Title Page
 Table of Contents
 List of Figures
 Experimental procedure
 Liquid phase
 Solid B-phase
 Solid a-phase
 Appendix A: Quadrupole relaxation...
 Appendix B: NMR spin-lattice relaxation...
 Appendix C: Linewidths and molecular...
 Biographical sketch

Title: Nuclear resonance of O17 in liquid and solid carbon monoxide /
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00099521/00001
 Material Information
Title: Nuclear resonance of O17 in liquid and solid carbon monoxide /
Physical Description: ix, 90 leaves : ill. ; 28 cm.
Language: English
Creator: Li, Funming, 1949-
Publication Date: 1979
Copyright Date: 1979
Subject: Carbon monoxide -- Molecular rotation   ( lcsh )
Oxygen   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 87-89.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Funming Li.
 Record Information
Bibliographic ID: UF00099521
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000097907
oclc - 06637177
notis - AAL3348


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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
        Page ix
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
    Experimental procedure
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Liquid phase
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    Solid B-phase
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Solid a-phase
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    Appendix A: Quadrupole relaxation through molecular tumbling motions in liquid CO
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Appendix B: NMR spin-lattice relaxation through the quadrupole interaction in solid B-CO
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
    Appendix C: Linewidths and molecular motions in a-CO
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
    Biographical sketch
        Page 90
        Page 91
        Page 92
        Page 93
Full Text






To my parents


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 77-0S658.




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 Spin-Lattice Relaxation .............................. 21

3. 2 Intramolecular Relaxation Mechanisms...................... 23

3. 3 Intermolecular Relaxation Mechanisms...................... 26

3. 4 Temperature Dependence of Spin-Lattice Relaxation Time.... 27

3. 5 Discussion..................... .......................... 29

CHAPTER IV SOLID 6-FHASE .... .......................... .. ..... 32

4.. 1 Quadrupole Perturbed N!,i Spectra ......................... 32

4. 2 TI Measurererts in a-C ................................ 40

CHAPTER V SOLID a-PhASE ....................................... 44

5. 1 Molecular Properties of a-CO.............................. 44

5. 2 Nuclear Quadrupole Resonance in a-CO...................... 44

5. 3 NQR Linewiidths and Molecular Motions...................... 47

5. 4 Discussion............................................. 55

MOTIONS IN LIQUID CO.................................... 58

INTERACTION IN SOLID a-CO................................ 71


LIST OF REFERENCES.................................................. 87

BIOGRAPHICAL SKETCH .............................. ................ 90


Figure Page

1-1 Crystal structure of a-CO. The molecules are aligned
parallel to the body diagonals ................................ 2

1-2 Crystal structure of 6-CO. 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

1-3 Phase diagram of CO in the P-T plane, The a-3 transition
temperature at equilibrium vapor pressure is 61.6 K. The
triple point is at 68.16 K..................................... 5

1-4 Heat capacity of CO .......................................... 6

2-1 Sample cell and temperature-controlled cryostat................ 15

2-2 Block diagram of the pulsed NMR/NQR spectrometer............... 18

3-1 Temparaturc dependence of the spin-lattice relaxation time T
of 0'7 in liquid and in S-CO................................ 28

3-2 Temperature dependence of the autocorrelation time for tumbling
rotational motion in liquid CO and for precessional motion in
S-CO.......................................... ............... 30

4-1 Definition of angular variables used in the text to discuss the
NMR rotation patterns in C-CO. The molecule is described
classically as processing rapidly about the crystal C axis..... 34

4-2 The i!;;R perturbeo spectia or -CO at difference ( ) angles,
4-3 Rotation pattern of the separations of the satellites of NMR
perturbed spectra in B-CO as a function of ([ )........... 38

4-4 Effective coupling constant as a function of temperature in
B-CO.......................................... ............... 41

5-1 Temperature dependence of the pure quadrupole resonance of 017
in a-CO...................................................... 4

5-2 Temperature behavior of the 017 NQR linewidths in a-CO. The
insertion shows some effective Tl measurements ............... 48

Figure Page

5-3 Three dimensional represencatior of h. 01'' NQR spectra in
a-CO. For clarity the linewidth has beer enlarged by a
factor of 2................................................. 49

5-4 The reduced temperature behavior of the reduced NQR frequency,. 55

A-1 Quadrupole induced transition probabilities among the Zeemanz
energy states for spin I = 5/2.............................. 59

A-2 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



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) spin-lattice

relaxation time T1 measurements in the liquid phase, T1 and quadrupole

perturbed NMR spectra in the S-phase and nuclear quadrupole resonance

(NQR) linewidth and Tl measurements in the a-phase -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 B-phase is also

obtained. The effective quadruple coupling constants are compared with

the static quadruple coupling constant obtained in the NQR measurement

of the c-phase. 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,


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 a-phase

were also obtained.

In the a-phase the quadruple interaction is sufficiently large enough

to perform a pure quadruple resonance experiment. The measurements in the

a-phase 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 a-phase. 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.


'. 1 Physical Properties of CO
Carbon monoxide is one of the most interesting diatomic molecular

crystals and has been studied extensively [1-12]. 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 dipole-dipole 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 end-over-end rotation in solid a-CO found the rate to

be unobservably slow [17].

