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'THSA

CRYSTALLOGRAPHIC STUDIES USING
GUADRUPOLE RESONANCE

Thesis {or II“ Degree of M. S.
MICHIGAN STAIE UNIVERSITY
Kenneth S. Robinson
I963

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3 1293 01743 0061 .

LIBRARY

Michigan State
University

 

1.14‘a

(I.

ABSTRACT

CRYSTALLOGRAPHIC STUDIES USING
QUADRUPOLE RESONANCE

By Kenneth S. Robinson

The interaction of the electric quadrupole moment of
a nucleus (if the particular nucleus possesses such a
moment) with the electric field due to surrounding charges
in the crystal lattice gives rise to discrete energy
levels, transitions between which energy levels can be
induced by subjecting a specimen of the crystalline material
to an oscillating magnetic field at a resonant frequency.
This was accomplished by a super—regenerative radio
frequency oscillator. If at the same time the crystal is
subjected to a constant magnetic field of low intensity
(Zeeman field) the degeneracy of the energy levels described
above is removed and the spectral lines as viewed on the
screen of an oscillosc0pe are split.

The separation of the components is a function of
both the intensity of the Zeeman field and of the angle
between the Zeeman field and the electric field gradient at
the nucleus, being zero for certain values of the angle.
Ifa.stereographic plot is made of the direction of the
Zeeman field at zero splitting of the spectral line, for

various orientations of the crystal in the radio frequency

Kenneth S. Robinson

coil, the direction of the electric field at the nucleus can
be determined. In this manner the number and orientations
of inequivalent molecules in the crystalline unit call can
be determined.

Studies were made of the splitting data for specimens
of sodium chlorate, paradichlorobenzene, and cuprite.
Both high and low temperature phases of paradichloro-

benzene were observed.

CRYSTALLOGRAPHIC STUDIES USING

QUADRUPOLE RESONANCE

By

Kenneth S. Robinson

,A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1963

DEDICATION

This thesis is dedicated to my wife, Ruth,
for her love and patience during this period of

study and research.

ii

TABLE OF CONTENTS

Page

DEDICATION . . . . . . . . . . . . . 11

LIST OF FIGURES. . . . . . . . . . . . iv
Chapter

I. THEORY. . . . . . . . . . . . 1

II. EXPERIMENTAL TECHNIQUES . . . . . . 10

III. EXPERIMENTAL RESULTS . . . . . . . 16

APPENDIX . . . . . . . . . . . . . . 29

BIBLIOGRAPHY. . . . . . . . . . . . . 32

iii

LIST OF FIGURES

Figure

1.

10.

11.

Energy level transitions for nuclei of spin 3/2,
after removal of degeneracy by application
of Zeeman field. . . .

Theoretical plot of separations of resonance
line components as a function of orientation
of the Zeeman field.

Superregenerative detector used to supply
radio frequency energy to crystal.

Theoretical stereo raphic plot of zero-
splitting for 5 nucleus in NaClO
showing 3- fold symmetry about a bod;
diagonal . . . . . .

Zero—Splitting data for NaClO, with theoretical
plot of fig. 3 superimposed. . . . . .

Stereographic projection of zero—splitting
data for paradichlorobenzene crystal, high
temperature phase. . . . . . .

Stereographic projection of zero-splitting
data for paradichlorobenzene crystal, low
temperature phase.

Theoretical plot of zero— splitting of cuprite
crystal, about axis parallel to edge of
unit cell. . . . . . . . . . .

Splitting data for cuprite crystal, with
theoretical plot of fig. 7 superimposed.

Splitting of a single line as a function of
Zeeman field intensity . . . . .

