Temperature.
300 '
-100
Fig
7
Qua dru po le S p l i t t i n g P a t t e r n s f o r R o t a t i o n about the
c C r y s t a l l o g r a p h i c Axis a t L i q u i d A i r Temperature.
-20TABLE I I
C o u p l i n g C o n s t a n t eqQ/h., Asymmet ry P a r a m e t e r n , and
the Angle
Formed Between t h e P r i n c i p a l
T e n s o r and t h e C r y s t a l
Indicated
in
Fig.
I .
Between t h e P r i n c i p a l
Axes, f o r
t h e Deuteron S i t e s
Is the Angle
Axis
Field
Between t h e N i t r o g e n
Bond.
(All
o f the
and P r o t o n
E s t i mated E r r o r s
eqQ/h
kHz
n
I
208 + 1
2
Bond
o f the efg
Also L is t e d
Vector
Temp.
0K
Axis
G r a d i e n t and t h e
Involved
f o r Angles
Formed
in the
Gi ven Ar e +1°
) .
*zz.
N-H
angl e
deg.
6 X
Gy
0 . I 6 ± 0 . 07
107
19
97
9
178+1
0.12+0.07
114
6
155
4
3
166 + 1
0.05+0.07
73
I 62
97
10
4
167 + 1
0.10+0.07
114
4
25
7
5
162 + 1
0.05+0.07
8
70
131
I
I
201 ±1
0.15+0.07
108
20.
98
7
2
178 + 1
0.08+0.07
114
5
155
4
6 Z
77
193
-21next
largest
as f i r s t
component s o f t h e e f g
principal
and second p r i n c i p a l
As can be seen f r o m T a b l e
lies
II,
the f i r s t
The a n g l e so f o r med r a n g e s
I O0 f o r
bonds I
and. 3.
be d e n ot ed
axes r e s p e c t i v e l y .
principal
n e a r l y a l o n g t h e N-H bond d i r e c t i o n
site.
about
tensor w i l l
component
a t each d e u t e r o n
f r o m I 0 f o r bond 5 t o
(See T a b l e
III
for
bond
Ie n g th s .)
If
the angles
direction
the u n i t
and t h e c r y s t a l
cell
axes f o r . o n e
are denoted
symmet r y r e l a t e d
angles,
f o r me d bet ween t h e e f g p r i n c i p a l
sites
6X, u - Q y ,
ex , 6»,
makes t h e e f g t e n s o r p r i n c i p a l
I
and s i t e
molecule
I
sim ila ritie s
bond s i t e s
Fortunately,
of s ite
I
This
o f molecule
and s i t e - 2
the evidence
o f the
sites
3,
normal
makes i t
to
alone.
of sites
possible
3,
4
to assign
I and 2 a t 193°K where t h e NMR
4,
The second p r i n c i p a l
ma tely along the
t h a t the deuterons
rotation
of sites
of structure
of
These
w o u l d make t h e a s s i g n m e n t o f e f g t e n s o r s
the efg tensors
The a n g l e s
axis
Q2 «
4 o f molecule 4 nearly p a r a l l e l .
and 5 u n d e r g o h i n d e r e d
spectra
Qz ; Tr-Qx , i r - Qy,
ambi guous on t h e b a s i s
of
have t h e c o r r e s p o n d i n g
3 o f molecule 4 almost p a r a l l e l
and s i t e
sites
and Qz , t h e n t h e t h r e e
o f the c e l l
6Z ; Tr-Qx , Gy,
o f the f i v e
axis
and 5 ar e merged.
component i s
found t o
to the plane d e fin e d
f o r med bet ween t h e
lie
approxi­
by N-j -Ng-D.
second p r i n c i p a l
component
-22and t h e nor mal
to
12° and 5° f o r
sites
estimated, e r r o r
principal
a xia lly
I
through
is' ±3°,
17 ° ,
symmetric.
somewhat I ar g.er t h a n
the
Sim ilar
results
O - D . . .0 b o n d .
tensor o rie n ta tio n
o f the I
bet ween 77°K and 19 30 K i s
16°,
5 respectively.
component because t h e ef g. t e n s o r s
C h i b a f o r
efg
t h e p l a n e o f N^j - N 2 -D ar e
14°,
The
for
the f i r s t
are n e a r l y
have been f o u n d by
The s l i g h t
and 2 s i t e
change i n
the
deuterons
s m a ll e r than the experim enta l
error.
The s p e c t r u m o b s e r v e d a t an o r i e n t a t i o n
with
c j_ Hq
and n e a r b j J Hq showed some e v i d e n c e o f a s p l i t t i n g
lines
corresponding
deuteron
of s ite
identical
I
corresponding
I.
This
may i n d i c a t e
can be f o u n d a t e i t h e r
positions.
spectrum i n which
is
to s i t e
The
this
only
s p littin g
region
almost along the
firs t
principal
of the
f o r which
axis
that
the
o f t wo a l m o s t
rotation
was e v i d e n c e d
t o maximum s p l i t t i n g ,
o f t he
is
that
the f i e l d
of the
H0
fie ld
gradient tensor.
It
was assumed t h a t
associated with
ferro e le ctric
structural
domai ns
domai ns o f o p p o s i t e
related
such a s p l i t t i n g
in
differences
the
crysta l.
p o larizatio n
by r e f l e c t i o n .
d e s c r i b e d would i n d i c a t e
in
c o r r e s p o n d i n g -to
O rd inarily
of a fe rro e le c tric
Observation
that
m i g h t be
t he
are
o f such a s p l i t t i n g
Li HzS domai ns
are not
as
-23related
by r e f l e c t i o n
ferroe lectric.
v / m ,, on t h e
cuttin g
S ilve r
form o f
noise
spectrum.
Tfie' c r y s t a l , was p r e p a r e d by
faces
p e rp e n d ic u la r to the c a x is .
wer e t h e n p a i n t e d
E in ord e r to minimize
ra tio
was so d e g r a d e d ,
rf
. A stifflc fe n tly
the
o f the
with
the
studies
coupling
that
into
the
to
the
line
The l ow t e m p e r a t u r e
o f domai n e f f e c t s .
t y p e O- H. . . 0 have been s t u d i e d
coupling
the
The s i g n a l
introduction
plates,
in
large.-..n.umbe , r h - y d - r oge-n^ boTi-d'ed sysf.'ems
c o n s t a n t eqQ/h,
t h e O-H s t r e t c h i n g
sim ilar
currents.
was no l o n g e r o b s e r v e d .
s p e c t r u m showed no o t h e r e v i d e n c e
of
on t h e s e f a c e s
by t h e
sampl e r e g i o n , o f t h e s e p o l a r i z i n g
sp littin g
a t t e m p t was made t o
o f e l e c t r i c f i e l d s , , o f the; o r d e r o f
and p o l i s h i n g
contacts
be c o n s i d e r e d a new t y p e o f
An u n s u c c e s s f u l
obs er v e- t h e e f f e c t
I
and m i g h t
with
0...0
and
' . The number o f f
is
small.
and o t h e r p e r t i n e n t
such s y s t e m s .
■ '
0 . . . 0 bond- l e n g t h s
correlation
bond l e n g t h s
frequencies
o f M-H...-.X s y s t ems
constant
to allow
Table
IV l i s t s '
inform ation
for
several
O
r a nge f r o m 2 . 4 0 A t o
O
a b o u t 3 A.
0.
The r a nge ( 2 6 ) 0 f . j \ | . . . 0 bond l e n g t h s i s a p p r o x i m a t e l y
O *
'
t o .3. 05 A._
A l t h o u g h a p l o t o f eqQ/h a g a i n s t b r i d g e
2 . 8 5 A.
length
• x These- e s t i m a t e s do n o t i n c l u d e weak o r b i f u r c a t e d
bonds f o r w h i c h t h e b r i d g e l e n g t h may be c o n s i d e r a b l y
longer.
'
- 24 -
'
TABLE I I I
N-H.;. .X Bond A n g l e s
from the
.
and I n t r a - b o n d ' L e n g t h s
Room T e m p e r a t u r e
Calculated
Dat a o f Padmanabhan and
Ba l a s u b r a ma n i a n - . ^ ^
N-H...X
O
A
H - bond D i m e n s i o n s ,
(H-bond)
Bond A n g l e ,
Deg.
N-H
H...X
N
X
I
11.8 . 2
1:02
CM
LO
CO
2.97
2
156.4
1.02
2.09
3.05
3
166.3
KO 2
1.84
2.84
4
146.1
I . 01
2.00
2.89
5
150.3
T . 05
2.03
2.99
for
0...0
correlate
bonds shows a s t r o n g
bridge
not s u c c e s s fu l.
length with
Within
dependence o f eqQ/h on
slope
of
such a p l o t
charge o f the
that
the
co rre la tio n ,
eqQ/h f o r
(l/r^„[))^
nucleus
no o t h e r n u c l e a r s i t e
N- H. . . 0
bonds was
NDg gr oup o f Li DzS t h e
is
can be used t o
nitrogen
an a t t e m p t t o
in
consistent.
The
estimate
effective-
the
the crude a p p r o x i m a t i o n
contributes
to
the efg
a t t he
TABLE IV
Coupling Constants
Bond
Compound
Urea d.
4
single crystal
Glycine
polycrystal Iine
( bond t y p e s ar e
conjectured)
N - D . . .0
N - D . . .0
ND3 a v e r a g e
ND3 a v e r a g e
Te m p e r a t u r e
298°K
2 98° K
298°K
I SO0 K
I 50°K
T>250°K
T >250°K
for
Some N-D Bonds.
eqQ/H ( k H z )
n
Reference
2 1 0 . 8+1
21 0.. 7 ± I
206.3+1.2
0.139+0.010
0.146+0.010
0.138+0.010
I9
168
188
80
71.2
0.00
0.00
0.24
0.40
20
.22
. 22
21
ND3 a v e r a g e
ND3 a v e r a g e
298°K
298°K
P r i ncipal
component
51 . 1 0
51 .46
N - D . . .0
298°K
I 30
0.0
22 .
180.1+1.0
0 . 00± 0.01
O
Q
uO
25“
Glycine
single crystal
( bond t y p e i s
conjectured)
N-D-i . . .0
N - D p . . .0
N- Dg. . . 0
and Asymmet r y P a r a me t e r s
ND4 D2PB4
crystal
ND4 Cl
single
crystal
ND3
. p o l y c r y s t a l I i ne
N - D . . .N
NDg a v e r a g e
O
Q
I
Z
(ND4 ) 2SO4
N - D . . . Cl
O
CO
single
75°K
185° K
77°K
156 ±7
72.5+.9
180
24
0,0
25
-26-
A
100
N-W -N
-
Fig. 8 .
C o u p l i n g C o n s t a n t P l o t t e d A g a i n s t t h e Square
o f t h e X--H S t r e t c h i n g F r e q u e n c y f o r O-H Bonds o f a Number
o f Compounds E x c e p t f o r t h e N-H Bonds o f Li HzS as I n d i c a t e d .
(A fte r Blinc & Hadzi) U
deuteron.
Such an e s t i m a t e y i e l d s
the e f f e c t i v e
Fig.
n u c l e a r charge o f
8 shows a p l o t
the
a v a l u e o f 0 . 45e f o r
nitrogen.
o f eqQ/h a g a i n s t
the
square o f
-27t h e O-H and
N
represented
in
- H
(27,28)
the
stretching
upper r i g h t
O - H . . , 0 bonds o f c r y s t a l
found in
phosphates
lower l e f t
hydrates
region.
to
nitrogen-hydrogen
plot.
o f the p l o t
while
ar e
t h e O - H . . .0 bonds
are r e p r e s e n t e d
in
the .
bonds o f t h e c r y s t a l
o
dependence on v
t h a n t h e bonds
phosphates
this
hydrates,
Bonds
The s t r o n g e r
show a l a r g e r
tra nsition
region
end s e l e n i t e s
o f the
the
frequencies.
and s e l e n i t e s . -
bet ween t h e
The p o i n t s
bonds o f Li HzS f a l l
two g e n e r a l
corresponding
near the
O-H bonds r e p r e s e n t e d
in
CHAPTER I I I
DEUTERON MOTION
CONDUCTIVITY A-ND- DIELECTRIC CONSTANT.
An e l e c t r o l y s i s
experiment
demonstrates
that
the c axis
to protons.
E lectrical
carried
conductivity
as VCP) show t h a t t h e c a x i s
L i HzS i s
(hereafter
conductivity
o f magnitude g r e a t e r than
dire ction .
in
due
c o n d u c t i v i t y measur ement s by
V a n d e r k o o y 9 C u t h b e r t and F e t c h e s )
orders
o u t by Vander k ooy
is
to
two t o t h r e e
conductivity
T h e i r e x p e r i m e n t s were c a r r i e d
referred
in
the a or b
o u t on c r y s t a l
sampl es o f a p p r o x i m a t e l y 8 mm t h i c k n e s s w i t h e l e c t r o d e s o f
2
a r e a a b o u t 0 . 5 cm .
A dc v o l t a g e o f 290 v o l t s was a p p l i e d
to the
sampl e and an e l e c t r o m e t e r was used t o measure t h e
current.
Although
po larizatio n
c a r e was t a k e n
currents
o f the v o lta g e
by w a i t i n g
found f o r
directions
Fig.
9.
It
i n T a b l e V.
o f temperature
in
in
application
temperature
the experiment.
t he a,
b and c a x i s
The c a x i s
f o u n d by VCP i s
was f o u n d t h a t t h e a and b a x i s
c o u l d n o t be r e p r e s e n t e d
this
circu it
the c o n d u c t i v i t y
are giv en
as a f u n c t i o n
a ti me a f t e r
b e f o r e m e a s u r i n g t h e c u r r e n t , no m e n t i o n i s
made o f t h e use o f a gu a r d r i n g
Values
to avoid measuring
by a s i n g l e
conductivity
displayed
in
conductivities
activation
energy ove r
range.
The c o n d u c t i v i t i e s
o f Li HzS and Li DzS r e p o r t e d
have been measur ed by a s l i g h t l y
diffe ren t
herein
technique.
The
-29TABLE V Conductivities
and A c t i v a t i o n
Deuterated Lith ium
E
A ctivation
(eV)
f o r Normal
and
Hydrazinium S u l f a t e
Compound
Axis
Measur ed R e f e r e n c e
Along
0.93+0.09x1Q -n
Li HzS
a
VCP
O
CO
O
1+
O
O
to
X
O
I
a(ft cm) "
Measur ed a t
Room Temp
Energies
Li HzS
b
VCP
Li HzS
C
VCP
0.23±0.09xl0"8
0.85
0 . 4 x l 0 “ 12
0.34
Li HzS
( I r r e g u l a r S a mp l e )
a
0 . I B x l O ' 12
0.58
Li DzS
a
0 . 4 x 1 0" 1 2
0.47
Li DzS
b
0 . 1 2x 1 0 " 9
0.65
Li DzS
C
0 . 10x 1 0 - 8
0. 51
Li HzS
C
experiment is
conductivities
given
described
for
i n T a b l e V.
in
Chapter IV.
The room t e m p e r a t u r e
deuterated
and u n d e u t e r a t e d c r y s t a l s
Activation
energies
are a l s o
are
given.
These ar e somewhat l o w e r f o r
Li HzS t h a n t h o s e f o u n d by VCP.
In a d d i t i o n
conductivities
t h e a and b a x i s
o f the
- 30
deuterated
compound were m e a s u r e d .
n o t be d e s c r i b e d ' by a s i n g l e
entire
temperature
temperature
fit
to
Two a a x i s
giving
the a c t i v a t i o n
axis
three
Fig.
firs t
curves
The c
has a c o n s t a n t s l o p e
to t h i s
c o n s t a n t o f Li DzS has been measured i n
The r e s u l t s
Figs.
ex ami ned t h e
f o u n d no anomal y i n
and 140° C.
vs I / T
Above
9.
laboratory.
in
i n T a b l e V.
o f more t h a n 1 . 5 eV.
vs 1 / T c u r v e
this
were u s e d,
t e m p e r a t u r e , r a n g e and t h e b e s t f i t
The d i e l e c t r i c
displayed
studies
indicated
and was b e s t
energy, f o r
log c o n d u c t i v i t i e s
energies
log c o n d u c t iv ity
shown i n
this
and one b a x i s
o f these
activation
over the e n t i r e
' is
t h e d a t a bet ween room
o f the a c t i v a t i o n
energies
they could
energy over the
and a b o u t I l O 0C a p p e a r e d l i n e a r
region.
indicate
activation
r a nge s t u d i e d ,
g e t an e s t i m a t e
170°C a l l
Although
10,
o f t h e s e measur ement s
11 and 12.
d ie le c tric
Pepinsky e t
con s ta n t o f the
the d i e l e c t r i c
constant
are
al,("*°)
who
crystal,
bet ween - 196° C
Our s t u d y has shown no d i e l e c t r i c
peak up t o
205° C.