1. 1. 1 Crystal Structure of CO

The-e are two solid phases, a nd ,'. of CO. At low temperatures.

X-ray diffraction studies showed that the crystal structure of the

a-phase 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 1-1. 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 a-CO is still in dispute [19].


c-I-- t./-

74 >'.-

0 K)5
(;i u

Figure 1-1. Crystal structure of a-CO. The molecules are aligned
parallel to the body diagonals.


In the B-phase wLich occurs above 61.6 K, the crystal structure

determined by Vegard [2] was that of B-N2, and subsequent X-ray diffrac-

tion studies of B-N2 [201 deduced this space group as P63/mc(D h). In

this regard one considers the crystal structure of B-CO, the same struc-

ture as 8-N2, to be hexagonal close packed. It was found that X-ray form

factors were equally consistent with a dynamic model where the molecules

process about the C-axis at an angle e' as shown in Figure 1-2, 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 1-3. The a,B phase transition is at

61.6 K at equilibrium vapor pressure, and the triple point is at 68.15 K.
Applying the Clausius-Clapeyron 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 1-4. The analysis of the results is compared with the entropy cal-

culated from the band spectrum of CO in the gas phase. The discrepancy


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'

\ Y '---------

v^^ iy --,,,,
f I

't\,~ (
~ t, I /

I 'K> j

Figure 1-2. Crystal structure of 8-CO. 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~lr-- ;` ~--- -~--- --r--

L /



26) Co

O6 100

Figure 1-3. Phase diagram of CO in the P-T plane. The a-B 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 b

S o

0 .I0 20 0 "40. 50 70 80
Figure 1-4. 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 end-for-end disorder [4]. Melhuish and Scott [15] calculated

the energy for the oriented and disoriented lattice and estimated the

Curie temperature for the order-disorder transition. They found Tc to be

,5 K for CO. However, heat capacity measurements on solid a-CO 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 (1-2)


XT/Xs = C /CV (1-3)

where XT is the ;sotiher'al c:omi-essibility, 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 a-CO 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, I-1,..., -(1-1), -(I).

The Hamiltonian can be written as

= --= ( (1-4)

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 (1-5)

-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(I-1)/1,.... -y(I-1)/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, (1-1),..., -(1-1), -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 (1-6)

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

spin-lattice 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 (1-7)

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 (1-8)

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 (1-8) 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 -
Sax ax
a a a
one has for the quadrupole term

SEQ VaBs (1-9)

In quantum mechanics, one considers p as an operator

( t =- e Kprotonis r (1-10)

and defines the operator Q p)as

Q (op) = f(3xax r2) p(op)()d3r
=P e r2)
S K=protons (3aKX3K a rK

From (1-9) one writes the Hamiltonian

H I V n(op) (1-12)
Q 6CaB as act

The Wigner-Eckart 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
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) (1-13)

Since Q can be written as combinations of irreducible tensors like

Q(I). Defining eQ, quadropole moment of nucleus, as

eQ = (1-14)
K=protons K -K I

and operating on the matrix element nim' in (1-13) for m = I, m' = I
where I is the nuclear spin, it follows that



In order for C to have a physical meaning it follows that one needs I>

to have a quadrupole interaction. From (1-12), (1-13), and (1-15) it
follows that

HQ = v [ I + I ) 12] (1-16)

In the principal axes of the field gradient, one defines eq = VZZ and the
asymmetry parameter


where VXX, Vyy, and VZZ are the principal field gradients. One then has
from (1-16)

H = e2 q- [31 2 + n ( I_)] (1-17)
Q 41(2I-I X Ii)]

One can also write (1-16) in the laboratory frame as

Q = 4- 1I[ V (3z2_-2)+V (I- z I )+V_( + +)+V+2( ) -2 +)2]


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 spin-lattice 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.


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 spin-lattice relaxation and quadrupole reso-

nance in the low temperature phase of CO, 017 is the best choice. 07
with nuclear spin.I = [34] has a nuclear quadrupole moment which,

interacting with the molecular electric environment, provides the domi-

nant mechanism of spin-lattice 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 2-1. The con-

densed sample was held in a Kel-F 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
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

Figure 2-1. Sample cell and temperature-controlled 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 feed-back 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 10-4 torr or less. A thin brass sheet

radiation shield held in place with smrll Kel-F 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 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.

2. 3 The Electromagnet

The electromagnet used in the rNR measurements was a Varian 4012-3B

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 10-4/hr. or less.

2. 4 The Spectrometer
The pulse spectrometer used in this experiment is depicted in

Figure 2-2. This set-up 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 t-3 D4

Figure 2-2. 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

photo-FET 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 cross-diode set

conducted strongly. The D3D4 diode-set was also conducting to protect

the receiver preamplifier and the photo-FET 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 cross-diodes 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 D-D2. 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&t-off region. A pulse

from the pulse generator turned on a LED diode which provides light to

turn on the Q switch. The source-drain 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


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 Fabri-Tek 1072 signal average. A PDP 8/E mini-

computer was interfaced with the Fabri-Tek 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

X-Y recorder.


