
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00099521/00001
Material Information
 Title:
 Nuclear resonance of O17 in liquid and solid carbon monoxide /
 Creator:
 Li, Funming, 1949
 Publication Date:
 1979
 Copyright Date:
 1979
 Language:
 English
 Physical Description:
 ix, 90 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Liquids ( jstor )
Magnetic fields ( jstor ) Magnetism ( jstor ) Molecular interactions ( jstor ) Molecules ( jstor ) Nuclear interactions ( jstor ) Quadrupoles ( jstor ) Temperature dependence ( jstor ) Tensors ( jstor ) Transition temperature ( jstor ) Carbon monoxide  Molecular rotation ( lcsh ) Dissertations, Academic  Physics  UF Oxygen ( lcsh ) Physics thesis Ph. D
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 8789.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Funming Li.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 023357607 ( AlephBibNum )
06637177 ( OCLC ) AAL3348 ( NOTIS )

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Full Text 
NUCLEAR RESO!LhANCE OF 0'" IN LIO!ID
AND SOLID CARBON riONOXIDF
BY
FUNNMING LI
A DTSSERTATION; PRESENTED O T:E GRADUATE COUNCiL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT tO THE REOUIPEi'.ETS
FOR THE [EGRE' E OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
To my parents
ACKNOWLEDGEMENTS
The author is greatly indebted to numbercus persons who have helped
to make his work in the Physics Department of the University of Florida
a rewarding experience. The excellent research facilities and the atmo
sphere for the stimulation of scientific interest were much appreciated.
Special thanks go to the late Dr. Thomas A. Scott, who was the
initial chairman of the author's supervisory committee and who suggested
the problem, providing essential support, guidance and assistance through
out the major phases of this research.
The author would also like to express his appreciation to the new
chairman, Dr. James R. Brookeman, for his patient assistance in preparing
this dissertation and for helpfiu!' discussions concerning the related
behavior of carbon monoxide and nitrogen. His gratefulness is also ex
tended to the other members of the supervisory committee, Drs. E. Dwight
Adams, Arthur A. Broyles,F. Eugene Dunnam and Charles P. Luehr for their
guidance and criticism throughout the research work. Thanks also go to
Mr. Paul C. Canepa for his ingenious technical aid ir helping to perform
these experiments.
The author wishes to acknowledge helpful discussions with Professor
A. Pigamonti during the year of his visit. His genuine interest in science
and voracious appetite for understanding the physics of matter have deeply
influenced this author. He also wishes to thank Professor E. Raymond Andrew,
whose critical reading and comrentiig on.this manuscript is very much
appreciated. The research presented in this dissertation was supported by
National Science Foundation Grant Number DMR 770S658.
1ii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS.................................. ............ ....... i
LIST OF FIGURES......... ...................................... vi
ABSTRACT............... .............................................viii
CHAPTER I INTRODUCTION........ ...... ............................ 1
1. 1 Physical Properties of CO.................................
1. 1. 1 Crystal Structure of CO........................... 1
1. 1. 2 Phase Diagram of CO............................... 3
1. 1. 3 Specific Heat and Other Thermodynamic
Properties of CO........... ............ ............ 3
1. 2 Nuclear Magnetic Resonance ............................... 8
1. 3 Nuclear Quadrupole Resonance.............................. 10
CHAPTER I! EXPERIMENTAL PROCEDURE................................ 14
2. 1 The Sample.............................................. . 14
2. 2 The Cryostat and Temperature Measurements................. 14
2. 3 The Electromagnet............................. ........... 17
2. 4 The Scectr, . r ................. ............... 1.
CHAPTER III LIQUID PHASE.......................................... 21
3. 1 NMR SpinLattice Relaxation .............................. 21
3. 2 Intramolecular Relaxation Mechanisms...................... 23
3. 3 Intermolecular Relaxation Mechanisms...................... 26
3. 4 Temperature Dependence of SpinLattice Relaxation Time.... 27
3. 5 Discussion..................... .......................... 29
Page
CHAPTER IV SOLID 6FHASE .... .......................... .. ..... 32
4.. 1 Quadrupole Perturbed N!,i Spectra ......................... 32
4. 2 TI Measurererts in aC ................................ 40
CHAPTER V SOLID aPhASE ....................................... 44
5. 1 Molecular Properties of aCO.............................. 44
5. 2 Nuclear Quadrupole Resonance in aCO...................... 44
5. 3 NQR Linewiidths and Molecular Motions...................... 47
5. 4 Discussion............................................. 55
APPEINIX A QUADRUPOLE RELAXATION THROUGH MOLECULAR TUMBLING
MOTIONS IN LIQUID CO.................................... 58
APPENDIX B NMR SPINLATTICE RELAXATION THROUGH THE QUADRUPOLE
INTERACTION IN SOLID aCO................................ 71
APPENDIX C LINEWIDTHS ANU MOLECULAR MOTIONS IN aCO ............... 79
LIST OF REFERENCES.................................................. 87
BIOGRAPHICAL SKETCH .............................. ................ 90
LIST OF FIGURES
Figure Page
11 Crystal structure of aCO. The molecules are aligned
parallel to the body diagonals ................................ 2
12 Crystal structure of 6CO. Open circles representing the cage
structure denote hcp positions of molecules. Only the center
molecule is illustrated which processes around the crystal C
axis with angle 6a near the magic angle........... ............ 4
13 Phase diagram of CO in the PT plane, The a3 transition
temperature at equilibrium vapor pressure is 61.6 K. The
triple point is at 68.16 K..................................... 5
14 Heat capacity of CO .......................................... 6
21 Sample cell and temperaturecontrolled cryostat................ 15
22 Block diagram of the pulsed NMR/NQR spectrometer............... 18
31 Temparaturc dependence of the spinlattice relaxation time T
of 0'7 in liquid and in SCO................................ 28
32 Temperature dependence of the autocorrelation time for tumbling
rotational motion in liquid CO and for precessional motion in
SCO.......................................... ............... 30
41 Definition of angular variables used in the text to discuss the
NMR rotation patterns in CCO. The molecule is described
classically as processing rapidly about the crystal C axis..... 34
42 The i!;;R perturbeo spectia or CO at difference ( ) angles,
where
43 Rotation pattern of the separations of the satellites of NMR
perturbed spectra in BCO as a function of ([ )........... 38
44 Effective coupling constant as a function of temperature in
BCO.......................................... ............... 41
51 Temperature dependence of the pure quadrupole resonance of 017
in aCO...................................................... 4
52 Temperature behavior of the 017 NQR linewidths in aCO. The
insertion shows some effective Tl measurements ............... 48
Figure Page
53 Three dimensional represencatior of h. 01'' NQR spectra in
aCO. For clarity the linewidth has beer enlarged by a
factor of 2................................................. 49
54 The reduced temperature behavior of the reduced NQR frequency,. 55
A1 Quadrupole induced transition probabilities among the Zeemanz
energy states for spin I = 5/2.............................. 59
A2 The Euler angles. First, a rotation of angle a about thle Z
axis. Secondly, a rotation of angle E about the Y' axis.
Finally, a rotation of angle y abouc the 2" xis.............. 62
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NUCLEAR RESONANCE OF 017 IN LIQUID
AND SOLID CARBON MONOXIDE
By
Funming Li
December 1979
Chairman: James R. Brookeman
Major Department: Physics
A study of molecular motions in various phases of carbon monoxide
has been performed through nuclear magnetic resonance (NMR) spinlattice
relaxation time T1 measurements in the liquid phase, T1 and quadrupole
perturbed NMR spectra in the Sphase and nuclear quadrupole resonance
(NQR) linewidth and Tl measurements in the aphase in a sample enriched
24.5% with 017
The tumbling rotational notion in the licuid phase and its temnrjr
ature dependence are obtained. An activation enthalpy of 198 3 ca'/moiie
is derived frmn the Tl measurements. Infori:ation on features of the
processional motion about the crystal C axis in the Bphase is also
obtained. The effective quadruple coupling constants are compared with
the static quadruple coupling constant obtained in the NQR measurement
of the cphase. The former is shown to be three orders of magnitude
smaller by precessional motion averaging. The temperature dependence
of the effective quadruple constant is also derived. It is shown that,
viii
as the temperature increases. tne average ie/iaticn of the processional
angles fr'om the magic angle increase from 3.23 =0.06' at 61.9 K to
3.36 0.06 near the triple point. T. measurements in the solid aphase
were also obtained.
In the aphase the quadruple interaction is sufficiently large enough
to perform a pure quadruple resonance experiment. The measurements in the
aphase provide the magnitude of the intramolecular quadrupole interaction.
Furthermore, the NQR 017 'inewidths exhibit a peculiar and unexpected be
havior with a maximuir, around 37 K and a divergence for T 28 K. The
effective T measurements in the temperature range frcm 49 55 K support
the assumption that the linewidth is determined by the molecular motions in
the aphase. This behavior has been analyzed on the basis of a model of
molecular reorientatiorial processes. These processes could account for the
residual entropy previously observed in the crystal. Comparison of the data
with the tneoretical picture allows one to derive information on the details
of the molecular reorientations and their temperature dependence.
CHAPTER I
INTRODUCTION
'. 1 Physical Properties of CO
Carbon monoxide is one of the most interesting diatomic molecular
crystals and has been studied extensively [112]. Besides its structural
simplicity and its similarity to other diatomic molecular crystals
[6,9,13,14], CO exhibits some peculiar behavior which is probably related
to the electric dipoledipole interaction of the molecules [5,15]. Since
the observation of residual entropy [4], interest has been focused on the
possibility of an ordering transition. Gill and Morrison extended the
studies to lower temperatures and found no transition occurring [10].
However, an anomaly in heat capacity near 18 K has been reported [16].
A theoretical calculation of the rate of ordering based on the model of a
twofold barrier for endoverend rotation in solid aCO found the rate to
be unobservably slow [17].
1. 1. 1 Crystal Structure of CO
Thee are two solid phases, a nd ,'. of CO. At low temperatures.
Xray diffraction studies showed that the crystal structure of the
aphase is primitive cubic with a basis of four molecules per unit cell
and that the molecules are aligned in the body diagonal directions as
shown in Figure 11. However, two space groups Pa3 (TE) [18] and
P213 (T ) [1] are compatible with this arrangement. The P213 space group
differs from Pa3 in that each molecular center is displaced by a small
amount 6 along the direction of the molecular axis. The problem concern
ing which is the correct structure for aCO is still in dispute [19].
,ceh
cI t./
74 >'.
0 K)5
(;i u
Figure 11. Crystal structure of aCO. The molecules are aligned
parallel to the body diagonals.
CO
In the Bphase wLich occurs above 61.6 K, the crystal structure
determined by Vegard [2] was that of BN2, and subsequent Xray diffrac
tion studies of BN2 [201 deduced this space group as P63/mc(D h). In
this regard one considers the crystal structure of BCO, the same struc
ture as 8N2, to be hexagonal close packed. It was found that Xray form
factors were equally consistent with a dynamic model where the molecules
process about the Caxis at an angle e' as shown in Figure 12, or with
a static disordered model in which the molecules are randomly distributed
among the 24 general positions of space group P63/mmc [20].
1. 1. 2 Phase Diaaram of CO
The phase diagram of CO has been determined by Fukushima, Gibson,
and Scott up to a pressure of 1.75 Kbar using the change in nuclear spin
spin relaxation time T2 of C13 as an indicator of the transition [21].
The phase diagram is shown in Figure 13. The a,B phase transition is at
61.6 K at equilibrium vapor pressure, and the triple point is at 68.15 K.
dP AS
Applying the ClausiusClapeyron equation, dT AV to the phase
transition boundaries the volume increments at zero pressure are deduced
to be AVm = 2.50 cm3/mole and AV = 0.92 cmr/mole [21]. While the change
of the volume in N2 is AVm = 2.5 cm3/mole [22], which is comparable to CO.
'2 1 .s n i ,) a r:_c ':, O. l hl c23,2'].
The compression experiments found no new phase at higher pressures [25].
This is unlike N2 where a new phase was found at higher pressures with a
tetragonal structure [26].
1. 1. 3 Specific Heat and Other Thermodynamic Properties of CO
The heat capacity of CO has been measured [3,4,10] and is shown in
Figure 14. The analysis of the results is compared with the entropy cal
culated from the band spectrum of CO in the gas phase. The discrepancy
C
r I;~I
7'c "% i \ \
5'( urlr; *' '.. / s
".\ nAI \A .,t" .L r
/"/ I E ;'6 < I 
//rI r 'g i
7 \ \ \ t .:;v'k f\ "' ^'
/ <, d4 / I SI ."
2 IK /
a r c d p p of o u O yt
c\ente i s 1 t w pees aon the
cyt a tv t'
/r
\ Y '
v^^ iy ,,,,
f I
't\,~ (
~ t, I /
I 'K> j
Figure 12. Crystal structure of 8CO. Open circles representing the
cage structure denote hcp positions of molecules. Only the
center molecule is illustrated which precesse; around the
crystal C axis with angle e' near the magic angle.
i.G
(3
1.0
CL
I~lr ;` ~ ~ r
L /
[.aI
LIQUID
26) Co
O6 100
Figure 13. Phase diagram of CO in the PT plane. The aB transition
temperature at equilibrium vapor pressure is 61.6 K. The
triple point is at 68.15 K. (Data are from Ref. [21]).
I a I I *
,1+ i
14 L L
+
5 h A 1
i
Iy4
*
i b
S o
0 .I0 20 0 "40. 50 70 80
TEMPEATURi' K
Figure 14. Heat capacity of CO. (The "+" data are from Clusius [3]. the "x'; from Clayton and
Giauque [4], and "o" from Gill and Morrison [10].)
was found to be approximately equal to R In 2. This was described as
frozen in endforend disorder [4]. Melhuish and Scott [15] calculated
the energy for the oriented and disoriented lattice and estimated the
Curie temperature for the orderdisorder transition. They found Tc to be
,5 K for CO. However, heat capacity measurements on solid aCO have been
carried to 2.5 K by Gill and Morrison with no indication of the occurrence
of the transition [10]. The adiabatic compressibility xs may be calculated
from the sound velocities and density using the equation
xs [p (V )1 (1 )
where V, and Vt are the longitudinal and transverse sound velocities, and
p is the density.
Voitovich et al. [27] measured Ve and Vt and thereby xs was obtained.
While from the thermodynamic equations one has
XT = Xs + ,2 TV/Cp (12)
and
XT/Xs = C /CV (13)
where XT is the ;sotiher'al c:omiessibility, s is the thermal expansion,
V is the molar volume and C and CV are the specific heats at constant
pressure and volume respectively. Krupsii et al. measured the thermal
expansion of aCO and applied the above equations using xs from [27] and
C from [4,10] to give a complete table for lattice parameters, molar
volume, volume expansion, specific heat at constant volume, and isothermal
compressibility [18].
1. 2 Nuclear Magnet o Pesonance
Consider an isolated nucleus in a steady magnetic field io and sup
pose that the nuclear spin number I is greater than zero, so that the
nucleus possesses a magnetic morn;nt. Quantum mechanically the angular
momentum of the nucleus is quantized along the applied field direction
with the magnetic number m taking values of I, I1,..., (11), (I).
The Hamiltonian can be written as
= = ( (14)
where u is the nuclear magnetic moment and Ho is an externally applied
field. The magnetic moment is related to the nuclear spin angular momen
tum by
vtl=t (15)
L h
where y is the gyrcmagnetic ratio of the nucleus, h equals where h is
Planck's constant. The components of the magnetic moment are thus given
by the (21+1) value of v, p(I1)/1,.... y(I1)/I, u. The energy levels
of the nuclear magnet in the magnetic field oi are therefore equally
spaced taking values of mpHo!I, where m = I, (11),..., (11), I.
From the classical point of view the nucleus may be regarded as a magnet
dipole t processing about the direction of the applied field io. The rate
of precession is given by the well known Larmor angular frequency
mL = yHo (16)
If an additional small magnetic field ti is applied at right angles to o,
the dipole 1 will experience a torque of p x 1 tending to increase the
angle a between a1 and no. If AI is made to rotate about io as axis, with
the sane frequency as the Larmor frequency ,L' the the angle 0 will
steadily increase. For rotation Frequencies differing from wo tne coup
ling between ? arid 1i will merely produce small perturbation of the pre
cessional motion with no net affect [28]. Hence the application of a
radio frequency field HI at a frequency mL will cause energy to be ab
sorbed by the nucleus. The experimental spectrum will be an infinite
sharp line at the Larmor frequency mL. However, the effect of the
neighboring nuclei will produce a local magnetic field at each nuclear
site. This will cause the line to be broadened. However, this broadening
is also effected by the distances and relative directions between the
neighboring nuclei as well as their motions. Nuclear Magnetic Resonance
(NMR) measurements thus provide useful information about the molecular
structure and the molecular dynamics in the solid and liquid phases.
For example, in a single crystal from the linewidth in the solid, it is
possible to calculate the second moment M2. Thus one will have the
information about the separations and directions between nuclei in a
unit cell. While for a powder sample it is not possible to determine
the directional relation between nuclei, it is still possible to find
out their separations. Moreover, a dramatic change of M2 may indicate
the onset of the rotation of the molecules in the solid and further
reduction of linewidth could be a clue to the diffusion of the molecules
in the solid. Other examples include chemical shifts in the liquid phase
for identification and analysis of the substance, Knight shifts for
studying information about the conducting electrons near the Fermi
surface, observation of chemical reactions processes from the growth of
the signal strength [28], and finally the molecular motions through the
spinlattice relation time T1 measurements as one will see later in this
experimental study of CO.
1. 3 Nuclear Ouadrupole Resonance
The interaction of a charge distribution with an electrical potential
due to the external sources is given by
E = fp()V(r)dr (17)
where V(r) is the electrostatic potential. Regarding the nucleus as a
charge distribution, and expanding V(r) about the origin of the center of
mass of the nucleus, one obtains
E = V(O)p()d3r + V p)r+ 1 s x x gpp()d r (18)
where x a = 1,2,3, stands for x,y,z, respectively and
V v 1= 0 V vV = 0
a ax W' ax ax
a a
The first term on the right hand side of (18) is the electrostatic energy
of the nucleus taking the nucleus as a point charge. The second term in
volves the electric dipole moment of the nucleus. Since the wave function
of the nucleus has definite parity, p(r) = p(r) the electric dipole
moment of the nucleus vanishes [29]. One is only interested in the third
term, the electric quadrupole interaction now. By introducing
a j= (3x a 6a r2)p ()d3r
and using the fact that
S 2V 
o0
Sax ax
a a a
one has for the quadrupole term
SEQ VaBs (19)
In quantum mechanics, one considers p as an operator
( t = e Kprotonis r (110)
and defines the operator Q p)as
Q (op) = f(3xax r2) p(op)()d3r
=P e r2)
(111)
S K=protons (3aKX3K a rK
From (19) one writes the Hamiltonian
H I V n(op) (112)
Q 6CaB as act
The WignerEckart theorem states that the set of matrix elements of an
irreducible tensor operator AP differs from any other irreducible tensor
B only by a constant factor [30]. The tensor operator Qn(P) then is
CfB
related to the tensor operator Q(I) as
n(op) = e
Q = e Kproton (3xoKx q 6 r2)
= CQ(I)
I I f II
= C (3 C_ F I 12) (113)
Since Q can be written as combinations of irreducible tensors like
Q(I). Defining eQ, quadropole moment of nucleus, as
eQ = (114)
K=protons K K I
and operating on the matrix element nim' in (113) for m = I, m' = I
where I is the nuclear spin, it follows that
(115)
r:
In order for C to have a physical meaning it follows that one needs I>
to have a quadrupole interaction. From (112), (113), and (115) it
follows that
HQ = v [ I + I ) 12] (116)
In the principal axes of the field gradient, one defines eq = VZZ and the
asymmetry parameter
VXX Vyy
VZZ
where VXX, Vyy, and VZZ are the principal field gradients. One then has
from (116)
H = e2 q [31 2 + n ( I_)] (117)
Q 41(2II X Ii)]
One can also write (116) in the laboratory frame as
Q = 4 1I[ V (3z2_2)+V (I z I )+V_( + +)+V+2( ) 2 +)2]
(118)
where
V 3 V
V 6 zz
V, = V2 i
V2 (Vzz yy + xy
and I+ Ix + iI I = Ix iy are the raising and lowering angular
momentum operators respectively.
The quadrupole interaction is therefore due to the interaction of the
nuclear quadrupole moment and the electric field gradient produced at the
nuclear site by the external electric sources. Intermolecular electric
sources usually provide only a small amount compared with those of intra
molecular origin. Nuclear quadrupole resonance (!'QR) therefore provides
a sensitive method to study the molecular motions as wall as the elec
tronic structure of the molecules. Comparison of the quadrupole coupling
constant (QCC) of the free molecule obtained from microwave studies and
that from NQR measurements in the solid phase provides information about
the alteration of the electronic distribution in the solid state. NQR
can also be used to study crystal structures, phase transitions, and im
purity effects [31]. Nuclear quadruple interactions usually provide a
very effective mechanism for spinlattice relaxation in the liquid phase.
Thus it permits a direct determination of certain correlation times char
acterizing the molecular dynamics. The study of the linewidth and reso
nance frequency also elucidates the molecular motions in the solid phase.
CHAPTER II
EXPERIMENTAL PROCEDURE
2. 1 The Sample
The predominant naturally occurring isotopes of carbon and oxygen do
not possess nuclear moments. Therefore an NMR study of CO has to resort
to enriched isotopes. C13 provides one of the possibilities [21,32,33].
For a comprehensive study of spinlattice relaxation and quadrupole reso
nance in the low temperature phase of CO, 017 is the best choice. 07
5
with nuclear spin.I = [34] has a nuclear quadrupole moment which,
interacting with the molecular electric environment, provides the domi
nant mechanism of spinlattice relaxation. From nuclear quadrupole reso
nance experiments which can only be performed with nuclear spin 1>1, the
quadrupole coupling constant of 017 in CO can be deduced. Therefore a
great deal of information about the CO molecules can be learned.