According
to
the
Debye
and e" = mT / ( I + M2T2 ) .
*
equations
Log e '
and l o g
e' ^ 1 / ( 1
e"
2 2
+ m T )
plotted
as
* A d i s c u s s i o n o f t h e d i e l e c t r i c l o s s may be f o u n d i n
C h a p t e r 1 1 ; D i e l e c t r i c B e h a v i o r and S t r u c t u r e by C h a r l e s
P h e l p s Smyt h, McGraw H i l l Book C o . , I n c . , New Y o r k , 1955.
Li HzS
Li HzS
Li DzS
Fig.
9.
Be s t F i t Cu r v e s o f t h e L o g a r i t h m o f C o n d u c t i v i t y
f o r Li HzS and Li DzS as a F u n c t i o n o f T e m p e r a t u r e .
( VCP
I n d i c a t e s R e s u l t s from Reference 29.)
A OeuTeRfiTE-D
o UuvevT ER RTtD
- -O -
- - -O"- O
—-O--
oy~
/r ^fQvezUdr' 5 Hz
F i g . 10.
L o g a r i t h m o f t h e Real and I m a g i n a r y P a r t s o f
t h e D i e l e c t r i c C o n s t a n t f o r Li HzS and Li DzS Measur ed A l o n g
t h e c A x i s vs F r equ enc y a t w h i c h t h e Measur ement Was Made.
-33f u r i c t i dns. o f w f o r
slopes
near - 0 . 5 ,
consistent with
several
as shown i n
the
with
ation
the
times
the
Fig.
relaxation
assumption
in
temperatures
10.
This
Debye e q u a t i o n s ^ w h i c h
assumption o f a s i n g l e
well
fixed
time
t
if
it
of a d is t r ib u t io n
T>T .
m
not
from the
agrees
y ( t ) of
quite
re la x­
w r y ( t ) dr
y ( t.) dr
I + W2T 2
1 + W2T 2
O
For t h i s
is
relations
JO
Y ( t ) = ( a - l ) T ma " l ' [ e ( 0 ) - e oo] T a " 1 f o r
for
behavior
result
.
00
eoo +
show n e g a t i v e
y ( t ),
if
0<T<Tm, and y ( t ) = O
corm>> l , t h e
m
d ie le c tric
constant
becomes
e.
TT( a- I ) [ e ( 0 )-eco]
£ 1- j e • = £ +
.__ ^
S i n 7^ f ------
2 ( “ Tm>2
The C o l e - C o l e
plot
Li DzS a l m o s t obeys
A more c o n c e n t r a t e d
of
this
by t h e
which
contribution
du ctivity
ation
A ctivation
curve
give
expression
for
of
to
Fig.
12.
for
s in is iiil
25°C f o r
relaxation
a = 0.5.
times
is
room t e m p e r a t u r e ,
out.
for
dc c o n ­
A single
c e n t e r e d on t h e
re la x­
e 1 axis.
the
relaxation
mechani sm ar e
For t h e
deuterated
compound two
are e v i d e n t ,
above 100°C.
.'' -• 3 ____
11 shows t h a t
e " made by t h e
a sem icircle
e V and a c o n t r i b u t i o n
region
at
Li HzS a t
a^/eow
energies
from Fig.
contributions
0.4
e 1 in
mechani sm has been s u b t r a c t e d
time would
estimated
versus
d istrib u tio n
exhibited
the
e"
___ ___ ’
a l ow t e m p e r a t u r e
o f 0.7
eV in
the
contribution
temperature
•
of
A Deuterated
(25 C)
o Undeuterated
F i g . 11.
EIl vs e '
(26.2
for
Li HzS and Li Dz S.
G 200
C^D e OTE. R R T BD
O U m D £ Ur e . R R T E D
G FlM D e
-
/00 W:
F i g . 12. L o g a r i t h m o f t h e Real and I m a g i n a r y P a r t s o f t h e
D i e l e c t r i c C o n s t a n t f o r Li HzS and L i p z S Measured A l o n g t h e
c A x i s as a F u n c t i o n o f T e m p e r a t u r e .
-36' PREVIOUS MMR STUDIES.
. Two p r e v i o u s
LiHzS.(1 2 ,1 3 )
the protons
me as ur ed.
j n the f i r s t
have been c a r r i e d
of these,
o f the f i r s t
wo r k were e x t e n d e d and t h e s p i n
t i m e o f t h e p r o t o n s was meas ur ed.
structure
s t u d y was d o n e .
second moments measur ed a t v a r i o u s
sampl es
the
o f L i ^ were
l a t e r w o r k , t h e second moment measur ement s
s t u d y o f t h e second moment was c a r r i e d
neutron
and i n
o u t on
t he second moment o f
and t h e q u . a d r u p o l a r s p l i t t i n g
In the
relaxation
NMR s t u d i e s
single
crystals
la ttice
The f i r s t
out before
t he
On t h e b a s i s
o f the
temperatures
in
at various
powder
orie n ta tio n s,
a r r a n g e m e n t o f t h e p r o t o n s was deduced and t h e NHg and
NHg m o t i o n s were h y p o t h e s i z e d .
second moment as a f u n c t i o n
s a te llite
separation.
s a te llite
separation,
crystal
Fig.
13 shows t h e p r o t o n
o f temperature along the L V
Because o f t h e change i n
the
C u t h b e r t and P e t c h ^ ^ ^ s u g g e s t t h a t
un d e r g o e s a g r a d u a l
structural
the
change w h i c h i s
complete at about I 65°C.
The r e d u c t i o n
temperature
rotation
in
t h e second moment bet ween l i q u i d
and SO0C was a t t r i b u t e d
o f t h e NH3 g r o u p .
a ir
to the on set o f hinde red
The f u r t h e r
reduction
between
20° and I SO0 C was t h o u g h t due t o w h a t e v e r mechani sm p r o v i d e d
ex c hange bet ween s i t e s
I
and 2.
The f i n a l
drop
in
second
>
moment t o
the very
l ow v a l u e o f a b o u t 0 . 5 gauss
a t 20Q°C
'I ---- ::—I - H
Mineral oil
-200
T E M P E R A T U R E ,* C
'
F i g . 13.
The Open C i r c l e s Gi v e t h e V a r i a t i o n w i t h
T e m p e r a t u r e o f t h e F r equ enc y S e p a r a t i o n Between t h e Two
Li 7 S a t e l l i t e s Whi ch Occur on t h e Same Si de o f t h e C e n t r a l
L i n e and C o i n c i d e i n t he Hi gh T e m p e r a t u r e Pol y mo r ph o f
L ith iu m Hydrazinium S u lfa t e .
The S o l i d C i r c l e s Gi ve t he
V a r i a t i o n w i t h T e m p e r a t u r e o f t h e Second Moment o f t he
P r o t o n Resonance i n Powdered L i Hz S.
( A f t e r Reference 12)
"38"
was t h o u g h t t o be due to. e i t h e r
protons
through
group.
the c r y s t a l
These a u t h o r s
p r o p e r t y o f the c r y s t a l
is
N-H.. . N - H . . . chains(Fig.
on t h e b a s i s
rapid d i f f u s i o n
or to
tumbling o f the e n t i r e
suggest th a t
the f e r r o e l e c t r i c
associated with
2).
It
ordering
o f the
was t h e n p o i n t e d o u t t h a t
of th is
m o d e l , ex c hange bet ween s i t e s
I
a loss
o f o r d e r o f t h e s e N-H d i p o l e s
and
would lead to
hence a f e r r o e l e c t r i c
general
and 2
tra nsition .
The second moment measur ement s p r e s e n t e d i n
stu dy are i n
o f t he
a g r e e me n t w i t h
the l a t e r
those o f the e a r l i e r ,
although
t h e knee i n
t h e c u r v e a t a b o u t 8 . 8 g a u s s 2 was n o t
repeated
(Fig.
The s p i n
14).
t h e p r o t o n s was me a s ur ed.
I/T .
la ttice
Fig.
relaxation
15 shows a p l o t
Bel ow a t e m p e r a t u r e o f a b o u t I 90°K t h e
attributed
to m olecular r e o r i e n t a t i o n
activation
e n e r g y 0 . 1 8 2 ± 0 . 0 1 3 eV.
energy i s
for
the decrease
found to
I n summary,
in
the
It
protons
I
relaxation
is
assumed t h a t
mechani sm i s
i n T-j and t h e a c t i v a t i o n
of sites
r o t a t i o n , and t h a t
of s ite s
is
t h e second moment s t u d i e s
temperature
vs
be 0 . 4 3 ± 0 . 0 9 eV.
suggest t h a t the protons
hindered
o f T]
o f t h e NHg gr oup w i t h
above a b o u t GO0 C t h e NH^ r e o r i e n t a t i o n
responsible
time o f
this
o f the p r o to n s
3, 4 and 5 under go a
m o t i o n becomes
r a n g e bet ween 100 and 1 5O0 K.
and 2 b e g i n t o
important
The
under go some s o r t o f
-39-
T CC )
F i g . 14.
T e m p e r a t u r e Dependence o f t h e Second Moment
o f t h e P r o t o n A b s o r p t i o n S i g n a l f r o m Powdered Li HzS.
( A f t e r R e f e r e n c e 13)
exchange bet ween s i t e s
either
a very
gr o u p s s t a r t s
to
rapid
in
the region
diffusion
o f protons
to occur or else
t u m b l e and t r a n s l a t e
measur ement s y i e l d
Li ^ q u a d r u p o l a r s p l i t t i n g
Above I SO0C
bet ween NgHg
t h e gr oups t h e m s e l v e s
through
activation
near O0C.
the c r y s t a l .
energies
studied
for
begin
The T^
the
as a f u n c t i o n
rotations.
of
-40-
125*K
37 Me/«
27 Mc/l
1 0 0 0 /T
C K " ')
F i g . 15.
T e m p e r a t u r e Dependence o f t h e P r o t o n Spi n
L a t t i c e R e l a x a t i o n Time T-j i n L i t h i u m H y d r a z i n i u m S u l f a t e .
( A f t e r Reference 13)
temperature
form which
implies
is
a structural
complete
change t o a h i g h e r symmet r y
ne a r I 6 5 ° C .
- 41 HIGH TEMPERATURE SPECTRUM.
If
the qu.adrupolar i n t e r a c t i o n
t h e Zeeman i n t e r a c t i o n
as i n
deuteron
changes s i t e s
its
if
is
there
a change i n
component p a r a l l e l
to
energy l e v e l
the e l e c t r i c
H0 .
sites
lines
When t h e r a t e
may a p p e a r ,
Evidence di s c u s s e d
protons
rotation.
it
in
has been p o s s i b l e
o f the m o tion,
in
place,
the previous
lines
in
t o ‘d e t e r m i n e
further
increased
These l i n e s
along w i t h
4 6 5° K.
n e a r I l O 0 K.
a new p a i r
The s p l i t t i n g
an e f g wh i c h i s
of
section
energies
(Fig.
it
is
sites
in
Li DzS
o f the frequency
involved.
16)
o f L i DzS i s
found t h a t
3,
4,
lines
lines
the
and 5
As t h e t e m p e r a t u r e
is
a p p e a r n e a r I 40° K.
and a r e s t i l l
of these
indicates
o f deuterons
estimates
o f temperature
are ver y s t r o n g
a new p a i r o f
unambiguously the type o f .
a spectrum corresp ond ing to
b r o a d e n and v a n i s h
the
u n der go h i n d e r e d
t h e NNR s p e c t r a
and t h e a c t i v a t i o n
as a f u n c t i o n
If
to
bet ween w h i c h t h e j u m p i n g o c c u r s .
When t h e d e u t e r o n NMR s p e c t r u m
studied
change
corresponding
corresponding to
t h e NHg and NHg s i t e s
By s t u d y i n g
motion t a k i n g
lines
changed
gradient
of s ite
becomes h i g h e r s t i l l ,
the average ove r the s i t e s
that
fie ld
on
when a
scheme i s
a r e b r o a d e n e d and d i s a p p e a r .
ex c hange f r e q u e n c y
spectral
a perturbation
these experiments,
becomes l a r g e e n o u g h , t h e s p e c t r a l
individual
is
in
evidence at
as a f u n c t i o n
o f the
143 K
F i g . 16.
NMR S p e c t r a Taken w i t h
b l l Hq a t T e m p e r a t u r e s
Indicated.
A y (KHz)
2
I OOi
3 4-5
-
-1 0 0
F i g . 17
Quadrupole S p l i t t i n g P a t t e r n s of the Two Sets
o f Averaged Li nes f o r R o t a t i o n about the a
C r y s t a l l o g r a p h i c Axis a t 95°C.
Dashed Li nes
I n d i c a t e Regions i n which t he S p e c t r a l Lines
Could not be Obser ve d.
ZH > I ) ^ V 3
3 4-5
-
Fig.
18 Quadrupole S p l i t t i n g P a t t e r n s o f the Two Sets
o f Averaged Li nes f o r R o t a t i o n about the b
C r y s t a l l o g r a p h i c Axi s at 95°C.
Dashed Li nes
I n d i c a t e Regions i n which the S p e c t r a l Li nes
Could not be Observed.
I
ZAv(KHz)
J
3 4-5
-
Q 11 H o
- IO O *
F i g . 19 Quadrupole S p l i t t i n g P a t t e r n s o f the Two Sets
o f Averaged Li nes f o r R o t a t i o n about the c C r y s t a l l o ­
g r a p h i c Axis a t 9 5 ° C .
Dashed Li nes I n d i c a t e Regions i n
which the S p e c t r a l Li nes Could not be Observed.
-46a n g l e bet ween c r y s t a l
Figs.
axis
17, .18, and 19.
Near 2 8 5 0 K, t h e
spectrum c o r r e s p o n d in g
disappear.
I
and 2,
appears
Figs.
in d ica ting
to s i t e s
I
A new p a i r o f l i n e s ,
not evident f o r
H0 .
and m a g n e t i c f i e l d
17,
orientations
the
due t o t h e a v e r a g e o f s i t e s
These l i n e s ,
however,
o f the c r y s t a l
18 and 19 show t h e
as w e l l
in
shown i n
and 2 , b r o a d e n and
a t a b o u t 330° K.
all
lines
is
sp littin g
the a n g u la r re g io n
re la tive
for
i n which
are
to
these l i n e s ,
the l i n e s
are
not observed.
The c o m p l e t e
studied
a t 193°,
rotation
298°,
s p e c t r u m o f t h e d e u t e r o n s was
338°,
368°,
spectrum i s
not s e n s i t i v e l y
a varia tion
o f ±5°C was a l l o w e d .
in
the spectrum w e ll
w ithin
and 43 8° K.
d e p e n d e n t on t h e
This
The v a l u e s
o f eqQ/h f o r
and 5 show a c o n s i s t e n t
I 65°C.
at the
If
all
parameter)""3 .
error.
in Table VI.
the averages over s i t e s
o f the u n i t
t h e e f g w o u l d v a r y as
cell
with
the
temperature
temperature y i e l d s
thermal
an e f g
Two p o s s i b l e
re mainder o f the decrease are the
in
4,
(la ttice
d e c r e a s e o f a b o u t 2.5% f r o m - S t i 0C t o 16 5 ° C .
for
3,
were t o expand
Using the average o f the a n i s o t r o p i c
expansion c o e f f i c i e n t s , ^ 3 0 ^ t h i s
origins
The r e s u l t s
d e c r e a s e o f a b o u t 10% f r o m - 80° C t o
the contents
same r a t e ,
temperature,
produced v a r i a t i o n s
experimental
o f t h e s e Measur ement s a r e t a b u l a t e d
Because t h e
the a m p litu d e o f the t o r s i o n a l
increase
TABLE
VI
C o u p l i n g C o n s t a n t e q Q / h , Asymmet r y P a r a m e t e r n , and t h e A n g l e s Between t h e F i r s t
P r i n c i p a l Axes and t h e C r y s t a l l o g r a p h i c Axes f o r t h e e f g s R e s u l t i n g f r o m
M o t i o n a l A v e r a g i n g o v e r S i t e s I and 2 and o v e r S i t e s 3, 4 and 5.