3. 1 NMR Spin-Lattice 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. (3-1)

where = Taking the average W over a statistical ensemble, and
introducing the correlation function G 6 as

a 3 (t-t) = < BRl(t)I a >< aH(t')IB > (3-2)

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) e-iWaSB dT (3-3)

Since one usually considers a time t >> it turns out that the limits
of the integration in (3-3) can be replaced by m and one has

WB 1 GB (T) e-iWaST (3-4)

In most cases H1 (t) can be expressed as a combination of two irre-
ducible tensors of rank k. Let

H1 (t) = A P (I) B (t) (3-5)

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 (3-4) can
be written as

Wa 1 = < =A(PI j > ( 0) e-aB1 dr (3-6)
p-k k

g(p)() = (P)(t) *(P)(t-r)

is the correlation function of B(p (t).

There are various possible mechanisms of relaxation, (1) the intra-
molecular quadrupole interaction, (2) the intramolecular spin-rotation

interaction, (3) the intramolecular anisotropic chemical shift interaction,
(4) the intermolecular magnetic dipole-dipole interaction and (5) the

intermolecular electric quadrupole-dipole interaction. However, an esti-
mate of the magnitudes shows that the spin-lattice 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

Consider (1) the intramolecular quadruple interaction. In Appendix
A it is shown that

S 3_ (3-7)
T !25 2 (3-7)

where e-- is the static quadrupole coupling constant, which one has

determined in the NQR measurement of CO. in the a-phase 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
while the experimental T1 at 77 K is 220 ms. This gives = 4.54 sec.
In (2) the spin-rotation 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. Assuming-a

Langevin equation for the angular momentum and a rotational diffusion of

Oebye type, Hubbard obtained [40]

Ss.r. 2C ) 21KTT + 2(Ci 21KT + 11 (3-8)

where C, and C,, are the components of the spin-rotation 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]:

ff2 =6KT


For linear molecules one has 3 C2 = 2 C + C = 2 C2 Assuming the
diffusion limit for the temperature of interest one has from (3-8) and

II eff 12
3h2 (3-10)
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 10-5 sec-] which is negligibly small compared
with (3-7).
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 (3-11)
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 10-0 and 7.i KG for the external magnetic
field H one calculates

f1- = 5.6 x 10-6 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 spin-lattice 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 dipole-dipole interaction

and (2) the intermolecular electric quadrupole-dipole interaction.
Consider first (1) the magnetic dipole-dipole interaction. Following

Bloembergen, Purcell and Pound [45] by assuming spherical molecules and a
Debye model of diffusion, one would have a contribution from inter-
molecular magnetic dipole-dipole interaction given by the following equa-

rl' 3_ y "h2No
T i.d. I al (3-12)

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 10-5 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 10-6 sec-1

Now consider (2) the intermolecular electric quadrupole-dipole 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(3-13)
S i.q. 1 2 (31Da)

where a, No, and D have the same definitions as discussed in the magnetic

dipole-dipole interaction. eQ is the quadrupole moment of nucleus 017,

andM is the dipole moment of-the CO molecule. Taking the value of Q

measured by Stevenson and Townes [45] as -0.026 x 1024cm3, one calculates

r1 = 4.4 x 10-4 sec-i

From the above considerations one immediately concludes that the

spin-lattice 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 Spin-Lattice Relaxation Time

Using the conventional - 7 pulse sequence, one measures the

spin-lattice relaxation time T1 in the liquid phase. Figure 3-1 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 -


00200o 00
E3 i L/. OL

_ 62.3
- --- -r



--BSOL!D .

E !92 CAL/MOLE .

I03/ T (* K")

Figure 3-1. Temperature dq;'nldence of the spin-lattice relaxation time T1 of O'7in liquid
and in s-CO. U!. is the activation enthalpy. The uncertainty at each
measurement is 10,.)

*. r
r- '

! 20-






2 =5.98 x 10- (3-14)

Using (3-14) and measurements of T!, the temperature dependence of T2

is shown in Figure 3-2. For comparison the T2 values of N2 are shown in

this picture [49]. Assuming an activation model [50], such as
2 = o e RT (3-15)

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 (3-14) inro (3-15) one has
T, = C eRT (3-16)

where C is some constant. From the thermodynamic relation

AG = AH + T [G (3-17)

it follows that

a ln T1
H = -R I (3-18)

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 0



P-Ir a,

60 65 70 75 00 65 90

3-2. Temperature dependence of the autocorrelation time for
tumbling rotational motion in liquid CO and for precessional
motion in B-CO. (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]).



I I _I I~J_ I


Amey [51] used the librational model suggested by Brot and compared

this with the far-infrared 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

Stokes-Einstein equation yields

2a2 (3-19)

from which it can be calculated that at 77 K, T = 2.8 x 10-12 sec. This

may be compared with the value of T = 2.7 x 10-13 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].