One liter of gaseous CO enriched to 24.5% in 017 was purchased from
Prochem U. S. Service, Inc., New Jersey. The sample was claimed to have
99.9 atom % of C1
2. 2 The Cryostat and Temperature Neasurements
The construction of the cryostat is shown in Figure 21. The con
densed sample was held in a KelF cavity of 0.43 irnch in diameter and
0.78 inch long. Surrounding the sample chamber was an rf coil wound from
30 turns of equally spaced #26 cotton covered copper wire. The induc
tance of the sample coil was approximately 10 pHi which was used for NMR
measurements operating at 4.098 iHz. Another sample coil made of n100
NN2 bath
Np orHe baih
VOcuUm jacket
radiation sicId shcc1
vacuum orheater
exchonger
brcss con
platjnum fherinonlejro
heater wire
samr"ple line
copper block
s lO F so !c h iu b r
sarnpfe churnter
*i1e gas heater
cxchangor
Figure 21. Sample cell and temperaturecontrolled cryostat,
turns of #36 enamel coated copper wire with an inductance of 'i100 1H was
used in the NQR measurements at 'u1.1 MHz. The sample coil was connected
to the spectrometer through the small stainless steel sample line. He
gas was admitted into the copper block on which a platinum resistance
thermometer was mounted. The He gas served as a heat exchanger between
the sample and the copper block. The platinum thermometer calibrated by
the National Bureau of Standards was used for two purposes. First, the
conventional temperature measurement was achieved by connecting four leads
to a Leeds and Northrup Mueller bridge. Secondly, the output of the gal
vanometer was phase shifted 180 then fed back to the power supply which
controlled the current to the heater. With the Mueller bridge the tem
perature can be measured accurately to 0.010 K; however, this experiment,
due to the uncertainty in the different readings of the N! and R mode in
the Mueller bridge (because of using feedback temperature control) a
0.050 K uncertainty should be assigned. Between the brass can and copper
block, He gas could be introduced to serve as a heat exchanger in case
temperature equilibrium of the sample with the liquid N2 or liquid He was
desired. To achieve a sample temperature higher than the surrounding
bath, this space was evacuated to 104 torr or less. A thin brass sheet
radiation shield held in place with smrll KelF pieces reduced addcticnal
radiation heat loss. Temperatures between 50 K and 77 K were achieved
by pumping the N2 bath and above 77 K the heater was applied. For tem
peratures below 50 K, liquid He was used as a cold bath with the heater
on to maintain the desired temperature. Before the sample was introduced
to the sample chamber, this space had been evacuated to 4 x 106 torr.
To avoid solidified sample blocking the line before it reached to the
sample chamber, a vacuum jacket was situated around the sample line.
2. 3 The Electromagnet
The electromagnet used in the rNR measurements was a Varian 40123B
rotatable electromagnet with a 12 inch diameter pole face and a 3.5 inch
gap. The magnetic field was regulated by a fieldial regulated system
which utilized a Hall effect probe inside the magnetic field. The max
imum magnetic field produced was around 9 KG. Taking long time stability
and maximum sensitivity of the NMR signals into consideration, a 7.1 KG
field strength seemed to be the best choice. This in turn produced a
4.098 MHz 017 NMR signal in liquid CO. The rotation angle of the magnet
could be adjusted to within 1 of accuracy. The inhomogeneity of the
magnetic field over the sample volume was determined in the liquid phase
to be ,0.1 gauss which gave a T2 of about 3.5 ms. After long hours of
operation (n 6 hrs.) the fluctuation of the magnetic field was found to
be 3 x 104/hr. or less.
2. 4 The Spectrometer
The pulse spectrometer used in this experiment is depicted in
Figure 22. This setup was used for the NMR experiments as well as the
NQR measurements with only a few modifications of the receiver pre
amplifier. The hicg stbiit: y General Radio 1061 frequency synthesizer
was the main rf source. Since it is a heterodyne system, the frequency
was set to 10 MHz higher than the resonance frequency to utilize a 10 MHz
amplifier and a phase sensitive detector. This signal when mixed with
the gated 10 MHz reference frequency was amplified to approximately 2V
peak to peak rf, before it entered the transmitter. The transmitter is
adequate to supply high power pulsed output and low power continuous
output. The magnetic field produced by the rf pulse was calculated to be
s30 gauss, which is large compared with the linewidths of the spectra being
Fh;i Tlk 1i072 Si"nc Averager
DI r
L Lfionjin 
I_ ,! o_ A ^ in
/4 \ cable I
02 J, r
D_ C / Tslype ASR
Sr C dDg f t3 D4
Figure 22. Block diagram of the pulsed NMR/NQR spectrometer.
studied. The same pulse which gated the 10 FiHz frequency controlled and
gated the final stage transmitter also.
In a pulse nuclear resonance experiment, the induction signal is
typically obscured for several microseconds after the transmitter pulse
by the ringing of the input turned circuit. This ringing can be reduced
by a weak coupling to the nonlinear input [35]. In this experiment, a
photoFET Q switch similar to the one described by Conradi [36] was used.
During the time the high power rf pulse was on, the D D2 crossdiode set
conducted strongly. The D3D4 diodeset was also conducting to protect
the receiver preamplifier and the photoFET Q switch. When the cross
diode set DsD4 is conducting, point A of the cable appears as a high
impedance. While the transmitter pulse is off, D3D4 ceases to conduct
and point B is a high impedance.
When the rf level reaches 0.5 volt, the crossdiodes D102 turn off.
This isolates the transmitter from the L. C. tuned circuit. Unfortu
nately, in practice the L. C. tuned circuit and the transmitter are still
weakly coupled by the capacitor C1 across diodes DD2. This coupling
ties the voltage across the sample coil to the slow ringing down of the
transmitter final stage [37], so the recovery time is prolonqeK. The
negative bias FET was adjusted just into the c&toff region. A pulse
from the pulse generator turned on a LED diode which provides light to
turn on the Q switch. The sourcedrain path appears as a low resistance
(02000) which provides a low impedance path to ground for the energy of
the ringing. The Q value of the input circuit hence was reduced during
the time the Q quenching pulse was on, causing it to ring down quicker.
Just.as the ringing decreases to the order of the free induction signal
(smV), the Q quenching pulse was turned off to restore a high Q value
20
in the receiver preamplifier for detecting the free induction signal. The
amplified signal was then mixed with the rf frequency (v + 10 MHz) split
from the same source. The combined signal was then phase sensitive de
tected by mixing with the 10 MHz reference signal.
The output signal was observed on an oscilloscope and also fed to
the Biomation 802 transient recorder which captured the fast signal for
slower transfer to the FabriTek 1072 signal average. A PDP 8/E mini
computer was interfaced with the FabriTek 1072 to perform the Fourier
transformation of the signal from the time domain (free induction decay)
into the frequency domain (spectrum). The result was then plotted on an
XY recorder.
CHAPTER III
LIQUID PHASE
3. 1 NMR SpinLattice Relaxation
In the liquid phase, due to the fast molecular tumbling motions, the
quadruple interaction is averaged to zero. The experimental NIMR signal
will, therefore, be a single line. The linewidths are mainly the result
of the inhomogeneity of the applied magnetic field. The random motions of
the molecules also provide the mechanisms for the nuclear spin system to
relax when perturbed from its equilibrium state. T measurements thus
can be used to study the molecular motions. In the liquid, the energy
differences between the Zeeman levels are all equal. The fast spin
exchange mechanism thus maintains a Boltzmann distribution of the nuclear
populations among the energy levels [38], i.e. a spin temperature can be
defined and the relaxation to the lattice temperature is governed by a
single exponential decay. In order to describe the macroscopic relaxation
time T1 through the microscopic molecular motions, one considers the time
dependent Hamiltonian : (t) caused by D m:olecular motions. This 1; (t)
represents a small perturbation and induces transitions between eigenstates
of the Zeeman Hamiltonian. The correlation function of H, (t) is then
introduced which expresses a correlation between the two different config
urations of a nuclear environment at two different times. The rate
equations among the energy levels are then considered. This connects the
relations between the experimental relaxation of the total magnetization
and the correlation time of the tumbling notions of molecules.
Consider H, (t) a small perturbation among the eigenstates
Ia >,...,I > etc. with eigenenergies a,6 etc. of a spin system S. The
transition probability per unit time for a transition from state Is > to
state a >, assuming the system S at time zero is in state i >, time
dependent perturbation theory gives [6]
= t <1 H1 (t)Ia >< al H (t')> e i t't)dt + C. C. (31)
where = Taking the average W over a statistical ensemble, and
introducing the correlation function G 6 as
a 3 (tt) = < BRl(t)I a >< aH(t')IB > (32)
One uses the properties that the random function < alH (t)I8 > is station
ary, i.e. GaG depends on t and t' only through the difference t t" = r.
It follows that
W 1 6t G (T) e o Wt dz + C. C.
I ft G B (r) eiWaSB dT (33)
Since one usually considers a time t >> it turns out that the limits
Wa3
of the integration in (33) can be replaced by m and one has
WB 1 GB (T) eiWaST (34)
In most cases H1 (t) can be expressed as a combination of two irre
ducible tensors of rank k. Let
k
H1 (t) = A P (I) B (t) (35)
where A is the spin part of the Hamiltonian arnd B is that part of the
spatial coordinates which is time dependent. It follows that (34) can
be written as
Wa 1 = < =A(PI j > ( 0) eaB1 dr (36)
pk k
where
g(p)() = (P)(t) *(P)(tr)
is the correlation function of B(p (t).
There are various possible mechanisms of relaxation, (1) the intra
molecular quadrupole interaction, (2) the intramolecular spinrotation
interaction, (3) the intramolecular anisotropic chemical shift interaction,
(4) the intermolecular magnetic dipoledipole interaction and (5) the
intermolecular electric quadrupoledipole interaction. However, an esti
mate of the magnitudes shows that the spinlattice relaxation is almost
completely dominated by the intramolecular quadrupole interaction.
3. 2 Intramolecular Relaxation Mechanisms
The interaction between the nuclear moments and the electrons in the
CO molecule drives the transition among the Zeeman levels of 0!7 nuclei.
In principle there are three relaxation mechanisms which are of intra
molecular origin. They are (1) the intramolecular quadrupole interaction,
(2) the spin rotation interaction and (3) the anisotropic chemical shift
interaction.
Consider (1) the intramolecular quadruple interaction. In Appendix
A it is shown that
S 3_ (37)
T !25 2 (37)
where e is the static quadrupole coupling constant, which one has
determined in the NQR measurement of CO. in the aphase to be 4.20 MHz
and [2 is the autocorrelation time of the molecular tumbling motions. If
one adopts the value of T2 from Raman anisotropic scattering [39] to be
1 1 1
2 3 x 10' sec. at a temperature of 77 K one has 1 = 4.17 sec
1
while the experimental T1 at 77 K is 220 ms. This gives = 4.54 sec.
1
In (2) the spinrotation interaction, the time fluctuation of the
interactions of the nuclear spins with the angular momentum of the mole
cule provides a relaxation mechanism for the nuclear spins. Assuminga
Langevin equation for the angular momentum and a rotational diffusion of
Oebye type, Hubbard obtained [40]
Ss.r. 2C ) 21KTT + 2(Ci 21KT + 11 (38)
where C, and C,, are the components of the spinrotation tensor perpendic
ular and parallel to the axis directed from the center of mass of the
molecule to the resonant nucleus. I is the moment of inertia about a line
through the center of mass and perpendicular to the molecular axis. I is
related to the angular momentum of the molecule I by It = LJ. T, is the
correlation time for the angular rromentum of the molecule and T2 is the
correlation time for molecular reorientation.
Liquids near their melting points are often described by a reorien
tation process called the "diffusion limit" in which the molecules turn
through a very small angle between collisions and T, and T are related
by the following equation [41]:
I
ff2 =6KT
(39)
For linear molecules one has 3 C2 = 2 C + C = 2 C2 Assuming the
diffusion limit for the temperature of interest one has from (38) and
(39)
C2
II eff 12
3h2 (310)
1 s.r. 2
Using the value Ceff = 23.1 KHz [42] and T2 from Raman experiments [39]
one finds l s.r = 3.6 x 105 sec] which is negligibly small compared
with (37).
Consider (3) the anisotropic chemical shift. The interactions of
the external D.C. magnetic field with the electrons around the resonant
nucleus in a molecule produces a magnetic field at the nuclear site. The
anisotropy of this coupling under the molecular motions may introduce
transitions between energy levels. In the fast motion regime one has [43]
a] .c. ^2 (o, 0)2 (311)
a.c. l 15 0
where Y is the gyromagnetic ratio of nucleus 017. H is the applied D.C.
magnetic field. r,, and a, are the parallel and perpendicular parts of the
chemical shift tensor in the principal reference frame respectively. T2
is the correlation time for molecular tumbling motions described before.
Using the value cited by Applemari and Dailey [44] that (o, o1) for
the 0 nucleus in CO is 4.6 x 100 and 7.i KG for the external magnetic
field H one calculates
f1 = 5.6 x 106 sec"
'IJ a.c
One notes from the small value that the relaxation due to the anisotropic
chemical shift interaction can be neglected. By neglecting the spin
rotation and the anisotropic chemical shift the calculated value of
4.17 sec for the intramolecular quadrlupole interaction is in good agree
ment with the experimental value of 4.54 sec"
3. 3 Intermolecular Relaxation Mechanisms
The spinlattice relaxation may also arise from the interactions of
the 017 nuclei with the intermolecular fields. There are two mechanisms
which, in principle, could induce transitions in the absence of a rf
field. They are (1) the intermolecular magnetic dipoledipole interaction
and (2) the intermolecular electric quadrupoledipole interaction.
Consider first (1) the magnetic dipoledipole interaction. Following
Bloembergen, Purcell and Pound [45] by assuming spherical molecules and a
1
Debye model of diffusion, one would have a contribution from inter
1
molecular magnetic dipoledipole interaction given by the following equa
tion:
rl' 3_ y "h2No
T i.d. I al (312)
where 2a is the closest approach of the neighboring molecules, D is the
diffusion coefficient of the liquid and No is the number of molecules
per cm3 Using D as 2.9 x 105 cmf/sec [4G] and the density of liquid
CO as 0.803 g/cm3, which gives N = 1.72 x 102l/cmr and 2a = 3,87 x 10"l cm,
one obtains
([ i.d v 4.5 x 106 sec1
Now consider (2) the intermolecular electric quadrupoledipole inter
action. It is well known that the CO molecule possesses a dipole moment
of O.11 x 10 esu [5]. The electric dipoles of neighboring CO molecules
thus produce a field gradient at the 017 nucleus site. This is similar to
the intramolecular quadrupole interaction except now the field gradient
is due to remote electric so.rces. Assuming again a Debye type diffusion
for the liquid following the calculations of Bonera and Rigamonti [47],
one has
29 e2Q2r,2]J
1 Q e 0(313)
S i.q. 1 2 (31Da)
where a, No, and D have the same definitions as discussed in the magnetic
dipoledipole interaction. eQ is the quadrupole moment of nucleus 017,
andM is the dipole moment ofthe CO molecule. Taking the value of Q
measured by Stevenson and Townes [45] as 0.026 x 1024cm3, one calculates
r1 = 4.4 x 104 seci
From the above considerations one immediately concludes that the
spinlattice relaxation in liquid CO is completely dominated by the intra
molecular quadrupole interaction. This is usually true for NMR spin
lattice relaxation of most substances possessing a nuclear quadrupole
moment in the liquid phase.
3. 4 Temperature Dependence of SpinLattice Relaxation Time
Using the conventional  7 pulse sequence, one measures the
spinlattice relaxation time T1 in the liquid phase. Figure 31 shows
the experimental T1 values at different temperatures. Data were taken
during a period of several months. Due to the uncertainty in the Tl
measurements, a 10% error should be assigned to each point. In view of
the cluster of points taken at different times, it shows no impurity is
effectively absorbed in the sample during the course of experiments.
By assuming the intramolecular quadrupole interaction is the only
contribution to T1 and using the NQR measurements extrapolated to zero
temperature, one would have e2 Qg 4.20 MHz. From this one has the
following relation for liquid
following relation for liquid CO
CSO. 3.3 7r.e S 71.4 GGGS.
' i
L 
i220
00200o 00
E3 i L/. OL
_ 62.3
  r
LIQUID 
ai
BSOL!D .
E !92 CAL/MOLE .
I03/ T (* K")
Figure 31. Temperature dq;'nldence of the spinlattice relaxation time T1 of O'7in liquid
and in sCO. U!. is the activation enthalpy. The uncertainty at each
measurement is 10,.)
*. r
r '
! 20
Il
_1
__
1
I
2 =5.98 x 10 (314)
'I
Using (314) and measurements of T!, the temperature dependence of T2
is shown in Figure 32. For comparison the T2 values of N2 are shown in
this picture [49]. Assuming an activation model [50], such as
AG
2 = o e RT (315)
where AG is the difference in the Gibbs function between the initial and
activated states, R is the gas constant and T is a constant and is ex
pected to be a relatively slowly varying function of temperature and
pressure [50]. Substituting (314) inro (315) one has
AG
T, = C eRT (316)
where C is some constant. From the thermodynamic relation
AG = AH + T [G (317)
it follows that
a ln T1
H = R I (318)
AH is the activation enthalpy, a measure of the energy required to re
orient a molecule under conditions of constant pressure. One notes that
it is a reasonable approximation to assume that the pressure along the
equilibrium vapor pressure curve is constant [50].
3. 5 Discussion
Ewing [7] compared the infrared spectra in the gas and liquid phases
of CO and suggested a potential barrier of V = V0 (1 cos 2e) with
V = 120 cal/mole in liquid CO.
a *~l
o
CO4
o 0
0
LIQUID CO
o
PIr
a,
60 65 70 75 00 65 90
TEMPERATURE K
32. Temperature dependence of the autocorrelation time for
tumbling rotational motion in liquid CO and for precessional
motion in BCO. (The "o" data are for CO while the "+" data
are for N2 from Ref. [49]. The autocorrelation time in
liquid CO and N2 is shown as "'" from experimental Raman
scattering [39]).
Figure
Figure
I I _I I~J_ I
i
Amey [51] used the librational model suggested by Brot and compared
this with the farinfrared spectrum of liquid CO and calculated a barrier
of 190 cal/mole. These are compared with our experimental activation
enthalpy calculated by least square fit to be 198 3 cal/mole. Consid
ering the different types of experiments, it can be regarded as satis
factory. In fact, NMR T1 measures the barrier height directly and this
value is more fundamental than the other parameters. The low value of
the barrier potential indicates that CO molecules in the liquid phase
undergo nearly free rotation.
If one uses the experimental results for the coefficients of diffu
sion in liquid CO [46] and assumes a Debye model of viscous fluids, the
StokesEinstein equation yields
2a2 (319)
9D
from which it can be calculated that at 77 K, T = 2.8 x 1012 sec. This
may be compared with the value of T = 2.7 x 1013 sec. derived from the
T1 measurements. In liquid N2 it was found also that use of the Stokes
formula for the reorientational correlation time overestimated the value
by cne order of magnitude [49].
CHAPTER IV
SOLID sPHASE
4. 1 Quadrupole Perturbed NMR Spectra
SThe hexagonal crystal structure of sCO has already been discussed
in 1. 1. 1. However, the inability of Xray diffraction to distinguish
between a static disorder and a free precession model has led to some con
fusion in the literature [20].
The experimental data presented here are consistent with the pre
cession model. It is concluded from these experimental NMR measurements
that the effective nuclear quadruple coupling constant of 0!7 in BCO is
reduced by a factor of 103 by motional averaging. This demonstrates
that the molecular motions in BCO are characterized by a time shorter
than the reciprocal of the static quadrupole resonance frequency,
Sl0sec., and that the reorientations are not isotropic because a non
zero coupling constant exists. The Hamiltonian of the 017 nuclei in sCO
can be written as
H = z + H (41)
where HZ represents the Zeeman energy while H the quadrupole Hamiltonian,
is small compared to HZ and can be treated as a perturbation. To a first
approximation, the shift in frequency of state m> can be written as
AV = I Oi (312 12 ) (42)
m n <'m 41 VT zz ( I 2) m> (42)
where m > are the eigenstates of Zeeman energy and m is the magnetic
quantum number of the nuclear spin 0"1. Vzz is the electric field
32
gradient at the 017 nuclear site in the laboratory frame. From (42),
knowing I = for 017, one has
1 = 1 (3m2 35) (43)
m h 40 zz
In order to relate the V in the laboratory frame to the static value of
the field gradient VZZ in the molecular frame, one considers the pre
cessional motion shown in Figure 41. In the case n = 0, the second
order irreducible tensor transformation enables one to have
Vzz : (3 cos2 0" 1) VZ (44)
where VZZ is the static electric field gradient in the molecular frame,
while 0' is the instantaneous angle between the molecular axis and the
external magnetic field o One also assumes the field gradient is of
intramolecular origin. if one applies the addition theorem, (44) can
be written as
Vzz (3 cos2 e 1) (3 cos2 e 1) VZ (45)
The effective Vzz would be given by the time average of the right hand
side of (45)
Vz = (3 cos2 6' 1) (3 cos' e 1) VZZ
= 4(3 cos e . 1) (3 sin2 y cos2 1) VZZ (46)
It turns out from (43) and (46) that
v = ie (3 cos2 e 1) (3 sin2 y cos2 t 1) (3m2 (47)
m 1h )60 4
Il ,
lN
Definition of angular variables used in the text to discuss
the N!R rotation patterns in sCO. The molecule is
described classically as processing rapidly about the
crystal C axis. h1 is the field produced by the rf coil.
o is tihe externally applied magnetic field. is the
angle between Ho and the projection of the C axis on the
plane perpendicular to Tl.
Figure 41.
and for various m one can immediately write
Av~ = e2Q (3 cos2 e 1) (3 sin2 y cos2 $ 1) (48)
3 = (3 cos2 e 1) (3 sin2 y cos2 0 1) (481)
E73 80h
Av~ 20h (3 cos2 B' 1) (3 sin' y cos2 6 1) (48)
The quadrupole splitting is of the order of KHz while the static
quadrupole coupling constant is of the order of MHz. This small nonzero
quadrupole frequency might result from a value of 8' close to the magic
angle e where 3 cos26 = 1. One supposes e6 is characterized by a
distribution function f(e), where c = e' 0e, which is an even function
of e and normalized. It follows that
3 cos2 e 1 = f2 f() [3 COS2 (90 + e) 1] de
J..o
2
1 T
= 7 f(e) [3 cos2 (eo + E)l + 3 cos2 (e c) 1] de
r
= f(E) e2de
= E2 (49)
where e is the root mean square of the angle E, i.e. 0" derivated from
magic angle 0 One defines
T AV+3 AVi5 = () (410)
3 2
2 2
And combining (48) (49), (410) and (410'), it follows that
(2) = 2 (3 sin' y cos2 C 1) (411)
801i o
and
v ( h Qo (3 sin2 y cos2 1) (411)
From the experimental perturbed spectra it is obvious that in several
cases a large portion of the sample grew as a single crystal of sCO.