Calculated
A v e r a g e s Based on e f g s Measur ed a t 78°K f o r I n d i v i d u a l S i t e s Ar e Gi v en f o r
Compari s o n .
Temperature
eqQ/h
kHz
9
n
6 X
C a l c u l a t e d average
o v e r s i t e s I and 2
± 0
48.9 ± 0.5
0.29
±
0.05
37.1
12 7 . 4
89.0
25°C
48.6 ± 0.5
0.22
±
0.05
38.0
12 7 . 5
89.9
65°C
4 7 . 5 ± 0 .5
0.18
±
0.05
38.4
128.3
88.4
95°C
46.3 ± 0.5
0.16
±
0.05
39.1
12 9 . 0
87.6
I 65°C
44.0 ± 0.5
0.20
±
0.05
40.1
13 0. 0
87.5
51.0
0.46
36.9
12 6 . 7
88.5
C a l c u l a t e d average
f o r s i t e s 3, 4 and
5
Measur ed f o r
a v e r a g e due t o
s i t e s I and 2
angles
■
(e rro r9 all
angles ± 2
95°C
91.0
±
3
0.90
±
0.15
64.0
52.0
131.0
I 65°C
89.4
±
3
0.75
±-
0.15
64.1
61 .3
139.5
64.2
45.8
124.8
97.8
0.98
-47-
Measur ed f o r
a v e r a g e due t o
s i t e s 3, 4 and 5
6 Z
I
CO
O
O
O
(e rro r, all
e y
- 48o scillatio ns
indicated
o f t h e ND3 g r o u p s , and t h e s t r u c t u r e
by the: d e c r e a s e i n
the d i f f e r e n c e
change
bet ween t h e
quadrupolar s p l i t t i n g s
o f t h e two L i ^
sites.
direction
principal
o f t h e ND3 gr oup
o f the f i r s t
a v e r a g e f o r ms
axis
an a n g l e o f 3 . 6 ° a t room t e m p e r a t u r e , 2 . 6 °
a t 95°C and a b o u t I 0 a t 16 50 C w i t h
calculated
The
t h e N-j -Ng d i r e c t i o n
from the neutron d i f f r a c t i o n
as
data obtained at
room t e m p e r a t u r e .
A somewhat s i m i l a r
glycine
(Table
opposed t o
for
IV).
parameter f o r
I and 2 d i d
at
resulting
0 . 2 2 as compared t o
in
a N-N bond i s
t h e N-D d i s t a n c e s
or a combination o f
from mo tion al
either
the
less
than
are g r e a t e r
these two.
a v e r a g i n g o v e r bonds
n o t change m e a s u r a b l y bet ween 65 and 95°C.
I 65°C showed a change and i t s
as y mmet r y p a r a m e t e r ,
Because o f t h e
could
is
The asymmet r y
This would suggest t h a t
glycine,
constant
bet ween 70 and 80 kHz as
glycine
an N-C bond o r t h a t
L i DzS t h a n i n
The c o u p l i n g
Li DzS a t 25°C.
gr oup i n
Li DzS.
is
in
o f t h e f o r m C-NDg as
L i Dz S . '
charge o f the n i t r o g e n
The e f g
efg
kHz i n
the
0.22 ± 0.05 in
in
gr o u p i s
NDg a v e r a g e
compared t o 4 8 . 6
th a t in
This
t h e N-NDg f o u n d i n
the g ly c in e
effective
ND3 gr o u p has been s t u d i e d
and a n g l e s
lim ited
be o b s e r v e d ,
coupling
are gi ven
in
constant,
Table VI.
a n g u l a r r a nge o v e r w h i c h t h e s e
the e s tim a te d e r r o r
is
The
larger
lines
than f o r
-49t h e D3- D 4 - D 5 a v e r a g e . '
A t 95°C t h e f i r s t
t h e D-]-Dg a v e r a g e l i e s
w i t h i n : 2° o f t h e p l a n e d e f i n e d
* " * ! V *"*2 anc^ w i t h i n
4° o f t h e
line
principal
bisecting
the.
axis
of
by
- N ^ -Hg
angle.
Values o f the c o u p l i n g
and t h e a n g l e s
crystal
constant,
bet ween t h e f i r s t
axes a r e g i v e n
as y mmet r y p a r a m e t e r ,
p rin cipal
i n T a b l e VI
for
and t he
a calculated
1 - 2 and o v e r s i t e s
average
is
o b t a i n e d f r o m an e f g whose component s i n
crystal
f r a me a r e t h e sums o f t h e component s o f t h e e f g
tensors
for
divided
by t h e number o f s i t e s .
calculated
individual
averages
rise
to
good.
through
and P e t c h ( T Z ) .
nitrogen.
possible
12 0 ° s t e p s
obtained at
t he
frame,
bet ween t h e
lowest
The t y p e o f m o t i o n wh i c h
The ND3 ex change t a k e s
This
I
as was s u g g e s t e d by C u t h b e r t
takes
p l a c e by
and 2 bonded t o
does n o t i n d i c a t e
as had p r e v i o u s l y
m o t i o n . ( T 3)
place
by means o f h i n d e r e d
The D1 , Dg a v e r a g i n g
bet ween s i t e s
bond d i r e c t i o n
the c r y s t a l
The ag r e e me n t
s y m m e t r i c a l l y a b o u t t h e N-N a x i s
interchange
calculated
t h e s e a v e r a g e s can be u n d e r s t o o d f r o m t h e
measur ement s d i s c u s s e d abov e.
rotation
in
and t h e v a l u e s
t e m p e r a t u r e measur ed i s
gives
sites
This
average
over s i t e s
the
3-4-5.
axis
a rotation
the
same
a b o u t t h e N-N
been s u g g e s t e d as a
-50EXCHANGE 'PROBABILITY..
Abr agamf
8 ) discusses
t h e ex c hange p r o b a b i l i t y
ex c hange bet ween s i t e s
Since
in
prin ciple
a general
method f o r
per u n i t t i m e ,
n, for
nuclear
f r o m a knowl edge o f t h e
the te ch n iq u e
is
calculating
line
applicable
t o t h e case
o f j u m p i n g bet ween any number o f n u c l e a r s i t e s ,
possible
to estim ate
o f temperature
t h e ex c hange p r o b a b i l i t y
and a l s o
ex change o f d e u t e r o n s
has a p p l i e d
the a c t i v a t i o n
it
is
as a f u n c t i o n
energy f o r the
o f b o t h NDg and NDg s i t e s .
a s i m i l a r method t o
width.
C h i ba (31)
t h e case o f DgO i n
B a ( C l O 3 ) 2 - D2O.
The c a l c u l a t i o n
ex change p r o b a b i l i t y
follow ing
o f the r e l a t i o n
and l i n e
assumptions.
+ At when i t
time t
is:
is
is
The p r o b a b i l i t y
frequency o f the n u c le u s ,
t 1 = t
width
bet ween i n t e r s i t e
based on t h e
that
t h e Lar mor
w, has a v a l u e w2 a t t h e t i m e
known t h a t
it
had a v a l u e w-| a t
1.
ind ep end ent o f the valu es o f w p r i o r to t
2.
d e p e n d e n t on t h e t i m e s t and t 1 o n l y t h r o u g h t h e
d i f f e r e n c e t ' - t = At and can a c c o r d i n g l y be w r i t t e n
W((l)l [ o i g g A t ) .
as
The a b s o r p t i o n
CO
I (to) = re. f G ( t ) e - i w t d t
O
is
obtained
by c a l c u l a t i n g
the
(3.1 )
relaxation
function
-51t
G( t ) = < e x p { i / ( o ( t ' ) d t ' }> .
O
It
is
(3.2)
found t h a t
G(t)=W"'exp{(iw+F)t}°T,
(3.3)
w h i c h makes
I (w)=re{W'A~^• I }
W is
a v e c t o r whose c o m p o n e n t s ,
that
a t any t i m e t
is
the
inverse
.
(3.4)
, ar e t h e
probabilities
t h e f r e q u e n c y o f t h e s y s t em i s
w .
o f the m a t r i x
A= i ( u - ojI )+ir
wher e w i s
a diagonal
3
(3.5)
matrix with
( irVae=i r^a)a sWB^ t h e p r o b a b i l i t y
ma and Wg9 T a
A- ^
u n it m atrix,
component s
of tra n sitio n
and w a c o n s t a n t .
n wher e n i s
t h e number o f d e u t e r o n
ing occurs.
The m a t r i c e s
sites
(w)
„ = oj 6 s
aB a aB
bet ween l e v e l s
A is
o f rank
among wh i c h j u m p ­
(3.5)
TT
Q
- Qj
and
where 26 i s
the l i n e
o)+ 6
o
o
w- 6
separation
(3.7)
( see F i g .
20)
give
for
t he
Merged
Unmerged
Ideal
Spec t r um
Merged
Unmerged
Unmerged
Merged
Unmerged
ND2
<----------- 4 6 . 5 kHz
— 28.5
kHz_^
Merged
Unmerged
ND3 I----------<-
39
<
Fig.
20.
66
Ideal
106 kHz
>
>
and Obser ved D3 and D2 NMR S p e c t r a .
-53ND2 case
i ( w+S+i A) - f i
fi
i (o)-6 + i a )-$2
(3.8)
y
-
where A i s t he unbroadened l i n e w i d t h .
The r e s u l t i n g
absorption
for
t h e ND2 case i s
C j f (-a)+6 - j
I(w )=rei
a) -
((0+ 6+ i A)1 -
( 6 2+ A2 -(u2 ) - 2 A f i + 2 i o j ( ^ - A ) J
(3.9)
For t h e ND^ case t h e m a t r i c e s
n/ 2
0/2
-O
(3.10)
j
and
/ (i)+ 6
(3.11)
\
0
j- 6
I
yield
(3.12)
I
i (co+6 + i a ) - o
Y
Y
i (uD+i a ) - o
Y
Y
i ( (D- 6 + i a ) - o
J .
- 54A is
is
the
unbroadened l i n e
then c a l c u l a t e d
to
width
and y = o / 2 .
The a b s o r p t i o n
be
(3. ] 3)
3 i M( f i +2A) - 3fiA- ( 3 f i / 2 ) ^ - 6 ^ + 3oj^-3A^
I.(.u)).= re
2.
.3
,2,
i j T - u ^ + 3 ( t i A ^ + o ) 6 ^ + ( i9 / 4 ) ^ ^ u ) + 6 f i o3A^ + j j B o) ^ A - A 0 - S ^ f i + 3 ^ 0 ) '
-3fi A2 - ( 9 / 4 . ) f i 2A
The l ow t e m p e r a t u r e
sists
o f two l i n e s
g r oup t h r e e
indicated
cases,
lines
in
fi <<6
spectrum observed f o r
s p lit
involved
Fig. 20•
H to) i s
(low te m peratu re)
and f r o m t h e
resulting
and t h e
line
approximated
and n >>6 ( h i g h
relations
the
line
For t h e ND^
separation
for
^LT-=
widths
temperature)
width
is
id e n ti­
Low T e m p e r a t u r e
Wht =■ ( 6 2/ 2 0 )
+ A
The NDq l i n e
widths
WfT =
wHT
are:
+ A
fl. + A
4/9
62/ f i + A
is
t wo l i m i t i n g
fied.
The NDg l i n e
V .
t h e ND^ gr oup c o n ­
by 75 kHz ( F i g . 20 )•
ar e
I
Hi gh T e m p e r a t u r e
are:
• Low T e m p e r a t u r e
Hi gh T e m p e r a t u r e
-55-.
These e x p r e s s i o n s
v a l u e , W, i s
are
obtained
(S/.N,) as; a. f u n c t i o n
to
be t h e
for
hence S/N w i t h
the
constant
the
line
is
is
both
used f o r
line
o f the q u a n t i t y
decrease i n
a temperature
width
for
the
the
high
for
the
value
calculation
o f the
for
6 = HO
kHz.
for
NDg g ro u p was t a k e n
unmerged l i n e s
plotted
against
three
to
( see F i g .
I/T
in
an a c t i v a t i o n
NDg gr oup
insure
The
no l i n e
width
o f th e .
20).
Figs.
is
is
the high
of
22.
the
so a
by t a k i n g
t h e average
giving
of
for
the
values
line
ND3 m o t i o n
the
t h e two
o f ft ar e
The NDg d a t a f o r
to
t ha n
the
the c a l c u l a t i o n
sp littin g
fitte d
energy f o r
larger
S used i n
The r e s u l t i n g
21,
The
temperature
unmerged l i n e s
the
cases.
unmerged l i n e s
S used i n
be h a l f
unmerged and merged r e g i o n s
yielding
the
The v a l u e
The v a l u e o f
and
t h e same so t h e same v a l u e o f " C
ND3 g r o u p was o b t a i n e d
o f the
assumed
where T / T 0
difference
l o w enough t o
o f C was used f o r
separation
is
ra tio
(2.3x1O^/sec"^) of
NDg g r oup merged l i n e s
l ow t e m p e r a t u r e .
noise
For t h e
value
width
width
and l ow t e m p e r a t u r e
( 3 . 9 x 1 OzV s e c - I ) t h a n t h a t
d iffe ren t
population
of in te re st.
approximately
to
C( T / T 0 ) ( S / N )
temperature T .
due t o t h e m o t i o n
merged l i n e
The l i n e
The l i n e
o b ta in e d from the
at
ft.
by m e a s u r i n g t h e s i g n a l
increasing
width
broadening
the
C is
for
of temperature.
reciprocal
c ompen s at e s
then sol ve d
the
shown
Ea = . 46eV .
SEC-
2.5
3.0
3.5
4.0
10°
TTt r K)"
F i g . 21.
T e m p e r a t u r e Dependence o f t h e Exchange
P r o b a b i l i t y O f o r Exchange Between D e u t e r o n S i t e s I and 2.
5
6
7
8
9
_IO3
T(OfC)
F i g . 22.
T e m p e r a t u r e Dependence o f t h e Exchange
P r o b a b i l i t y fi f o r Exchange Bet ween D e u t e r o n S i t e s I , 2 and 3.
-58The NDg d a t a f o r
slope, than
shown i n
t h e merged l i n e s
f o r the
Fig.
shows a much
unmerged r e g i o n .
22 c o r r e s p o n d s
. 19 e V., and a p p r o x i m a t e s
to
The met hod d i s c u s s e d
in
The s t r a i g h t
an a c t i v a t i o n
a best f i t
this
steeper
to the
section
line
energy of
d a t a shewn.
is
n o t as
dependable a te c h n iq u e
for
for
t h e measur ement o f t h e d e u t e r o n s p i n -
such m o t i o n
la ttic e
I/T
is
width
as i s
relaxation
time,
T-].
d e p e n d e n t on b o t h
the
The
A.
sp littin g
is
value
such t h a t
case o f t h e ND3 m o t i o n t h e
spectrum d i f f e r
markedly
advantage o f t h i s
an e s t i m a t e
on t h e b a s i s
is
sp littin g
6 is
the
o f a much
(see F i g .
less
fa lls
on t h e
unmerged l i n e s .
idealized
activation
6 and t h e
unambi guous o n l y
ft v e r s u s
line
is
the
real
In the
s p e c t r u m and t h e r e a l
20).
t y p e o f meas ur ement i s
o f the
required
line
the averaged l i n e
of
energy
The s l o p e o f t h e c u r v e
chosen f o r
average o f the p o s i t i o n s
than
determing the a c t i v a t i o n
energy f o r
The
primary
that
it
may p r o v i d e
a p a r t i c u l a r motion
c o m p l e x and e x p e n s i v e
apparatus
f o r T-j m e a s u r e m e n t s .
DEUTERON SPI N-LATTI CE RELAXATION.
The
spin -lattice
Li DzS was measur ed
relaxation
by a d o u b l e
t i m e T-j
90° p u l s e
for
deuterons
technique.
The
in
- 59
firs t
pulse
is
applied
to
the system i n
the n u c l e a r response a given
p u l s e has a m a g n i t u d e Sq .
from the f i r s t
pulse
equilibrium
t i m e a f t e r t h e end o f t h e
The second p u l s e
by a t i m e
t.