4. 1 Quadrupole Perturbed NMR Spectra

SThe hexagonal crystal structure of s-CO has already been discussed

in 1. 1. 1. However, the inability of X-ray 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 B-CO is
reduced by a factor of 103 by motional averaging. This demonstrates

that the molecular motions in B-CO are characterized by a time shorter

than the reciprocal of the static quadrupole resonance frequency,

Sl0-sec., and that the reorientations are not isotropic because a non-

zero coupling constant exists. The Hamiltonian of the 017 nuclei in s-CO
can be written as

H = z + H (4-1)

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 ) (4-2)
m n <'m 41 VT zz ( I- 2) m> (4-2)

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


gradient at the 017 nuclear site in the laboratory frame. From (4-2),
knowing I = for 017, one has

1 = 1 (3m2 35) (4-3)
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 4-1. In the case n = 0, the second
order irreducible tensor transformation enables one to have

Vzz : (3 cos2 0" 1) VZ (4-4)

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, (4-4) can
be written as

Vzz (3 cos2 e 1) (3 cos2 e 1) VZ (4-5)

The effective Vzz would be given by the time average of the right hand
side of (4-5)

V-z = (3 cos2 6' 1) (3 cos' e 1) VZZ

= 4(3 cos e -. 1) (3 sin2 y cos2 1) VZZ (4-6)

It turns out from (4-3) and (4-6) that

v = ie (3 cos2 e 1) (3 sin2 y cos2 t 1) (3m2 (4-7)
m 1h )60 4

Il ,


Definition of angular variables used in the text to discuss
the N!R rotation patterns in s-CO. 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 4-1.

and for various m one can immediately write

Av~ = e2Q (3 cos2 e 1) (3 sin2 y cos2 $ 1) (4-8)

3 = (3 cos2 e 1) (3 sin2 y cos2 0 1) (4-81)
E73 80h

Av~ -20h (3 cos2 B' 1) (3 sin' y cos2 6 1) (4-8-)

The quadrupole splitting is of the order of KHz while the static

quadrupole coupling constant is of the order of MHz. This small non-zero

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

1 T
= 7 f(e) [3 cos2 (eo + E)-l + 3 cos2 (e c) 1] de


= f(E) e2de

= E2 (4-9)

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 = () (4-10)

3 2
2 2

And combining (4-8) (4-9), (4-10) and (4-10'), it follows that

(2) = 2 (3 sin' y cos2 C 1) (4-11)
801i o

v ( h Qo (3 sin2 y cos2 1) (4-11)

From the experimental perturbed spectra it is obvious that in several
cases a large portion of the sample grew as a single crystal of s-CO.
The evidence comes from the rotation patterns of the separations of the
satellites as shown in Figure 4-2, in this case at 64.7 K. The perturbed
spectra at different angles are also shown in Figure 4-3. 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 (4-12)


2 ) = .2 3 sin2 cos2 -1 (4-12')

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 E-more would give 13 sin2 y cos2 { -11 = 13 sin2 y -11. It is obvi-
ous for symmetry reasons that the rotation pattern of (4-12-) 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 (4-11) and (4-11-) can be simplified as

Figure 4-2. The NMR perturbed spectra of B-CO at different (4 o)
angles, where io is a laboratory reference angle.

)----------: li:!-~~

. 9 1 I d

- I

0 20 43 60 00 100

Figure 4-3. Rotation pattern of the separations of the satellites of NMR
B-CO as a function of (* o).

120 140 160 1I0

perturbed spectra in








9 I I i ,






i2(2)!- s 6e2Qq -2
2, 40 1h 2 (4-13)

2v( ) 3 o (4-13)
I40 h 0

In fact the values of 13 sin2 y cos2 1 1I as a function of have two
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 .(4-14)

It is clear that one can also have an angle ^' such that

3 sin2 y cos2 I' = 1

From (4-14) one determines

1 1
sin = co (4-15)
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
a-CO one can determine the value of e If one defines the effective
quadrupole coupling constant as
eQ> 1
< eff h e (3 cos2 1) (4-16)
= e2g E2
2h 0

one .would have from (4-12), (4-12') and (4-14)

2 =2 -0 h >, (4-17)
e f


2( 0) h eff (4-17')
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 (4-18)
Seff h

which is three orders of magnitude smaller. Figure 4-4 shows the effec-

tive quadrupole coupling constant for several temperatures of B-CO. 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 s-CO
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 (4-19)

W2 =25 1 (4-19-)
2 2f




61 6; 63 64 65 66 67 68 69


Figure 4-4. Effective coupling constant as a function of temperature in --CO. (A typical error
bar is shown .t T = 68 K.)

then the spin-lattice relaxation is governed by a single exponential


1 4
T1 gW (4-20)

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 (4-21)

where T now is the autocorrelation time for the precessional motion

around the crystal C axis. Using e2g from NOR measurements in the
a-phase one has the following relation between T, and T:

= 3.93 x 1014 1 (4-22)
2 T (4-22)

Shown in Figure 3-2 is the temperature dependence of 1^ and 2'. Although
there is a different meaning for T' and T2, they specify the correlational

motions in B-phase and liquid CO respectively. Figure 3-2 shows that

there is a discontinuity at the transition temperature. It is reasonable
that the characteristic time of the motions in B-CO takes a longer period
than that in liquid CO, while the discontinuity in the opposite sense of


B-N2 near the transition temperature is not clear at this moment [52].

For e-N2 the same assumptions were made as one did in B-CO, namely that

I = W2 and that all lines are saturated. One therefore has

1 (4-23)
T1 1

in liquid N2 as well as in g-N2.