The evidence comes from the rotation patterns of the separations of the
satellites as shown in Figure 42, in this case at 64.7 K. The perturbed
spectra at different angles are also shown in Figure 43. One notes that
the linewidths are mainly due to the inhomogeneity of the applied magnetic
field. Since the separation between m = + and m = + and that between
5 3
m = and m 2are symmetric, from the spectra one can only determine
(2) = 3 sin Y cos2 1 (412)
and
2 ) = .2 3 sin2 cos2 1 (412')
Noting that 12v( 2) = 2 12v(i)l while one measures 12(')1, the measure
ment of 12v() car be used as a s'ipler;ent to check the accuracy of the
12u(')i measurement. However, the rotation of the H field at an angle
S= wouid give 13 sin2 y cos2 1 II = 1, while at an angle of shifted
by Emore would give 13 sin2 y cos2 { 11 = 13 sin2 y 11. It is obvi
ous for symmetry reasons that the rotation pattern of (412) as shown in
Figure 4.2 is a periodic function of T, having two local maxima. As in
Figure 4.2 it is at io = 320, where o is a laboratory reference angle,
that gives = At this angle (411) and (411) can be simplified as
Figure 42. The NMR perturbed spectra of BCO at different (4 o)
angles, where io is a laboratory reference angle.
): li:!~~
. 9 1 I d
 I
0 20 43 60 00 100
0
Figure 43. Rotation pattern of the separations of the satellites of NMR
BCO as a function of (* o).
120 140 160 1I0
perturbed spectra in
I
I I
1.1
I
.1.2
i.
.2
S'2
i.
9 I I i ,
!
'

I
i
i2(2)! s 6e2Qq 2
2, 40 1h 2 (413)
and
2v( ) 3 o (413)
I40 h 0
In fact the values of 13 sin2 y cos2 1 1I as a function of have two
2
local maxima. For < sin2 y 1 the first (larger) maximum located at
4 = 0 while for 0 < sin2 y it is at = 90 [49].
Comparing with different single crystals, it is possible to determine
which maximum corresponds to ( = 90 so that
13 sin2 y cos2 Il = 1 .(414)
It is clear that one can also have an angle ^' such that
3 sin2 y cos2 I' = 1
From (414) one determines
1 1
sin = co (415)
v3 COS #
From the experimental value of 12v ,() and the value of the static
quadrupole coupling constant which one found in tne O0R measurements in
aCO one can determine the value of e If one defines the effective
quadrupole coupling constant as
eQ> 1
< eff h e (3 cos2 1) (416)
eff
= e2g E2
2h 0
one .would have from (412), (412') and (414)
2 =2 0 h >, (417)
e f
and
2( 0) h eff (417')
elf
The experimental data from the separations of the satellites give
< > = 7.04 KHz at T = 64.7 K
h eff
Comparing this data with the static quadrupole coupling constant of
3.83 MHz one finds that
< e > = 1.67 x 103 (418)
Seff h
which is three orders of magnitude smaller. Figure 44 shows the effec
tive quadrupole coupling constant for several temperatures of BCO. With
increased temperature one expects E to increase, causing the effective
quadrupole coupling to increase with temperature. This is consistent with
the experimental results where E = 3.23 0.060 at T = 62 K and
E = 3.36 0.060 at T = 68 K near the triple point.
4. 2 Tl Measurements in sCO
It is shown in Appendix B that if all lines in the perturbed spectrum
are saturated and one assumes W1 = W2 where
W W5 3 (419)
W2 =25 1 (419)
2 2f
N
Cr
*w
v
61 6; 63 64 65 66 67 68 69
TEMPERATURE K
Figure 44. Effective coupling constant as a function of temperature in CO. (A typical error
bar is shown .t T = 68 K.)
then the spinlattice relaxation is governed by a single exponential
decay
1 4
T1 gW (420)
T1
In the course of the experiment a sequence of 16 pulses was applied
before a 7  pulse sequence was employed to measure the spin
lattice relaxation time. It was found that within experimental error the
nuclear magnetization recovered as a single exponential. Furthermore the
normal  pulse sequence would give an almost identical result
except for very small r. This may come from the small separations between
all lines and a pulse would suffice to saturate all the lines. The
single exponential decay supports our assumption of V = W2. Following
the same step as in liquid CO, and assuming the relaxation is dominated
by the intramolecular quadruple interaction, one has
T1 25 2 2 (421)
where T now is the autocorrelation time for the precessional motion
around the crystal C axis. Using e2g from NOR measurements in the
aphase one has the following relation between T, and T:
= 3.93 x 1014 1 (422)
2 T (422)
Shown in Figure 32 is the temperature dependence of 1^ and 2'. Although
there is a different meaning for T' and T2, they specify the correlational
motions in Bphase and liquid CO respectively. Figure 32 shows that
there is a discontinuity at the transition temperature. It is reasonable
that the characteristic time of the motions in BCO takes a longer period
than that in liquid CO, while the discontinuity in the opposite sense of
43
BN2 near the transition temperature is not clear at this moment [52].
For eN2 the same assumptions were made as one did in BCO, namely that
I = W2 and that all lines are saturated. One therefore has
1 (423)
T1 1
in liquid N2 as well as in gN2.
CHAPTER V
SOLID aPHASE
5. 1 Molecular Properties of aCO
Solid aCO is one of the simplest diatomic molecular crystals and
has been an interesting subject for studying the lattice dynamics of
molecular crystals [5,11,12,5355]. The crystal structure has been dis
cussed in 1. 1. 1. One notes that in the face center cubic crystal
structure the locations of the molecules and the different orientations
along the diagonal can be related to the molecular dipoledipole and
quadrupolequadrupole electrostatic energies for the various pair orien
tations [15]. The specific heat study of CO by Clayton and Giauque [4]
observed a residual entropy of approximately R In 2 at low temperatures
of aCO. Gill and Morrison [10] extended the measurements down to 2.5 K
with no indication of the occurrence of an orderdisorder transition.
However, it can be mentioned that a small anomaly has been observed in
the heat capacity of aCO near 18 K and ascribed to the freezingin of
molecular headtotail reorientation [16].
5. 2 Nuclear Quadrupole Resonance in aCO
To study the molecular dynamics in aCO, one can employ the 017
nuclear quadrupole resonance, since the quadruple coupling constant is
large enough to be able to observe the NQR frequency directly. For 017
in aCO intermolecular contributions to the field gradient are very
small compared to the intramolecular interactions. Each molecule can
therefore be regarded as an isolated spin system. The electric field
gradient (EFG) tensor at the nuclear site is thus axially symmetric and
the molecules in aCO differ little from that in the gas phase. The quad
rupole Hamiltonian can be written as
e= 2Q() [3 (51)
where eq (T) is the time average of the principal component of the EFG
tensor taken over the vibrational states at temperature T and eQ is the
nuclear quadruple moment. It follows that
8 3 et Q(T) (52)
5 3 10
+ +++ n
3 = 3 e2Qq(T) (53)
3 1 20
The zero temperature NQR frequency would in turn determine the static
quadrupole coupling constant (QCC). Since by extrapolating the NQR fre
quency to 0 K one gets
O5 3 = 1.15 MHz
2 +
With zero point correction estimated from Raman spectra [55], one obtains
eOq 4.20 MHz (54)
h
The measurements of the NQR frequency versus the temperature are shown in
Figure 51. The static QCC was also used to derive T2 and T2 in the
liquid and sphases respectively. The microwave experiment gave 4.43 MHz
for e2Q. This was refined by Flygare and Weiss to be 4.48 MHz by con
sidering in addition the spinrotation interaction in the microwave
spectrum [56]. Since the same averaging factor due to molecular stretch
ing motions may exist in both solid and liquid phases, the experimental
~F T~I1TfI
11501
1050
ao
T u/
iOOL
IOoqY
F I ~
Figure 51. Temperature dependence of the pure quadrupole resonance
of 017 in aCO.
0 20 50 40 50
T I
~p'"c~~m~e
"
I
, I
!
NQR determined QCC from the aphase should be more appropriate for deriving
the correlation time of the molecular tumbling motions in the liquid phase.
5. 3 NQR Linewidths and Molecular Motions
The most interesting thing that occurs in the NQR measurements of
the aCO is the anomalous behavior of the linewidths. Figure 52 shows
the temperature dependence of the linewidths and some T1 measurements,
while Figure 53 shows a three dimensional representation of the spectra.
As one can see there is a maximum at 37 K and a divergence below 28 K.
One might speculate the broadening of the linewidths is related to static
effects such as anomalies in the expansion coefficient, metastable states,
strains, etc. However, the thermal expansion of aCO shows no anomalous
behavior in this region [18] and the broadening persists for long periods
of time (, 5 hours). As one will see later, the spinlattice relaxation
measurements in the temperature range 48 60 K seem to support the hypo
thesis that the iinewidth is actually dominated by the motional contri
bution.
The occurrence of orientational disorder in the aCO crystal has been
expected since the first estimate of the residual entropy was made [4].
However, the frozenin static disorder cannot explain the peculiar be
havior of the linewidth observed. In the light of the temperature be
havior of the 017 NQR linewidth and spinlattice relaxation rates, it
will be assumed that the disorder in the orientations of the CO dipoles
is of a dynamic nature at least for T '28 K.
In the dynamic model one assumes that the molecules execute libra
tional motions around the equilibrium positions plus sudden reorientations
among the various directions which are, in principle, permitted by the
crystal structure. In order to explain the peculiar behavior of the
S2. r" r  "
Li
7 c 2.0
0
D
1.5
1.0 
0.5
o 0
0
0
0
0 0
T a 
50 60
T K
._..1~
T a/
0
O
o ol
00 0o,
Io o o o I
I i !
50 60
10 20 30
T
Figure 52.
Temperature behavior of the 017 NQR linewidths in aCO.
The insertion shows some effective T1 measurements.
6C
4T INTENSiTY
//
T (K)
45
VQ KHz
)/ iiOO
/7, / 7 /7
~i~c~
________~~~~~c~
IL ~ _~~
__~_ ~, ~__~~~
____
__ ~~~
~__
~Z~=
~=s~l~,~~_,_TCZ__
=ZTc~
 ~2_5 _
C__~__

~c"~__7~:

j ; 
Figure 53. Three dimensional representation of the 017 NQR spectra in aCO. For clarity
the linewidth has been enlarged by a factor of 2.
linewidth, that the quadruple resonance is not affected (normal Bayer
type) by the reorientation, one assumes the molecule executes libra
tional motion plus a sudden near 7T reorientation [57]. In presence
of motions, the quadruple Hamiltonian can be divided into two parts.
HQ = < HQ > + [H < i >] (55)
The average Hamiltonian < HQ> gives the NQR frequency while the fluc
tuating part drives the relaxation process and causes the line broad
ening [58]. For 017 in the CO molecule, one assumes a rigid EFG tensor
of intramolecular origin with cylindrical symmetry which, as a conse
quence of motions, moves its principal axes in the laboratory frame. A
straightforward tensor transformation gives for the instantaneous EFG
components (using s(t) as defined in Appendix C)
Vx (t) = t (1 32) (56)
V (t) = e [1 3 (1s(t)) 32 (1 (t))]
yy 2 2.. 2
Vz (t) eq [2 + 3 + 39 + 3 ( s(t)) + 3 ( s(t))]
xy (t) = I [ (1 s(t)) + ," s(t)]
Vxz (t) = [ (1 s(t)) + s(t) + (1 s(t))]
Vyz (t) _3 [ 2 (1 s(t)) + ,, ]
where n,, and 4, are the inplane and outofplane librational angles with
respect to the 7 reorientation respectively. * is a random variable of
I
order < 42 >2 which allows reversal of an angle which is not exactly r as
discussed in [57]. Taking the librational average of (56) and assuming
no correlation between t,,, 4, and 4* one has
_ C1 3< >) (527)
32 2
S y > = e [1 < s(t) > ) 3 s > ]
< Vzz > = eq [1 < 42 > <2 > 3 (1 < s(t)> )]
and
< V > < Vz > =
In the case of disorder, where < s(t) > = 0, one has
5 3
< Vzz> = eq (I 2<*2> 2<2 > ) (58)
The resonance frequency between m =c and m = will be
3 sQ R (I 3 < 3 3 *2)
S < > h < 4 > 2 *2) (59)
This is the usual Bayer type NQR frequency. One also notes that for the
order phase < s(t) > f 0
3 Eo h [1 23 < > (1 < s(t) >)]
VQ !0 h 2 2 4
(510)
which is higher than in (59).
One also notes that if reorientation is sufficiently different from
r, i.e. s* is a large angle, ,Q would be significantly affected. In
order to evaluate the linewidth one considers the fluctuating parts of the
components of the EFG tensor from (56) and (57) and gets
V < Vx> 2(4L < >) (511)
xx xx 2
vyy < vy > = [4 < 2 > (s(t) < s(t) > )]
V ~ < V > 3e[+ 2 < 2' >
zz zz 2
+ (s(t) < s(t)> )]
V < > = (1 st)) s(t)]
xy xy 2 2
V < V > = L s(t)
xz xz 2
V V = E [ (1 s(t)) + 4n]
yz yz 2 2
There are two parts contributing to the linewidth of the quadruple
resonance, i.e. the adiabatic terms and the nonadiabatic contribution due
to the lifetime of the energy levels [58]. One writes
21 5 + (512)
T2 T+2 5 T 3
where
1 = (t)C(tT) dz
Jo
is the adiabatic contribution and m'(t) is the departure of the instan
taneous resonance from its average value and
1 7 W (513)
T 5
T5 5 + ,m
t m 2
1 = 3 (514)
13 3,
"2 22
where W is the transition probability per unitatime from state m to
mm
state m by the fluctuation terms in (511).
By neglecting for simplicity the terms which do not involve the re
orientations one would have
r q 12 3 < J (0) (515).
'2 20h J 16
where Jl (0o) is the Fourier transform of s(t)s(t ). Assuming a har
monic oscillator equation for ,(t) and an exponential correlation function
s(t) with characteristic time T 2, one has
1 9eQq 1 < Y T 2" 20 < ^^2> T
T5 20 9 1 + (& + 2
2 2
+ 5 <*4 3
4 J Z ) (516)
where 1" is the correlation time for reorientation. w is the resonance
5 3
frequency between m = and m = +. is the frequency for the harmonic
oscillator ,(t) which is much higher than 0, while
1 f 9e_ q 2 5 *2 2< 4*2> 2 +
2 I *~ ~ 2 (518)T 2
9+ C+ (517)
Combining (515), (516) and (517) one obtains
12 [ 9e2Qq23 < *> JII (0) + 9 > < I
T2 2A 16"I 4 AI T
5 < > f3w +1 1 < *2> T
9 1 + "2
1 +I
In the fast motion regime, were Jn (0) ~ Ji (I ) = Jj () = J ( ),
one can neglect terms involving < ,4*>, so that (518) reduces to
'= 2 Ie2,1 5c 4Ti
T + 2 4 + (519)
2 20 2
From (519), which has a maximum at T2' = 1.188 one notes that
at b37 K the correlation time for reorientation of CO is
2 = 1.064 x 10 sec. The spinlattice relaxation for NQR with I = is
not governed by a single exponential decay [59]. One can define an
effective relaxation rate which in light of (518) one would
IU eff
have
[ 1I ]e T j1 (520)
TI I T2 T
where a is a constant of the order of unity. The measurement of
[,ij eff
in the temperature range of 49. 60 K supports the hypothesis that the
linewidth is due to the motions. In fact, for T = 55 K one has
S eff = 0.25 x 103ec1 and according to (520) the linewidth should
be, for a = 1, 0.25 KHz in satisfactory agreement with the experimental
result. One notes that a background contribution to the linewidth can be
expected from dipolar interactions, strains, and defects, etc. For
T s 37 K the linewidth seem to point out the changeover to a slow motion
regime where &v decreases on cooling. Forr'>> one would have from (519)
v < > 1 (521)
2r2
It should be pointed out that in the slow motion regime the perturba
tion approach implicit in the above derivation breaks down. Alexander and
Tzalmona [60] showed for slow motions in pure quadrupole resonance, spin
lattice relaxation time and therefore the reversal of the linewidth is
directly proportional to the emotional correlation'times. It can be spec
ulated that the lack of symmetry of av versus T around T = 37 K is related
to the breakdown of the perturbation approach in the slow motion regime.
For temperatures lower than 32 K, 6u exhibits a dramatically increased
broadening in a few degrees up to 10 KHz and the NQR signal is no longer
detectable below %28 K. Careful attempts for various temperatures down
to 4.2 K over a wide range of frequency searching have been unsuccessful.
A possible origin for the behavior of the linewidths could be a
change of nature of the reorientations. In particular, the slowing down
of motion could favor reorientations from a body diagonal to an equiv
alent one. As discussed by Rigamonti and Brookeman [CG], for a reorien
tation effectively different from r, one would have
I 6v 2 (522)
T1
T2
where T;' is the fluctuation time between the diagonal equilibrium
positions.
This provides one of the explanations of not being able to observe the
NQR signal below u28K. Another possible origin can be found as in aN,
where the broadening of lines was characteristic of all mixtures of isotopes
and is a consequence of the statistical randomness of the environments of
a given type of molecule that produces a dynamic disorder and a local
inhomogeneity in the average field gradient [62].
5. 4 Discussion
As shown in Figure 54, the temperature dependence of resonance fre
quency of aCO is nearly like that of aN2, furthermore the reduced NQR
frequency (T) / VQ (0) has the value of 0.84 for CO as compared with
0.82 for N2 at the transition temperature,[49].
+ 0
409
+ 0
0
+ o
+ 0
+ 0
.9 
.1 .2 .5 .4 .5 .6 .7 .8 .9 1.0
T/T
Figure 54. The reduced temperature behavior of the reduced NQR frequency. ("o" represents
aCO and "+" represents aN2. Data for aN2 from Ref. [49]).
1~C`7.'~IT .I 1 r
. Ci
The peculiar behavior of the linewidth in the aphase NQR measurements
elucidates the possibility of the motions in aCO. This is consistent with
the observation of the residual entropy. Some (T )eff measurements also
support the assumption that motional effects determine the linewidth.
The temperature behavior of av between 40 52 K is satisfactorily
fitted by the law
E
Sv = e KT (523)
and suggests an activated temperature behavior for Ti', with
E 1.2 K cal/mole.
The assumption of hindered rotation can be ruled out since it would
affect resonance frequency by a factor of 1, and this would conflict with
the microwave measurements of QCC = 4.4 MHz.
It could also be noted that no reorientation by an angle significantly
different from could be present in the fast motion regime where
(T a 37 K), since the averaging effect would change vq, and this is not
observed in the experiment. However, for temperatures T < 32 K, it is
possible that the reorientation involves angles different from ir since in
the slow motion regime 2 >> and no average is taken.
The dynamics driving the molecular reorientation should have a co
operative character through the molecular dipole and quadrupole interac
tions. The correlation time T" introduced above has to be considered
a local correlation time driven by the cooperative relaxational time at
a given wavevector. Gill and Morrison [10] measured the dielectric con
stant of aCO down to 6 K and found no critical effect. This seems to
suggest that the characteristic wavevector of the cooperative excitation
should be far from zero and the possible phase transition, if any, should
be of antiferroelectric type.
APPENDIX A
QUADRUPOLE RELAXATION THROUGH MOLECULAR
TUMBLING MOTIONS IN LIQUID CO
In the liquid phase for nuclear spin i = , the Zeeman energy levels
are all equally spaced. In the absence of an rf field, transitions intro
duced by the quadruple interaction are shown in Figure A1.
The actual values of the transition probabilities depend on the de
tailed forms of the spectral densities of the field gradients. However,
all the upper transition probabilities for Am = 1 are simply related to
each other. This is also true for Am = 2. The quadrupole Hamiltonian
can be written as in (118)
H4 . r Vo( (3 2) + V+ (I Iz + T I
Q 41(211) L 3 o z + ^ z z 
+ V_ (I1 Iz + Iz 1+) + V+2 (I_)2 + V_2 (I+)2 (A!)
where I+ and I are the raising and lowering angular momentum operators
respectively. V represents the irreducible field gradient tensor com
ponents which are related to the cartesian tensor components by
V [3Vzz (Vxx + V yy (A2)
V1 = Vzx izy
+2 = Vxx Vyy) iVxy
One notes that the Vij are symmetric and V + V + V = 0. For the
Srasxx yy zz
raising and lowering operators one has
_
2
3
2
SI
SW
W VI VA(ihA)
I I
JW W,(I+A)
,w Wz 2,.A
Figure AI. Quadrupole induced transition probabilities among the
Zeeman energy states for spin I 5/2.
I ( I+2A)
2
2;
I2A)
iQ
I
60
I,1 I,m > = m /iTIin(mei TT l,I,1 >
and
Iz+ I,m > = (mrl) /I+TTY) m(mn I ,rrl > .
One then calculates the matrix elements from (A3) and (A.3')
< Iz + I+ > 2 = 80
1< 1+ Iz + 1,1; 2 >2 = 80
and'
< 5, +1 I+lz
< 5, I+ i
2' +2 I z I
and also
o< th t s1 11TI
For the transitions
< +1 ( ) 2
< + (1 )
2' 2 
1.+' > 2 = 32
5, >12 = 32
f '
z+ I >2 = 0
involving Am =2, one obtains
S5 12
> = 40
_I = >140
< 1 (I )2 2'T >i2= 72
< I4 (It) t > 2' 72
(A3)
(A3)
(A4)
(A4')
(A4")
(A5)
(A5 )
Fron: (32), (34), (A4) and (A5) it follows
0ao eiLt 1T + (t ) dr (A6)
2 40O e i2L V+2(t) V;2 (t T) dr (A6')
where 1, = W5 and W2 = 215 1. One will need to calculate
2' 2 2' 2
V+1(t) VI (t T) and V2 VtY2 (t TT. To do so one considers the
irreducible tensor transformation as following. For irreducible tensors
Vk of rank k, the 2k + 1 components of Vk are transformed according to
the.irreducible representation Dk of the rotation group
Vq= V p D (a, B, ) (A7)
where (a, P, y) are the Euler angles of the rotation taking the unprimed
reference frame into the primed frame.
The Euler angles are defined by three consecutive rotations: (1) A
rotation of angle a about the z axis, (2) A rotation of angle B about the
y'axis and (3) A rotation of angle y about the z" axis as shown in
Figure A2.
The unitary operator D (a, 3, y) can be written as
D (a, B, y) = eiYJ'"eiJJy eiaJz (A8)
Using the fact that
Jy = eiaJz J e i
Jy y
(A9)
Z, ZZ
Y, /
\ y, Y"
\ /
x i .
yX \
Figure A2. The Euler angles. First, a rotation of angle a about the
Z axis. Secondly, a rotation of angle S about the Y' axis.
Finally, a rotation of angle y about the Z" axis.