= S0e - t / T ]
is
separated
The r e s p o n s e S ( t )
t h e second p u l s e was measur ed and f i t t e d
S0 - S ( t )
and
to
to
the r e l a t i o n
_
(3.14)
*
It
can be shown
(m = - I ,
carried
0,
that
the three
+1),
Eq.
(3.14)
out w ith
the
lines
bet ween m = ( 0*--*+!)
population
m = -I
for
only f o r
corresponding to
are s y m m e t r i c a l l y
populations
single
constant,
are the p r o b a b i l i t i e s
are
then
s y s t em
the experiment
transitions
bet ween t h e m = 0, m = I
equilibrium
time
deuteron
and m = ( - 1^ 0 ) merged so t h a t
differences
levels
holds
level
the
and m = Os
reduced to z e r o .
The
re-established
with a
T 1 = ( P 1 + 2P2 ) " 1 , wher e P1 and P2
for
a
Am = I
and Am = 2 t r a n s i t i o n
.
respectiv e ly.*
F i g . 23
shows t h e v a l u e s
met hod p l o t t e d
the spectrum in
carry
against
the
1/T.
region
o f ! / T 1 obtained
by t h i s
Because o f t h e c o m p l e x i t y o f
b e l o w O0C i t
was n o t p o s s i b l e
o u t t h e e x p e r i m e n t on t h e ND3 l i n e s
at
a crystal
^App end ix B c o n t a i n s a devel opment o f t h e e q u a t i o n s
p e r t i n e n t to t h i s s e c tio n .
to
SEC
F i g . 23.
Temperature
L a t t i c e R e l a x a t i o n Ti me.
I O 3Z T ( 0 K)
Dependence o f t h e
Deuteron Spi n-
-61o rie nta tion
f o r which
t o do so r e s u l t e d
in
the
line
s p littin g
erroneous
response,
was z e r o .
Attempts
I
because t h e F o u r i e r
s p e c t r u m o f t h e 90° p u l s e c o n t a i n e d component s a t t h e
qu enc y o f n e i g h b o r i n g
lines
and c o n s e q u e n t l y t h e
fre ­
response
was m o d u l a t e d a t a f r e q u e n c y d e t e r m i n e d by t h e s e p a r a t i o n
bet ween t h e s e
lines.
measur ement s o f T-j
zero c r o s s i n g s .
The same p r o b l e m was e n c o u n t e r e d when
f o r t h e NDg l i n e s
C o n s e q u e n t l y t h e NDg and l o w t e m p e r a t u r e
NDg e x p e r i m e n t s were done w i t h
The e x p e r i m e n t a l
tra nsition
single
T -j .
a
m = -I
relaxation
The v a l u e s
single
results
for
time which
is
o f T-j measur ed
crystal
oriented with
a j j Hq and f o r
taken to
(Fig. 23)
by s a t u r a t i n g
and a n a l y z i n g
relaxation
the
the
t o m = 0 a p p e a r t o depend on o n l y a
[ j H0 were o b t a i n e d
transition
were a t t e m p t e d a t t h e
the
t i m e T-j .
be t h e e x p e r i m e n t a l
for
the o r i e n t a t i o n
the m = 0 to m = - I
response to de te rm in e t h i s
*
The t h e o r e t i c a l r e s p o n s e
for
t h e e x p e r i m e n t i n w h i c h o n l y one o f t h e t r a n s i t i o n s
saturated
is
So " S ( t )
is
(3.15)
= 1
S0 [ 3 e x p ( - 3 P 1 ) t + exp - ( P 1 + 2 P g ) t ]
*Appendix B contains..a
p e r t i n e n t to t h i s s e c tio n .
development o f the eq ua ti o ns
.*
T A B L E VII
V a l u e s o f E l e c t r i c F i e l d G r a d i e n t T e n s o r Components i n t h e C r y s t a l
•System M u l t i p l i e d by K = 3 e Q / 2 h , i n k H z , f o r t h e F i v e D e u t e r o n . S i t e s i n
L i N 2D5SO4 .
-71 .2
-115.8
187.0
221 .8
-128.0
O
I
51 .9
0.0
-152.3
-122.5
90.3
15.1
62 . 2
186.0
0.0
136.2
-17.7
46.0
75.7
-186.3
LO
- 28.8
CO
1
CO
LO
VO
O r
50.0
I
5
CO
CO
4
Ktjjyz
O
3
CO
2
VO
-167.9
I
264.6
VO
-96.7
K*XZ
O
LO
I
K*xY
O T-
K*ZZ
ro
K^ yy
I
Si t e
T
A
B
L
E VIII
V a l u e s o f E l e c t r i c F i e l d G r a d i e n t T e n s o r Components i n t h e System
z ' l l a Mul t i p i i e d by K = 3eQ/2h i n kHz f o r t h e F i v e D e u t e r o n S i t e s i n
L i N 2H5SO4 .
K<i>y 1y 1
K^z 1z '
264.6
-167.9
- 96.7
2
-115.8
187.0
3
221.8
-122.5
4
-128.0
186.0
5
- 96.5
46.0
Site
I
K<f>x' x ‘
. .
Ki1X ' z '
Ki’y 1z 1
51.9
-126.9
- 29.8
-7 1.2
- 50.0
0.0
-152.3
- 99.3
62.2
90.3
15.1
17 . 7
0.0
13 6. 2
- 84.3
75.7
-186.3
Ki1X 1y 1
- 58
-
50.5
.
_
T
A
B
L
E IX
V a l u e s o f t h e E l e c t r i c F i e l d G r a d i e n t T e n s o r Components i n t h e System
z ' i n t h e Pl a n e o f a , b and 18° f r o m z 1 || a M u l t i p l i e d by K = 3eQ/2h i n kHz
f o r t h e D e u t e r o n s o f S i t e s 3, 4 , and 5 S e p a r a t e l y .
A l s o I n c l u d e d Ar e t h e
D3 A v e r a g e e f g T e n s o r V a l u e s (Same O r i e n t a t i o n ) .
Site
3
Kt j j y i y i
K(J)z I z I
-122.5
381.7
-259.2
-120.9
46.0
•
x 1y 1
x 1z 1
K(j)yiz i
- 54.5
33.6
- 94.4
- 65. 1
- 58.9
12 4 . 3
20.6
-126.8
80.8
- 22.7
-203.2
43.2
36.7
28.5
- 65.2
- 16.3
7.0
10.7
36.7
- 36.5
-
:
- 15.2
- 10.7
-64
K(J)x ' x •
186
. 4
5
ND3
(+72°)
ND3
(-72°)
•
0.2
9.1
-65The v a l u e s
o f the
component s o f t h e
<f>-jjs were o b t a i n e d
re la tive
z axis
crystal
parallel
to the f i e l d
an a n g l e
Hq .
tensor
a b o u t t he
e to
Table V I I
in
the a n a l y s i s .
tensor r e la t iv e
Table V I I I
through
b r in g the
gives
the
t h e c r y s t a l system
f o r the
The component s o f t h e
t o a s y s t em i n w h i c h
z'
Jja
tensors,
te n s o r expressed
s y s t e m by a r o t a t i o n
axis
component s o f t h e e f g
bonds used i n
gradient
by. t r a n . s f o r m i ng f r o m t h e
to the c r y s t a l
appropriate
fie ld
efg
are g i ve n i n
and t h e component s o f t h e e f g t e n s o r r e l a t i v e
a system in which
the
c ry s ta lis
through
72°
from z 1
the
angle
The t r a n s i t i o n
rotated
|J b
probabilities
about
are given
the
c axis
in Table
were c a l c u l a t e d
assumption t h a t
t h e exchange c a u s i n g r e l a x a t i o n
associated with
rotation
to
I X.
on t h e
is
o f t h e NDg gr oup o r NDg g r o u p , as
opposed t o ex change bet ween n e i g h b o r i n g NgDg g r o u p s .
Us i n g
the values
the
o f P-] and Pg ( see T a b l e
a || H0 d a t a ,
values
of
(S0 - S ( t ) )
X)
calculated
can be p l o t t e d
logarithm ic
paper a g a i n s t t .
these p l o t s
are almo st l i n e a r which e x p l a i n s
the experim ental
relaxation
a single
time.
By f i t t i n g
relaxation
time T ] e is
NDg g r o u p s w i t h
on s e m i -
Fo r v a l u e s o f Sq - S ( t ) > 0 . 1 5
the f a c t t h a t
r e s p o n s e seems t o depend on a s i n g l e
effective
relaxation
for
a
[I H 0 .
a straight
time
calculated
is
for
line
found.
to these p l o t s
This
effective
b o t h t h e NDg and t h e
The r e l a x a t i o n
time f o r
t h e NDg
- 66 TABLE
X
C a l c u l a t e d V a l u e s o f t h e A m = I and Am = 2 T r a n s i t i on
P r o b a b i l i t i e s f o r t h e Cases' o f I n t e r e s t
P
P
2
I
ND3 (H0 Il a)
a)Tc <<!
cot
0.845X1
>>1
C
0. 845x1 Ql I
Oi^T s e c 2
c
NDg(720 f r o m H0
toTc < < l
Tc / s e c ^
2 . 0 2 x l 0 11Tc / s e c 2
0.505X1011
2
to T
sec
2
b about c a x i s )
0. 9 93 x 1
Tc Z s e c 2
I .48x1
Tc Z s e c 2
ND2 (H0 Il a)
wTc » l
0 . 1 5 2 x 1 O ^ Z w 2Tc Sec'2
0 . 3 5 5 x 1 O^ I Zw2Tc Sec2
- 67group w i t h
obtained
z 1 in
t h e ab p l a n e 72° f r o m t h e b a x i s
dire ctly
relaxation
times
The f r e q u e n c y w =
experiment.
so t h a t
from
= - (P-j + 2 Pp ) .
depend on Tc , t h e c o r r e l a t i o n
2
x 14. 020: MHz was empl oyed f o r
The a s s u m p t i o n
the
These c a l c u l a t e d
all
tt
calculated
is
is
time.
the e n t i r e .
made t h a t Tc = T0 e x p ( Ea/ k T )
relaxation
times
depend on t h e
p a r a m e t e r s . Ea and T0 .
In
the
temperature
a maximum i n
t h e I /T-j
curve
regions
in
the
magnitude,
region
curve
b e l o w 250°K ( I / T
is
evident.
The s l o p e s
regions
proportional
I / T c and M2Tc r e s p e c t i v e l y .
the
made t h a t a t t h e
l ow t e m p e r a t u r e
variables
for
w and Tc .
Tc and i t
comparison
(Fig.
is
The a s s u m p t i o n
peak t h e h i g h t e m p e r a t u r e T-j must equal
T1 .
This
equation
Since m i s
through
is
this
contains
only
k n o wn , t h e e q u a t i o n
Tc = 1 . 0 2 x 10 ^ s e c .
ob taine d from the
temperature
23)
in
Eg and T0 .
above and b e l o w t h e peak T-j
found t h a t
the value
ments a t t h i s
curve
is
o f the
Tc on t h e p a r a m e t e r s
For t e m p e r a t u r e
is
^ / 0 K)
above and b e l o w t h e peak a r e e q ual
and depend t h r o u g h
to
> 4 x 1 0
Tc = 4 . 3
region
is
line
width
x 10"^ sec.
obtained
t he
is
solved
By
me as ur e­
The s o l i d
by
calculating
I / T 1 = ^ ^ I + Pz
f o r T i n the r e g i o n o f
2
1 . 6 x I O- 8 s e c .
T h i s e x p r e s s i o n was o b t a i n e d f r o m t h e
in itia l
slope
follow ing
of
the
return
a 90° p u l s e .
to
equilibrium
The v a l u e
of
the p o pula tion
o f T1 c a l c u l a t e d
from the
-68value
o f Tc a t
best f i t
to
the i n t e r s e c t i o n
the data gives
is
T-j = 7 . 6 x I O " 4 s e c .
an a c t i v a t i o n
energy
Ea = 0 . 2 0 ± O. Ol eV and T 0 .= 1 . 6 x I O-4 4 s e c .
e n e r g y f o u n d by t h e
comparison
in
line
width
technique
Eg = 0 . 1 8 2 ± 0 . OlSeV f o r
LiHzs(IZ)
and T0 = S x
motion
in
KDgPO4 . ^ )
occurs
because o f t h e
is
The a c t i v a t i o n
Ea = 0 . I 9eV.
t h e NHg h i n d e r e d
10 " ^ 4 sec f o r d e u t e r o n
A displacement in
difference
d e p e n d i n g on t h e o r i e n t a t i o n
in
The
the
rotation
intrabond
best
the values
By
fit
of
line
T]
a t w h i c h t h e measur ement was
made.
As t h e
temperature
NDg g r o u p b e g i n s
to
must be e x p l a i n e d
explanations
there
P intar
the
is
3.
in
differences
times
NDg a v e r a g e
This
in
large
fie ld
mechani sm.
it
entire
NgDg g r o u p .
is
This
Th r ee
possible
that
McCl e m e n t ,
such a m o t i o n m i g h t cause
frequency
o b s e r v e d f o r I / T x 1 0 ^ / T ( 0 K)
the
fact
ar e s t i l l
implies
f o r the
= 2.80 x 10- ^ / ° K .
second moment and t h e
Howev er ,
appear broadened,
10 " 5 s e c .
o f the
value o f l/T-j
F irst,
suggest , t h a t
relaxation
due t o t h e
near I / T
ar e c o n s i d e r e d .
reduction
s m a l l e r than
the
by some d i f f e r e n t
and Pe t c h ( 1 3)
dependent
larger
increase
reorientation
large
increased,
for
gradient
t h a n t h o s e me a s u r e d )
in
the s p e c t r a l
observed at
a Tc f o r
value
that
T
2.2
such a m o t i o n
leads
values
to
lines
and do n o t
larger
impossibly
( o f the o r d e r
o rd e r to achieve
I
the
than
large
o r 10
short
-69relaxatfon
groups
is
times
observed.
n o t t h e mechani sm f o r
Secondly, d i f f u s i o n
If
such a m o t i o n
values
for
oriented
values
Ti
at
±72°)
value
the
I
I CT 3 / °
/Ti
K
versus
p l a c e ,between ND^ g r o u p s .
groups
is
T c
is
the
region.
value
calculated
IX e f g axes
1/ T c u r v e o b s e r v e d
Us i n g
observed a t
t o be 1 . 6 x 1 0 " ^ s ec .
t o wTc << I .
t o wTc >> I w h i c h - m a k e s
leaves
(Table
gradient
d e t e r m i n e Tc f r o m t h e T-j
temperature
which
an u n a c c e p t a b l e
This
hydrazine
o f Tc c o r r e s p o n d s
corresponds
groups
may t a k e
this
;
relaxation.
can be used t o
measur ed i n
hydrazine
assumed t h e NDg- a v e r a g e f i e l d
neighboring
1/T - 2/3 x
of
is
= 100 m i l l i s e c o n d s
This
Thus t u m b l i n g o f t h e
Howev er ,
in
this
diffusion
the
slope
region
bet ween NDg
explanation.
the p o s s i b i l i t y
o f jumping
bet ween NDg and
NDg s i t e s .
The T-j measur ement s
with
the
crystal
be h a v io r the
same as t h a t
and P2 a r e g i v e n
these
values
obtained.
times
relaxation
It
is
in
o f P-j
The T-j
shorter
oriented
Table
for
values
for
the
NDg gr oup w i t h
X and b y . b e s t f i t t i n g
for
a || Hq .
Eq.
(3.15)
dependence o f T -J on T c i s
the
values
t i m e on t h e b a s i s
that
out
a Jj H0 , mak i ng t h e r e l a x a t i o n
and P g , t h e
than those
reasonable
t h e ND2 gr oup were c a r r i e d
ND2 gr oup a r e
predicted
of jumping
the jumping
for
10 t o
the
100
ND3
bet ween ND3 s i t e s .
bet ween ND2 s i t e s
is
the
P -j
for
-70prfmary
relaxation
When a 9 0 ° . p u l s e
stays
mechani sm f o r
is
approximately
applied
in
has a much s h o r t e r
to
the deuterons
the
ND3 s y s t e m ,
e q u ilib riu m with
t h & ND2 s y s t em
the l a t t i c e
t h a n t h e ND3 s y s t e m .
t h e r e I a. x a t t o n t i m e measur ed f o r
o f these s i t e s .
This
since
it:
means t h a t
t h e MD3 s y s t e m i s
really
a
measur e o f
t h e t i m e f o r m i x i n g bet ween ND^ and ND3
*
populations.