5. 1 Molecular Properties of a-CO

Solid a-CO is one of the simplest diatomic molecular crystals and

has been an interesting subject for studying the lattice dynamics of

molecular crystals [5,11,12,53-55]. 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 dipole-dipole and

quadrupole-quadrupole 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 a-CO. Gill and Morrison [10] extended the measurements down to 2.5 K

with no indication of the occurrence of an order-disorder transition.

However, it can be mentioned that a small anomaly has been observed in

the heat capacity of a-CO near 18 K and ascribed to the freezing-in of

molecular head-to-tail reorientation [16].

5. 2 Nuclear Quadrupole Resonance in a-CO

To study the molecular dynamics in a-CO, 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 a-CO 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 a-CO differ little from that in the gas phase. The quad-

rupole Hamiltonian can be written as

e= 2Q() [3 (5-1)

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) (5-2)
5 3 10
+ +++ n

3 = 3 e2Qq(T) (5-3)
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 (5-4)

The measurements of the NQR frequency versus the temperature are shown in

Figure 5-1. The static QCC was also used to derive T2 and T2 in the

liquid and s-phases 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 spin-rotation 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



T u/


F I ~

Figure 5-1. Temperature dependence of the pure quadrupole resonance
of 017 in a-CO.

0 20 50 40 50




, I


NQR determined QCC from the a-phase 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 a-CO is the anomalous behavior of the linewidths. Figure 5-2 shows

the temperature dependence of the linewidths and some T1 measurements,

while Figure 5-3 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 a-CO 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 spin-lattice 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-


The occurrence of orientational disorder in the a-CO crystal has been

expected since the first estimate of the residual entropy was made [4].

However, the frozen-in 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 spin-lattice 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--- -- "


7 c 2.0-


1.0 -


o 0


0 0

T a -

50 60


T a/


o ol
00 0o,
Io o o o I

I i !

50 60

10 20 30

Figure 5-2.

Temperature behavior of the 017 NQR linewidths in a-CO.
The insertion shows some effective T1 measurements.



T (K)


)/- iiOO
/7--, / 7 /--7

----I-L- ~ _~-~
__~_ ~--, ~__~-~~
__ ------~---~----~-------

---- ~-2_5 _

j-------- ---;--- --------------------------------

Figure 5-3. Three dimensional representation of the 017 NQR spectra in a-CO. 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 >] (5-5)

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) (5-6)

V (t) = e [1 3 (1-s(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 in-plane and out-of-plane librational angles with
respect to the 7 reorientation respectively. * is a random variable of
order < 42 >2 which allows reversal of an angle which is not exactly r as
discussed in [57]. Taking the librational average of (5-6) and assuming
no correlation between t,,, 4, and 4* one has

-_ C1 3< >) (52-7)
32 -2

S y > = e [1 < s(t) > ) 3 s > ]

< Vzz > = eq [1 < 42 > <2 > 3 (1 < s(t)> )]

< V > < Vz > =

In the case of disorder, where < s(t) > = 0, one has

5 3
< Vzz> = eq (I- 2-<*2>- 2<2 > ) (5-8)

The resonance frequency between m =c and m =- will be

3 s-Q R (I 3 < 3 3 *2)
S < > h < 4 > 2- *2) (5-9)

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
which is higher than in (5-9).
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 (5-6) and (5-7) and gets

V < Vx> 2(4L < >) (5-11)
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 + (5-12)
T2 T+2 5 T 3


1 = (t)C(t-T) dz
is the adiabatic contribution and m'(t) is the departure of the instan-
taneous resonance from its average value and

1 7 W (5-13)
T 5
T5 5 + ,m
-t m 2

1 = 3 (5-14)
-13 3,
"2 22

where W is the transition probability per unitatime from state m to
state m- by the fluctuation terms in (5-11).
By neglecting for simplicity the terms which do not involve the re-
orientations one would have

r q 12 3 < J (0) (5-15).
'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 ) (5-16)

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 (5-18)T 2
9+ C+ (5-17)

Combining (5-15), (5-16) and (5-17) one obtains

12 [ 9e2Qq23 < *> JII (0) + 9 > < I
T2 2A 16"I 4 A-I 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 (5-18) reduces to

'= 2 Ie2,1 5-c 4Ti
T + 2 -4 + (5-19)
2 20 2
From (5-19), 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 spin-lattice 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 (5-18) one would
IU eff

[ 1I ]e T j1 (5-20)

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 103ec-1 and according to (5-20) 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 (5-19)

-v < > 1 (5-21)
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 (5-22)

where T;' is the fluctuation time between the diagonal equilibrium


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 a-N,

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 5-4, the temperature dependence of resonance fre-

quency of a-CO is nearly like that of a-N2, 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

+ 0

+ o

+ 0

+ 0

.9 -

.1 .2 .5 .4 .5 .6 .7 .8 .9 1.0


Figure 5-4. The reduced temperature behavior of the reduced NQR frequency. ("o" represents
a-CO and "+" represents a-N2. Data for a-N2 from Ref. [49]).

1------~-----C`7.'~-I--T .-I 1 r

. Ci-

The peculiar behavior of the linewidth in the a-phase NQR measurements

elucidates the possibility of the motions in a-CO. 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
Sv = e KT (5-23)

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 a-CO 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 anti-ferroelectric type.