.
which can be proven as follows. Let eiJZ a > b > where {la>} is
a complete set of eigenkets before the rotation of a about the z axis,
{Ib>} is another complete set of eigenkets in the coordinate system
after rotation. One has
< b' J b > = < a' J a > (A10)
= < a' eiaz J .eia a >
It .follows that
Jy eiaJz Jy eiaJ (A11)
Using (All) and by expanding, it can be proven that
e 1y = eiaJz eiJ eiaz (A12)
Following the samp procedure, one has
eiYJz" = eiaJ e i' eiYJz ei JY eiaJz (A13)
From (A8), (A12), and (A13), it can be shown that
D (c, B, y) = eiaJZ iJy eiYJ (A14)
Let Ij,m > be a simultaneous eigenket for Jd and Jz. the matrix elements
D then will be written as
m,m
DJm = < j,m D (a, B, y) j,m" >
= < j,m Ieiaz eiJy eiYJz j,m >
S iem eiYm' d m, (A15)
where the definition of dj is obvious.
m rn,
64
From the table [30] one has for elements of d2 ,m..) as
2 = d2 = cos4 () (A16)
22 2 2
d21 = d2 d2 d2 s (tCos
21 1= 2 21 = sin (1 + cos
do = d = d2 d2 =/3/8 sin2
2 02 20 02
d2 d2 =d2 = d2 1
21 2 21 12 2 )
d2_2 = d2 = sin4 ( )
22 22
d2 = d (2 cos B 1) (cos + 1)
11 1 d 2
di = d = (2 cos B + 1) (1 cos B)
dIo = dt = d2 1 = d2 /= 37/ sin B cos B
do (3 os2 1)
In the principal reference frame since V.. = 0 for i A j one would
have from (A2) that
V 3 V
o = ZZ
V., = 0
V2 = (VXX Vyy
2 2 *XX *YY'
65
Transformation from the principal frame of the field gradient to the
laboratory reference frame, yields
V+ 3v Z eiY /372 sin 5 cos 6
1 iec ,j 1
+ (VX Vyy) ei2 eiY [ sin B (U + cos 6)]
+ I (VX yy) e2 eiY [ sin B (cos 1 1)] (A17)
3 iY
V1 VZZ e ( ,3/2 sin B cos B)
v6
+ Vyy) ei e [ sin B (cos 6 1)]
( Vy) ei2a iY [ sin B (cos B + 1)] (A17')
+ (VXX YY e e
V 3 V ei2Y /3/8 sin2 B
+2 =rVXX e
1 i2 i2Y 4 B
+ XX yy) e e cos4 )
+ (VXX Vyy) ei2z ei2Y sin4 () (A18)
S _3 V 2y e2 3/8 in2 B
2 ZZ
1 2V i2 e iY
+ (Vxx Vy) e e 2e) sin4(
+ (VXX Vyy) ei2a ei2 cos4() (A18)
2 (XX YY) 2ia 17Cs( A1
If one assumes isotropic motions in the liquid phase and takes the
average over all possible orientations one obtains
3
V (t) V ( = T ( + ) V (A19)
V1l (t) v* (t) (3 +4n)V2
1 TO (1 + 3 ZZ
and
3 n
+ n2 2
2 (t) V*2 (1 + ) VZZ
where n = is the asymmetry parameter, Assume
ZZ
T
Vi (t) V' (t r) = IVi (t e" 2 (A21)
for i = 1, 2 where 2 characterizes the correlation function of Vi (t).
Using (A6) and (A6'). one finds
Ie___ 1 1 2 2r2
w,  f 24 (1 + ) (A22)
2 401 3(1 + 7+
24 (1 + ) 2 (A.22')
In the fast motion regime uLT2 <
W1 e 48 (12 (A23)
1 S 4) 'J
W2 = [ 48 (1 + q ) T~ (A23')
The relationship between the experimental T1 measurement and W' W2
are obtained by considering the rate equations governing the populations
among the energy levels.
The number of nuclei in state m has the following time dependence:
dNm
S m (N Nm Wm;m ) (A24)
where W ,m is the transition probability per unit time from state m to
state m', and Nm are the number of nuclei in state m. One can write down
all the equations
dN5/2 1 +
dt N5/2 1 2 3/2 + N3/2 (1 +
+ N/2 W2 (1 + 2A) (A25)
3/2 2 9
dN3/2 2 N W N1/2
dt 5 3/2 N3/2 W1 (1 + A) 10 N/2
2 0
+ N12 W1 (1 + A) + N3/2 l + T N/2 W2 (1 + 2A)
dN
1/2 2 1 9
d N1/2 1 (1 +A) N1/2 N (1 + 2A) T /2 2
d142 N112 N7l Y 11 N22 W2
dt 5 /2 N (I + 2)
5 N3/2 W1 2 N5/2W2 10 N3/2 2 (1 + 2&)
dN
1/2 2 1 9
dt = 1/2 1 2 N1/2 12 10 N1/2 2 (1 + 2A)
+  (2 + 2) ++&
+ N3/2 1 (1 + A) + N5/2 W2 (1 + 2 N3/2
d3/2 =2 9
dt 3/2 1 5 N3/2 1 + ) N3/2 2 1 + 2
+ N5/2 U1 (1 + A) + N1/2 "1 + N,/2 2
dN5/2 1
dt = 2 (1 + A) N5/2 W2 (1 + 2A) + N3/2 1+ N /2W2
where the downward transition probability per unit time for Am = 1 and
Am = 2 are given by the upward transition probabilities per unit time
multiplied by the Boltzmann factor 1 + A and 1 + 2A respectively, where
A = and one notes hTL << KT in the temperature range of interest.
Let N 5/2, No3/2, and N 0/2 be the number of nuclei in state
5 3 1
m 2' 2' 2, respectively, at equilibrium with lattice bath. To the
first approximation one would have
N32 N A = N5/2 N3/2 (A26)
N
where no = and N is the total nuLmber of nuclei. Similarly,
N12 (N / 0 2n o o (A27)
N0 /2 1N/2) = 2n 5 N/2 N1/2 (A27)
1/2
It follows that
dN 5/2 o
dt (N5/2 N5/2) (N 5/2 5 N/2) 2
+ (N32 N3/2) W1 + 1 (N1/2 N/2) 2 (A28)
Defining nm = N N one can write the following equations from (A25):
dn5/2 1 1
dt = "5/2 W1 2 n5/2 2 + n3/2 W1 + 2 n1/2 2 (A29)
dn3/2 9
dt =  "3/2 W n3/2 W1  "3/2 W2 + n5/2 W1
S2 W + 9 l
n5 1/2 1 10 1/2 2
dn/2 2 1
dt "1/2 1 2 "1/2 "2
dn1/2 2 1
dt "1/2'1 2 n1/2
dn3/2 2
Sdt n3/2 W1 5 n3/2 "1
9 2
0 "1/2 W2 + n3/2
1 9
+ n5/2 t n"3/2 2
9 2
2 0 n1/2 2+ 5 n3/2
1 9
+ n5/2 2 10 n3/2 I2
9
10 n3/2 W2 + n5/2 1
5 n1/2 1 + T n1/2 W2
dn5/2 1 1
dt = n5/2 1 2 n5/2 W2 + "3/2 W + 2 n1/2 2
By multiplying each equation with the corresponding m and summing all
together, cn2 obtains the following equation:
fc 2 (A3C)
d
dt z z o
+ [ n3/2 + n "1/2 n1/2 "3/2 ](141
io a first approximation and assuming that the spin temperature can be
defined, one notes
6 3 3 6 6 3 3 6
n + n n n n n + n  n  n
5 3/2 +5 1/2 5 1/2 5 3/2 5o 5 5 o a o0
+ (i+ 3 + +
10 1 10 lOj o
(A31)
where Ks and Ts is the spin temperature which is generally greater
than the lattice temperature. One then concludes that
d
S> (W1 + W2) (A32)
which implies
1 (2 1 2) (A33)
and finally, it follows from (A23), (A23 ) and (A33) that
1' e 2q 1 + ) 2 (A34)
in the fast motion regime.
APPENDIX B
NMR SPINLATTICE RELAXATION
THROUGH THE QUADRUPOLE INTERACTION IN SOLID BCO
In solid BCO, the time averaged quadruple interaction represents
a small perturbation term to the total Hamiltonian and the Zeeman energy
levels are unequal. In the absence of spinexchange transitions, the
Boltzmann distribution of the populations among the energy levels cannot
be maintained. In such a case the spinlattice relaxation is, in general,
not characterized by a single relaxation rate. In addition, the relative
importance of the different relaxation rates depends on the nature of the
relaxation processes and on the initial conditions [38]. However, since
the quadrupole splitting are small compared with the Zeeman energy levels,
this simplifies the problem. Considering for spin I = , from (A25)
one has
(B1)
d(N5/2 N3/2) 2
d(N /t N (5/2  3/2)( + (3/2 2) 1+ W2
4
+ (N2 N_1/2) D7 + nA (21 W2)
d(N3/2 N 12)
dt3/ (N5/2 N3/2 1 ( 2) W (N3/2 N1/2)(4W1 + W2
+ V (N/12  N32 2 nA (W 1 2)
d(N 12 N_ 12)2 +
dt 1/ /2 N3/2) 2 + ~ (N3/2 N12) (I1 W2)
9 2
S (N1/2 N/2) + l(N1/2 N3/2( 
+ 1 (N3/2 N_5/2) W2 + n (2W2 1
d(N_2 N32) (N_3/2 N_5/2)(W 2)
dt
(N1/2 3/2) (41 + 7W2
+ (N3/2 N1/2) W2 nA (W1 5W2)
d(N N )
d(N3/2 5/2 = (N N 2)(2 
+ (N_1/2 N3/2) (W1 + W2)
+ T (1/2 1/2 W2 + o (21 J2
where W1, W2 and A have the same definitions as in Appendix A. If one
now defines Nm1/2 = Nm+1 N from the symmetrical equations above, it
follows (B2)
N2 = N2 (2W1 + I W2) + 2 n ( + nA (W1 2)
N1 = N2 ^ W2) 5 N1 (4W + 72) + T Y2 n (W 5 )
1 2 2 9 2
No 2N22 + N (1 W2) NO 2 + N1 (W1 W2)
1 4
w+ Nt_2m+ 2 n (W1 + 2W2)
with two symmetric equations for N1 and N2.2
The solutions of these three equations are characterized by only
three relaxation times. To simplify the equations in (B2), one defines
Np = N n (B3)
p p 0
where no and A have the same meanings as in Appendix A. (B2) then
becomes
N2 = N2 1 "2 ) + Ni (W. +2) + N N6 W1 2 (B4)
NI = N2 W" 2) 1 N (4W1 + 7W2) + N1 W2
1 2 9 2 1
No 2 N2 + w Ni (W1 W2) No + N1 (W1 2 ) + 2 2 N
Now assuming a precessional motion in BCO and for simplicity an
axially symmetric field gradient, one can calculate IVi12 from (A17),
(A17), (A18) and (A18') for i = 1 and i = 2
IV112 = sin2 B cos2 B V2 (B5)
IV;212 = 6 sin4 B VZ
In chapter III it is shown that
3 cos2 B 1 0
so that, (B5) can be reduced to
+V'1 I2 VZZ (B6)
IV2 V 1 V2
v 2 4 = ZZ
V6 12 = 0
where the V'terms are the EFG tensor components in the crystal frame,
while the tensor transformation from the crystal reference frame to the
laboratory frame follows the same procedure as in (A17) and (A18).
V, = V Di + V2 D_2 + V Di, + V'1 D_1 (B7)
41 2 2 1 11 1 2
V2 = V D22 + V 22+ V D12 + V_^ D 12. (B7')
For a powder sample using (A16) and (A18) and taking over all the
possible directions, one calculates
iv+1 2 vz (B8)
V (B8')
IV212 = 3 v (B8)
The same equations will hold for IV_112 and !V_212 assuming again
T
Vi (t) V (t T) = IVi (t)!2 e 2
for i = 1, 2 where T2 characterizes the correlational precessional
motions. Using (A6) and (A6') one obtains
W1 e2 2 48 2 (B9)
1 40Aj L ^
W,2 2 48 +41 (B91)
From (B9) and (B9') it is obvious that W1 W2. In the fast preces
sional motion regime ,L 2T <<1. It follows from (B9) and (B9') that
W1 = 48 feQ 2 T (B10)
W2 8 T24oJ2 (
Lh40J
That is, 14 = W2 in the fast motion regime.
75
Now from (B4) it is possible to write
S. 1 2 51N 1 1N (B1 1 )
1 2 1 15 'Ili +T N1
1 9 1
N 2 1Te 5 1NO0 2 1N 2
Since one considers the initial condition that all lines are saturated,
it follows from the symmetric equations that Nl = N and N '2 = N;.
(B11) is further reduced to
N= I 11 10 +4NO (B12)
N' = WN I
N6 1N 5 N6
The general solutions of (812) will be
N' = a eX2t+ap e3t (B13)
p p p2 p3
where p = 0, 1, 2. Substituting (B13) into (B12) and comparing the
e1t, eX2t and e3t terms one has
( 2 W1 A) a + 5 + 9 a l = 0 (B14)
1 a 13
2 121 + (0 1+ + X1) a = 0
l a21 + ( 5 + X1) ao0 = 0
and similar equations for eX2t and eX terms
75
In order that all aj / 0 for i = 0, 1, 2; j
*0
5 +
2*'
1
2"1
13
10 1 + X
= 1, 2, 3 one needs
10 1
0
9
 W1 + I
(B15)
=0
This is reduced to
(5x 4W1)(2A 3W1)(10 33 W.1) 0
i.e., with solutions
4 3 33
S=4 W 1 ', 1 w]
In the case that all the lines are saturated at time zero, one has
from (B3)
(B16)
N2 = N1 = NO = 0
and (B17)
N2 = Ni = N6 = noA
4 3 33
Let Xi = 1 Hi 2 = 11 and A3 = T10 I Comparing with terms of
e (B12) gives, at t= 0
(818)
17 8 4 9
( a? a22 + a23) + (an + + a + 313) + 1 (ao: + a02 + a03) = 0
1 a2 a )+ ) 10U
2 (a21 + a22 + 23 + ( all + a12 + 2a13) = 0
3 3
(a21 + a22 + a23) + (a0l  a02 + a03) = 0
From (817) and (B13) one has at t = 0
n A = a21 + a22 + a23 (B19)
n = all + a12 + a13
n A = a01 + a02 + a03
(B18) and (B19) enable one to deduce
a2! = aol = all (B20)
3 2
a22 = 10 02 = a12
3
a23 = 4 ai3 = 2 a23
which gives
a21 = ail = a01 = no
and all other a.i = 0 for i = 0, 1, 2 ; and j = 2, 3. The solutions will
then be
N.= n0a (1 e "/SA t) (822)
for p = 0, 1, 2.
78
Therefore, in the case of complete saturation and W1 = W, the spin
lattice relaxation is governed by a single exponential decay having
14
Tl W1 (B23)
From (B10) and (B23) one has the same equation as in the liquid phase
in (A34)
1 3 e2Qqq2
1 J '2 (B24)
where T' is the correlation time for the molecular precessional motions
around the crystal C axis.
APPENDIX C
LINEWIDTHS AND MOLECULAR MOTIONS IN aCO
One assumes that the molecules in aCO execute librational motions
and sudden reorientations. The tensor components in the laboratory frame
and those in the principal reference frame are related by the rotational
operator R (a, t)
V = R (a, 4,) Vj R (a, A) (C1)
where R (a, R ) = R () R (a). The V.j and V.. terms are the EFG ten
y x 1i 1j
sor components in the principal and laboratory reference frames, respec
tively. i, is the outofplane librational angle and a is the reorien
tational angle.
 0 0
2
Vj = eq 0 1 0 (C2)
0 0 1
In (C2) a symmetric field gradient has been assumed. It so happens
that by assuming a small angle for 4$,
Vxx e (I 32 ) (C3)
V =  (1 + 32 sin2 a 3 sin2 a)
yy 2
V = (1 + 32 Cos2 a 3 cos2 a)
Vzz sin a2
=Y
Vxz e2 34, cos a
xz 2
V =y 3 sin a cos a
yz 2
For a sudden reorientation of an angle from 0 to i as is the case
discussed in NaNO2 by Ambrosetti et al. [57].
S= ( + ) (2+ s(t) + (C4)
where s(t) may take values +1 or 1, and 4* is introduced to account for
jumps not exactly equal to r. 4,, is the inplane librational angle. By
assuming small angles for 4,, and 4* and expanding sin a and cos a, one has
sin a = (1  2 + s(t)) [ (1 4 s (t)]
2 2 _4 4) SW
sin2 ( (t)) + + ( s(t))
cos a (1 _) [ + (1 ) s(t)] ( s(t))
4*2
cos2 1+ s(t) 1) (1 (t)) ,,2
From (C3) and (C5) it follows that
Vx = e f1 34,) (C6)
V = (1 [I 3*2
yy = 3 2 (1 S(t)) 342, + (1 S(t))]
zz= [2 + 3? + 34 + 3 2 ( s(t)) + 3 (1 s(t))]
= 3ei g>
Vxy g ( + s s(t) ,, s(t))
V = [ 3 + s(t)(l (1 s(t))
Vyz 3 q (~ s(t) + 2+ ') +
yz 2 2 2
Assuming 4,, and * are not correlated and also that < >= 0, the
librational average of (C6) is given by
< V > = (1 3 < .2 > ) (C7)
xx 2
< V > f [1 3*2 (1 < s(t) >) 3 < e,2 >]
yy 2 2
< Vz > = [2 + >+< 2> 3 < 2>+3 (1 < s(t) >)]
< V > = < Vxz > =< Vyz > = 0
xy xz yz
The quadruple Hamiltonian can be divided into two parts
HQ < Q > + [H < H >] (C8)
The average Hamiltonian < H > gives the NQR frequency, while the second
part drives the relaxation process and causes the line to broaden [58].
The fluctuating terms of the components of th2 EFG tensor can be
written as
V < V > 3 I ( 22 < 2 > ) (C9)
V < V = < 2 > (s(t) < s(t) >)
yy yy 2 2
23 (1 s(t))]
Vz < V > = 3 2 < 2 >+ .2 < ,,2 >
zz zz 2
+ (s(t) < s(t) > ) + 3 (1 s(t))]
V < V > q * (1 s(t)) + ,, s(t)]
xy xy 2 2
V < V > 3e s(t)
xz xz 2
Vyz < V >yz (1 s(t)) + @,
yz yz 2 2
where in (C9) consideration is given only to the second order of r,,
1 and (*.
For spin I = the NQR frequency between m = and m = 3, T
is given by [58].
1 = + + 1 (C10)
T2 T2 T5/2 + 3/2
where
I= "(t) '(t T' dr
2 o
o'(t) is the departure of the instantaneous resonance frequency from its
average value and
1 1 (Cll)
T5/2 m +5/2 w5/2, m (
1 3/2, m (C:12)
T 2 m/ +3/2
3/2 m 3/2
where T5/2 and T+3/2 are the lifetimes of the states m =5/2 and m =+3/2
respectively. From (A1) and (C9) one will have, assuming order phase
< s(t) >= 0
'(t) = 9e 2 3 [ '2 < 2 + _2 <"2 > (c13)
2 3
+ I s(t) + ,,* s(t)]
Since p,, and $, occur at higher frequencies, for simplicity one may neglect
these terms. One would then have
fI "(T r dr = 2 * s(t) s(t ) dT (C14)
Jod 8ot 0 J
Defining
T sT r eiT dT = JII (W) (Ci15)
it follows from (C14) and (C15) that
1 9e Qq 3' < *4> (C 6
T2 20 16 JI
Consider now for terms. From (C1l), one writes
T,5/2
T52 = W5/2,3/2 4 W5/2,11/2
T5/2
Using (A1), (A4), (A4'), (A6) and (A6) one obtains
I = 4 0 L f V (t) V(t ) e dr (C17)
r5/2 I 401 1 L
+ 4 40 V +2TTTy (T ei d
4+ f L {[i(TY'YT e1t dr
4 V_2(t) V*2(t tT ei1'e d
Now one needs to consider
A V (t) V*l (t ) eiT dr
where
2 r
A = 2 ) s(t) t ) s (t ) e d (C18)
+ s(t) s(t ) e dr + ,(t),,(t r) ei dr
and where no correlation between the value of * and s(t) has been
assumed.
If one assumes an harmonic oscillator for , and ,,, and for s(t)
an exponential correlation function with characteristic time Ti, it follows
from (C'i) that
(C19)
A f 3e 2 r< 2 > 2T < *2 > 2 i
S+ ( + W)2 2^2+ 4 Tr 2
One notes that w, >> and .,T'>> 1.
3ecf)2 2> 2T" 2 2 7
S f3eq) .2 > < **2 > 24
A = j 2
(C20) can be reduced to
f3eq2 < ,*2 > T"
A = *2 (C21)
A ) 2
While for
B t +T_ T  T e dT
Since V.. involves V2, one needs only to consider, assuming = 0,
V 1 ' 3*2 3 *
Vy < > e 3 s(t) ( s(t)] (C22)
and
Vxy y > (1 s(t)) + ,,, s(t)]
If one further assumes that no correlation exists between i,, (., (*
and s(t),(C22) can be simplified as
y V < > 3 s(t) (C22
yy yy 2 2 s(t) (C22
V < > = 0
xy xy
It follows that
B 3 2 9 s(t) s(t) e dr (C23)
2 J ( 4
34 119 2
From (C17), (C21) and (C23)
(C24)
15/ 3e2 2 o< 2 2 + 18o0<* >J (I W)
5/2 80 7 2i
Considering now the term and using (A1), (A4), (A4'), (A5),
3/2
(A5'), (A6) and (A6 ), one writes
T13/ = 80 V+l(t) V l(t z) e1T dr (C25)
T  +
3/2 40 J _
+ V_ 7(t) V*1(t'T ei02 dr 
+ j (T r V*ETi Tr e1' dr
+reQ 2 F .) t  dT
+ V_(t) V*2(t ) e 2 dr
+ < p.2 > + < .4 > Jl ( 1() (C25)
1 + '2 1 2
4 2
From (C10), (C24) and (C26), one has
=2 2 <* > J 0)+ > 2
+ 3< > < > I + mT
+2 ,*2 2 (C 27)
S 2 3
If one assumes fast motion, i.e. J11 (0) = Jil (Y m) = JI (o) = JII (y ),
one can neglect terms involving < ,*'+ > since p* is a small angle. One
then has (C28)
1_ = 9e2Qq @2 > 2 L2L.
T 2 L + W2 2
2 20_ 1 + 2 9 + 2
These terms on the right hand side of (C28) came from the lifetime of
5 3 1
the states m = 2and rm , while the adiabatic term has been neg
lected. One can also show that for
T, = 1.188 1 (C29)
(C28) has a maximum.
One thing worth noting in (C7) is that < Vzz > shows the usual
Bayertype dependence for the NQR resonance frequency even though it is
derived from the small angle (*. One also notes that for large i*, i.e.
significantly different from r reorientation, < Vzz > will be affected.
In the fast motion regime, rT' <<1, (C28) can be further reduced to
1 [2 9eq 2 2 (C30)
'2 Lj [ 20t1 J
which relates the linewidth and the correlation time of the molecular
reorientational motion.
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BIOGRAPHICAL SKETCH
Funming Li was born on October 1, 1949, in Taipei, Taiwan. He
entered the National Tsing Hua University in September, )967, and
received a Bachelor of Science degree in physics in June, 1971.