L e t t h i s t i m e be d e n o t e d Tm and assume t h a t
Tm = Tmoe Ea/ k T .
gives
Ea = - . 7 5
erature
mixing
The b e s t f i t
time,
a 90° p u l s e ,
the
stays
o f the m ixin g
time
the
ND3 s i t e
nearly
that
the
in
of
the
this
1/T-|
for
the
for
*See A p p e n d i x B f o r
exc h ang e p r o c e s s .
larger
ND2
subjected
3/2 the
1/T-]
t h e ND3 s yst em
3/2
the s h o r t
is
in tro ­
to
t he
Th er e a r e o n l y
involves
than the f r a c t i o n
since
discussion
t e mp ­
o c c u p y i n g ND3 s i t e s
so t h e j u m p i n g
ND2 s i t e s ,
is
proportional
o f ND2 s i t e s .
3/2
o f the
during
The f a c t o r
t h e ND3 i s
as ND3 s i t e s
o f t h e NDg s i t e s
The l / T - j
la ttice
number o f d e u t e r o n s
deuterons
in fin ite
ND.2 s y s t e m i s
due t o m i x i n g
with
then
s ec.
can be assumed t h a t
equilibrium
as many ND2 s i t e s
sites.
it
long
relaxation
When t h e
ND2 s y s t e m r e l a x e s .
exchanging w i t h
fraction
on t h e
contribution
since
duced because t h e
fra ction
ND3 r e l a x a t i o n
Tmo = 1 . 3 x 10” ^
g r o u p c a n n o t be n e g l e c t e d .
for
the
± 0 . 0 5 eV and an u n u s u a l l y
The e f f e c t
to
for
it
is
2/3
a
o f ND3
proportional
of equations
to*
governing
-71the f r a c t i o n
o f ND2 sui tes j u m p i n g ,
3/2. the
for
the
1/T-]
result
time
is
NDg s i t e s .
Whe-n t h i s
t h e dashed l i n e
due t o NDg h i n d e r e d
must be p r o p o r t i o n a l , t o
in
Fig.
rotation
is
energy o b ta in e d
23.
sec.
made,
energy
By c o m p a r i s o n t h e '
by means o f t h e
0". 46eV.
is
The c o r r e l a t i o n
has a c t i v a t i o n
0 .'50. ±' 0 . 02eV and T 0 = 2 . 2 x 10™^
activation
correction
.
line
width
study
■
DISCUSSION..
The measur ement s
indicate
certain
described
motions
in
this
chapter c le a r ly
o f the deuterons
in
the
crysta l.
The d e u t e r o n s
o f t h e NDg g r o u p u n d e r g o . a h i n d e r e d
This
provides
rotation
relaxation
o f these deuterons
co rrelatio n
time f o r
width
is
study
time
basis
over the
tion
of
the
o f 10 ~ 5 sec i s
temperature.
this
mec h ani s m f o r
up t o a b o u t 65°C.
motion
-the
The
as i n d i c a t e d f r o m t h e l i n e
1 0 “ ^ Stec a t IOO0 K.
The d e u t e r o n s
On t h e
the p r i n c i p a l
ro ta tio n .
o f t h e ND2 gr o u p exchange a t
line
w i d t h me a s u r e me n t s ,
predicted
The merged l i n e s
NDg s i t e s
mechani sm f o r
for
this
a lower ra te
a correlation
ex c hange n e a r room
o b s e r v e d due t o
t h e av er age
a p p e a r n e a r 50°C and t h e p r i m a r y r e l a x a ­
these
deuterons
ex c hange bet ween MDg s i t e s
over the
is
provided
entire
by t h e .
range
studied,
namel y f r o m T = BO0C t o T - I l O 0 C.
Measur ement o f t h e
d ie le c tric
constant
reveals"two
-72
contributions
temperature
comparable
to
the
relaxation
contribution
to
temperature
that
for
mechani sm.
has an a c t i v a t i o n
the
contribution
The l ow
e n e r g y ~0 . 4 0 e V
exchange, w h i l e
has an a c t i v a t i o n
the
high
energy o f about
0.70' eV.
Jumpi ng bet ween
relaxation
lim it
of
o f NDg s i t e
is
activation
larger
activation
energy f o r
d ie le c tric
this
than
large
relaxation
c axis
with
high
in
activation
constants
the a p p a r a t u s .
0 . 7 5 ± 0. 05eV
dc
contribution
for
the
d ie le c tric
As i s
and t w o f o l d
this
energies
and t h o s e
rotation
plane o f a u n i t
the
cell
in
Fig.
axes
in
crysta l,
I
for
the
the
t h a t the
t h e mechani sm r e s p o n s i b l e
there
the
to
the
unreflected
This
ar e n g l i d e s ,
crystal
re flection
gi ve' s r i s e
spectrum i d e n t i c a l
cell.
for
relaxation.
indicated
sy mmet r y o f
to
good a g r e e m e n t .
N D 2 e x c h a n g e , and t h e N D 2 t o N Dg j u m p i n g i n d i c a t e s
N D 2 m o t i o n may be a s s o c i a t e d w i t h
up p e r
However , t h e
temperature
o f 0 . 7 OeV i s
d ie le c tric
the
the quoted c a x i s
e n e r g y o f 0 .6 Se V.
the
explains.the
65°C t o
mechani sm i s
The a g r e e me n t bet ween t h e
very
from
range a v a i l a b l e
energy f o r
slig h tly
conductivity
the
deuterons
the tem perature
The a c t i v a t i o n
which
t h e NDg and NDg s i t e s
to
cell.
through
a deuteron
Because o f
t h e ab
sp littin g
spectrum observed f o r
coupled w it h
b glides
the
t h e absence o f any
- 73pdsitive
or the
high
l i n e St, i s
place
indication
either
temperature
the
by way o f r e f l e c t i o n
is
d e finite
the deuteron
possibly
transition
idea
obtained
spectrum
of spectral
t h a t domai n r e v e r s a l
through
takes
t h e ab p l a n e .
s p e c t r u m above - SB0C shows' change,
change i n
can be made as t o
C ertainly
is
l ow t e m p e r a t u r e
due t o s t r u c t u r a l
conclusions
the change.
the
spectrum o f a d o u b lin g
con sistent with
. Although
which
in
the c r y s t a l ,
the e x a c t n a tu re
no i n d i c a t i o n
of a fe rro e le c tric
be l ow 19 0 ° C .
This
is
in
no
of
ag r e e me n t
I
with
t h e absence o f an anomal y i n
b e l o w ZOO0C , w h i c h s u b s t a n t i a t e s
ferro e le ctric
temperature
transition
of the
takes
crystal .
the d i e l e c t r i c
e,arl i er- c l ai ms H 0)
place
.
constant .
that.no
bel ow t h e d e c o m p o s i t i o n
.
CHAPTER IV
EXPERIMENTAL
CRYSTAL PREPARATION.
Single
dissolving
crystals
o f L i N2 H5 SO^ were p r e p a r e d by
stoichiom etric
amounts o f L i 2CO3 ( r e a g e n t g r a d e )
and (NH2 ) 2 * H2SO4 ( r e a g e n t g r a d e )
by s l o w e v a p o r a t i o n .
single
faces
crystals
L a r ge
w ith well
can be grown by t h i s
deuterated
samples,
in water,
and c r y s t a l l i z i n g
( 2 . 9 5 c m x 3. 05cm x 5 . 6 c m,
developed
( 100) ,
technique.
the c r y s t a l s
d e s c r i b e d above were d i s s o l v e d
( 1 1 0 ) and ( 1 0 1 )
In o r d e r to o b t a i n
o f Li NgH3SO^ o b t a i n e d
in
34g.)
as
90% D2O and r e c r y s t a l I i zed
An e s t i m a t e o f t h e p e r c e n t o f d e u t e r a t i on o f t h e s e c r y s t a l s
was made by c o m p a r i n g p r o t o n
sampl es o f e q u a l
crystal
size.
signal
strengths
f r o m two
The r e s u l t s
indicated
t h a t the
was a b o u t 50% d e u t e r a t e d .
A second r e c r y s t a l I i z a -
t i on f r o m 99% D2O was t h e n c a r r i e d
d i s h was p l a c e d
in
out.
The c r y s t a l l i z i n g
a d e siccator during th is
o r d e r t o m i n i m i z e w a t e r v a p o r exchange w i t h
process
in
the atmosphere.
NMR SPECTRA.
A R o b i n s o n ( 3 2 ) c i r c u i t employing frequency modulation
bet ween 900 and 1500 Hz and o p e r a t i n g
obtain
of this
most o f t h e NMR s p e c t r a .
a t 14 MHz was used t o
A solid
c i r c u i t ( 3 3 ) was used t o o b t a i n
state m odification
t h e room t e m p e r a t u r e
data.
The p e r ma n e n t m a g n e t i c f i e l d
f o r t h e e x p e r i m e n t was
“ 75"
p r o v i d e d by a 15" V a r i a n
F ie ld ial
control.
t h e p e r ma n e n t f i e l d
crystal
axis
e l e c t r o m a g n e t w i t h Model
The c r y s t a l
was o r i e n t e d
H0 by. v i s u a l l y
parallel
aligning
s p e c t r o m e t e r base p l a t e ,
its
w h i c h was c a l i b r a t e d
such a f a s h i o n t h a t
holder ro tate d
crystal
axis
it
with
base i n
rotating
in
The f i e l d
perpendicular to
with
fie ld
was t y p i c a l l y
dial
the s p e c t r o m e t e r i n
the f i e l d .
The r o t a t i o n
obtained
in
crystal
t h e sweep c e n t e r c o i n c i d e d
was swept o v e r t h e r a n g e i n c l u d i n g
orientation
angle
the f o l l o w i n g f a s h i o n .
t h e c e n t e r o f t h e NMR s p e c t r u m and t h e n
crystal
about the
base p l a t e .
was s e t so t h a t
sweep r a n g e v a r i e d
degrees.
a circula r
The s p e c t r u m c o r r e s p o n d i n g t o a p a r t i c u l a r
o rie nta tion
a
The
t h e t u b e and hence t h e c r y s t a l
was r e a d f r o m t h e c i r c u l a r
to
pe rp e n d ic u la r to the c i r c u l a r
The s p e c t r o m e t e r t h e n r e s t e d on i t s
holder in
re la tive
to the spectrometer tube.
s p e c t r o m e t e r t u b e was f i x e d
V- FR- 2703
f r o m 50 gauss t o
and t e m p e r a t u r e .
the magnetic
the spectrum.
The
500 gauss d e p e n d i n g on
Sweep t i m e s bet ween I
and 25 m i n u t e s were used.
The room t e m p e r a t u r e
s p e c t r u m were o b t a i n e d
(P.A.R.)
Model
source f o r
am plifier.
s p e c t r u m and p a r t
using a P rince ton A p plie d
JB-4 l o c k - i n
the s p e c tro m e te r,
Al I
the
o f t h e 77°K
Research
a m p l i f i e r as t h e m o d u l a t i o n
and s i g n a l
d e t e c t o r and
r e m a i n i n g d a t a was o b t a i n e d
using a
-76P.A.R.
Model. HR- 8 l o c k - i n
i n was o r d i n a r i l y
chdrt
recorder
fed d i r e c t l y
(Fig.
for
F ie ld ial
the
(Fig.
counter with
Poor s i g n a l
to
noise
averaging
for
orie n ta tio n s.
was f e d t o
a V i d a r Model
by an o u t p u t
by a H e w l e t t
ra tio
Temperatures
Model
obtaining
in
in order to
obtain
spectra,
the
detector
converter
in
a Technical
am plifier.
storage
The
rate
of
The a c c u m u l a t e d s t o r a g e o f f r o m
ou t o f the m u l t i - c h a n n e l
a form s u i t a b l e
this
the s p e c t r a
lock-in
4 0 1 D 400 c h a n n e l
experiment
for
o f the
t h e empl oyment
240 v o l t a g e - t o - : f r e q u e n c y
3 t o 10 sweeps was t h e n c a l l e d
diagram o f the
from the
frequency
made n e c e s s a r y
The o u t p u t o f t h e
analyzer.
a n a l y z e r and r e c o r d e d
The X
Pac k ar d 5245L
sweep was c o o r d i n a t e d w i t h
the m u ltichannel
study.
spectra).
o u t p u t o f t h e c o n v e r t e r was t h e n s t o r e d
magnetic f i e l d
for
o r a Mos el ey
The o p e r a t i n g
technique
Measur ement C o r p o r a t i o n
block
spectra)
t i m e base
5261A p r e a m p l i f i e r .
of a signal
certain
either a B ristol
16 shows t y p i c a l
magnet sweep c o n t r o l .
electronic
The o u t p u t o f t h e l o c k -
X-Y p l o t t e r was p r o v i d e d
s p e c t r o m e t e r was m o n i t o r e d
and t h e
to
4 shows t y p i c a l
7000 AR X-Y r e c o r d e r
dfive
am plifier.
is
for
shown i n
analysis.
Fig.
24.
e x p e r i m e n t were h e l d w i t h i n
and w i t h i n
± 10 C f o r
The 77°K s p e c t r u m was o b t a i n e d
the
A
line
± 50C
width
by s u b m e r g i n g ,
in
a .
■
E lectronic
Count
V i deo A m p l i f i e r
5261 A
G. R.
I 204B
Modulation
Nobatroi
Q . B . 6-2
Signal
Signal tc]'
Chart
t Recor d t o r t o
Signal
Averaging
System
Chart
Recorde r
6 VDC
Sol a
RegulatorTr a n s f o r m e r
Spectrometer
I
i
I
i
S p e c t r o m e t e r Tube
3 H
ing T r i g g e r
Generator
Synchronizing
F i g . 24.
ampl e i n
Slampl e C o i l
Field
Pr o b e
Pulse from F i e l d
B l o c k Di agr am o f NMR S p e c t r o m e t e r .
Dial
Control
. -78Dewar o f Ti.q u i d
sealed
point
in
a ir,
a brass
of liquid
the. sampl e and sampl e c o i l
c a n . . The t e m p e r a t u r e
a ir
a t Bozeman i s
s p e c t r u m was o b t a i n e d
brass
can i n
me t h a n o l
was c o n t r o l l e d
bath.
t h e measur ement s
o f an e l e c t r i c
in
this
at elevated
the
control
The r e m a i n d e r o f
contact with
A stainless
heating
thermal
o f t h e NDg
c o il,
iso la tio n
steel
for
empl oy e d t o o b t a i n
bro ad en in g data a t
±1° t o l e r a n c e
l o w t e m p e r a t u r e was
integral
part
a ir
connector
t h e s y s t e m.
by means o f a s p e c t r o m e t e r t u b e b u i l t
of a liq u id
a brass
insulating
thermal
effected
with
A
t e m p e r a t u r e wer e made by.,means
good t h e r m a l
sample.*
and sampl e t o p r o v i d e
separating
air.
s i I i cone,: o i l
broadening
same f a s h i o n .
t u b e was p l a c e d a r o u n d t h e
The t e m p e r a t u r e
u s i n g as a b a t h
The l i n e
heater.in
can s u r r o u n d i n g
line
of liq u id
o f t he
Some o f t h e measur ement s t a k e n a t
a re sisto r.
g r o u p was s t u d i e d
the
The t e m p e r a t u r e
by t h e a d d i t i o n
65° and 95°C were o b t a i n e d
for
The 193°K
t h e r m o c o u p l e was used t o measur e t h e
te m p e r a tu re o f the
heated w i t h
7.7 . 3' ° K . .
boiling
by s u b m e r g i n g t h e sampl e s e a l e d i n a
a Dewar o f m e t h a n o l .
c h r o m e l - a l umel
at the
w h i c h were
re servo ir.*
as an
The neck
t h e sampl e c o i .,1 f r o m t h e r e s e r v o i r was wound
a heating
e l e m e n t and c o n t r o l
* The two t e m p e r a t u r e
d e t a i l i n r e f e r e n c e 34.
control
is
exercised
devices
by means o f
are p i c t u r e d
in
“ 79"
t h e c u r r e n t , passed t h r o u g h
the
heating element.
CONDUCTIVITY.'AMD DIELECTRIC 'CONSTANT.
Sampl es s u i t a b l e
for
both c o n d u c t i v i t y
c o n s t a n t measur ement s were o b t a i n e d
shaped s e c t i o n s
from l a r g e
saw.