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 A-1.

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 (1-18)

H4 -. r Vo( (3 2) + V+ (I Iz + T I
Q 41(21-1) 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 (A-2)

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






W VI VA(ihA)

JW W,(I+A)

,w Wz 2,.A

Figure A-I. Quadrupole induced transition probabilities among the
Zeeman energy states for spin I 5/2.

I ( I+2A)







I,1 I,m > = m /iTI---in(mei TT l,I,1 >
Iz+ I,m > = (mrl) /I+TTY)- m(mn I ,rrl > .

One then calculates the matrix elements from (A-3) and (A.-3')

< Iz + I+ > 2 = 80

1< 1+ Iz + 1,1; 2 >2 = 80


< 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







(A-5 )

Fron: (3-2), (3-4), (A-4) and (A-5) it follows

0ao e-iLt 1T + (t- ) dr (A-6)

2 40O e i2L V+2(t) V;2 (t T) dr (A-6')

where 1, = W5 and W2 = 215 1. One will need to calculate
2' 2 2' 2
V+1(t) V-I (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, -) (A-7)

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 A-2.
The unitary operator D (a, 3, y) can be written as

D (a, B, y) = e-iYJ'"e-iJJy e-iaJz (A-8)

Using the fact that

Jy = e-iaJz J e i
Jy y



Y, /

\ y, Y"
\- /

x i .

yX \

Figure A-2. 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 e-iJZ 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 > (A-10)

= < a' eiaz J .e-ia a >

It .follows that

Jy e-iaJz Jy eiaJ (A-11)

Using (A-ll) and by expanding, it can be proven that

e -1y = e-iaJz e-iJ eiaz (A-12)

Following the samp procedure, one has

e-iYJz" = e-iaJ e -i' e-iYJz ei JY eiaJz (A-13)

From (A-8), (A-12), and (A-13), it can be shown that

D (c, B, y) = e-iaJZ -iJy e-iYJ (A-14)

Let Ij,m > be a simultaneous eigenket for Jd and Jz. the matrix elements
D then will be written as

DJm = < j,m D (a, B, y) j,m" >

= < j,m Ie-iaz e-iJy e-iYJz j,m >

S -iem e-iYm' d m, (A-15)

where the definition of dj is obvious.
m rn,


From the table [30] one has for elements of d2 ,m..) as

2 = d2 = cos4 () (A-16)
22 --2 2

d21 = -d2 d2 d2 s (t--Cos
21 1= -2 -2-1 = sin (1 + cos

do = d = d2 d2- =/3-/8 sin2
2 02 -20 0-2

d2 d2 =-d2 = -d2 1
2-1 -2 -21 -12 2 )

d2_2 = d2 = sin4 ( )
2-2 -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 (A-2) that

V 3 V
o = ZZ

V., = 0

V2 = (VXX Vyy
2 2 *XX *YY'


Transformation from the principal frame of the field gradient to the
laboratory reference frame, yields

V+ 3v Z e-iY /372 sin 5 cos 6

1 -iec ,-j 1
+ (VX Vyy) e-i2 e-iY [- sin B (U + cos 6)]

+ I (VX yy) e2 e-iY [- sin B (cos 1 1)] (A-17)

3 iY
V-1 VZZ e (- ,3/2 sin B cos B)

+ Vyy) e-i e [ sin B (cos 6 1)]

( Vy) ei2a iY [- sin B (cos B + 1)] (A-17')
+ (VXX YY e e

V 3 V e-i2Y /3/8 sin2 B
+2 =rVXX e

1 -i2 -i2Y 4 B
+ XX yy) e e cos4 )

+ (VXX Vyy) ei2z e-i2Y sin4 (-) (A-18)

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() (A-18-)
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

V (t) V ( = T ( + ) V (A-19)

V1l (t) v* (t) (3 +4n)V2
1 TO (1 + 3 ZZ


3 n
+ n2 2
-2 (t) V*2 (1 + ) VZZ

where n = is the asymmetry parameter, Assume
Vi (t) V' (t r) = IVi (t e" -2 (A-21)

for i = 1, 2 where -2 characterizes the correlation function of Vi (t).
Using (A-6) and (A-6'). one finds

Ie___ 1 1 2 2r2
w, -- f 24 (1 + ) (A-22)
2 401 3(1 + 7+

24 (1 + ) 2 (A.-22')

In the fast motion regime uLT2 <

W1 e- 48 (12 (A-23)
1 S 4) 'J
W2 = [ 48 (1 + q ) T~ (A-23')

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:

S m (-N Nm Wm;m ) (A-24)

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) (A-25)

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)

1/2 2 1 9
d N1/2 1 (1 +A) N1/2 N (1 + 2A) T /2 2
d-142 N112 N7l Y 11 N-22 W2

dt 5 /2 N (I + 2)
5 N3/2 W1 2 N5/2W2 10 N3/2 2 (1 + 2&)

-1/2 2 1 9
dt = -1/2 1 2 N-1/2 12 10 N1/2 2 (1 + 2A)