Splitting of resonance lines resulting from
two or more physically inequivalent nuclear
sites in the unit'cell of the crystal of
sodium chlorate. . . . . . . . . .

iv

Page

12

17

19

21

23

25

27

29

30

CHAPTER I
THEORY

The potential of a distribution of charged nuclear
matter may be expressed in the series,
O (r,e) = _§_. [Z +-_£_.cos 9 + 1/2 _Q;.(3 cos2 9—1) +....]1 (l)

r r r2

In the first term ze represents the monOpole charge
distribution and determines the coulomb potential of
the nucleus. P represents the electric dipole moment
of the nucleus. It can be shown, from parity considera-
tions, backed by experimental evidencegl)that the electric
dipole moment of any nucleus is zero. The quantity Q
specifies the electric quadrupole moment of the average
nuclear charge distribution. This quantity is zero for
Spherically symmetric charge distributions, as well as
for nuclei of Spins zero and one-half. Charge distribu—
tions represented by a prolate ellipsoid possess positive
nuclear electric quadrupole moments, and a negative Q
corresponds to an oblate ellipsoid distribution. For
example, the electric quadrupole moment of the deuteron
has a value of + 2.73 x 10‘27 cm2, showing that the
ground state of this nucleus is not a pure s state of

zero orbital angular momentum.

1

A strong coupling between the electric quadrupole
moment of a nucleus and the electric field gradient at the
nucleus due to surrounding charges produces quantized energy
levels between which transitions may be observed under
prOper circumstances. This phenomenon is known as pure
nuclear quadrupole resonance.

It should be emphasized that these energy levels are
caused by a coupling of two electric systems present with-
in the atom, in contrast to nuclear magnetic resonance,
which results from the application of a strong external
magnetic field, which then interacts with magnetic dipole
moment of the nucleus.

To observe the quadrupole resonance Spectra, transi-
tions between the energy levels must be brought about by
absorption of energy supplied from some external source,
and for maximum probability of such transitions, energy
must be supplied at a frequency such that the condition,

I} L/ = Em + l - Em is satisfied. In principle, several
mechanisms for supplying such energy to the nucleus
present themselves, but practical considerations eliminate
all but one. Since nuclei do not possess electric dipole
moments, the only way an oscillating electric field could
be used is to couple it with the quadrupole electric moment
of the nucleus. However, to induce a sufficient number of
transitions by this method, electric field gradients too large

to generate in the laboratory would be required. In practice,

an oscillating magnetic field is coupled with the magnetic
dipole moment of the nucleus, to ”shake” the nucleus and
cause transitions between energy levels resulting from
coupling of the electric quadrupole moment of the nucleus
and the field gradient at the nucleus due to surrounding
charges. It is evident that definite energy levels will
exist only if the average field gradient at the nucleus
is constant in time, hence nuclear quadrupole resonance is
observed only in crystalline solids.(2)

.If the electric field at the nucleus has cylindrical
symmetry, quantum-mechanical theory gives the following
expression for the above energy levelS—-(3)

2
_qu 2
Em _ AI (21_l) [3 m — I (I + 1)] (3)

 

where I represents the nuclear spin number, m the mag-

2

netic quantum number. The expression €3<fiQ is called

the quadrupole coupling constant, where eqss sz is the
component of the field gradient tensor in the direction
of the symmetry axis and eQ represents the electric
quadrupole moment of the nucleus.
For equation (3), it can be seen that the states
9V+ m and HV—m possess equal energies, since only even
powers of m appear, and that therefore for nuclei of half-

integral spin there are I + 1/2 doubly degenerate energy

levels, since m may take on values I, I — l, l - 2,. . . -I.

If now a weak constant magnetic field HO, (Zeeman field)
is applied at an angle 9, this degeneracy is removed
and for a given |m |>l/2, there are two energy levels—-(3)

2
8 Q 2 , q._
E+ = 41%2I—l [3m — I (I+l); + m 'h i 2 cos a (A)

where_I1.= XII represents the angular frequency of the
Larmor precession of the nucleus in the magnetic field.
For [ml = l/2, a Special quantum mechanical case arises,

in which mixing of the 9V+l/2 and (7‘j-l/2 states forms new

(3)

states, +, having energies given by —-
E+ = A [3/4 - I (I+l) ': _%;_ 'fi 47L cos 6] (5)
2
where A = —§—33;- and
AI (2I-l)
2 ”/2
f = [l + (I + l/2)2 tan O]i

Since (4) represents energy states for lml >il/2, and
(5) represents energy states for lm I: l/2, transitions
can occur between two energy levels, one being either
one of the levels (A) and the other being either one

of the levels (5). For the case I = 3/2 the resulting

four transitions are shown in the accompanying diagram.