Care was t a k e n
large
surfaces
crystal
to
single
guarantee
or faces
then p a i n t e d
thickness
for
with
that
normals to the
the
a string
g u a r d r i n g was
a
contact.
diagram o f the c o n d u c t i v i t y
25.
bS
The c o n d u c t i v i t y
The a v e r a g e
thickness,
the
crystal
equivalent
c irc u it
current
through
through
Rc and Rs .
to
the
A is
is
the area o f t h e
resistance
the c r y s t a l
ring
experiment is
x V
c o n t a c t , and Rq i s
bet ween t h e
a. damp
One c o n t a c t was a
a v e r a g e ar e a o f 4 x 1 0' ^ ' c m ^ .
= crystal
crystal
to a
t h e sampl es was a b o u t 0 . 1 5 cm.
Fig.
° =?
waf er-
crystals
on them.
around t h e c i r c u l a r
A block
the
small
a x i s . ■ These, f a c e s were t h e n p o l i s h e d w i t h
round dot w i t h
wher e t
by c u t t i n g
o f t h e s e w a f e r s were p a r a l l e l
p a p e r and c o n t a c t s were p a i n t e d
shown i n
and d i e l e c t r i c
is
is
shown i n
the
Rg r e p r e s e n t s
guard r i n g ,
and t h e
o f the
Fig.
26).
sum o f t h e
the
(The
The
current
the re s is ta n c e
including
contact
crysta l.
circula r
through
surface
on t h e o p p o s i t e
crystal
J . F l u k e 881A
Diff e r e n t ia l
Voltmeter
H e a t e r Supp l y
G. R. 1230A
Electrometer
or
K e i t h l e y 148
Nano
v o l tmeter
Oil Bath(Methanol
Low T e m p e r a t u r e )
Dressen
Barnes
5-300F
Regulated
Power
Suppl y
Sample
Probe
Y Axis
M o s e l ey
7000 A R
X-Y
Pl o t t e r
X Axis
Thermo
couple
Lead
Insulating
Container
Heater R e s i s t o r
Fig.
25.
B l o c k Di agr am o f C o n d u c t i v i t y
E x p e r i me n t
for
-81face.
Rs i s
the
resistance
c e n t e r c o n t a c t on t h e
bet ween gu ar d r i n g
same f a c e , and Rc i s
and t h e
the c r y s t a l
resi stance.
Fig.
If
R is
through
26.
Crystal
made s m a l l
R$ i s
the c u r r e n t
Equivalent
compared t o
ne gligible
through
C ircuit
Rq and Rg , t h e n
compared t o
the c u r r e n t
voltage
dropped almost e n t i r e l y
Re =
voltage
through
t o a good a p p r o x i m a t i o n ,
and I i s
For v o l t a g e s
Dr essen
5-300F regulated
s u p p l y V0 .
Rc and s i n c e
across
where V0 i s
t h e c u r r e n t measur ed t h r o u g h
electrom eter.
Barnes
through
the e l e c t r o m e t e r r e s i s t a n c e
approximately
V0 i s
that
the c u r r e n t
Rc , so
R is
IR<<IRC the
Rc .
Thus
the a p p l i e d
the
bet ween 10 and 300 v o l t s
a
power s u p p l y was used t o
The v o l t a g e was measur ed w i t h
a J . F l u k e Model
- 82-
8 8 T d i f f e r e n t i a l • v o l t m e . t e r . - ,Bel ow 10 v o l t s
v o l t m e t e r was used as t h e s o u r c e o f Vn .
the c r y s t a l
resistance
Typical
values f o r
r a nge bet ween I O7 and I O15 ohm's f o r
t h e a s s o r t m e n t o f sampl es
and t e m p e r a t u r e s
E l e c t r o m e t e r r e s i s t a n c e was h e l d a t
t h a n Rc .
the d i f f e r e n t i a l
least
In o r d e r to accomplish t h i s ,
studied.
100 t i m e s
a Ge n e r a l
1 230A e l e c t r o m e t e r , was used as a K e i t h l e y
Model
smaller
Radi o Model
148
nanovoltm e te r.
Elevated
temperatures
sampl e p r o b e w i t h i n
t h e a s s e mb l y i n
were o b t a i n e d by p l a c i n g
the
a l e a k - p r o o f c o n t a i n e r and s u b me r g i n g
a bath o f m i n e ra l
The b a t h t e m p e r a t u r e
was c o n t r o l l e d
by c o n t r o l l i n g
in
Lower t e m p e r a t u r e s were O b t a i n e d by u s i n g
the bath.
me t h a n o l
the
o il.
current
as a b a t h and a d d i n g l i q u i d
The t e m p e r a t u r e was measur ed w i t h
air
for
crystal
This
current
the y axis
allowed a continuous
vs t e m p e r a t u r e .
sampl e was r e v e r s e d p e r i o d i c a l l y
up o f h i g h r e s i s t a n c e
(electrode
layers
"p o lariza tion").
small
the x axis
t h e Mo s e l e y 7000 AR X-Y p l o t t e r .
t h e e l e c t r o m e t e r was f e d i n t o
plo tte r.
in
amount s .
a chrome!-alumel
t h e r m o c o u p l e , w h i c h was used t o p r o v i d e
voltage
through a r e s i s t o r
drive
The o u t p u t o f
i n p u t o f the
measur ement o f t h e
The v o l t a g e
a p p lie d to
the
in order to minimize b u i l d ­
a d j a c e n t to the e l e c t r o d e s
Upon r e v e r s i n g
the v o l t a g e .
-83large
transient
indicated
after
representative
c u r r e n t was o b s e r v e d .
the
decay o f t h e
o f the c r y s t a l
The a c d i e l e c t r i c
holder
541A o s c i l l o s c o p e
is
signals
in
d e te c t the
lock-in
balanced b r id e
d ie le c tric
detector with
mode was u s e d .
Wi t h
and r e s i s t a n c e
o f the
parts
ar e t h e n
for
Fig.
28.
Fig.
preamp i n
the
o f the
For measur ement s
c o n s t a n t t h e P. A . R.' Model
sampl e a r e
to
used t o
in
the
balanced the
in
calculate
difference
capacitance
t o Cy
The f o l l o w i n g
the real
E = E 1 S-
is"
and i m a g i n a r y
:
= -E 0TOjA1
Rc
a r e t h e sampl e c a p a c i t a n c e ,
A is
HR8
t h e same r a t i o s
and Rg.
constant
respectively,
27*
A Tektronix
superposition
condition.
=■ = Ec
E qA
Here Cc ,' Rc , and t
a bridge
o f t h e pr eamp, was used t o
the bridge
o f the d i e l e c t r i c
and t h i c k n e s s
shown i n
the p r e a m p l i f i e r
and Rv as C-j and R-j a r e
equations
in
is
a Type I A2 dual
t h d two c h a n n e l s
o f the a axis
c irc u it
provides
be
conductivity.
pictured
with
s u b t r a c t mode, w h i c h
t r a n s i e n t was t a k e n t o
c o n s t a n t was measur ed w i t h
c i r c u i t . . A diagram o f the
The c r y s t a l
The a v e r a g e c u r r e n t
resistance
t h e ar ea o f c o n t a c t ,
■ * The c r y s t a l h o l d e r was d e s i g n e d and c o n s t r u c t e d
P r o f . V . H. S c h mi d t .
and
by
l_Fajia_d_ax l.hj_eJ_d_P xoj/ided _by_CoEPe_r^ Scr een Cage
H. P.
200CD
Wi de
Range
Osci I ) a t o r
Tektronix
541A
Osci 11oscoDe
1A2 Dual
Tr a c e Preamp i n
Subtract
Mode
PAR-HR- 8 L o c k - i n A m p l i f i e r w i t h
Type A Preamp i n D i f f e r e n c e
Mode
M o n i t o r Out
R e f e r e n c e Out
F i g . 27.
C ircuit.
Di agr am o f D i e l e c t r i c
C o n s t a n t Measur ement
Br ass
Frame
Ceramic
Supports
S p r i ng - 1 o a d e d
Center
C o n t a c t - --------
S t a i nless
Lead S h i e l d
S p r i n g - 1 oaded
Guard Ri ng
_ Contact
Base
Contact
Adjustable
-Ceramic
Support
Fig.
28. Sampl e Probe Empl oyed f o r
C o n s t a n t Me as ur ement .
the
D ielectric
- 86co t h e a n g u l a r f r e q u e n c y o f t h e ac v o l t a g e
measur ement i s
made.
The t e m p e r a t u r e
placing
in
o f the c r y s t a l
t h e sampl e h o l d e r i n
a bath o f f i n e
sand.
its
t h e sand b a t h .
was c o n t r o l l e d
protective
brass
container
through a r e s i s t o r buried
To l o w e r t h e t e m p e r a t u r e ,
c o o l e d by a d d i n g s m a l l
by
T e m p e r a t u r e s above room t e m p e r a t u r e
were o b t a i n e d by p a s s i n g a c u r r e n t
in
a t which the
quantities
t e m p e r a t u r e was measur ed w i t h
of liq u id
the
b a t h was
air.
a chrome!-alumel
The
thermocouple.
RELAXATION TIME.
The s p i n - l a t t i c e
with
relaxation
t i m e measur ement s were made
a d o u b l e 90° p u l s e s e q u e n c e .
supplied
sec t o
by a p u l s e g e n e r a t o r s u p p l y i n g
5 mi n i n t e r v a l s .
length
as w e l l
single
t r i g g e r was
pulses a t 1.3
The t r i g g e r was f e d t o a T e k t r o n i x
Type 162 wa v e f o r m g e n e r a t o r ,
variable
The i n i t i a l
which produces a sawtooth o f
as a p u l s e .
. The p u l s e and s a w t o o t h
were f e d s e p a r a t e l y
t o two T e k t r o n i x Model
generators
t h e 163 t r i g g e r e d by t h e s a w t o o t h t o
fire
to allow
a t a time a f t e r
t h e 163 t r i g g e r e d
163 p u l s e
from the pu lse.
Thi s ti me was c o n t r o l l e d by changi ng the p o i n t along the
sawt oot h a t which the v o l t a g e had r i s e n t o a val ue
s u f f i c i e n t t o t r i g g e r the pu l se g e n e r a t o r .
Pul se l engt hs
and s e p a r a t i o n were measured w i t h a H e w l e t t Packard 5245- L
- 87electronic
counter with
.H-. P'.. 53 21A t i m e
interval
pu l se g e n e r a t o r s
an r f
5262A t i m e
unit.
interval
The o u t p u t s
whi c h' d e r i v e s
rf
c o n t r o l le d . 14.0-20 MHz o s c i l l a t o r .
pulse
an r f
pulse
transm itter
is
fed to
The n u c l e a r decay s i g n a l
Minneapolis
diagram o f
pulse
o f the
head. (34)
sampl e h o l d e r
to a pulse
the o u t p u t o f which i s
Honeywell
potentiometer.
the apparatus
A typical
is
measur ement
two 90° p u l s e s
nuclear signal
second p u l s e
during
The o u t p u t
up i n t h e
and t h e n
is
receiver.
separated
By r e p e a t i n g
ta il
this
Fig.
o f the
into
is
the p o t e n t i o m e t e r .
29.
t .
the
The
follow ing
pulse
the
procedure
is
A block
by s u b j e c t i n g
the t a i l
sequences, noise
measur ed w i t h
2730.
by a t i m e
the o u t p u t
p e r i o d o f the
in
a
measur ed by a
Model
obtained
measur ed d u r i n g
by f e e d i n g
a b rie f
integrator.
is
pictured
to
o f the r f
o u t p u t o f t h e r e c e i v e r was measur ed w i t h
boxcar i n t e g r a t o r ,
sampl e t o
picked
and t h e n
from a c r y s t a l '
The o u t p u t
a crossed c o i l
is
and f e d t o a p r e a m p l i f i e r
The s i g n a l
input
tra nsm itte r.*
and an
from the
were f e d t o 'a p u l s e a m p l i f i e r
g a tin g .c irc u it
gate d r i v e s
unit
the
receiver
boxcar
f o r many p u l s e
a v e r a g e d and t h e o u t p u t o f
The t i m e
the boxcar
t
separating
* The pul s-e a p p a r a t u s was c o n s t r u c t e d by R o b e r t
P a r k e r and Fr ed J . BI an k e n b u r . g .
The b a s i c d e s i g n i s
C l a r k e t a I . ( 35,36)
after
O
Cr ossed C o i l
F i g . 29.
P u l s e Head
B l o c k Di agr am o f P u l s e A p p a r a t u s
A - Trigger
a -
Pulse Gener at or
B - Tektronix
Model
j n
_
162 Waveform G e n e r a t o r
b -
C - Tektronix
Model
163 P u l s e G e n e r a t o r
D - Tektronix
Model
163 P u l s e G e n e r a t o r
E - Pulse A m p l i f i e r
F -
c -
d -
_TL
Radi o F r e q u e n c y Gate
e -
G - Crystal
Controlled
O scillator
H -
Radi o F r e q u e n c y P u l s e T r a n s m i t t e r
I -
Radi o F r e q u e n c y P r e a m p l i f i e r
J -
Pulse A m p l i f i e r
f
-
9 -
h K -
Box Car I n t e g r a t o r
L -
Potentiometer
Key f o r
F i g . 29
- f x
n
_
t h e ’ two p u l s e s
repeated.
is
changed and t h e e n t i r e
The s l o p e
o f a semilog
. v e r s Us t makes p o s s i b l e
relaxation
the
into/the
directed
a ir
pulse
uniform ly
by b o i l - o f f
obtained
by p a s s i n g
containing
of. t h e
sp in -la ttice
i s ' a c h i e v e d by p a s s i n g h o t o r co.l.d,,
head i n
such a f a s h i o n
over the c r y s t a l .
air
that
Col d a i r
liq u id
air.
Physics
the a i r
is
to
then throug h
the
is
B u i l d i n g - compressed:
a tube
h e a t i n g e l e m e n t and f i n a l l y
attached
is
provided-
Heat ed a i r
The t e m p e r a t u r e was measured w i t h
thermocouple
for
from the
a regulator,
an e l e c t r i c
p u l se h e a d .
varia tion
o f signal, voltage
calculation
from a c o n t a i n e r o f
system th r o u g h
a l u me l
is .
t i m e T-,.
Temperature c o n t r o l
a ir
plot
procedure
crystal.
into
a
a chromelTemperature
t h e measur ement s above room t e m p e r a t u r e was
± 10 C and b e l o w room t e m p e r a t u r e
± 1 . 5°C.
•
APPENDIX A
Nuclei
moment' .*
o f spin
If
I
I may po s s es s an e l e c t r i c
such a n u c l e u s
vanishing e l e c t r i c
fie ld
is
placed at a s i t e
gradient
(efg),
quadrupole
o f no n­
and i n t e r a c t i o n
bet ween t h e n u c l e a r q u a d r u p o l e moment and t h e e f g may be
observed.
given
The H a m i l t o n i a n
o f such an i n t e r a c t i o n ,
Hg,
is
by t h e t e n s o r p r o d u c t o f t h e n u c l e a r q u a d r u p o l e moment
tensor with
states
the efg t e n s o r .
Transitions
bet ween t h e e n e r g y
o f t h e s y s t e m can be i n d u c e d by c o u p l i n g
component o f a r e s o n a n t r a d i o
frequency f i e l d
t h e m a g n e t i c moment o f t h e n u c l e u s .
possible
t o measur e t h e i n t e r a c t i o n
In t h i s
the magnetic
( E = hv) with
fashion
it
is
energy c h a r a c t e r i z e d
by
Hg.
T h r e e i m p o r t a n t cases a r e d i s t i n g u i s h e d w i t h
this
problem.
The f i r s t
case,
t h a t t e r me d p u r e n u c l e a r
quadrupole resonance,
is
called
o r Zeeman c a s e ,
t h e weak f i e l d
the system o f n u c l e i
p e r ma n e n t m a g n e t i c f i e l d
f o r the system is
outlined
at sites
interaction
is
treated
inte ractio n
energy.
abov e.