Following two years of service as a second lieutenant in the Chinese
Armored Force, he worked as a science teacher in Langchu High School
in Taipei for one year. He came to the United States and entered the
University of Florida for graduate study in physics in September, 1974.
He held a teaching assistantship from September, 1974, to June. 1977,
and a research assistantship from June, 1977, to December, 1979. He
married the former Sumay Chen in December, 1976.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
James R. Brookeman, Chairman
Associate Professor of Physics
and Physical Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
E. Dwight Adams
Professor cf Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Arthur A. Broyles
Professor of Physics and
Physical Sciences

Full Text 
PAGE 1
NUCLEAR RESONANCE OF G 1 " IN LIQUID AND SOLID CARBON MONOXIDE sv RJMMING LI A DISSERTATION PRESENTED TO THE GRADUATE COUNCil OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Of DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1979
PAGE 2
To my parents
PAGE 3
ACKNOWLEDGEMENTS The author is greatly indebted to numbercus persons who have helped to make his work in the Physics Department of the University of Florida a rewarding experience. The excellent research facilities arÂ»d the atmosphere for the stimulation of scientific interest were much appreciated. Special thanks go to the late Dr. Thomas A. Scott, who was the initial chairman of the author's .supervisory committee and who suggested the problem, providing essential support, guidance and assistance throughout the major phases of this research. The author would also like to express his appreciation to the new chairman, Dr. James R. Brookeman, for his patient assistance in preparing this dissertation and for helpfu 1 discussions concerning the related behavior of carbon monoxide and nitrogen. His gratefulness is also extended to the other members of the supervisory committee, Drs. E. Dwight Adams, Arthur A. Broyles, F. Eugene Dunnam and Charles P. Luehr for their guidance and criticism throughout the research work. Thanks also go to Mr. Paul C. Canepa for his ingenious technical aid in helping to perform these experiments. The author wishes to acknowledge helpful discussions with Professor A. Pigamonti during the year of his visit. His genuine interest in science and voracious appetite for understanding the physics of matter have deeply influenced this author. He also wishes to thank Professor E. Raymond Andrew, whose critical reading and commenting on this manuscript is very much appreciated. The research presented in this dissertation was supported by National Science Foundation Grant Number DMR 770SG5S. i n
PAGE 4
TABLt" OF CONTENTS Page ACKNOWLEDGEMENTS , 111 LIST OF FIGURES vi ABSTRACT , , vi i i CHAPTER I INTRODUCTION 1 1. 1 Physical Properties of CO 1 1. I. 1 Crystal Structure of CO 1 1. 1. 2 Phase Diagram of CO 3 1. 1. 3 Specific Heat and Other Thermodynamic Properties of CO. 3 1. 2 Nuclear Magnetic Resonance........ 8 1 . 3 Nuclear Quadrupole Resonance 10 CHAPTER II EXPERIMENTAL PROCEDURE 14 2. 1 The Sample 14 2. 2 The Cryostat and Temperature Measurements 14 2. 3 The Electromagnet 17 2. 4 The Scectrr :e?r , ' 7 CHAPTER III LIQUID PHASE. 21 3. 1 NMR SpinLattice Relaxation 21 3. 2 Intramolecular Relaxation Mechanisms 23 3. 3 Intermolecular Relaxation Mechanisms .,. 26 3. 4 Temperature Dependence of SpinLattice Relaxation Time.... 27 3. 5 Discussion , 29 TV
PAGE 5
Page CHAPTER IV SOLID pPHASE. ... . 32 4.. 1 Quadrupole Perturbed NMR Spec tra 32 4. 2 Tj Measurements in s CO , 40 CHAPTER V SOLID oPHASE 44 5. 1 Molecular Properties of aCO 44 b. 2 Nuclear Quadrupole Resonance in aCO Â„., 44 5. 3 NQR Linewidths and Molecular Motions 47 S. 4 Discussion 55 APPENDIX A QUADRUPOLE RELAXATION THROUGH MOLECULAR TUMBLING MOTIONS IN LIQUID CO , . , 58 APPENDIX 8 NMR SPINLATTICE RELAXATION THROUGH THE QUADRUPOLE INTERACTION IN SOLID BCQ 71 APPENDIX C LINEWIDTHS AND MOLECULAR MOTIONS IN aCO 79 LIST OF REFERENCES , , 87 BIOGRAPHICAL SKETCH 90
PAGE 6
LIST Or FIGURES Figure Page 11 Crystal structure of aCO. The molecules ere aligned parallel to the body diagonals 2 12 .Crystal structure of pCO. Open circles representing the cage structure denote hep positions of molecules. Only the center molecule is illustrated which processes around the crystal C axis with angle 9" near the magic angle , 4 13 Phase diagram of CO in the PT plane, The 4 )... 38 44 Effective coupling constant as a function of temperature in BCO 41 51 Temperature dependence of the pure ouadrupole resonance of 17 in aC0 ' Â„ 46 52 Temperature behavior of the 17 NQR linewidths in aCO. The insertion shows some effective T. measurements 48 Vi
PAGE 7
Mqur^ Page 53 Three dimensional representation of the r; NQR spectra in aCO. For clarity the linewidth has been enlarged by a factor of 2 , 49 54 The reduced temperature behavior of the reduced NQR frequency,. 56 Al Quadrupole induced transition probabilities among the Zeemafi energy states for spin I = 5/2... "... 59 A2 The Euler angles. First, a rotation of angle a about the Z axis. Secondly, a rotation of angle 6 about the V axis. Finally, a rotation of angle y about the 1" axis 62 vn
PAGE 8
Abstract cf Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUCLEAR RESONANCE OF 17 IN LIQUID AND SOLID CARBON MONOXIDE By Funming Li December 1979 Chairman: James R. Brookeman Major Department: Physics A study of molecular motions in various phases of carbon monoxide has been performed through nuclear magnetic resonance (NMR) spinlattice relaxation time T, measurements in the liquid phase, T, and quadrupole perturbed NMR spectra in the 3phase and nuclear quadrupole resonance (NQR) linewidth and T, measurements in the aphase in a sample enriched 24.5% with 17 . The tumbling rotational motion in the liquid phase and its temperature dependence are obtained. An activation enthalpy of 198 Â±3 cal/mole is derived from the T, measurements. Information on features of the precessions! motion about the crystal C axis in the 8phase is also obtained. The effective quadrupole coupling constants are compared with the static quadrupole coupling constant obtained in the NQR measurement of the aphase. The former is shown to be three orders of magnitude smaller by precessional motion averaging. The temperature dependence of the effective quadrupole constant is also derived. It is shown that, viii
PAGE 9
as the temperature increases, the average deviation of the processional angles from the magic angle increase from 3.23 t0.06Â° at 61.9 K to 3.36 Â±0.06Â° near the triple point. T. measurements in the solid 3 phase were also obtained. In the aphase the quadripole interaction is sufficiently large enough to perform a pure quadrupole resonance experiment. The measurements in the aphase provide the magnitude of the intramolecular quadrupole interaction. Furthermore, the NQR i7 iinewidths exhibit a peculiar and unexpected behavior with a maximum around 37 K and a divergence for T < 28 K. The effective T measurements in the temperature range from 49 * 55 K support the assumption that the linewidth is determined by the molecular motions in the aphase. This behavior has been analyzed on the basis of a model of molecular recrientational processes. These processes could account for the residual entropy previously observed in the crystal. Comparison of the data with the theoretical picture allows one to derive information on the details of the molecular reorientations and their temperature dependence. IX
PAGE 10
CHAPTER I INTRODUCTION 1 . 1 Physical Properties of CO Carbon monoxide is one of the most interesting diatomic: molecular crystals and has been studied extensively [1121. Besides its structural simplicity and its similarity to other diatomic molecular crystals [6,9,13,14], CO exhibits some peculiar behavior which is probably related to the electric dipoledipole interaction of the molecules [5,15]. Since the observation of residual entropy [4], interest has been focused on the possibility of an ordering transition. Gill and Morrison extended the studies to lower temperatures and found no transition occurring [10]. However, an anomaly in heat, capacity near 18 K has been reported [16]. A theoretical calculation of the rate of ordering based on the model of a twofold barrier for endoverend rotation in solid a C0 found the rate to be unobservably slow [17]. 1. 1. 1 Crys tal Structure of CO There are two solid phases, a and . of CO. At low temperatures, Xray diffraction studies showed that the crystal structure of the aphase is primitive cubic with a basis of four molecules per unit cell and that the molecules are aligned in the body diagonal directions as shown in Figure 11. However, two space groups Pa3 (T?) [IS] and P2.j3 (T ) [1] are compatible with this arrangement. The P2,3 space group differs from Pa3 in that each molecular center is displaced by a small amount 6' along the direction of the molecular axis. The problem concerning which is the correct structure for a C0 is still in dispute [19]. 1
PAGE 11
\_i VO V Â»3Â« ^^tl f\ j q! 8K o A* 00 X O G) n ... CS^ KJ 1 f.,.^HS) V,. Q ) o Figure 11. Crystal structure of aCO. The molecules are aligned parallel to the body diagonals.
PAGE 12
In the 3phase which occurs above 61.6 K, the crystal structure determined by Vegard [2] was that of BN 2 , and subsequent Xray diffraction studies of bN 2 [20] deduced this space group as P6 3 /mmc(D!ih). In this regard one considers the crystal structure of 3CO, the same structure as SN 2 , to be hexagonal close packed. It was found that Xray form factors were equally consistent with a dynamic model where the molecules precess about the Caxis at an angle e' as shown in Figure 12, or with a static disordered model in which the molecules are randomly distributed among the 24 general positions of space group P6 3 /mmc [20]. 1.1.2 P hase Diagram of CO The phase diagram of CO has been determined by Fukushima, Gibson, arid Scott up to a pressure of 1.75 Kbar using the change in nuclear spinspin relaxation time T" 2 of C 13 as an indicator of the transition [21]. The phase diagram is shown in Figure 13. The a, 6 phase transition is at 61.6 K at equilibrium vapor pressure, and the triple point is at 68.15 K. Applying the ClausiusClapeyron equation, Â— = Â— , to the phase transition boundaries the volume increments at zero pressure are deduced to be AV m =2.50 cm 3 /mole &nd AV Qg =0.92 cm 3 /mole [21]. While the change of the volume in N 2 is av^ = 2.5 cm 3 /rnole [22], which is comparable to CO. However, iV , in Â•'.',, has a The compression experiments found no new phase at higher pressures [25], This is unlike N 2 where a new phase was found at higher pressures with a tetragonal structure [26]. 1.1.3 Specific Heat and Other Thermodynamic Properties of C O The heat capacity of CO has been measured [3,4,10] and is shown in Figure 14. Tfie analysis of the results is compared with the entropy calculated from the band spectrum of CO in the gas phase. The discrepancy
PAGE 13
mzzzzi^ij Figure 12. Crystal structure of 3CQ. Open circles representing the cage structure denote hep positions of molecules. Only the center molecule is illustrated which processes around the crystal C axis with angle o' near the magic angle.
PAGE 14
j r T~ i r i.S 1.4 r i . Â• r 1.0 hi m Id .4 L J L__L_, ?,0 40 60 GO TEMPERATURE 100 K Fiyure 13. Phase diagram of CO in the PT plane. The ct6 transition temperature at equilibrium vapor pressure is 61.6 K. The triple point is at 63.15 K. {Data are from Ref. [21]).
PAGE 15
T is s +* + Â•< +* e 4. K. J.. h. MX Â•**Â• G s <* a. 13 I Â— I Â— O O rÂ— I I E o c so a; i. JSÂ« c , 1 Â«J "*" to m vf+ rÂ— <5 JC E HO Â» Â— S4' O T5
PAGE 16
was found to bo approximately equal to R In 2. This was described as frozen in endforend disorder [4]. Mslhuish and Scott [15] calculated the energy for the oriented and disoriented lattice and estimated the Curie temperature for the orderdisorder transition. They found Tc to be ^5 K for CO. However, heat capacity measurements on solid aCO have been carried to 2.5 K by Gill and Morrison with no indication of the occurrence of the transition [10]. The adiabatic compressibility x s may be calculated from the sound velocities and density using the equation X s Cp (V* f^)]1 (!_,) where V Â£ and V t are the longitudinal and transverse sound velocities, and p is the density. Voitovich et al . [27] measured V Â£ and V t and thereby x was obtained. While from the thermodynamic equations one has X T = x s + H 2 TV/C p (12) and x T /x s = C p /C v (13) where x T is the isothermal compressibility, Â°, is the thermal expansion, V is the molar volume and C p and C y are the specific heats at constant pressure and volume respectively. Krupsii et al . measured the thermal expansion of aCO and applied the above equations using x from [27] and C p from [4,10] to give a complete table for lattice parameters, molar volume, volume expansion, specific heat at constant volume, and isothermal compressibility [18].
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1 , 2 Nuclear Magnetic Resonance Consider an isolated nucleus in a steady magnetic field fto and suppose that the nuclear spin number I is greater than zero, so that the nucleus possesses a magnetic moment. Quantum mechanically the angular momentum of the nucleus is quantized along the applied field direction with the magnetic number m taking values of I, 11,..., (11), (I). The Hamiltonian can be written as *"Â« t % (14) where y is the nuclear magnetic moment and Ho is an externally applied field. The magnetic moment is related to the nuclear spin angular momentum by vtit = y (15) where y is the gyrcmagnetic ratio of the nucleus, h equals Â£where h is C.T\ Planck's constant. The components of the magnetic moment are thus given by the (21+1) value of y, y(Il)/l,. . . , y(Il)/I, v . The energy levels of the nuclear magnet in the mag?ietic field Ttes are therefore equally spaced taking values of myHo/I, where m ^ I, (11),..., (11), I. From the classical point of view the nucleus may be regarded as a magnet dipole v precessing about the direction of the applied field fio. The rate of precession is given by the well known Larmor angular frequency u L = yHo (16) If an additional small magnetic field rh is applied at right angles to "fio, the dipole y will experience a torque of y x % tending to increase the angle e between y and 3o. If T\i is made to rotate about to as axis, with
PAGE 18
the same frequency as the Larmor frequency m, , then the angle will steadily increase. For rotation frequencies differing from u>o the coupling between jj and Hi will merely produce small perturbation of the precessiona"; motion with no net affect [28]. Hence the application of a radio frequency field Hi at a frequency u>. will cause energy to be absorbed by the nucleus. The experimental spectrum will be an infinite sharp line at the Larmor frequency
PAGE 19
10 1. 3 Nuclear Quadrupole Resonance The interaction of a charge distribution with an electrical potential due to the external sources is given by E = [p(r)V(r)d 3 r (17) where V(r) is the electrostatic potential. Regarding the nucleus as a charge distribution,, and expanding V(r) about tne origin of the center of mass of the nucleus, one obtains E = V(0) P (r)d 3 r + EY a J^pfr)d 3 r + \ \ V^jx^pCr^r (18) where x , a = 1,2,3, stands for x,y,z. respectively and v = 3* lr o v = 2SL. tf = o o 3X U ' V.6 3x 3X P U Â• The first term on the right hand side of (18) is the electrostatic energy of the nucleus taking the nucleus as a point charge. The second term involves the electric dipole moment of the nucleus. Since the wave function of the nucleus has definite parity, p(r) = p(r) , the electric dipole moment of the nucleus vanishes [29]. One is only interested in the third term, the electric quadrupole interaction now. By introducing and using the fact that L SX 3X a a a one has for the quadrupole term In quantum mechanics, one considers p as an operator
PAGE 20
(ao) > W) & ' * K=p,StÂ„Â„ S ^ V 010) and defines the operator 0/Â° p ' as ^ P) =i^VaV 2 ^ (0P) (^ v 2 (1 " 11} = e K=protons (3x aK X 3K ' S a& r â€¢ From (19) one writes the Hamiltonian jr = 11 V ( Â°P) M12) The WignerEckart theorem states that the set of matrix elements of an irreducible tensor operator A Â£ differs from any ether irreducible tensor B only by a constant factor [30]. The tensor operator Q^ op ^ then is related tc the tensor operator Q(I) as C = e K=protons (3x ,K x ,3< " 6 aS r2) = CQ(I) I I, tI.I C (3 a 2 2 5 ag l2) . (113) Since Q^Â° p can be written as combinations of irreducible tensors like Q(I). Defining eQ, quadropole moment of nucleus, as e ^" <1 1 l e K=prLns (3^r K *)l,I> (114) and operating on the matrix element mm' in (113) for m = I, m' =Â• I where I is the nuclear spin, it follows that
PAGE 21
12 In order Tor C to have a physical meaning it follows that one needs I> j to have a quadrupole interaction. From (112), (113), and (115) it follows that o,3 ! Q = 6TT2tnT ,1 v ae C ! Ci B i B + V a } Â«.b I2] (1 " 16 > In the principal axes of the field gradient, one defines eq = V ?7 and the asymmetry parameter V V . V XX V YY whers V XX' V YY' and V ZZ are the P ri ' nci P a " ! f*eld gradients. One then has from (116) "q = 4iM?ry ^ 3I z I? + ^ T x " *?Â» 017) One can also write (116) in the laboratory frame as (118) where VÂ„ = V V Â±l = V xz * ^yz V Â± 2 4^zz" V yy )Â±iV xy and I + = I x + il , I_ = I x il are the raising and lowering angular momentum operators respectively. The quadrupole interaction is therefore due to the interaction of the nuclear quadrupole moment and the electric field gradient produced at the nuclear site by the external electric sources. Intermodular electric
PAGE 22
13 sources usually provide only a small amount compared with those of intramolecular origin. Nuclear quadrupole resonance (NQR) therefore provides a sensitive method to study the molecular motions as well as the electronic structure of the molecules. Comparison of the quadrupole coupling constant (QCC) of the free molecule obtained from microwave studies and that from NQR measurements in the solid phase provides information about the alteration of the electronic distribution in the solid state. NQR can also be used to study crystal structures, phase transitions, and impurity effects [31]. Nuclear quadrupole interactions usually provide a yery effective mechanism for spinlattice relaxation in the liquid phase. Thus it permits a direct determination of certain correlation times characterizing the molecular dynamics. The study of the linewidth and resonance frequency also elucidates the molecular motions in the solid phase.
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CHAPTER II EXPERIMENTAL PROCEDURE 2. 1 Th e Sam ple The predominant naturally occurring isotopes of carbon and oxygen do not. possess nuclear moments. Therefore an MMR study of CO has to resort 13 to enriched isotopes. C provides one of the possibilities [21,32,33]. For a comprehensive study of spinlattice relaxation and quadrupole reso1 7 17 nance in the low temperature phase of CO, 0' is the best choice. with nuclear spin I = j [34] has a nuclear quadrupole moment which, interacting with the molecular electric environment, provides the dominant mechanism of spinlattice relaxation. From nuclear quadrupole resonance experiments which can only be performed with nuclear spin I>1 , the quadrupole coupling constant of in CO can be deduced. Therefore a great deal of information about the CO molecules can be learned. One liter of gaseous CO enriched to 24.5% in v/as purchased from Prochem U. S. Service, Inc., New Jersey. The sample was claimed to have 99.9 atom % of C 1 ". 2. 2 The Cryostat and Temperature Measurements The construction of the cryostat is shown in Figure 21. The condensed sample was held in a KelF cavity of 0.43 irich in diameter and 0.78 inch long. Surrounding the sample chamber was an rf coil wound from 30 turns of equally spaced #26 cotton covered copper wire. The inductance of the sample coil was approximately iO yH which was used for NMR measurements operating at 4.098 MHz. Another sample coil made of ^100 14
PAGE 24
15 Â£N 2 both N 2 orHebath vacuum jacket radiation shield shcei vacuum or heater exchanger 1 Â— brass can _ Â— , platinum thermometer heater wire sample line copper block KclF sample holder scir.pl.; coil sample chamber i'ie gas heater exchanger Figure 21. Sample cell and temperaturecontrolled cryostat. Â•V
PAGE 25
turns of #35 enamel coated copper wire with an inductance of ^100 yH was used in the NQR measurements at ^1.1 MHz. The sample coil was connected to the spectrometer through the small stainless steel sample line. He gas was admitted into the copper block on which a platinum resistance thermometer was mounted. The He gas served as a heat exchanger between the sample and the copper block. The platinum thermometer calibrated by the National Bureau of Standards was used for two purposes. First, the conventional temperature measurement was achieved by connecting four leads to a Leeds and Northrup Mueller bridge. Secondly, the output of the galvanometer was phase shifted 130Â° then fed back to the power supply which controlled the current to the heater. With the Mueller bridge the temperature can be measured accurately to 0.01Â° K; however, this experiment, due to the uncertainty in the different readings of the N and R mode in the Mueller bridge (because of using feedback temperature control) a 0.05Â° K uncertainty should be assigned. Between the brass can and copper block, He gas could be introduced to serve as a heat exchanger in case temperature equilibrium of the sample with the liquid K ? or liquid He was desired. To achieve a sample temperature higher than the surrounding bath, this space was evacuated to 10" ' torr or less. A thin brass sheet radiation shield held in place with small Kel F p : eces reduced additional radiation heat loss. Temperatures between 50 K and 11 K were achieved by pumping the N 2 bath and above 77 K the heater was applied. For temperatures below 50 K, liquid He was used as a cold bath with the heater on to maintain the desired temperature. Before the sample was introduced to the sample chamber, this space had been evacuated to 4 x 10" 6 torr. To avoid solidified sample blocking the line before it reached to the sample chamber, a vacuum jacket was situated around the sample line.
PAGE 26
17 2. 3 The Electromagnet The electromagnet used in the NMR measurements was a Varian 401233 rotatable electromagnet with a 12 inch diameter pole face and a 3.5 inch gap. The magnetic field was regulated by a fieldial regulated system which utilized a Hall effect probe inside the magnetic field. The maximum magnetic field produced was around 9 KG. Taking long time stability and maximum sensitivity of the NMR signals into consideration, a 7.1 KG field strength seemed to be the best choice. This in turn produced a 4.093 MHz NMR signal in liquid CO. The rotation angle of the magnet could be adjusted to within 1Â° of accuracy. The inhomogeneity of the magnetic field over the sample volume was determined in the liquid phase to be M).l gauss which gave a T of about 3.5 ms. After long hours of operation (^ 6 hrs.) the fluctuation of the magnetic field was found to be 3 x 10~ 4 /hr. or less. 2. 4 The Spectrometer The pulse spectrometer used in this experiment is depicted in Figure 22. This setup was used for the NMR experiments as well as the NQR measurements with only a few modifications of the receiver preamplifier. The high stability General Radio 1051 frequency synthesizer was the main rf source. Since it is a heterodyne system, the frequency was set to 10 MHz higher than the resonance frequency to utilize a 10 MHz amplifier and a phase sensitive detector. This signal when mixed with the gated 10 MHz reference frequency was amplified to approximately 2V peak to peak rf, before it entered the transmitter. The transmitter is adequate to supply high power pulsed output and low power continuous output. The magnetic field produced by the rf pulse was calculated to be ^30 gauss, which is large compared with the linewidths of the spectra being
PAGE 27
IB Figure 22. Block diagram of the pulsed NMR/NQR spectrometer.