 

 

 

 

 

- — 3/2
The lengths on the I
, + 3/2
diagram, labeledm,«, p, )3'
” I
bear no relationship to the 0‘ I3 B'
intensities of the spectral w -
L +

 

 

lines resulting from these Fig. 1.--Energy level transitions

for nuclei of spin 3/2, after re-
moval of degeneracy by application
of Zeeman field.

transitions.

If each of the energies in (A) is, in turn, subtracted
from each of the energies in (5) and if the resulting

expression is divided by‘h, to give angular frequency

(3)

(L), the following frequencies are obtained--

 

(1)0‘ 6 A _ .2L;;£:_ _f2_ cos 6

 

 

 

fi. 2
MP = 61‘1A __:_;;_£_—rLcose
Lkhifl = éfiA +-—§—§_£_ .f7. cos 6
Cljfy == €65 —§L%%£_._fd— cos 6

Here f represents the absolute value of the radical,
and the «and )3 components differ only in the sign
assigned to f. In this sense there is no real physical
difference between the g: and D, and the“ and {3'
components.

The splitting of the dicomponents is expressed by

CU“, — we: = (3—f)_n_. cos 6

Setting this quantity equal to zero, and assuming cos a

 

 

, 2 l 2
¢ (3 we get .f = (l + (I + l/2) tan2 a] / = 3 ,
tan 26 = 8 2: and
(I + 1/2)5
a = arc tan 2 A 2/ arc tan V 2 for I = 3/2 . (8)
I + 1 2

Cos 6 can be assumed unequal to zero, for zero splitting,
since for 6 = 90° the splitting is (I + l/2)I:L, as can

be seen by substituting the expression for f in (7), and

 

 

 

 

 

simplifying.
2
1 B
c l
O
H
4.)
CO
(2..
CO
3 L 1 l OJ
m 110‘” 150
c o
g 25 1
Q)
Q) '1
N -_L
B!
-2 _

 

 

 

 

Zeeman Field Angle

Fig. 2.-—Theoretical plot of separations of
resonance line components as a function of
orientation of the Zeeman field.
Separations are in units ole = 3 H

The Splitting of the fi component is expressed by
(«UP — U)? = (3 + f).§l cos a. This expression vanishes for
f = -3, which occurs for e = 125°16'. In summary of the
above discussion, it is stated that, for the case I = 3/2
zero Splitting of the «component of the quadrupole reson-
ance line occurs when the angle between the Zeeman field and
the field gradient at the nucleus is approximately 54°44”,
and zero splitting of the P component occurs at e = l25°16‘.

Quantum-mechanical theory gives the ratio of the in-
tensities of the P component to theg(,component as f-l / f+l,
so that the dpair is always more intense than the P pair.
Note that here f is an absolute value. For zero splitting
of the ngine, this ratio becomes l/2, making the zero
splitting condition relatively easy to determine experi-
mentally, since generally there will be a strong line
formed by the coalescing of the eacomponents.

It can be seen that, for zero splitting, the
direction of the Zeeman field is an element of a right cir-

3/2). The

cular cone of semivertical angle 54°44’ (for I
axis of the cone of zero Splitting is a symmetry axis of
the electric field gradient at the nucleus. Hence the
directions of the symmetry axes of the electric fields in
the Specimen can be determined experimentally, and if the
symmetry axis of the crystal is known by other means, such
as crystallographic measurements, then the orientations of
the electric field gradients within the unit cell of the

crystal can be determined.

The Hamiltonian for the interaction of the electric
quadrupole moment of the nucleus with the electric
field at the nucleus due to the surrounding charges
in the crystal lattice can be expressed as
HQ = eQ {V22 (31: - 12) + (Vxx " Vyy) (Ii " 15)]
4I(2I—l)
It can be seen from the above that only two para-
meters are needed to characterize the derivatives of the
potential--VZZ and Vxx — Vyy’ The asymmetry parameter,

V .
ZZ , or what is equivalent,

VXX . V22

1)

 

 

 

aEx _ QEy

71:: (a)( 3Y7
Lb Ez
’az

is defined to express the departure of the charge con—
figuration of the nucleus from cylindrical symmetry. A
nucleus may originally possess a symmetrical field but
this symmetry may be destroyed by intermolecular forces
in the crystal lattice. Thus in the free tetrahalide
molecule there is three-fold symmetry about each halogen
bond, but in the crystalline state intermolecular forces

destroy this symmetry.