The second c a s e ,
consists
o f nonzero e f g ,
H0 ("' IO g a u s s ) .
t h e n Ht o t
regard to
of subjecting
to
a weak
The H a m i l t o n i a n
= Hg + Hmag, where t h e m a g n e t i c
as a p e r t u r b a t i o n
The t h i r d
on t h e q u a d r u p o l a r
case c o n s i s t s
*General d i s c u s s io n o f the t o p i c
f o u n d i n r e f e r e n c e s 37 and 38.
of th is
of subjecting
appendix i s
-92a system o f n u c l e i
re la tiv e ly
sm all,
wherein
to
The. Hami l t o n i an i s
*
pertu.rbati on. .
the quadrupolar
interaction
a s tro n g magnetic f i e l d . ("IO ^
then
Ht o t - .= Hmgg + Hq w i t h
- I n t h e p r e s e n t wor k t h e pr o b l e m: was t h e
o f the efg
this,
deuterons
protons
3.
tensor at
the hydrogen s i t e s .
( Q = 2 . 7 7 x 10“ ^
(Q = O ) .
The e x p e r i m e n t
gauss).
Hg t h e
determination
I n o r d e r t o do
cm^) .were s u b s t i t u t e d
fa lls
in
is
for
t h e c a t e g o r y o f case-
The c a l c u l a t i o n
representation
diagonal.
o f t h e e n e r g i e s p r o c e e d s by c h o o s i n g a
p
in which I
and I z ar e s i m u l t a n e o u s l y
The b a s i s
states
are de noted,
I 2 I I,m,n)
= 1(1 + I ) I I , m, n )•
I z I I ,m,n)
= m] I , m , n ) .
The e n e r g y t o z e r o t h
order
is
-
|I,m ,n),
•
the magnetic
energy
interaction
■- •
E0 = ( I » m|
where
'
- YH0 I z I I ,m)
(A.I )
yH0m
wher e H0 i s
z axis,
o f the
Y is
the
the
p e r ma n e n t m a g n e t i c
gyromagnetic
fie ld
ra tio ,
directed
I z is
the
z component
n u c l e a r a n g u l a r momentum o p e r a t o r , and m i s
magnetic
quant um n u m b e r .
The p e r t u r b a t i o n
along the
the
is
• * T h i s case i s c o n s i d e r e d by V o l k o f f ( ^ S ) and t h e
e x p e r i m e n t i s d e s c r i b e d i n d e t a i l i n t h e wor k o f V o l k o f f ,
F e t c h and Smel l i e . H O )
- 93 (A.2)
Hn = - / e g •
■Q - r n m r
+
^
e is
]
( 3I z 2 -
+
*
1Z 1- + 1 U ( V E )
+ ' 4 I I Z(VE) „ + ' 4 1
2,1 ' " ' + I ,
2 ‘+
'
2
the e le c tr o n
and I
is
operators
are d e f i n e d
component s o f t h e e f g t e n s o r
by t h e m a g n e t i c f i e l d
The r a i s i n g
(with
the z ax is
H0 ) a r e g i v e n
established
by
■ ( A . 3)
( v E ) ±1 = ± —L - (<j,zx ± i <j>yZ )
/ 6“
The f i r s t
El
^^
order c o rre ctio n
= 2 I ( i? _
-
O'
'CU
11
C
U.
V
I
(21
2
-
[ 2 1 ( 1
L
-
I )
2mZ
(«>0
•
■
( A . 6)
-
8m { ] ) ] ( V E ) ± 1 I 2
is
[ 4 1 ( 1
1
(m
'
+ ')]
+
I )
CM
+1
+
(m
" I
LU
l>
QBH0
(A.5)
I
'
I ) /
•
is
r—
3
QBH0
( A . 4)
( tJ5XX ” ^yy ~ ^ ^ tJ5x y )
The second o r d e r c o r r e c t i o n
6
and
I + = (I
± i I ) and t h e
—
X
y
vEo = - J - t z z
(vE)±2 -
2(vE)
c h a r g e , Q t h e n u c l e a r q u a d r u p o l e moment
t h e n u c l e a r a n g u l a r momentum.
lowering
i
(A . 7)
-94"
P
In general
|v.Ey|
eq = :<PZ ' z
(y = ±1.,2). can be e x p r e s s e d
re la tive
IVEy I
.=
and s e t t i n g
t o z |j H0 w i t h
and
hK
TT i i I -
is
I)
21( 21
I )2
c o n s t a n t ) and
the n u c le u s )
L3"2 " 1(1 + 1U
to
E-j .
l]
The L a r mo r f r e q u e n c y
' ( A . 10)
is
14 MHz.
and Ep down some two o r d e r s
^
order e ffe c ts .
o f the p a tte r n
resulting
t h e n be o f t h e
same m a g n i t u d e as t h e
The s h i f t
in
constant
This
of
the c e n t e r
f r o m t h e second o r d e r e f f e c t w o u l d
a b o u t t h e same s i z e
line
widths
observed.
as t h e e s t i m a t e d e r r o r
hence t h e second o r d e r e f f e c t
can be seen t h a t
)l(vi ) ±1
,2
) | ( vE)
The maximum v a l u e f o r t h e c o u p l i n g
magnitude from f i r s t
experiment,
- 8m2
K / v ^ measur es t h e m a g n i t u d e o f Eg
makes - J i - = - 2 ^ x - ■
?■■■,
vL
T . 4 x 10 7
It
( A . 9)
<’ E ) o
vL
+ I ) - 2m2 -
a b o u t 200 kHz.
is
ggH0 = hv.
3 ( m [ 41( 1 + 1 ) - 1
-
can be seen t h a t
This
x , y a r b i t r a r y . . Then
j^3m2
E.
re la tive
of
(A.8 )
-e^ tI5. = ! ( ( c o u p l i n g
3(m ^ 2 1 ( 1
It
units
e^q ^ IVEy J^
( V[_s t h e La r mo r f r e q u e n c y o f
E1 '
in
the f i r s t
is
not s tud ied.
order co rrectio n
* z " denotes the a x i s o f the
efg t e n s o r expressed in dia gonal
in the
depends on*
l a r g e s t component o f t h e
form.
- 95the
nuclear spin
the q u a n t i t y
I 5 t h e n u c l e a r q u a d r u p o l e moment Q5 and
( V E ) q = I f zz .where f zz i s
o f the e l e c t r o s t a t i c
parallel
a t the n u c le a r s i t e ,
with
z
t o Hq .
For d e u t e r o n s
the energy i s
E = - Y H 0 B1 + - | 9
The m a g n e t i c
quadrupole
levels
ar e s h i f t e d
causing the s in g le
two l i n e s
lin e ,
as p i c t u r e d
a
line
in
Fig.
30.
These
b
YH0 + ^
n
<0;
_eQ
- 2V l t z z
\
- v H„ ----------
m-1
absorption
3 e Q / h ( VE0 ) .
yHn
m=0
by t h e
spaced e q u a l l y on bo t h s i d e s
a r e s e p a r a t e d by a f r e q u e n c y
m=-l
by
( A . 11)
o f the deuteron
into
o f the unperturbed
given
( 3nl2 „ 2) I z z
inte ractio n,
a t y H0 t o s p l i t
lines
potential
the. second d e r i v a t i v e
■ y H 0
+
a
hv0
= yH0
y ho"
hv->
YH0+|eQ<i).
|
hv->
F i g . 30.
a)
Zeeman L e v e l s and T r a n s i t i o n Fr equ enc y
b)
Zeeman L e v e l s S h i f t e d by 1 s t O r d e r Q u a d r u p o l e
P e r t u r b a t i o n and C o r r e s p o n d i n g T r a n s i t i o n F r e q u e n c i e s .
- 96C h a ngi n g t h e o r i e n t a t i o n
the constant f i e l d
t e n s o r a.long. Hq .
Hq s - changes
The r e s u l t
bet ween t h e s p e c t r a l
is
symmetric
five
lines
( vxE = 0)
independent
the form
o f t h e sampl e r e l a t i v e
t h e component o f t h e e f g
is
that
changes.
the s p l i t t i n g
( v • E = 0)
it
has o n l y
The t e n s o r can be p u t i n t o
,
I -e q ( I - n )/2
<Hj =
wher e eq i s
I
0
\
0
the
0
0
-e q (l+ n )/2
( A . 12)
0
0
eq
l a r g e s t and - e q ( I - n ) / 2 t h e s m a l l e s t
I U x 1X 1 -
r a n g e 0 <, n <= I .
The v a l u e s
o f n > eq 9 and t h r e e
dire ction
specifying
the o r i e n t a t i o n
cosines
re la tive
to
4» y ' y ' )
the c r y s t a l
^z l Zl
I =
c o m p o n e n t , ma k i n g
tensor
observed
Because, t h e e f g t e n s o r
and t r a c e l e s s
components.
to
71
lie
1* n t h e
independent
o f the diagonal
axes c o n s t i t u t e
five
independent q u a n t i t i e s .
A transform ation
and l a b o r a t o r y
s y s t ems w h i c h r e l a t e s
system f i x e d
in
and C o f Eq.
(2.1)
tensor
Ai
in
can be c a r r i e d
the c r y s t a l ,
are r e l a t e d
the c r y s t a l
'SeQ
( JJ
2h
through
o u t bet ween t h e c r y s t a l
the efg t e n s o r t o
Eq.
( 2 . 1).
the
The A, B
t o t h e component s o f t h e e f g
s y s t em as f o l l o w s :
+kk)
-SeQ
2h m
( A . 13)
-97-
Bt *
(+.1.1 j i <;
As was d i s c u s s e d
s p littin g
crystal
in
( A . T4)
+kk)
i,j,k
Chapter
frequency versus
axes d e t e r m i n e
( A . 15}
= x,y,z
I , the experim ental
angle f o r
values
for
the
rotation
This
the c o u p lin g
of
is
constant,
about the th r e e
the
t h e n d i a g o n a l i z e d and t h e
as y mmet r y p a r a m e t e r and t h e
the efg t e n s o r p r i n c i p a l
obtained.
S
tensor
axis
system i n
,
o.f .
A 9 B and C f r o m w h i c h
a r e f o u n d t h e compon ent s o f t h e e f g t e n s o r i n
system.
curves
crystal
values
of
orientation
the c r y s t a l
ar e
APPENDIX B
The. d e u t e r o n s y s t e m i s
relaxation
proba bility
is
level
p r o c e s s e s may t a k e p l a c e
Am = 2 ( P2 ) t r a n s i t i o n s .
it
a three
s y s t e m so t h a t
by way o f
In o r d e r to c a l c u l a t e
of a tr a n s itio n
bet ween l e v e l s
is
inte ractio n
diffe ren t
the
fluctuation
in
the
o f the syste m ,
assumed t h a t t h e mechani sm r e s p o n s i b l e
tra nsition
Am =. T ( P 1 ) o r
for
t he
the n u c le a r q u a d r u p o le
due t o d e u t e r o n j u m p i n g bet ween s i t e s
ele ctric
fie ld
gradient.
p r o c e e d s ( 9)
by t a k i n g
a tra nsition
bet ween s t a t e s
of
The c a l c u l a t i o n
as t h e p r o b a b i l i t y
per u n i t
ti me o f
k and m
(B . I )
where t h e c o r r e l a t i o n
Gm k ( T )
the
=
two c a s e s ,
possible
t
) | m) ,
for
Gm k ( T )
times
( B. 2)
sites
for
Pmk f o r
the
d istrib utio ns
and
Having determined
two c a s e s ,
it
two c a s e s .
are then c a l c u l a t e d
S ubstitution
(ND2 g r o u p )
(NDg g r o u p ) .
these
values o f
must be d e t e r m i n e d f o r
ex change bet ween two s i t e s
the p o p u la t io n
pulse.
+
The v a l u e o f G m k ( T )
to c a l c u la t e
relaxation
defined
an ensembl e a v e r a g e o v e r a l l
ex c hange among t h r e e
the values
is
(mI H g ( t ) ] k ) ( kI H g ( t
bar de notin g
(m ] H q(t)Ik ).
function
is
t he n
The
f r o m a k nowl edge o f
immediately f o l l o w i n g
o f the values
for
a 90°
P1 and Pg c a l c u l a t e d
- 99above y i e l d
expressions
o n l y on t h e c o r r e l a t i o n
for
the r e la x a t i o n
times
dependi ng
t i m e Tc , and t h e La r mo r f r e q u e n c y
o f the n u c l e u s .
Consider f i r s t
the
calculation
j u m p i n g bet ween two s i t e s .
site
2 corresponding
Chapter I I .
(m -
to the deuteron
k ) = (£EL_^__Ek) = a).
Let there
bet ween s i t e s
I
occupy s i t e
I
positions
equally
and c o n s i d e r n u c l e i
o r b) o c c u p y s i t e
P1 ( T ) F 1 + P2 ( T ) F 2J
2.
to a is
wher e F1 ( i
the H am ilto nian
representative
of site
I which
from s i t e
exc h ang e f r o m s i t e
changes.
I
to
site
in
P1- ( T )
nucleus w i l l
be f o u n d a t s i t e
is
P1 ( T ) F 1 + P2 ( T ) F 2
place
th a t at T = O
then
o f the environment
as t h e n u c l e i
2 the i n t e r a c t i o n
I.
in
d istrib ute d
2 , so t h a t
the p r o b a b i l i t y
t h e ensembl e a v e r a g e i s
F2
2
indicated
energy
F1 r e p r e s e n t s
t h e m a t r i x e l e m e n t o f Hg c o n t a i n i n g
components.
and
The e l e m e n t o f t h e
For t h e case u n d e r c o n s i d e r a t i o n ,
the f a c t o r
I
= 1 , 2 ) denotes t h a t
part of
diffe rs
as s i t e
Assume t h a t exchange t a k e s
ens embl e a v e r a g e c o r r e s p o n d i n g
I
t h e case o f
so t h a t
be n n u c l e i
and 2,
for
Denot e t h e s i t e s
L e t -fim = Em and -frk =
as f o l l o w s .
a)
of
that
the efg
at time T , the
The r e m a i n i n g
element o f
-100corresponding
t o b.
To d e t e r m i n e
(t )
nuclei
l
in
site
and P g ( T ) ,
and ng n u c l e i
and d e n o t e t h e p r o b a b i l i t y
bet ween s i t e s
p roba bility
equations
nI
I
(B.3)
Then
and 2,
choose an ensembl e w i t h
in
site
of rotation
2.
n]
L e t n-j + ng = n
t h r o u g h +180°
P_p+ = Pp _ = ^ r 5 Pyi b e i n g
2
o f ex change due t o + o r - 180° r o t a t i o n .
The
describing
Pr + so t h a t
the p o p u l a t i o n s
a r e , d e n o t i n g _Jl by « ,
at
= " p r n I + Pr n2 = “ n2
( B. 4)
n 2 = - P r Hg + Pr n-j
which are s a t i s f i e d
+ ^2
by
H1 = B . + ^ e ~ 2Pr T and
I ( I - O - 2 pV t ) F 1 + I ( H e - 2V ) F 2
-e
( B. 5 )
The c o n s t a n t
t e r m does n o t c o n t r i b u t e
i s dropped. . This
to the
relaxation
and
leaves
^ ( o ) Ir ( T ) =. Tf ( FI ~ ^ 2 )
Ty/Tc
, where 2 Pr - I /-Tc .
Then f o r t h e Am = I t r a n s i t i o n .
6 Al
!41(21
-
I )
( 2m *
^ )
(I
± m) ( I ± m + I )
! ( F 1 - F2 ) 2e " T / T c
Here Fi
I o r 2.
= ^ U xx. -
and
Pl
to
( B. 6 )
I = I makes
2n2
- fi-9
32 ( F 1 " r 2 !’ e
2 » Q2n2
/
= h2
32
, 2B2Q2
,
^ y y i ) ±1 ^ x y i 3 ( 1 = 1 ’ 2 ) d e n o t i n g s i t e s
Specializing
Gi l ( T)
h,Z
-
( A<f>x z )
wher e AcfJi j - means (J)i j ( s i t e
■T/T.
( B 07 )
F2* ,
+ (M yz)
2 ) - (j>i j ( s i t e
(B.8)
!+Oi2Tc 2
I).
For t h e Am = 2
t r a n s i t i on
G6 2 (T ) = e £- ^£-((FF ,l
which
gives
for
the
-
F^J2B
F2 ) ‘ b "- t1Zt
/ f cC
tra nsition
probability
( B- 9)
-IO
Z-
The same a r g u m e n t s a r e t h e n
among s i t e s
for
I,
2 and 3.
ex change among t h e
p I
=
applied
Transition
sites
( 4 1 2 *
x z
>2
t o t h e case o f j u m p i n g
proba bilities
obtained
I , 2 and 3 a r e
+ U 13 Ox z ) 2 + U 3 3 Ox z ) 2
+ U 1 2 Ox y ) 2 + ( A 13Oyz ) 2 + ( A23Oy z ) 2
■■ T;
J I +Oi^T£
' (B.12)
+
( B. I 3)
where Aa ^ Cfr1- j
=
" - U i j l b s a and b d e no t i ng s i t e
(<fr-jj)a
numbers.