+ -- (2 + 2) ++&
+ N3/2 1 (1 + A) + N-5/2 W2 (1 + 2 N3/2

d-3/2 =2 9
dt -3/2 1 -5 N3/2 1 + ) N3/2 2 1 + 2

+ N-5/2 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 (A-26)

where no = and N is the total nuLmber of nuclei. Similarly,

N12 (N / 0 -2n o o (A-27)
N0 /2 1N/2) = 2n 5 N/2 N1/2 (A-27)
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 (A-28)

Defining nm = N N one can write the following equations from (A-25):

dn5/2 1 1
dt = -"5/2 W1 2 n5/2 2 + n3/2 W1 + 2 n1/2 2 (A-29)

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

dn-1/2 2 1
dt "-1/2'1 -2 n-1/2

dn-3/2 2
Sdt -n-3/2 W1 5 n-3/2 "1

9 2
0- "1/2 W2 + n3/2
1 9
+ n5/2 t- n"-3/2 2

9 2
2 -0 n-1/2 2+ 5 n-3/2

1 9
+ n-5/2 2 10 n3/2 I2

10 n-3/2 W2 + n-5/2 1

5 n-1/2 1 + T n-1/2 W2

dn-5/2 1 1
dt = -n-5/2 1 2 n-5/2 W2 + "-3/2 W + 2 n-1/2 2

By multiplying each equation with the corresponding m and summing all
together, cn2 obtains the following equation:

fc 2 (A-3C)

dt z z o

+ [ n3/2 + n "1/2- n-1/2 "-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


where Ks and Ts is the spin temperature which is generally greater
than the lattice temperature. One then concludes that

S> (W1 + W2) (A-32)

which implies

1 (2 1 2) (A-33)

and finally, it follows from (A-23), (A-23 ) and (A-33) that

1' e 2q 1 + ) 2 (A-34)

in the fast motion regime.


In solid B-CO, 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 spin-exchange transitions, the
Boltzmann distribution of the populations among the energy levels cannot

be maintained. In such a case the spin-lattice 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 (A-25)
one has
d(N5/2 N3/2) 2
d(N- /t-- N (5/2 - 3/2)(- + (3/2 2) 1+ W2

+ -(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(N-1/2 N3/2( -

+ 1 (N3/2 N_5/2) W2 + n (2W2 1

d(N_2 N32) (N_3/2 N_5/2)(W 2)
(N1/2 3/2) (41 + 7W2

+ (N3/2 N1/2) W2 nA (W1 5W2)

d(N N )
d(N-3/2 -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 (B-2)
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 + N-1 (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 (B-2), one defines

Np = N n (B-3)
p p 0

where no and A have the same meanings as in Appendix A. (B-2) then

N2 = -N2 1 "2 ) + Ni (W. +2) + N N6 W1 2 (B-4)

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 B-CO and for simplicity an
axially symmetric field gradient, one can calculate IVi12 from (A-17),
(A-17-), (A-18) and (A-18') for i = 1 and i = 2

IV112 = sin2 B cos2 B V2 (B-5)

IV;212 = 6- sin4 B VZ

In chapter III it is shown that

3 cos2 B 1 0

so that, (B-5) can be reduced to

+V'1 I2 VZZ (B-6)

IV-2 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 (A-17) and (A-18).

V, = V Di + V2 D_2 + V Di, + V'1 D_1 (B-7)
41 2 2 1 11 -1 2

V2 = V D22 + V 22+ V D12 + V_^ D 12. (B-7')

For a powder sample using (A-16) and (A-18) and taking over all the
possible directions, one calculates

iv+1 2 vz (B-8)

V (B-8')
IV212 = 3 v (B-8)

The same equations will hold for IV_112 and !V_212 assuming again
Vi (t) V (t T) = IVi (t)!2 e 2

for i = 1, 2 where T2 characterizes the correlational precessional
motions. Using (A-6) and (A-6') one obtains

W1- e2 2 48 2 (B-9)

1 40Aj L ^
W,2 2 48 +-4-1- (B-91)

From (B-9) and (B-9') it is obvious that W1 W2. In the fast preces-
sional motion regime ,L 2T <<1. It follows from (B-9) and (B-9') that

W1 = 48 feQ 2 T (B-10)

W2 8 T24oJ2 (

That is, 14 = W2 in the fast motion regime.


Now from (B-4) it is possible to write

S. 1 2 51N 1 1N (B-1 1 )

1 2 1 15 'Ili +T- N-1
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;.
(B-11) is further reduced to

N= I 11 10 +4NO (B-12)

N' = WN I

N6 1N 5 N6

The general solutions of (8-12) will be

N' = a e-X2t+ap e-3t (B-13)
p p p2 p3

where p = 0, 1, 2. Substituting (B-13) into (B-12) and comparing the
e-1t, e-X2t and e-3t terms one has

(- 2 W1 A) a + 5 + 9 a l = 0 (B-14)

1 a 13
2 121 + (-0 1+ + X1) a = 0

l a21 + (- 5 + X1) ao0 = 0

and similar equations for e-X2t and e-X terms


In order that all aj / 0 for i = 0, 1, 2; j

5 +


10 1 + X

= 1, 2, 3 one needs

10 1


- W1 + I



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 (B-3)


N2 = N1 = NO = 0

and (B-17)

N2 = Ni = N6 = -noA

4 3 33
Let Xi = 1 Hi 2 = 11 and A3 = T10 I Comparing with terms of

e (B-12) gives, at t= 0

17 8 4 9
(- a? a22 + a23) + (an + + a + 313) + -1 (ao: + a02 + a03) = 0
1 a2 a )+ ) 10-U

2 (a21 + a22 + 23 + (- all + a12 + 2a13) = 0

3 3
(a21 + a22 + a23) + (-a0l - a02 + a03) = 0

From (8-17) and (B-13) one has at t = 0

n A = a21 + a22 + a23 (B-19)

n = all + a12 + a13

n A = a01 + a02 + a03

(B-18) and (B-19) enable one to deduce

a2! = aol = all (B-20)

3 2
a22 = 10 02 = a12

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) (8-22)

for p = 0, 1, 2.