In all the previous discussion it was assumed that an
axially (or cylindrically) symmetric electric field
existed at the nucleus(asymmetry parameter = O).
In many cases, including those studied here,'r\is small
enough that it can be disregarded. In fact, one piece
of information obtained from nuclear quadrupole
resonance measurements is an upper limit to the value

of the asymmetry parameter.

CHAPTER II
EXPERIMENTAL TECHNIQUES

Since the electric field gradient at the nucleus is
principally due to the valence electrons of the atom,
determination of this gradient can provide useful infor-
mation about the binding of the atom in the crystal.

While in principle the gradient can be determined by
measurements of the intensity ratio of them and P
components, f - l/f + I, it is more feasible experimentally
to measure the dependence of the splitting of theoc
component on the direction of the magnetic field. This
requires the following apparatus:

1. A means of subjecting the sample to a radio
frequency magnetic field whose frequency can be
swept through the desired region.

2. A means of detecting the resonant energy absorption.

3. A method of applying a constant magnetic field in
the Zeeman region to the Specimen.

4. A method of analyzing, visualizing, and
unraveling the loci of zero splitting for each
of the symmetry axes of the unit cell.

With respect to one and two above, considerably more

radio frequency power is required than in nuclear magnetic

resonance experiments, because of the smaller lattice

10

11

relaxation times. The sensitivity of the detection apparatus
must be sufficient to give a measureable signal above the
noise level of the oscillator and amplifier. In nuclear
magnetic resonance it is possible to hold the radio

frequency field at a constant frequency and search for

resonance by varying the Larmor frequency of the nucleus

' J

by means of slowly varying magnetic field. In quadrupole
resonance, the nuclear frequencies are fixed by the

crystalline structure and provision must be made for sweep-

 

ing the applied radio frequency back and forth over the

,i
1..

resonant frequency of the nucleus. If this is done at a
constant rate, absorption will occur each time the resonant
frequency is passed through.

The above requirements are fulfilled by the self-
quenctdxmgsuper—regenerative oscillating detector, a form
of which is shown in the accompanying diagram. The sample
is placed in a radio frequency coil, L1 and the voltage
level of oscillation becomes a function of energy absorp-
tion from the tuned circuit by the nuclear spin system.
The oscillations in the tank circuit are quenched by
the charging of the capacitor 04’ the rate of quenching
being determined by the R104 time constant. This causes
the specimen to be subjected to bursts of radio frequency
energy. If energy is absorbed from the circuit by the
specimen, the discharge of C2 will be hastened and the

radio frequency bursts will be initiated earlier by nuclear

.m
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13

signal. This results in a still greater absorption of energy
and provides the high signal to noise ratio inherent in this
detector. The voltage appearing across R2 is applied to

the vertical deflection plates of a cathode ray oscil—
lOSOOpe, and if, periodically, energy is absorbed from the
circuit, a signal is seen on the screen of the oscillo—
scope. In the arrangement used the quench frequency was
found to be of the order of 50 kilocycles.

Provisions is made for sweeping back and forth over
the quadrupole resonant frequency of the nucleus by the
vibrating condenser, 02 which is driven by the line
voltage and which frequency modulates the oscillator.

Thus 120 times per second energy is absorbed from the
circuit by the spin system of the nucleus, provided the
oscillator is tuned to the proper range.

A problem which arises in connection with the use
of the self-quenched superregenerative detector is that of
the side bands occuring at integral multiples of the
quench frequency on either side of the fundamental
frequency. Resonance can occur at any of these side
band frequencies, if it possesses sufficient energy, as
the oscillator is swept through its range, thus leading
to confusion concerning the correct resonant frequency.
This confusion can be overcome by varying the quench
frequency and observing the resonant line which does

not shift. The author has also found that one can vary

‘- 13.37277; 11:: M'". £1

14

the oscillator frequency and count the resonances appear-
ing, and choose the central one of these as the fundamental.