These e x p r e s s i o n s
proba bilities
times
is
for
Am = 1 , 2 t r a n s i t i o n
now be used t o
calculate
t h e cases u n d e r c o n s i d e r a t i o n .
saturation
three
w ill
f o r the
levels
o f the
two l i n e s
The f i r s t
at a crossing
are p o p u l a t e d a t e q u i l i b r i u m
the r e la x a tio n
point.
with
N _ - | ,
o f these
The
N q
and
-103N+1 n u c l e i
in
respectively.
t h e m = - I 5 m = 0 and m = +1 l e v e l s
Because t h e
s p a c e d , one can t o
and N+ i
are a l m o s t e q u a l l y
good a p p r o x i m a t i o n
- Nq =
from e q u i l i b r i u m
levels
let
Nq - N_-j = nQ
L e t n-j 5 n 2 and n^ d e n o t e v a r i a t i o n s
o f the p o p u la tio n s
and +1 r e s p e c t i v e l y .
of levels
These p o p u l a t i o n s
m = - I ,0
t h e n may be p i c t u r e d
\
as i n
Fig.
time t
is
n 3
=
30, where n-j = - n g - n ^ .
proportional
The s i g n a l
observed a t
t o Bng., where ng = n^ - ng+nQ.
- n 3 P I - I1 3 P 2 + n 2 P l ~
( n 3 + n 2 ^ P 2
= - n 2 ( P 1+2P2 ) + n 2 ( P v P2 )
b u t n 2 = 0 and n2 = 0 so t h a t
has t h e s o l u t i o n
.
= S0e - t / T l
s and by symmet ry
Then ns = Oq - O qB- ( pl +2P2
= Oq U - e - t / T ] ) where I / T ]
S0- S ( t )
n3 + ( P 1+2P2 ) n 3 = O5 w h i c h
n 3 = - n g e " ^p I +^ p2
O1 = Oq O " ^ p I + 2P2
( B. I 4)
= P1+2P2 and t h u s
where Sq i s
equilibrium
Now f o r t h e case w h e r e i n o n l y one o f
S a tu ra te d 5 the po pula tions
Fig.
31.
The s i g n a l
are d i s t u r b e d
at time
t
is
signal
the
height.
lines
is
as p i c t u r e d
in
proportional
t o ns and
ns = n 2- ( - n 0+ n l )•
n 1 = - n 1 ( P 1+2P2 ) + n 2 ( P1 - P 2 )
and
n 3 = - B P 1O2 + P1O1 - P1 (.n2+ n 1 ) = - SP1 n 2 .
(B.15)
( B. I 6 )
m= + l
n o+ n 3
F i g . 31.
R e l a t i v e P o p u l a t i o n s o f N u c l e a r L e v e l s as a
F u n c t i o n o f Ti me f o r S y mme t r i c S a t u r a t i o n .
A t t = 0, a 90°
P u l s e S a t u r a t e s t h e Syst em.
m= + l____________ np+n 3
F i g . 32.
R e l a t i v e P o p u l a t i o n s o f N u c l e a r L e v e l s as a
F u n c t i o n o f Ti me f o r S a t u r a t i o n o f S i n g l e L i n e .
At t = 0
90° P u l s e S a t u r a t e s m = (0+-+-1 ) T r a n s i t i o n .
These e q u a t i o n s
"I
=
are s a t i s f i e d
by n 9 = - U p e - 3p I t
^
2
, ^ ■ ( P l + 2P2 ) t
maki ng
n0 ( l - | e - 3 P , t . i £ - ( P 1+2P2 ) t )
4
S0- S t t )
-
and
'Oe-3pI t + e-(P,«P2)t)s ;
Finally
deuterons
by ex c hange t o ND^ s i t e s
as p i c t u r e d
in
Fig.
e v e r y two NDg s i t e s
appropriate
(3.15)
t h e mechani sm p r o p o s e d f o r
The p o p u l a t i o n
is
32.
Since t h e r e
the p o p u l a t i o n s
factors.
guaranteeing
levels
are denoted
are m u l t i p l i e d
The e x p e r i m e n t
is
n 33 = - n g i .
Then t h e
equation
carried
j umps
g o v e r n i n g ng-j
is
given
( B . l 7)
the p r o b a b i l i t y
t h a t a d e u t e r o n a t an NDg
t o an ND2 s i t e .
The r a t e
deuterons
o f the
t , n 2g = - n 2 i » and
.
n31 = -P(ng-j-n2-|)
wher e p d e n o t e s
to
by t h e
symmetric s a t u r a t i o n
T h i s makes ngg = O f o r a l l
site
t h e NDg
ar e t h r e e NDg s i t e s
NDg s y s t e m .
rate
relaxing
considered.
o f t h e m = - I sO9+!
numerical
out at a c ro s s in g ,
by
and
equation
for
the corresponding
level
of
o f t h e ND^ gr oup i s
O2 T = - ( P 1+2P2+ | p ) n 21+ | p n 3 1 .
(B.18)
-106P1+2 P2 i s
ide ntified
case o f s y m m e t r i c
s y s t e m and w i l l
saturation
Og1 i s
o f the
three
level
the f o l l o w i n g
d iffe re n tia l
No c o n f u s i o n w i l l
levels
be e n c o u n t e r e d
t h e NDg as w e l l
I
t he
deuteron
equation
obtained
n31+| P n 31+Pn31+PPn 31 = 0 •
since
for
be d e n o t e d P.
By s u b s t i t u t i o n ,
for
as ! / T 1 g o v e r n i n g r e l a x a t i o n
(B.19)
by d r o p p i n g
as t h e NDg r a t e s
the s u b s c r i p t
I
are symmet ri c i n
and 3.
3nD+ 3 n 33
3n32
3 n o+ 3n31
t =0
ZnQ+Zngg
2n 22
^ 2n a ± 2n2j _____
F i g . 33.
R e l a t i v e P o p u l a t i o n s o f N u c l e a r L e v e l s as a
F u n c t i o n o f Time f o r S y mme t r i c S a t u r a t i o n o f t h e NDg Sy s t em,
I n d i c a t i n g C o u p l i n g w i t h NDg Syst em.
-107Assumi ng t h e s o l u t i o n
and t h a t
P>>p,*
Now s i n c e
sites
n3 = naec l t + n j:)ec 2' t , w i t h
a solution
n3 = nae - P t + n be - P t -j s o b t a i n e d .
P>>p, I . e . , t h e j u m p i n g bet ween ND3 and NDg
o c c u r s much more s l o w l y
deuterons
na + nb = n0 ,
o f the
ND3 g r o u p ,
than the r e l a x a t i o n
the
relaxation
by p , because na =0 and n b =nQ, w i t h
rate
the r e s u l t
of
is
lim ited
that
n3 " noe " p t If
t h e NDg s y s t e m i s
p rim a rily
n -|3 = 0 i t
for
t o j u m p i n g bet ween t h e
follows
the r e tu r n
experiment f o r
crossing,
f r o m Eq.
two ND2 s i t e s .
( B. 18)
to e q u ilib riu m
is
t h e ND2 gr o u p i s
that
the
(3.15),
r a t h e r than
3_p due t o j u m p i n g
due
time constant
( P] + 2Pg ) +^-P•
not c a r r i e d
is
Si nce
However,
the
out at a
so t h e I -j due t o ND2 h i n d e r e d r o t a t i o n
g o v e r n e d by Eq.
correction
s a t u r a t e d , the r e l a x a t i o n
(PytZP2 ) ,
is
plus
the
bet ween ND2 and ND^ s i t e s .
* The l a t t e r a s s u m p t i o n i s j u s t i f i e d s i n c e t h e b a s i s o f
t h e p r o p o s e d mechani sm i s t h e e x p e r i m e n t a l o b s e r v a t i o n t h a t
d e u t e r o n s o f t h e ND2 g r oup r e l a x v i a P] and Pg much f a s t e r
t h a n t h e d e u t e r o n s o f t h e ND3 gr o u p
relax.
APPEN-DIX. C
Table
structural
indicated
XI and. F i g .
the r e c e n t l y
i n f o r m a t i o n (4%) 0 n L i N2-DgSO^.
i n T a b l e XI
N-H d i s t a n c e s
features
34 summ,a r i z e
included
are not i n
in
of the s t r u c t u r e
Table
available
The N-D d i s t a n c e s
good a g r e e me n t w i t h
III.
the
However, the general
ar e much t h e same.
-109TABLE XI
Bonded D i s t a n c e s
Bond
Di s t a n c e
Bond
Distance
S -0(1)
1.484
N(I)-D(I)
1.018
S -0(2)
1.487
N(1 ) - D (2 )
I . 030
S -0(3)
1.467
N( 2) - D( 3 )
1.015
S -0(4)
1.476
N( 2) - D ( 4 )
1.022
N(1 ) - N ( 2 )
1.425
N( 2) - D (5 )
1.024
Non-Bonded D i s t a n c e s
Li-O (I)'
1. 9 4 2
D (2 ) - N ( I ) '
2.042
Li - 0 ( 2 ) "
1.993
D(3)-0(l )
1.907
Li-O(S)
1.949
D(4)-0(2)
1.932
Li - 0 (4)
1.922
D( 5 ) - 0 ( 2 )
2.064
and " = r e l a t e d
by symmet r y o p e r a t i o n
Fig. 3 4 .
S t e r e o s c o p i c Vi ew o f a U n i t C e l l
c Axis w ith P o s i t i v e a Axis to Right o f Viewer.
o f LiN^DrSO*.
Vi ew I s Down
( A f t e r Ross and Hami I t o n ( 4 0 ) )
LITERATURE CITED
T.
F. Uona and G. S h t r a u e 5 F e r r o e l e c t r i c
M a c M i l l a n C o . , New Y o r k , 1962.
C r y s t a l s , The
2.
V. H. S c h m i d t ,
(1961) .
o f Wa s h i n g t o n
3.
J . L . B j o r k s t a m and E. A.
9-61 ( 1 9 5 9 ) .
4.
V. H. S c h m i d t and E. A.
(1962) .
5.
H. B. Si I s b e e , E. A. U e h l i n g ,
P h y s . Rev. 1_33, Al 65 (1 9 6 4 ) .
6.
0.
7.
V. H. S c h m i d t ,
630 (1 9 6 8 ) .
8.
A. A b r a g a m, The P r i n c i p l e s o f N u c l e a r M a g n e t i s m , O x f o r d
U n i v e r s i t y P r e s s , L o n d o n , 1961.
9.
C. P. S c h l i c t e r . P r i n c i p l e s o f M a g n e t i c R e s o n a n c e ,
H a r p e r and Row, New Y o r k , 1963.
PhD T h e s i s ,
U niversity
Uehl i n.g, P h y s . Rev.
Uehling,
L . B j o r k s t a m , P h y s . Rev.
P h y s . Rev.
114,
P h y s . R e v . 1 26 , 447
and V.
1 5 3 , 599
1_64, 749
H. S c h m i d t ,
(1967).
(1 9 6 7 ) ;
ibid.
173,
10.
R. P e p i n s k y 5 K.
P h y s . Rev. TJM ,
Vedam, Y. O k a y a 5 and S. Hoshi no,
1467 (1 9 5 8 ) .
11.
V. M. Padmanabhan and R. B a l a s u b r a m a n i a n , A c t .
22 , 532 ( 1 9 6 7 ) .
12.
J . D. C u t h b e r t and H. E. P e t c h , Can.
(1963) .
13.
W. D. Mac Cl e me n t , M. P i n t a r 5 and H. E . P e t c h , Can.
Phys,.. 45 , 3257 (1 9 6 7 ) .
14.
N. N i i z e k i
(1964) .
15.
I.
16.
J . H.. Van den Hende and H-. B a u t i n ,
660 ( 1 9 6 4 ) .
17.
T, . C h i b a ,
and H. K o i z u m i ,
D. Br own,
Acta.
Cryst.
Cryst.
J . P h y s . 4J_, 1629
J . P h y s . Soc.
Japan 1_9_, 132
1_7, 654 (1 9 6 4 ) .
Acta.
0.
Cryst.
J . Chem. P h y s . 4J_, 1 352 (1 9 6 4 ) .
TT^
18.
R. B T i n c and D.
Hadzi , N a t u r e 21 2 , 1 307
19v
I.
Chem.
20.
R. Vi j a r a g h a b a n and V.
20, 2290 ( 1 9 6 5 ) .
21.
Chiba,
Bull.
S o c . Japan 38^, 259 (1 9 6 5 ) .
S a r a s w a t i , J.. P h y s . S o c . Japan
W. D e r b y s h i r e , T i G o r v i n and D. W a r n e r ,
.. Phys.. T2, 299 ( 1 9 6 7 ) .
Bull.
Cfiem.
(19. 66).
S o c , Japan 38,
Molecular.
22.
T . Chiba,
23.
M. L i n z e r and R. Fo r eman, J . Chem.
(1967).
24..
S. Rabi deau and 'P. 'Walds.te.i-n, J . Chem.
(1966) .
25.
D. O ' R e i l l y
(1967) .
26.
W. C. H a m i l t o n and I b e r s , Hy dr ogen B ond i ng i n S o l i d s ,
W. A. B e n j a mi n Co. I n c . , . New Y o r k , 19 68, pp. 2 6 0 - 2 6 5 .
27.
A. V.
Acad.
28.
R. S . K r i s h n a n and K.
61 A, 1 22 (1 9 6 5 ) .
29.
J . V a n d e r k o o y , J . C u t h b e r t , and H. E. F e t c h ,
PTiys. 42 , 1 871 ( T 9 6 4 ) .
30.
S. Dev a nar ay an an and K. R. K.
Acad.. Sci . 64A, 1 73 (1 9 6 6 ) .
31.
T . Chiba,
32.
F . N. H. R o b i n s o n , . J . Sci .
33.
F . J . B I a n k e n b u r g , R. R. K n i s p e l , and V.
Rev . Sci . I n s t . . 37, 1 020 (1 966 ) .
34.
R. K n i s p e l , PhD Thesi s- ,
(1969).
35.
W. "G. C l a r k ,
P h y s . 45 , 4600
P h y s . 46 , .1 291
R. W a r r i e r and P. S . N a r a y a n a n ,
SCi . 64A,' 254 (1 9 6 6 ) .
J.' Chem.
(1965).
P h y s . _46, • 4690 -
and T . T s a n g , J . Chem.
Krishnan,
490
Proc.
Indian
P r o c . I n d i a n A c a d . Sci
Can.
Easwaran, P r o c . In d i a n
P h y s . 89 , 947
(1 9 6 3 ) .
I n s t r . 36_, 481
(1 9 5 9 ) .
H. S c h m i d t ,
Mont ana S t a t e U n i v e r s i t y
Rev . SCi . I n s t .
J.
35^, 31 6 (1 963) .
-11336.
W. G. C l a r k and A.
1593 ( 1 9 6 7 ) .
37.
I . P . Das and E. L . Hahn5 N u c le a r Q u a d r u p o l e Resonance
S p e c t r o s c o p y 5 Ac ademi c Pr ess I n c . 5 New Yor k 5 1 958.
38.
M. H. Cohen and F . R e i f 5 S o l i d S t a t e
Pr es s I n c . , New Yor k 5 1 957 , Vol . 5.
P h y s i c s , Academi c
39.
G.
(1 9 5 3 ) .
40.
G. M. V o l k o f f 5 H. E . F e t c h ,
J . P h y s . 30, 270 (1 9 5 2 ) .
41.
F . Ross and W.
M.
I.
K e r l i n 5 Rev.
V o l k o f f 5 Can. J-.
Hamilton
Sci . I n s t .
P h y s . _31_, 820
and D. W. I .
(private
_38s
S m e l l i e 5 Can.
communication)
MONTANA CT 4-rr-____
w "OC WUU5644
•
M
D 378
H 839
c o p .2
H o w e ll, F r a n c is L .
H y d r o g e n b o n d in g i n
l i t h i u m h y d r a z in iu m
s u lfa te
^AMK ANP AOQRgga
3 ;a r
.
£ -
,*
ir
3)371
/V /J ?