PAGE 28
19 studied. The same pulse which gated the 10 MHz frequency controlled and gated the final stage transmitter also. In a pulse nuclear resonance experiment, the induction signal is typically obscured for several microseconds after the transmitter pulse by the ringing of the input turned circuit. This ringing can be reduced by a weak coupling to the nonlinear input [35]. In this experiment, a photoFET Q switch similar to the one described by Conradi [36] was used. During the time the high power rf pulse was on, the D,D ? crossdiode set conducted strongly. The DgD. diodeset was also conducting to protect the receiver preamplifier and the photoFET Q switch. When the crossdiode set D~D, is conducting, point A of the x X cable appears as a high impedance. While the transmitter pulse is off, DD* ceases to conduct and point B is a high impedance. When the rf level reaches 0.5 volt, the cressdiodes D,D 2 turn off. This isolates the transmitter from the L. C. tuned circuit. Unfortunately, in practice the L. C. tuned circuit and the transmitter are still weakly coupled by the capacitor C, across diodes D,D 9 . This coupling ties the voltage across the sample coil to the slow ringing down of the transmitter final stage [37], so the recovery time is prolonge:;. The negative bias FET was adjusted just into the cutoff region. A pulse from the pulse generator turned on a LED diode which provides light to turn on the Q switch. The sourcedrain path appears as a low resistance (^200ft) which provides a low impedance path to ground for the energy of the ringing. The Q value of the input circuit hence was reduced during the time the Q quenching pulse was on, causing it to ring down quicker. Just as the ringing decreases to the order of the free induction signal ('MnV), the Q quenching pulse was turned off to restore a high Q value
PAGE 29
20 in the receiver preamplifier for detecting the free induction signal. The amplified signal was then mixed with the rf frequency (v + 10 MHz) split from the same source. The combined signal was then phase sensitive detected by mixing with the 10 MHz reference signal. The output signal was observed on an oscilloscope and also fed to the Biomation 802 transient recorder which captured the fast signal for slower transfer to the FabriTek 1072 signal averager. A POP 8/E minicomputer was interfaced with the FabriTek 1072 to perform the Fourier transformation of the signal from the time domain (free induction decay) into the frequency domain (spectrum). The result was then plotted on an XY recorder.
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CHAPTER III LIQUID PHASE 3. 1 NMR SpinLattic e Relax atio n In the liquid phase, due to the fast molecular tumbling motions, the quadrupole interaction is averaged to zero. The experimental NMR signal will, therefore, be a single line. The linewidths are mainly the result of the inhomogeneity of the applied magnetic field. The random motions of the molecules also provide the mechanisms for the nuclear spin system to relax when perturbed from its equilibrium state. T, measurements thus can be used to study the molecular motions. In the liquid, the energy differences between the Zeeman levels are all equal. The fast spinexchange mechanism thus maintains a Boltzmann distribution of the nuclear populations among the energy levels [38], i.e. a spin temperature can be defined and the relaxation to the lattice temperature is governed by a single exponential decay. In order to describe the macroscopic relaxation time T, through the microscopic molecular motions, one considers the time dependent Hamiltonian Â«, (t) caused by the molecular motions. This ;/, (tj represents a small perturbation and induces transitions between eigenstates of the Zeeman Hamiltonian. The correlation function of h, (t) is then introduced which expresses a correlation between the two different configurations of a nuclear environment, at two different times. The rate equations among the energy levels are then considered. This connects the relations between the experimental relaxation of the total magnetization and the correlation time of the tumbling motions of molecules. 21
PAGE 31
Consider h^ (t) a small perturbation among the eigenstates  a >,..., e > , etc. with eigenenergies a, 3 etc. of a spin system S. The transition probability per unit time for a transition from state  3 > to state (a >, assuming the system S at time zero is in state JB >, time dependent perturbation theory gives [6] Wpza !*<Â»!*! (t)a><Â« fl ] (f)B>e 1 M t 't) dt + C. C. (31) where W oB a a Taking the average W over a statistical ensemble, and introducing the correlation function G Â„ as ctB G a3 ^ t " t '' = < BlV 1 ^ a >< a ! ff l( t ')l 8 * (32) One uses the properties that the random function < a#,(t)e > is stationary, i.e. G aC depends on t and V only through the difference t V = t. It follows thai W o6 = , j* G o6 ( T ) e1 W fc + c. C. 1 J t G _ (t) e' 1w Â«S T di aB (33) Since one usually considers a time t >> Â— , it turns out that the limits of the integration in (33) can be replaced by + Â» and one has W 1 + '2 Â«8 ^j G (t) e" lw aB T cIt aB (34) In most cases h, (t) can be expressed as a combination of two irreducible tensors of rank k. Let *, (t) I A ( P) (I) B ( P } (t) pk (35)
PAGE 32
23 where A is the spin part of the Hamiltonian and B is that part of the spatial coordinates which is time dependent. It follows that (34) can be written as KÂ« I h\\ < Â«A (P) 6 > V P) (r) e~ i: W d, (36] aB pk V where g(P)( T ) = B (p) (t) B* (p) (tt) is the correlation function of B^ p; (t), There are various possible mechanisms of relaxation, (1) the intramolecular quadrupole interaction, (2) the intramolecular spinrotation interaction, (3) the intramolecular anisotropic chemical shift interaction, (4) the intermodular magnetic dipoledipole interaction and (5) the intermolecular electric quadrupoledipole interaction. However, an estimate of the magnitudes shows that the spinlattice relaxation is almost completely dominated by the intramolecular quadrupole interaction. 3. 2 In tramolecular Relaxation f'echanisms The interaction between the nuclear moments and the electrons in the CO molecule drives the transition among the Zeeman levels of 17 nuclei. In principle there are three relaxation mechanisms which are of intramolecular origin. They are (1) the intramolecular quadrupole interaction, (2) the spin rotation interaction and (3) the anisotropic chemical shift interaction. Consider (1) the intramolecular quadrupole interaction. In Appendix A it is shown that T ] 125 { + ' l 2 t 3 7 '
PAGE 33
cA e 2 0a . where Â— ,^ is the static quadrupole coupling constant, which one has determined in the NQR measurement of CO. in the aphase to be 4.20 MHz and %2 1S tne autocorrelation time of the molecular tumbling motions. If one adepts the value of t r from Raman anisotropic scattering [39] to be x 2 = 3 x 10" 3 sec. at a temperature of 77 K one has i =4.17 sec" 1 while the experimental T, at 77 K is 220 ms. This gives i = 4.54 sec 1 1 In (2) the spinrotation interaction, the time fluctuation of the interactions of the nuclear spins with the angular momentum of the molecule provides a relaxation mechanism for the nuclear spins. Assuming a Langevin equation for the angular momentum and a rotational diffusion of Debye type, Hubbard obtained [40] gh 22IKTt + 2(CÂ„ C, ) : 9h: II.KT L + 1 r J T 2 (38) V s.r. where C, and CÂ„ are the components of the spinrotation tensor perpendicular and parallel to the axis directed from the center of mass of the molecule to the resonant nucleus. I is the moment of inertia about a line through the center of mass and perpendicular to the molecular axis. I is related to the angular momentum of the molecule 3 by iZ = tvJ. t, is the correlation time Tor the angular momentum of the molecule and T? is the correlation time for molecular reorientation. Liquids near their melting points are often described by a reorientation process called the "diffusion limit" in which the molecules turn through a very small angle between collisions and t and t, are related c J by the following eouation [41]: _I J T 2 6KT (39)
PAGE 34
25 For linear molecules one has 3 C 2 ,. = 2 Ci + Cm = 2 C 2 . Assunrinq the ST"! " * J diffusion limit for the temperature of interest one has from (38) and (39) C; s.r. 'ML 3h2 If T 2 (310) Using the value C ff = 23.1 KHz [42] and t 2 from Raman experiments [39] one finds Ls r ~ 3.6 x 10~ 5 sec" 3 which is negligibly small compared with (37). Consider (3) the anisotropic chemical shift. The interactions of the external D.C. magnetic field with the electrons around the resonant nucleus in a molecule produces a magnetic field at the nuclear site. The anisotropy of this coupling under the molecular motions may introduce transitions between energy levels. In the fast motion regime one has [43] h a.c. , .0 (311) where y is the gyromagnetic ratio of nucleus 0. H is the applied D.C. magnetic field. oÂ„ and o x are the parallel and perpendicular parts of the chemical shift tensor in the principal reference frame respectively. t is the correlation time for molecular tumbling motions described before. Using the value cited by Appleman and Dai ley [44] that (oÂ„ oj for the nucleus in CO is 4.6 x 10"Â° and 7,1 KG for the external magnetic field H one calculates = 5.6 x 10" b sec" a.c One notes from the small value that the relaxation due to the anisotropic chemical shift interaction can be neglected. By neglecting the spinrotation and the anisotropic chemical shift the calculated value of
PAGE 35
26 4.17 sec" for the intramolecular quadrupole interaction is in good agreement with the experimental value of 4.54 sec . 3. 3 Intermolecular Relaxation Mechanisms The spinlattice relaxation may also arise from the interactions of the 17 nuclei with the intermolecular fields. There are two mechanisms which, in principle, could induce transitions in the absence of a rf field. They are (1) the intermolecular magnetic dipoledipole interaction and (2) the intermolecular electric quadrupoledipole interaction. Consider first (1) the magnetic dipoledipole interaction. Following Bloembergen, Purcell and Pound [45] by assuming spherical molecules and a Debye model of diffusion, one would have a ~ contribution from inter'l molecular magnetic dipoledipole "interaction given by the following equation: fll 3,t ^ h % [rj i.d. r it air (3 ' 12) where 2a is the closest approach of the neighboring molecules, D is the diffusion coefficient of the liquid and N is the number of molecules per cm 3 . Using D as 2.9 x 1C~ 5 cm ? /sec [46] and the density of liquid CO as 0.803 g/cm 3 , which gives N = 1.72 x 10 22 /cm 3 and 2a Â« 3.87 x 10" 8 cm, one obtains ] . d 4.5 x 10" 6 sec" 1 . Now consider (2) the intermolecular electric quadrupoledipole interaction. It is well known that the CO molecule possesses a dipole moment Â— 1 8 of 0.11 x 10" esu [5J. The electric dipoles of neighboring CO molecules thus produce a field gradient at the 17 nucleus site. This is similar to the intramolecular quadrupole interaction except now the field gradient
PAGE 36
21 is due to remote electric soirees. Assuming again a Debye type diffusion for the liquid following the calculations of Bonera and Rigamonti [47], one has 9tt e 2 Q ? M 2 N i n " to f2rrÂ£ < 3 i 3 ) i .q. h Da where a, N , and D have the same definitions as discussed in the magnetic dipoledipole interaction. eQ is the quadrupole moment of nucleus 17 , and M is the dipole moment of the CO molecule. Taking the value of Q measured by Stevenson and Townes [43] as 0.026 x 10~ 2l+ cm 3 , one calculates '1 ft (y r I 4.4 x 10 H sec 1 . l V i.q. From the above considerations one immediately concludes that the spinlattice relaxation in liquid CO is completely dominated by the intramolecular quadrupole interaction. This is usually true for NMR spinlattice relaxation of most substances possessing a nuclear quadrupole moment in the liquid phase. 3. 4 Temper ature Dependence of SpinLattice Relaxation Time Using the conventional j x Â« pulse sequence , one measures the spinlattice relaxation time T, in the liquid phase. Figure 31 shows the experimental T, values at different temperatures. Data were taken during a period of several months. Due to the uncertainty in the T, measurements, a 10% error should be assigned to each point. In view of the cluster of points taken at different times, it shows no impurity is effectively absorbed in the sample during the course of experiments. By assuming the intramolecular quadrupole interaction is the only contribution to T, and using the NQR measurements extrapolated to zero temperature, one would have ~& ^ 4. 20 .MHz. From this one has the following relation for liquid CO
PAGE 37
28 Â°jef^
PAGE 38
29 5.98 x 10' Â•H l (314) Using (314) and measurements of T, , the temperature dependence of x_ is shown in Figure 32. For comparison the to values of N are shown in this picture [49]. Assuming an activation model [50], such as RT (315) where AG is the difference in the Gibbs function between the initial and activated states, R is the gas constant and t is a constant and is expected to be a relatively slowly varying function of temperature and pressure [50]. Substituting (314) into (315) one has T, = C e i AG RT where C is some constant. From the thermodynamic relati on AG = AH + T f3AG (316) (317) it follows that AH (318) [ 3 In T] ] Â•R Â— ~yt \ '.. J AH is the activation enthalpy, a measure of the energy required to reorient a molecule under conditions of constant pressure. One notes that it is a reasonable approximation to assume that the pressure along the equilibrium vapor pressure curve is constant [50]. 3. 5 Discussio n Ewing [7] compared the infrared spectra in the gas and liquid phases of CO and suggested a potential barrier of V = i V (1 cos 2e) with V Q = 120 cal/mole in liquid CO.
PAGE 39
30 u Uf
PAGE 40
31 Amey [51] usee the librational model suggested by Brot and compared this with the farinfrared spectrum of liquid CO and calculated a barrier of 190 cal/mole. These are compared with our experimental activation enthalpy calculated by least square fit to be 198 Â±3 cal/mole. Considering the different types of experiments, it can be regarded as satisfactory. In fact, NMR Tj measures the barrier height directly and this value is more fundamental than the other parameters. The low value of the barrier potential indicates that CO molecules in the liquid phase undergo nearly free rotation. If one uses the experimental results for the coefficients of diffusion in liquid CO [46 J and assumes a Debye model of viscous fluids, the StokesEinstein equation yields T W(319) from which it can be calculated that at 77 K, x ^ 2.8 x 10~ 12 sec. This may be compared with the value of t = 2.7 x 10~ 13 sec. derived from the T^ measurements. In liquid ,\' 2 it was found also that use of the Stokes formula for the reorientational correlation time overestimated the value by one order o x " magnitude [49].
PAGE 41
CHAPTER IV SOLID BPHASE 4. 1 Quadrupole Perturbed N.MR Spectra . The hexagonal crystal structure of 3CO has already been discussed in 1. 1. 1. However, the inability of Xray diffraction to distinguish between a static disorder and a free precession model has led to some confusion in the literature [20]. The experimental data presented here are consistent with the precession model. It is concluded from these experimental NMR measurements that the effective nuclear quadrupole coupling constant of 17 in 8 CO is reduced by a factor of ^ 10 by motional averaging. This demonstrates that the molecular motions in BCO are characterized by a time shorter than the reciprocal of the static quadrupole resonance frequency, * 10~ G sec, and that the reorientations are not isotropic because a nonzero coupling constant exists. The Hamiltonian of the 17 nuclei in BCO can be written as H Â« Â«2 + ^ (41 ) where h 7 represents the Zeeman energy while h the quadrupole Hamiltonian. is small compared to #, and can be treated as a perturbation. To a first approximation, the shift in frequency of state rn> can be written as % = f HwftT v Z2 (3i 2 z n in > (42) where m> are the eigenstates of Zeeman energy and m is the magnetic quantum number of the nuclear spin 1 / . VÂ„, is the electric field zz 32
PAGE 42
33 gradient at the O 17 nuclear site in the laboratory frame. From (42) 5 2 knowing I = * for 17 , one has %, 1 Â§ v z Z < 3 * 2 *> Â• < 4 3 > In order to relate the V in the laboratory frame to the static value of the field gradient V zz in the molecular frame, one considers the precessional motion shov/n in Figure_41. In the case n 0, the second order irreducible tensor transformation enables one to have V zz = 1 (3 COs2 6 " " 1J V ZZ (4_4) where V 7 , is the static electric field gradient in the molecular frame, while e'is the instantaneous angle between the molecular axis and the external magnetic field H . One also assumes the field gradient is of intramolecular origin. If one applies the addition theorem, (44) can be written as I (3 cos 2 e' 1) (3 cos 2 9 1) V 77 Â• (45) The effective V would be given by the time average of the right hand side of (45) V_ z = \ (3 cos 2 61) (3 cos 2 8 1) V zz = \jz cos 2 e1) (3 sin 2 Y cos 2 $ 1) V z7 (46) It turns out from (43) and (46) that Av = r4Â§&~(3 cos 2 e1) (3 sin 2 Y cos 2 * 1) (3m 2 2) (47) m n ibu 4
PAGE 43
H& H Â© Figure 41. Definition of angular variables used in the text to discuss the NMR rotation patterns in 3CQ. The molecule is described classically as precessing rapidly about the crystal C axis. Tl] is the field produced by the rf coil. H is the externally applied magnetic field. is the angle between H and the projection of the C axis on the plane perpendicular to fl ] .
PAGE 44
and for various m one can immediately write Av +(; = ^jjjj(3 cos 2 o1) (3 sin 2 Y cos 2 $ 1) e 2 Qq Av 2 , 30h (3 cos 2 6' 1) (3 sin 2 Y cos 2 * 1) (48) (48') (48") AvT ^tM(3 cos 2 e' 1) (3 sin'y cos 2 <Â£> 1 ) . The quadrupole splitting is of the order of KHz while the static quadrupole coupling constant is of the order of MHz. This small nonzero quadrupole frequency might result from a value of 6' close to the magic angle 9 , where 3 ccs 2 e = 1. One supposes 9' is characterized by a distribution function f(e), where e = 9' 9 , which is an even function of e and normalized. It follows that 1} 3 cos 2 9' 1 = f(e) [3 cos 2 (e + e) 1] dt I o J..2L 2 f f(e) [3 cos 2 (e + e)l + 3 cos 2 (e e) Â•1] d< f(e) e 2 de where e is the root mean square of the angle e, i.e. magic angle 8 . One defines and T Av Â±3 * Av Â±5 7. 2 ? Av Â±1 Â± Av Â± 3 2 2 Â±v< 2 > * Â„<Â» (49) '" derivated from (410) (41 0)
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36 And combining (48) (49), (410) and (410'). it follows thai (2) 6e 2 Qq ?/>Â•? ? , ,x v = "RnTh Â£ l ( 3 sin Y cos 2 9I) and 80h (1) 3e 2 Qq /, Â• ? > ,> v ~80h e o ^ sin Y cos " ^ (411) (411 From the experimental perturbed spectra it is obvious that in several cases a large portion of the sample grew as a single crystal of 3CO. The evidence comes from the rotation patterns of the separations of the satellites as shown in Figure 42, in this case at 64.7 K. The perturbed spectra at different angles are also shown in Figure 43. One notes that the linewidths are mainly due to the inhomogeneity of the applied magnetic r q field. Since the separation between m ^ +^ and m a +y and tnat between 5 3 m = j and m = j are symmetric, from the spectra one can only determine and 1 2v 2v (2) 0) 6 e 2 0q 40 3 e 2 Qq _ 2 40 h e o 3 sin 2 v cos 2 6 1 3 sin 2 y cos 2 9 1 (412) (412') Noting that 2v ment of 2v^ ( 2 ) ,(1) ,(i) 2 \?.v { ! \ while one measures [2\T '[, the measurean be used as a supplement to check the accuracy of the 2 measurement. However, the rotation of the H field at an angle 3 $ = would give 3 sin 2 y cos 2 9 1 1 = 1 , while at an angle of 9 shifted by j more would give J3 sin 2 Y cos 2 9 1  = 3 sin 2 y 1  . It is obvious for symmetry reasons that the rotation pattern of (412') as shown in Figure 4.2 is a periodic function of it, having two local maxima. As in Figure 4.2 it is at 9 9 Q = 32Â°, where
PAGE 46
1\lk **. i v \ * V /' jVaIAa to Figure 42. The NMR perturbed spectra of 8 CO at different ($ angles, where
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c O (3 O i o
PAGE 48
and : 2v ( 2 ) *.Â£*!& ' } J 40 h 40 h ff o (413) (413) In fact the values of [3 sin 2 y cos 2 6 1 as a function of have two local maxima. For ~ < sin 2 y < 1 the first (larger) maximum located at = 0Â° while for < sin 2 y < [t is at * 90Â° [49]. Comparing with different single crystals, it is possible to determine which maximum corresponds to = 90Â° so that 3 sin 2 y cos 2 4> lj = 1 (414) It is clear that one can also have an angle y such that 3 sin 2 y cos 2 ' 1 From (414) one determines sin y = Â± COS <}>' (O (415) From the experimental value of 2v v '\ and the value of the static quadrupole coupling constant v/hich one found in the NOR measurements in aCO one can determine the value of e . If one defines the effective o quadrupole coupling constant as <<*!&>. . SO Â« 1 "" h ' pff "F V ZZ 7 < 3 cos2 e ' 1 ) eft = eiQa Â£ 2 2h Â£ o (416) one would have from (412), (412") and (414)
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40 .(0 _6 e 2 Qq 20 h > eff (417) and .(i) 1 .eiQa > >0 " h eff (417*) The experimental data from the separations of the satellites give < =tP > = 7. 04 KHz at T = 64.7 K eff Comparing this data with the static quadrupole coupling constant of 3.83 MHz one finds that e!Qg.> . 1.67 x 10 eff Â•3 e 2 Qq h (418! which is three orders of magnitude smaller. Figure 44 shows the effective quadrupole coupling constant for several temperatures of BCO. With increased temperature one expects e to increase, causing the effective quadrupole coupling to increase with temperature. This is consistent with the experimental results where e = 3.23Â° Â± 0.06Â° at T = 62 K and e Q = 3. 36 Â± 0.06Â° at T = 68 K near the triple point. 4. 2 T] Measurem ents in s CO It is shown in Appendix B that if all lines in the perturbed spectrum are saturated and one assumes W, = WÂ« where W l W 5 W 2 =2W 5 1 2 2 (419) (419")
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4 1 o I CO. t o o CJ O f' N" Q. o m
PAGE 51
42 then the spinlattice relaxation is governed by a single exponential decay 1 4 T7 T, " 5 W 1 (420) In the course of the experiment a sequence of 16 pulses was applied before a ^ T j P u ' se sequence was employed to measure the spinlattice relaxation time. It was found that within experimental error the nuclear magnetization recovered as a single exponential. Furthermore the normal t 5pulse sequence would give an almost identical result except for very small t. This may come from the small separations between all lines and a Â£ pulse would suffice to saturate all the lines. The single exponential decay supports our assumption of W, = WÂ«. Following the same step as in liquid CO, and assuming the relaxation is dominated by the intramolecular quadrupole interaction, one has \2 3 125 e2Qc (421) where t' r now is the autocorrelation time for the precessional motion around the crystal C axis. Using Â— ip from NOR measurements in the aphase one has the following relation between T, and t': T 2 = 5. S3 x 10 14 1 422) Shown in Figure 32 is the temperature dependence of x ' and T Â„. Although C. L. there is a different meaning for "C and T ? , they specify the correlational motions in Bphase and liquid CO respectively. Figure 32 shows that there is a discontinuity at the transition temperature. It is reasonable that the characteristic time of the motions in eCO takes a longer period than that in liquid CO, while the discontinuity in the opposite sense of
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43 3N ? near the transition temperature is not clear at this moment [52]. For 3NÂ« the same assumptions were made as one did in eCO, namely that W, = W ? and that all lines are saturated. One therefore has 1 = 5W, (423) 'l ' in liquid N ? as well as in gNp.