Therefore, in the case of complete saturation and W1 = W, the spin-
lattice relaxation is governed by a single exponential decay having

Tl W1 (B-23)

From (B-10) and (B-23) one has the same equation as in the liquid phase
in (A-34)

1 3 e2Qqq2
1 J '2 (B-24)

where T' is the correlation time for the molecular precessional motions
around the crystal C axis.


One assumes that the molecules in a-CO 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) (C-1)

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 out-of-plane librational angle and a is the reorien-

tational angle.

- 0 0
Vj = eq 0 1 0 (C-2)

0 0 1

In (C-2) a symmetric field gradient has been assumed. It so happens

that by assuming a small angle for 4$,

Vxx e (I 32 ) (C-3)

V = - (1 + 32 sin2 a 3 sin2 a)
yy 2
V = (1 + 32 Cos2 a 3 cos2 a)
Vzz sin a2


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) + (C-4)

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 in-plane 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))
cos2 1+ s(t) 1) (1 -(t)) ,,2

From (C-3) and (C-5) it follows that

Vx = e f1 34,) (C-6)
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 (C-6) is given by

< V > = (1 3 < .2 > ) (C-7)
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 >] (C-8)

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 > ) (C-9)

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 (C-9) 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 (C-10)
T2 T2 T5/2 + 3/2


I= "(t) '(t --T' dr
2 o
o'(t) is the departure of the instantaneous resonance frequency from its
average value and

1 1 (C-ll)
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 (A-1) and (C-9) one will have, assuming order phase
< s(t) >= 0

'(t) = 9e 2 3 [ '2 < 2 + _2 -<"2 > (c-13)

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 (C-14)
Jod- 8ot 0 J


T- sT r e-iT dT = JII (W) (C-i15)

it follows from (C-14) and (C-15) that

1 9e Qq 3' < *4> (C- 6
T2 20 16 JI
Consider now for terms. From (C-1l), one writes

T52 = W5/2,3/2 4- W5/2,11/2
Using (A-1), (A-4), (A-4'), (A-6) and (A-6-) one obtains

I = 4 0 L f V (t) V(t -) e- dr (C-17)
r5/2 I 401 1 L

+ 4 40 V +2TT-Ty (T e-i d
4+ f L {[i(TY'YT e-1t dr

4 V_2(t) V*2(t -tT e-i1'e d-

Now one needs to consider

A V (t) V*l (t -) e-iT dr
2 r
A = 2 ) s(t) t -) s (t ) e- d (C-18)

+ s(t) s(t -) e dr + ,(t),,(t r) e-i dr

and where no correlation between the value of * and s(t) has been

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
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

(C-20) can be reduced to
f3eq2 < ,*2 > T"
A = --*2 (C-21)
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)] (C-22)


Vxy y > (1 s(t)) + ,,, s(t)]

If one further assumes that no correlation exists between i,, (., (*
and s(t),(C--22) can be simplified as

y V < > --3 s(t) (C-22
yy yy 2 2 s(t) (C-22

V -< > = 0
xy xy

It follows that

B 3 2 9 s(t) s(t-) e- dr (C-23)
2 J ( 4
34 119 2

From (C-17), (C-21) and (C-23)
15/ 3e2 2 o< 2 2 + 18o0<* >J (I W)
5/2 80 7 2i
Considering now the term and using (A-1), (A-4), (A-4'), (A-5),
(A-5'), (A-6) and (A-6 ), one writes

T13/ = 80 V+l(t) V l(t z) e-1T dr (C-25)

T- -- +
3/2 40 J _

+ V_ 7(t) V*1(t--'-T e-i02 dr -

+ j (T r V*ETi Tr e-1' dr

+r-eQ 2 F .) t- -- dT

+ V_(t) V*2(t ) e- 2 dr

+ < p.2 > + < .4 > Jl ( 1() (C-25)
1 + '2 1 2
4 2

From (C-10), (C-24) and (C-26), 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 (C-28)

1_ = 9e2Qq @2 > 2 L2L.
T 2 L- + W2 2-
2 20_ 1 + -2 9 + 2

These terms on the right hand side of (C-28) 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 (C-29)

(C-28) has a maximum.

One thing worth noting in (C-7) is that < Vzz > shows the usual
Bayer-type 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, (C-28) can be further reduced to

1 [2 9eq 2 2 (C-30)
'2 Lj [ 20t1 J

which relates the linewidth and the correlation time of the molecular
reorientational motion.


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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

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