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CHAPTER V SOLID aPHASE 5. 1 Mgjecular Propert ies of aCO Solid aCO is one of the simplest diatomic molecular crystals and has been an interesting subject for studying the lattice dynamics of molecular crystals [6,11,12,5355]. The crystal structure has been discussed in 1. 1. 1. One notes that in the face center cubic crystal structure the locations of the molecules and the different orientations along the diagonal can be related to the molecular dipoledipole and quadrupclequadrupole electrostatic energies for the various pair orientations [15]. The specific heat study of CO by Clayton and Giauque [4] observed a residual entropy of approximately R In 2 at low temperatures of aCO. Gill and Morrison [10] extended the measurements down to 2.5 K with no indication of the occurrence of an orderdisorder transition. However, it can be mentioned that a small anomaly has been observed in the heat capacity of aCO near 18 K and ascribed to the freezingin of molecular headtotail reorientation [16]. 5. 2 Nuclear Quadrupole Resonan ce in aCO To study the molecular dynamics in aCO, one can employ the 17 nuclear quadrupole resonance, since the quadrupole coupling constant is large enough to be able to observe the NQR frequency directly. For 17 in aCO intermolecular contributions to the field gradient are \ery small compared to the intramolecular interactions. Each molecule can therefore be regarded as an isolated spin system. The electric field 44
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13 gradient (t'FG) tensor at the nuclear site is thus axially symmetric and the molecules in aCO differ little from that in the gas phase. The quadrupole Hamiltonian can be written as VIi^Bt) l" 31 !: 2 ] (51) where eq (T) is the time average of the principal component of the EFG tensor taken over the vibrational states at temperature T and eQ is the nuclear quadrupole moment. It follows that Â»5 3 4*m (52! Â±2" ^Â±2 " = J e 2 Qg(T) 4 Â«4 20 % (5 3) The zero temperature NQR frequency would in turn determine the static quadrupole coupling constant (QCC). Since by extrapolating the NQR frequency to K one gets ^ 5 3=1 .15 MHz Â±2 <Â± 2 With zero point correction estimated from Raman spectra [55], one obtains e 2 0q 4.20 MHz r = u '^ : mz (54) The measurements of the NQR frequency versus the temperature are shown in Figure 51. The static QCC was also used to derive t ? _ and T x in the liquid and 3phases respectively. The microwave experiment gave 4.43 MHz f 0r __Â£ This was refined by Flygare and Weiss to be 4.48 MHz by considering in addition the spinrotation interaction in the microwave spectrum [56]. Since the same averaging factor due to molecular stretching motions may exist, in both solid and liquid phases, the experimental
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46 Figure 51. Temperature dependence of the pure quadrupole resonance of 17 in aCO.
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47 NQR determined QCC from the aphase should be more appropriate for deriving the correlation time of the molecular tumbling motions in the liquid phase. 5. 3 NQR Linewidt.hs a nd Molecular Motions The most interesting thing that occurs in the NQR measurements of the aCO is the anomalous behavior of the linewidths. Figure 52 shows the temperature dependence of the linewidths and some T, measurements, while Figure 53 shows a three dimensional representation of the spectra. As one can see there is a maximum at ^ 37 K and a divergence below % 28 K. One might speculate the broadening of the linewidths is related to static effects such as anomalies in the expansion coefficient, metastable states, strains, etc. However, the thermal expansion of aCO shows no anomalous behavior in this region [18] and the broadening persists for long periods of time K 5 hours). As one will see later, the spinlattice relaxation measurements in the temperature range 48 * 60 K seem to support the hypothesis that the linewidth is actually dominated by the motional contribution. The occurrence of orientational disorder in the aCO crystal has been expected since the first estimate of the residual entropy was made [4]. However, the frozenin static disorder cannot explain the peculiar behavior of the linewidth observed. In the light of the temperature behavior of the 17 NQR linewidth and spinlattice relaxation rates, it will be assumed that the disorder in the orientations of the CO dipoles is of a dynamic nature at least for T Â£28 K. In the dynamic model one assumes that the molecules execute librational motions around the equilibrium positions plus sudden reorientations among the various directions which are, in principle, permitted by the crystal structure. In order to explain the peculiar behavior of the
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48 Figure 52. Temperature behavior of the 17 NQR linewidths in aCO. The insertion shows some effective T^ measurements.
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49 o o 3 ctmz. o Q. a c SQJ x: c 1Â— >
PAGE 59
50 1 ine.vidth, that the quadrupole resonance is not affected (normal Bayer type) by the reorientation, one assumes the molecule executes Vibrational motion plus a sudden near tt reorientation [57]. In presence of motions, the quadrupole Hamiltonian can bo divided into two parts. H Q = < H Q > + ^Q " < H Q ^ ' ^'^ The average Hamiltonian < H r > gives the NQR frequency while the flues' tuating part drives the relaxation process and causes the line broadening [58]. For 17 in the CO molecule, one assumes a rigid EFG tensor of intramolecular origin with cylindrical symmetry which, as a consequence of motions, moves its principal axes in the laboratory frame. A straightforward tensor transformation gives for the instantaneous EFG components (using s(t) as defined in Appendix C) V xx (t) = 3J (1 3*) (56) Vyy ( t ) . S [i ltÂ£ (ls(t)) 3*2 ^f(1 sit))] V zz (t) ^ [2 + 3*i + 3*?i + 2Â£l (l S (t)) + ^ (1 s(t))] V xy (t) = 'H 1 ^ CT (1 " s(t)) + "" s(t)] v xz (t) = t 9 ^ [ V (1 " s(t)) + s(t) + ~r (1 ' s(t))] Vyz (t) = ^9. [ 4J (i _ S (t)) + Â„ ] where 4>Â„ and (j> x are the inplane and outofplane librational angles with respect to the n reorientation respectively. Â£* is a random variable of i order < <{>Â£ > 2 which allows reversal of an angle which is not exactly n as discussed in [57]. Taking the librational average of (56) and assuming no correlation between Â„, 4> x and * one has
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and 51 ^ S0 3<$Â£>) C57J < V > = e 4 [1 &JÂ£ (1 < s(t) > ) 3 < tf > ] < V zz > eq [1  < tf >  <^ > Â• 34J1 ( I < s(t) > )] < v xy > < v xz > < v y2 > (58) In the casa of disorder, where < s(t) > = 0, one has < V zz > = eq (1 j<$l > 2 < *" > " aT ^ Â• 5 3 The resonance frequency between m ~ Â± * and m = Â± j will be 3 e 2 Qa M 3 x2 3 2 3 * 2 % /eg* V Q = TO h (1 " 2 * * S * " 1 K ^ > 4 * j ' lb y; This is the usual Bayer type NQR frequency. One also notes that for the order phase < s(t) > f v Q tÂ§ ^ga [i f < Â«2 > f < a > 3j f )] (510) which is higher than in (59). One also notes that if reorientation is sufficiently different from ir, i.e. * is a large angle, v Q would be significantly affected. In order to evaluate the linewidth one considers the fluctuating parts of the components of the EFG tensor from (56) and (57) and gets
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'">2 V < V > ^ g (*i < *i > ) xx xx 2 rx rx ' V < V yy > = ^ [? < *?, > *Â£ (s(t) < s(t) > )] (5n) V < V > = zz zz 2 [ + *?. <<{'"> + C(s(t) < s(t) > )] V xy" < V xy > = ^*x C^(l s(t)) . S(t)] v < v > = M ^ s(t) xz xz V < V > = yz yz 3eq I 9 [ *2 (1 S(t)) + Â„] . There are two parts contributing to the linewidth of the quadrupole resonance, i.e. the adiabatic terms and the nonadiabatic contribution due to the lifetime of the energy levels [58]. One writes i .i +; T 2 T' 2 2 1 +1 T + 5 T + 3 Â± 2 _ 2 (512) where 1 T 2 )'(t)t0'(tT) dT is the adiabatic contribution and w*(t) is the departure of the instantaneous resonance from its average value and 7 W P (513) 5 ,5 Â±75, m tj mfj 2 \ 1 W. (514) '3 +3 + injt 3  , m
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53 where W is the transition probability per unit "time from state m to state rrr by the fluctuation terms in (511). By neglecting for simplicity the terms which do not involve the reorientations one would have 9e 2 Qq 20h 1 nf^ h <Â°> (515) where J n (to) is the Fourier transform of s(t)s(t t). Assuming a harmonic oscillator equation for ^(t) and an exponential correlation function s(t) with characteristic time t^', one has 1 f 9e Qg I 2 T 5 <$* 2 > ty . 20 < x l' + 1 J n (j T ; n T Â— Â— Â— Â— Â— 1 + to z iy l l + io 2 Tr 2 r , 2 9 < J n n 4 U H i 2 W J Combining (515), (516) and (517) one obtains (517) T, 1 = f 9e 2 Qq ' 20"fi 5 < 4>*S 3 < $*"> i / 01 + 9 < fr*^> , j 1 TT J TT VU; + ^Â— Jit 5 T5 U II II I 2 4 2 < 6*2: f
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1 3 In the fast motion regime, where JÂ™ (0) ~ JÂ™ [j a) Jj (w) J^ (tw), one can neglect terms involving < $**+>, so that (518) reduces to 2
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directly proportional to the motional correlation' times. It can be speculated that the lack of symmetry of 5v versus T around T 37 K is related to the breakdown of the perturbation approach in the slow motion regime. For temperatures lower than 32 K, 6v exhibits a dramatically increased broadening in a few degrees up to 10 KHz and the NQR signal is no longer detectable below ^28 K. Careful attempts for various temperatures down to 4.2 K over a wide range of frequency searching have been unsuccessful. A possible origin for the behavior of the linewidths could be a change of nature of the reorientations. In particular, the slowing down of motion could favor reorientations from a body diagonal to an equivalent one. As discussed by Rigamonti and Brookeman [CI], for a reorientation effectively different from ir, one would have I Mv^ w ? x'(522) 'l Â£ where tÂ«' is the fluctuation time between the diagonal equilibrium positions. This provides one of the explanations of not being able to observe the NQR signal below ^2SK. Another possible origin can be found as in aNÂ„ where the broadening of lines was characteristic of all mixtures of isotopes and is a consequence of the statistical randomness of the environments of a given type of molecule that produces a dynamic disorder and a local inhomogeneity in the average field gradient [62]. 5. 4 Discussion As shown in Figure 54, the temperature dependence of resonance frequency of aCO is nearly like that of aN ? , furthermore the reduced NQR frequency v Q (T) / v Q (0) has the value of 0.84 for CO as compared with 0.82 for Np at the transition temperature. [49].
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r Â— "T o CO El Â— I ai cr. CP~J
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57 The peculiar behavior of the linewidth in the aphase NQR measurements elucidates the possibility of the motions in aCO. This is consistent with the observation of the residual entropy. Some (T^^. measurements also support the assumption that motional effects determine the linewidth. The temperature behavior of Sv between 40 * 52 K is satisfactorily fitted by the law _E_ and suggests an activated temperature behavior for tÂ«% with E * 1.2 K cal/mole. The assumption of hindered rotation can be ruled out since it would affect resonance frequency by a factor of 1 and this would conflict with the microwave measurements of QCC = 4.4 MHz. It could also be noted that no reorientation by an angle significantly different froni it could be present in the fast motion regime where (T > 37 K), since the averaging effect would change v Q , and this is not observed in the experiment. However, for temperatures T < 32 K, it is possible that the reorientation involves angles different from rr since in the slow motion regime utÂ£' >> 1 and no average is taken. The dynamics driving the molecular reorientation should have a cooperative character through the molecular dipole and quadrupole interactions. The correlation time t~ introduced above has to be considered a local correlation time driven by the cooperative relaxational time at a given wavevector. Gill and Morrison [10] measured the dielectric constant of aCO down to 6 K and found no critical effect. This seems to suggest that the characteristic wavevector of the cooperative excitation should be far from zero and the possible phase transition, if any, should be of antiferroelectric type.
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APPENDIX A QUADRUPOLE RELAXATION THROUGH MOLECULAR TUMBLING MOTIONS IN LIQUID CO 5 In the liquid phase for nuclear spin I = j, the Zeeman energy levels are all equally spaced. In the absence of an rf field, transitions introduced by the quadrupole interaction are shown in Figure Al . The actual values of the transition probabilities depend on the detailed forms of the spectral densities of the field gradients. However, all the upper transition probabilities for Am = 1 are simply related to each other. This is also true for Am = 2. The quadrupole Hamiltonian can be written as in (118) *Q 41(211) L 3 VX z l ' ' V + U l z + A z l > .+ V_ (I + I z + I z I + ) + V +2 (IJ2 + V_ 2 (I + ) 2 ] (Al) where I + and I_ are the raising and lowering angular momentum operators respectively. V represents the irreducible field gradient tensor components which, are related to the cartesian tensor components by VÂ„ = [3V (V + V + V )] (A2) o ygL zz v xx yy zz' J v ' V Â±l V zx * 1V zy V Â±2 ' 7 < V xx " V * 1V xy ' One notes that the V. . are symmetric and V + V + V =0. For the raising and lowering operators one has 58
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59 rn 5 2 a T 3 2 .1 2 W, W,(!+a) *  f Wi W,(I+A) f w, gWi(lJA) Wi Wi(l+A) SO W; tW S Â£W2(l+2Â£) V>1 IqWa(I*2A) ,W g *W*tl + 2A) 10 2 Figure Al . Quadrupole induced transition probabilities among the Zeeman energy states for spin I 5/2.
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I+I, I I,m > = m / I(I+ll m(mÂ±l) I.wttl > and II+ I I,m > = (mÂ±l) /ITL+VJ ~m[mÂ±T7 I,mÂ±l > . z j  (A3) (A3') One then calculates the matrix elements from (A3) and (A3') 2 I 2' ~2 <5 + 3 2' ~2 Vz + ^ ^z + V* Â£+1 Â„ 2'~2 2'~2 80 = 80 (A4) ana' i + 1 2' ~2 5 ! 2' "2 Vz + vÂ± hh + l zh 5 1 I 2 2Â»Â±2 > = 32 5 ,3 2' z 2 = 32 (A4') and also 2' "2 Vz + yÂ± 14 = (A4") For the transitions involving Am =Â±2, one obtains 2 5 + 1 2' "2 (U 2 5 + 5 . 4Q 2' ~2 4U (A5) 2' "2 j u Â± ; I 2' "2 and < I, 4 2' 2 2' + 2 40 72 72 (A5)
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From (32), (34), (A4) and (A5) it follows n40 j >L . f & W. ti40 J 30 SO e" u L T OTTVJ, : (t t) dt e i2 'Â°L T V +2 (t) V* 2 (t t) dt (A6) (A61 Jo where W ] = Wg 3 and W 2 2W g r One will need to calculate V 1 V 2 VjTtTY^ (t t) and \^PJ* 2 (t~~7F. To do so one considers the irreducible tensor transformation as following. For irreducible tensors k t V of rank k, the 2k + 1 components of V are transformed accordinq to P 3 the. irreducible representation D. of the rotation group VÂ« = I vj D k (a, 3, y) q t p pq v , p, tv (A7) where (a, B, y) are the Euler angles of the rotation taking the unprimed reference frame into the primed frame. The Euler angles are defined by three consecutive rotations: (1) A rotation of angle a about the z axis, (2) A rotation of angle 6 about the y'axis and (3) A rotation of angle y about the z" axis as shown in Figure A2. The unitary operator D (a, 8, y) can be written as D (<*, B, y) = e Using the fact that iYjÂ•i6 J. iaJi _ iaj 7 , Â„iaJ 7 V~ y (A8) (Ag;
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X v A X ,// Fiyure A2. The Euler angles. First, a rotation of angle a about the Z axis. Secondly, a rotation of angle 3 about the Y' axi< Finally, a rotation of angle y about the T" axis.
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53 which can be proven as follows. Let e~ a z j a > 'b > where {ja>} is a complete set of eigenkets before the rotation of a about the z axis, {b>} is another complete set of eigenkets in the coordinate system after rotation. One has < b' J. b > = < a" J a> = < a' a > (A10) It follows that V y e (All) Using (All) and by expanding, it can be proven that eiBV = eiaJz e " lBJ y e 1aJz (A12: Following the same procedure, one has iYJ z" = e " iaj 7 _1$J v iYJz Â„i6Jw iaJ J y e e ,MU y e (A13) From (AS), (A12), and (A13), it can be shown that D (a, e, y) = e" Â•iaJ z iBJ y iYJ 2 (A14) Let j,m > be a simultaneous eigenket for J" and J . the matrix elements D J , then will be written as m,m D J , = < j,m m,m Â° = < J 5 m D (a, B, Y ) J.rrT > iaJ 7 iaJw iYj, e z e Ye z j,m lam iYm ,j e e a . m m (A15) where the definition of dji , is obvious,
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64 From the table [30] one has for elements of d 2 .(g) as mm C = d% = cos 4 {%) 22 22 c (A16) d 21 = ~ d 12 = " d 2l = d l2 = I Sil1 3 (i + C0S & ) d ?0 d 02 = d 20 = d 02 =/3 / Â§ Si " 2 S d 2l = d ?2 = " d 21 = d 12 = \ S1 ' n B (C Â° S B " 1} d 2 = d 2 = sin 4 () 22 22 X
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65 Transformation from the principal frame of the field gradient to the laboratory reference frame, yields v n = V 77 e" 1Y /3/2 sin 3 cos 3 +1 /Â§ ZZ + 1 (V n V yY ) e* i2a e" iY [ sin 3 (1 + cos 6)] + 1 (V xx v YY ) e i2a e" iT [\ sin 3 (cos 3 !)] (A17) V , = V 7V e 1Y (/3/2 sin 3 cos 3) _1 /6~ ZZ + \ (V xx V Yy ) e" i2a e iY [1 sin 3 (cos 31)] a. 1 fv \i ^ J 2a p 1y fy sin s ( co3 B + ^ (A17) 2 ^ XX " V YY ; e V + 2 = f V XX e_1 ' 2Y W* Sin2 ^ vb 2 XX " V YY e l"2a i2Y _t ,3\ e cos H vTsj + 1 (Vxx V yY ) e i2a e" i?Y sin" (f) (A18) V 2 " f V ZZ v6 e l2Y /3/8 sin 2 3 + Â£ (V xx V yY J e e sin^ () + I (V XXV YY> ^^ e1?Y COsh{ 2 ] (A18')
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ob If one assumes isotropic motions in the liquid phase and takes the average over all possible orientations one obtains 3 v +1 (t) v* 1 (t; TO (1 + 3 ) V !z (A19) U It) ^ (t) tÂ§ (1 + 3j ) 7Z and %Z (t) Vfe It) t (1 =3) V z 3 :a2o: Â»2 (t) V !z (t) ^ (1 + \ ) . zz ,2 2 V uv V y Y YY where n = Â— % Â— is the asymmetry parameter. Assume '11 V i (t) V* (t t) =  VTTtp^l e" x 2 (A21) for i = Â±1, Â±2 where tÂ« characterizes the correlation function of V^ (t) Using (A6) and (A6'). one finds V r e!Qci^ 40ti 2 2x 9 24 (1 + \ ) ^r43 1+ cofx L 2 2 ! 401. 24 2 \ "o u 3 1 + 4cofx* In the fast motion regime m.Tp << 1 (A22) (A~22') (A23) + , , , , (A23) [ 40h J J * The relationship between the experimental T, measurement and W, , Wp are obtained by considering the rate equations governing the populations among the energy levels. V '^] 2 48 (1 + 4> T ? 40h j J * W 2 [^] 2 43 (1 + 4) x 2 .
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67 The number of nuclei in state m has the following time dependence: dN m I a? = r* ( " N m \ m * V w m:ra ' (*24 Â» where W is the transition probability per unit time from state m to state m' t and N are the number of nuclei in state m. One can write down all the equations dN 5/2 1 "dP = " N 5/2 W l " 2 N 3/2 W 2 + N 3/2 W l ^ + A > + \ N 1/2 W 2 (1 + 2A) (A25) dN iF = " I N 3/2 W l " N 3/2 W 1 < 1+A > W N 1/2 W 2 +  N 1/2 W ] (1 + a) + N 3/? W, + ^ N_ 1/2 W 2 (1 + 2A) dN l/2 dt N 1/2 W 1 (1 ,A) 2N 1/2 W 2 (1 + 2A) tn 1/2 W 2 + I N 3/2 W l + 1 N 5/2 W 2 + if N 3/2 W 2 ^ + 2A > dN "dT^ = " I N l/2 W l 7 N l/2 W 2 TO N l/2 W 2 ^ + 2A > +  N_ 3/2 W] (1 + A) + \ N_ 5/2 W 2 (1 + 2A) + ^ N 3/2 W 2 ^ N. 3/2 W]  N 3/2 W, (1 + A) ^ N _ 3/2 W 2 (1 + 2A) + N 5/2 W l < ] +A ) + I N 1/2 W 1 + W N 1/2 W 2 dN '5/2 dt >5/2 W ] (1 + A) 1 N_ 5/2 W 2 (1 + 2A) + N_ 3/2 W] + 1 N. 1/2 W 2
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68 where the downward transition probability per unit time for Am = 1 and Am = 2 are given by the upward transition probabilities per unit time multiplied by the Boltzmann factor 1 + a and 1 + 2a respectively, where A = ~Yf and one notes nco. << KT in the temperature range of interest. Let N 5 , 2 , N 3 , 2 , and NÂ°, /2 be the number of nuclei in state 5 3 1 2' 2' 2' first approximation one would have N H/2 N where n Q = g, and N is the total number of nuclei. Similarly. m = t, o", p respectively, at equilibrium with lattice bath. To the N P K/2 N 3/2> ~%'~N 5/2 " N 3/2 ^ ^ (Â«5/2 " N ?/2> ~~ 2l1 o A * N ?/2 " N ?/2 ^ ^1/2 It follows that dp=(N 5/2 NÂ° /2 ) W] 1 (N 5/2 NÂ° /2 ) W 2 + (N 3/2 NÂ° /2 ) W 1 + 1 (N 1/2 NÂ° /2 ) W 2 . (A28) Defining n m = N N^ , one can write the following equations from (A25): ~dF = n 5/2 W l " \ n 5/2 W 2 + n 3/2 W l + \ n l/2 W 2 ^" 29 ) 3/2 2 9 "St = " 5" n 3/2 W l n 3/2 W l " TOn 3/2 W 2 + M 5/2 W l + I n l/2 W l + if n l/2 W 2
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69 dF= " I n l/2 W l " 1 n l/2 U 2 " ll n l/2 W 2 + I n 3/2 W l 1 9 + 2 n 5/2 W 2 + TO n 3/2 W 2 dn dt " l/Z = I n 1/2 U l Â• \ n l/2 U 2 " if n l/2 W 2 + I n 3/2 W l + 2" rl 5/2 W 2 + W n 3/2 W 2 dn_ "dt" 3/2 2 9 J ' ' ' n 3/2 W l " 5 n 3/2 W l " TO n 3/2 W 2 + n 5/2 W l + f n l/2 W l + TÂ§ n "i/2 W 2 dn 1 1 ftÂ— = " n 5/2 W l " i n 5/2 W 2 + n 3/2 W l + t n l/2 W 2 By multiplying each equation with the corresponding m and summing all togather, ens obtains the following equation: (A3C) dt I K> Â«V.Pl I 1 z oJ w I n 3/2 + I n l/2 " I n l/2 " I n 3/2 (W 1 " W 2 } lo a first approximation and assuming that the spin temperature can be defined, one notes Â£ 4. 3 1 Â§. 5 n 3/2 5 r 'l/2 " 5 n l/2 " 5 n 3/2 6^3 3 6 = n + n Tn n 5o 5o 5o do ,.18 . 3 3 , 1L, + i To + 10 + To + To A n o (A31)
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70 ha. Â• Â° where A' = 77^and T is the spin temperature which is generally greater KT S s :han the lattice temperature. One then concludes that dt which implies I ^v ^J ^1 + V (A32) 1 = Â§ (W, + w 2 ) (A33) and finally, it follows from (A23) , (A23 ) and (A33) that r a 2 T ] " 125 e 2 Qq 1 (1 + n 2 ) t, (A34) in the fast motion regime.
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APPENDIX B NMR SPINLATTICE RELAXATION THROUGH THE QUADRUPOLE INTERACTION IN SOLID BCO In solid eCO, the time averaged quadrupcle interaction represents a small perturbation term to the total Hamiltonian and the Zeeman energy levels are unequal. In the absence of spinexchange transitions, the Boltzmann distribution of the populations among the energy levels cannot be maintained. In such a case the spinlattice relaxation is, in general, not characterized by a single relaxation rate. In addition, the relative importance of the different relaxation rates depends on the nature of the relaxation processes and on the initial conditions [38]. However, since the quadrupole splittings are small compared with the Zeeman energy levels, this simplifies the problem. Considering for spin I = j, from (A25) one has (Bl) dt = < N 5/2 " N 3/2 )( ' 2W 1 " K> + f (N 3/2 " N I/2> (W 1 + V d ( N 5/2 N 3/2> + TO < N l/2 " N l/2> W 2 + I V ^1 " W 2 : d ^dt" " V = (N 5/2 " N 3/2> < W 1 " \V~ VS/2 ~ N 1/2 )(4W 1 + *%> + 4 (N l/2 " N 3/2 } W 2 " I n o A (W 1 5W 2 } 71
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72 '^dt = 1 (N 5/2 " N 3/2> U 2 + I < N 3/2 " N l/2 } ^ " W 2> f (N 1/2 N_ 1/2 ) W 2 + f (N. 1/2 N_ 3/2 )(W 1 W 2 ) d < N l/2 N 3/2 } = < N 3/2 " N 5/2>< W l " 1 W 2 } dt " I < N l/2 ~ N 3/2> < 4W 1 + 7W 2 ) + TO < N 3/2 " N l/2> W 2 " I V {W 1 " 5W 2 ) d(N ^ /9 N /9 ) , ' 3/ dt ' " < N 3/2 " NB/a"' 2 "! " ? W 2> + I < N l/2 " N 3/2> < W 1 + V + TO ("1/2 " N V2> W 2 + 5 V (W 1 H 2> where W, , WÂ„ and A have the same definitions as in Appendix A. If one now defines N ., ,p = N^i N from the symmetrical equations above, it follows (B2) N 2 = N 2 (2W 1 + 1 W 2 ) +  N] (^ + W 2 ) + ^ N Q W 2 +  n Q A ( W] J W 2 ) N 1 = N 2 (W 1 gWg) \ N 1 (4W ] + 7W 2 ) + ^ N_ 1 Wg J n Q A (W ] 5Wg) N Q = 1 N 2 W 2 +  N] (W, W 2 ) f N Q W 2 + f N., (W 1 W 2 ) + I N ,W, + 4 n A (W, + 2W 9 ) 22 2 5 o 1 2 with two symmetric equations for N , and N ? .
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75 The solutions of these three equations are characterized by only three relaxation times. To simplify the equations in (B2), one defines N p = N p n Q (B3) where n and a have the same meanings as in Appendix A. (B2) then becomes N 2 = NJ (2W ] + 1 W 2 ) +  Nf (W 1 + W 2 ) + ^ N5 W 2 (B4) N 1 = N^ (W ] ^W 2 ) \Nf (4W 1 + 7W 2 ) + ^ N^ W 2 N Q = 1 N 2 W 2 + f N; (W 1 W 2 ) f N 6 W 2 + f N :i (W, W 2 ) + 1 N 2 W 2 . Now assuming a precessional motion in 3CO and for simplicity an axially symmetric field gradient, one can calculate Vrj 2 from (A17), (A17'), (A13) and (A18') for i = Â±1 and i = Â±2 9 V^l 2 = J sin 2 b cos 2 p V 2 Z (B5) v; 2 ! 2 =re sin " 3 v zz In chapter III it is shown that 3 cos 2 81*0 so that, (B5) can be reduced to i v Â±l I 2
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where the V "tennis are the EFG tensor components in the crystal frame, while the tensor transformation from the crystal reference frame to the laboratory frame follows the same procedure as in (A17) and (A18). V 4l = V 2 D 21 + V : 2 D 21 + V l D ll + V 'l D ll V +2 T 2 D 22 + V; 2 D_ 22 + Vf D 12 + V :i D. i2 (B7) (B7') For a powder sample using (A16) and (A18) and taking over all the possible directions, one calculates V 2 v 77 H ! 10 "Z V IV 2 _1 V 2 ,v +2' 10 ; ZZ (B8) (B8') The same equations will hold for Iv_j 2 and [V_ 2 I 2 , assuming again V i (t) V (t t) = V i (t)! 2 e for i = Â±1 , Â±2 where xX characterizes the correlational precessional motions. Using (A6) and (A6') one obtains T 2 u = \* 2 M 48 1 ^ 40TiJ J L T 2 WÂ„ fe!Qgl 2 48 ~ 2 Â— (B9) (B9') From (B9) and (B9') it is obvious that W, = WÂ«. In the fast precessional motion regime u, xX Â«1 . It follows from (B9) and (B9') that W ] = 48 W 2 = 48 r e 2 Qg ^ h40, ^e 2 _Qol 2 ti40 (B10) (B10') That is, w\ = W ? in the fast motion regime.
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75 Now from (B4) it is possible to write N' = _ f. U N+ Â— W N' + UN' (Bll) ,N 2 2 T7 5 Tl 10 10 K ' ^0 2 T N 2 5 1 H 2 12 Since one considers the initial condition that all lines are saturated, it follows from the symmetric equations that N', = NC and N' 2 = NÂ«. (Bl'l) is further reduced to N 2 = " I W 1 N 2 + f W 1 N 1 + TO" W 1 N 6 (B " 12) W' = 1 W N' Â— W N' "i 2 i IN 2 io ri Q N' = W N' Â— W N* M 1 N 2 5 w ro The general solutions of (B12) will be where p = 0, 1, 2. Substituting (B13) into (B12) and comparing the Ait A o t i X a t j. i e x , e * and e d terms one has (Â§ w 1 + \ x ) a n + Wj an + ^ W ] a 01 = (B14) ^ w 1 a 21 + ( W, + A x ) ail = W ] a 2 i + (w ] + A!) a i = and similar equations for e~ 2 Â° and e~ 3 terms.
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75 In order that all a, f for i = 0, 1, 2; j = 1, 2, 3 one needs 1 %j 2 W l + X JrW, 5 1 i W 10 w l "W w i +A Â° lW, + A o 1 :bi5) = This is reduced to (5A 4W.)(2A 3Wj)(lQX 33 Wj) i.e. , with solutions 4 3 33 X = 5 W l Â• 2 W l ' TO W l Bie: In the case that all the lines are saturated at time zero, one has from (B3) and N 2 N 1 N Q N^ = N, = Ni = n A c 1 o (B17) 4 3 33 Let \ 1 = FUj , x 2 = o" V7, and ;\ 3 = tq W, . Comparing with terms of e" At , (B12) gives, at t=
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77 (B18) (~ a 2 i a 22 + jf a 23 ) +  (an + a 12 + a 13 ) + ^ (a o: + a 02 + a 03 ) = i i i 2 ( 3 2i + 322 + a 23/ + (J a n + 5 a i2 + 2a 13 ) = 3 3 (a 2 i + a 22 + a 2 3) + (aoi ygao2 + jf a os) = 0. From (B17) and (B13) one has at t = n A = a 2 i + a 2 2 + a 23 (B19) n Q A = an + a i2 + a i3 n Q A = a i + ao2 + a 3 (B18) and (B19) enable one to deduce a 2 i = a i = an (B20) 3 2 a 22 " TO 3 2 " " 5 a 12 a Â„ 3 = _4 a , 3 =  a 23 which gives a 21 a Il _ a 0i " _n Q A and all other a. = for i = 0, 1 , 2 ; and j = 2, 3. The solutions will then be N p = n Q A (1 e V5XW,tj (B . 22) for p = 0, 1, 2.
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Therefore, in the case of complete saturation and W, = VL, the spinlattice relaxation is governed by a single exponential decay having 1 4 TT T, " 5 W 1 (B23) From (B10) and (B23) one has the same equation as in the liquid phase in (A34) 1 _ 3 T 1 125 r eÂ£Sa ;B24) where x' is the correlation time for the molecular precessional motions around the crystal C axis.
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APPENDIX C L1NEWIDTHS AND MOLECULAR MOTIONS IN aCO One assumes that the molecules in aCO execute librational motions and sudden reorientations. The tensor components in the laboratory frame and those in the principal reference frame are related by the rotational operator R (a, x ) ]/.. = R" 1 (a, k) V(j R (a, *) (Cl) where R (a, .) = R ($.) R (a). The Vr. and V.. terms are the EFG teny x ij tj sor components in the principal and laboratory reference frames, respectively. $1 is the outof plane librational angle end a is the reorientations! angle. V{. = eq Â£00 o 1 1 ;c2) In (C2) a symmetric field gradient has been assumed. It so happens that by assuming a small angle for <$> Â±i vÂ„ 2Â§ 3*5 ) V =  (1 + 3+1 sin 2 a 3 sin 2 a) V zz = " ~2 (1 + 3?t cos2 a " 3 cos2 a) V xy = " ~2 3K sin a (C3) 79
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89 v., sa 3 K X2 COS a V w , = Â§ 3 sin a COS a yz ^ For a sudden reorientation of an angle from to u as is the case discussed in NaNCL by Ambrosetti et al . [57]. Â« (f + *Â£) (f + *Â£) s(t) + Â„ (C4) where s(t) may take values +1 or 1, and * is introduced to account for jumps not exactly equal to it. Â„ is the inplane librational angle. By assuming small angles for Â„ and * and expanding sin a and cos u, one has *Â£)<$*5.(tÂ» + *. [*Â£+ + Mr (1 " s(t))] V__ = . Q [2 + 3*5 + 3*2 + 3iji (1 _ s(t)) + 3 ? ^ (1 _ s(t))] \y = ^** (^ + ^S(t) + *Â„ S(t)) v xz = ^U^ +S (t)(l *Ji) + iif (1 _ s(t))] v.Â«^(*Js(t) +*Â£+;,) . yz
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81 Assuming i> H and $* are not correlated and also that < $Â„ > = 0, the librational average of (C6) is given by < V xx > = 3J. (T3 < 9, 2 > ) (C7) < V > = S [1 3^i (1 < s(t) >) 3 < 9Â„ 2 >] < V > = Â£Â§. [_ 2 + 3<^ 2 >+ 3 < + ^ (1 < s(t) >)] ===0 xy xz yz The quadruoole Ha/nil Ionian can be divided into two parts H Q = * H Q > + [A Q " * H Q >] * (C " 8) The average Hamiltonian < h q > gives the NQR frequency, while the second part drives the relaxation process and causes the line to broaden [58]. The fluctuating terms of the components of the EFG tensor can be written as v 7Z < v xx > 2fa ($ x 2 . < 9x 2 > ) (c9) v yy < v yy > = 3 f [; < 9Â„ 2 > ^f ? (s(t) < s(t) >) ^ (1 s(t))] V Z7 < V^ > = ^ > A 2 < K 2 >+ 9n 2 < 9Â„ 2 > + iÂ£(s(t) < s(t) > ) +2*jÂ£(l s(t))] V xy " * V xy * = " ^ * C " *? (1 " S(t)) + " S(t)] V xz< V xz> a ^xs(t) v < v > 3Â§a[4o s (t)) + *^ yz yz c c
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82 where in (C9) consideration is given only to the second order of $Â„, j, and $*. For spin I = j , the NQR frequency between m = Â±j and m = Â±j , T 2 is given by [58]. 1.1+11 + 1 T 2 T 2 2 I T Â±5/2 T Â±3/2 J (C10) where r'(t) u*(t t) dx ii'(t) is the departure of the instantaneous resonance frequency from its average value and 5/2 T Â±3/2 m* Â±3/2 itf Â±5/2 W Â±5/2' m I w 3/2 , m (Oil) (C12) where T +5 , 2 and T +3/ ~ are the lifetimes of the states m %c/? an 'J m %o/ ? respectively. From (Al ) and (C9) one will have, assuming order phase < s(t) >= <Â°'(t) 9e Qq 2oti /6 [*i 2 < <$>Â± 2 > + u 2 < 4>h 2 > (C13) i*2 + ^s(t) + I*,,** s(t)] Since $Â„ and ^ occur at higher frequencies, for simplicity one may neglect these terms. One would then have r(t)o>(t t) dr = 9e 2 Qq 2dfi 2 , f. s(t) s(t t) di (C14) Jo
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83 Defining s(t) sTT^T]" e~ 1ClJT dr = <]Â„ (w) it follows from (C14) and (C15) that f "l 2 1 . 9e Qq T 2 [ 20n Consider now for terms. From (Cll), one writes 3 < * k > 16 J n (0) Â±5/J + w Â±^/: T Â±5/2 Â±5 / 2 > Â±3 / 2 Â±5/2, Â±1/2 Using (Al), (A4), (A4'h (A6) and (A6') one obtains f Â±5/2 ( 40"n t 2 r 80 I L Jo T^tTlt^t t) e" 1WT dx _1Q
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Â«' If one assumes an harmonic oscillator for ^ and 6.,, and for s(t) :ponential cc from (CiS) that an exponential correlation function with characteristic time xX\ it follows A = 2Â§a 2 2 2tX' < 4>x > ^ (C19) i*2 1 + (wr~ + oj)2 t Â— : 2 r+ A2 J One notes that o^ >> u and ia x rX^Â» 1 I J t,*2 "1 + a)" 2 ""^ (C20) can be reduced to A f 3e 3. < ft* 2 \ " J TT~3^f :c2D While for v; 9 TFj V*Tt ~T7 e" 1 ^ 1 dx Â±z Since V^ involves V + Â«Â» one needs only to consider, assuming = 0, v yy " < v 'yy > ' " ~f [ ^r 1 Â— s!t Â» " %^ i 1 " Â«<*Â» (c 22 > and V < V xy xy [ s(t)) + K4H s(t)] If one further assumes that no correlation exists between <+>Â„, ^, (j>* and s(t),(C22) can be simplified as V < v > . M 3<>* 2 sft) yy yy z 2 su; (C22) V < V > Â« xy xy
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35 It follows that 3 9 < 2 i iTtTTr^Tye" 1 ^ dx (C23) 3eq 2 9 < ** h > . { 3 x Â— 4 J H ( 2 u) From (C17) , (C21) and (C23) Â•Â±5/2 { 80n J i_ 2 (C24) 1 Considering now the y term and using (Al), (A4)Â» (A4')Â» (A5) 'Â±3/2 (A5'), (A6) and (A6'), one writes 3/2 40h L 80 V +1 (t) V^(t t) e" ,aT dT r^rv^Tt'^T eiuT dx 1 (C25) 4dti V +1 (t) V*^Tr^~TTe"'2 T dx .0) V_ 1 Ttrvf 1 Tc xT e" j 2 T dx h f eO  2 40n J 72 UT . V +2 (t) V* 2 (t t) e"'2 T d 1*T V_ 2 (tTV* 2 (t x) e 1 ? 1 dx Employing the A and B terms as discussed above, one writes Â±3/2 9e 2 Qg 20n + f < **2 > T 2' < * > t~+ Â— z* Â— ^T 1 + jf w 2 xÂ£ +f<^> J n ( 1.) (C2i
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Rfi From (C10), (C24) and (C26), one has 3 1 V
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88 23 24 30. 31. 32. V. M. Cheng, W. B. Daniel and R. K. Crawford, Phys. Rev. B11 . 3972 (1975). W. F. Giauque and J. 0. Clayton, 0. Am. Chem. Soc. 55, 4875 (1933). J. R. Brookeman and T. A. Scott, J. Low Temp. Phys. 1_2, 491 (1973). 25. R. Stevenson, J. Chem. Phys. 27, 673 (1957). 25. R. L. Mills and A, F. Schuch, Phys. Rev. Letters 23, 1154 (1969). 27. E. I. Voitovich, A. M. Tolkachev, and V. G. Manzhelii, J. Low Temp Phys. 5, 435 (1971). 28. Â£. R. Andrew, Nuclear Magnetic Resonance (Cambridge at the University Press, London, 196977 29. C. P. Slichter, Princi ples o f Magnetic R esonance (Harper and Row Publishers, New York, 196377" D. M. Brink and G. R. Satchler, Angular Momentum (Oxford at the Clarendon Press, London, 1962). T. P. Das and E. L. Hahn, "Nuclear Quadrupole Resonance Spectroscopy", Solid State Physics , Supp. 1, (Academic Press Inc., New York, 1958). E. Fufcushima, A. A. V. Gibson, and T. A. Scott, J. Chem. Phys. 66, 4811 ^ 1977) . Â— 33. A. A. V. Gibson, T. A, Scott and E. Fukushima, J. Mag. Res. 27, 29 (1977). Â— 34. F. Alder and F. C. Yu, Phys. Rev. 81_, 1067 (1951). 35. W. G. Clark, Rev. Sci. Instrum. 35, 316 (1964). 36. M. S. Conradi, Rev. Sci. Instrum. 48, 359 (1977). 37. A. N. Garroway and D. Ware, Rev. Sci. Instrum. 46, 1342 (1975). 38. E. R. Andrew and D. P. Tunstall, Proc. Phys. Soc, IS, 1 (1961). 39. J. Bruining and J. H. R. Clarke, Molec. Phys. 31_, 1425 (1976). 40. P. S. Hubbard, Phys. Rev. 131, 11 55 (1963). 41. D. L. Vanderhart, J. Chem. Phys. 50, 1858 (1974). 42. W. N. Lipscomb, Adv. Mag. Res. 2, 137 (1966). 43. A Abragam, Principles of Nuclea r Magnetism , Chap. VIII (Oxford at the Clarendon Press, London, 197377 44. B. R. Appleman a>vd 3. P. Dailey, Adv. Mag. Res. 7, 231 (1974).
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45. M. Bloembergert, E. M. Puree! 1 and R. V. Pound, Phys. Rev. 73, 679 (1948). ~ 46. G. C. Castagnoli, Physi'ca M, 953 (1964). 47. G. Bonera and A. Rigamonti, J. Chem. Phys. 42_, 175 (1965). 48. M. J. Stevenson and C. H. Townes, Phys. Rev. 1_07, 635 (1957). 49. A. S. DeReggi, P. C. Canepa and T. A. Scott, J. Mag. Res. 1, 144 (1969). 50. R. L. Armstrong and P. A. Speight, J. Mag. Res. 2, 141 (1970). 51. R. L. Amey, J. Phys. Chem. 78, 1958 (1974). 52. T. A. Scott, Phys. Rep. 27C, 90 (1976). 53. T. Shinoda and H. Enokido, J. Phys. Soc. Japan 26, 1353 (1969). 54. A. Anderson, 7. S. Sun and M. C. A. Oonkersloot, Can. J. Phys. 48, 2265 (1970). 55. D. A. Goodings and M. Henkelman, Can. J. Phys. 49, 2898 (1971). 56. W. H. F'iygare and V. W. Weiss, J. Chem. Phys. 45, 2785 (1956). 57. R. Ambrosetti, R. Ange'lone, A. Cclliqiani and A. Rigamonti, Phys. Rev. B15, 4318 (1977). 58. A. Abragam, Pr inciples of N uclear Magn etis m Chap. X (Oxford at the Clarendon Press, London, 1973]. 59. A. C. Daniel and W. G. MouHon, J. Chem. Phys. 41_, 1333 (1964). 60. S. Alexander and A. Tzalmona, Phys. Rev. A138 , 845 (1965). 61. A. Rigamonti and J. R. Brookeman, submitted for publication in Physical Review B. 62. J. R. Brookeman, M. M. McEnnan and T. A. Scott, Phys. Rev. B4, 3661 (1971). Â—
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BIOGRAPHICAL SKETCH Funming Li was born on October 1, 1949, in Taipei, Taiwan. He entered the National Tsing Hua University in September,, 1967, and received a Bachelor of Science degree in physics in June, 1971. Following two years of service as a second lieutenant in the Chinese Armored Force, he worked as a science teacher in Langchu High School in Taipei for one year. He came to the United States and entered the University of Florida for graduate study in physics in September, 1974. He held a teaching assistantship from September, 1974, to June, 1977 s and a research assistantship from June, 1977, to December, 1979. He married the former Sumay Chen in December, 1976. 90
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. vK>vj a< bmki^tJames R. Brookeman, Chairman Associate Professor of Physics and Physical Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (VI K / !' E . Dwight Adams Professor cf Phvsics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Arthur A. Broyles 2/ Professor of Physics and Physical Sciences
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully dequate, in scope and quality, as a dissertation for the degree of a Doctor of Philosophy F. Eugene Dunnam Associate Dean, College of Liberal Arts and Sciences and Professor of Physics and Physical Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ( /! ' n ,r> w? L'Jicl ii.C:i _ ..;'. y~4ujjjx Charles P. Luehr Associate Professor of Mathematics This dissertation was presented to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1973 Dean, Graduate School
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