The Zeeman field of three above, is supplied by a
double Helmholtz coil of radius about 11 cm. and having
about 500 turns on each coil. This provides a uniform
field at the center of about 36 gauss when one ampere of
current exists in the coil. The coil is arranged so that
it can be rotated about a vertical axis. Since the
crystal is arranged in the cavity of the oscillator coil
in such a way that it can be rotated about a horizontal
axis, it becomes possible to sweep the magnetic field
over the crystal from all directions. This obviates the
inconvenience of actually orienting the magnetic field
over the entire sphere.

The problem of four above is met in the following
manner. The zero-splitting data will now consist of a
set of ordered pairs of numbers, representing the angular
settings of the crystal in its rotation about a horizontal
axis, and of the Helmholtz coil about a vertical axis,
respectively. This data can best be analyzed, and the
zero—splitting can best be visualized, by plotting this
data on a stereographic net, which, in effect, accomplishes
stereographic projection.

In the stereographic projection, a point is projected
from the surface of a sphere onto a plane which is per-

pendicular to an axis of the sphere and passes through

15

the center of the sphere, by passing a line from the point
to the pole of the Sphere which lies on the Opposite side
of the plane, the intersection of the line with the plane
being the projection of the point. The distinguishing
characteristic of the stereographic projection is that a
small circle on the surface of the Sphere projects as a
circle (or the limiting case of a circle, a straight line).
In effect the cone of zero-splitting intersects the
surface of the imaginary Sphere in a circle, which is
then projected onto the plane of the paper. It then
becomes a simple matter to draw the best arcs through the
projected points, or to fit a theoretical plot to the
data, if the unit cell structure is known from other
sources. The center of the projected circle is the symmetry

axis of the electric field gradient present at the nucleus.

CHAPTER III

EXPERIMENTAL RESULTS

The structure of crystalline sodium chlorate is well
known, as a result of x-ray diffraction work. Sodium
chlorate has a cubic structure, with the unit cell
containing four molecules, each molecule containing

3

with the three oxygen atoms at the vertices of the base,

a Na+ion and a C10- ion. The Clog ions form a pyramid

the chlorine atom at the apex, and the lines connecting

the chlorine and the sodium nuclei being parallel, res-
pectively to each of the four body diagonals of the unit
cell. Therefore a three-fold symmetry exists about each
body diagonal. Since the C135 nucleus has Spin 3/2, and
since the electric field gradient is symmetric about the
sodium chlorine axis, equation (7) gives 54° 44' for the
value of the semivertical angle of the cone of zero-
splitting about each field gradient. Figure 4 is a
theoretical stereographic plot of the loci of zero-splitting

for the chlorine nucleus in NaClO when the crystal is

3)
rotated about a body diagonal. There are four physically
inequivalent sites in the unit cell, for which the symmetry
axes of the electric field gradients are represented by

points labeled A, B, C, the fourth symmetry axis being
16

17

 

 

 

 

Fig. 4.--Theoretical Sterggfiraphic Blot of Zero-
Splitting for Cl ucleus in NaClO ,
Showing 3-fold Symmetry about a Boéy
Diagonal. '

18

represented by the dot at the center, coinciding with the
north-south axis of the projecting sphere.

Points A, B, C, are separated from the center by
70° 32' on the stereographic net, which is the angle
formed by the intersection of the body diagonals of a
cube, and are separated from each other by 109° 28’. Since
the cone of zero splitting has a semivertical angle of
54° 44', its intersection with the projecting Sphere
will be a circle, the major arc of which lies above the
projection plane, or primitive circle, and the rest below.
The smaller arc about each point represents a projection
from below the plane.

Since the cone of zero-splitting has two surfaces,
or nappes, there will be associated with each of the points
A, B, C, a point A', B', C' representing the other end
of the field gradient symmetry axis, surrounded by pro-
jected arcs representing the other nappe of the cone.
In this case the larger arc is projected from below and
the smaller from above. It should be pointed out that
the points representing the field gradient axes are at
the physical centers of the arcs, as measured on the
stereographic net, and not at the geometric centers.

Figure (5) represents the zero—splitting data for
sodium chlorate rotated about a 3—fold axis, with the theoret—
ical plot of figure (3) superimposed. Regions of the curve

which were deficient in data points were searched again

l9

 

 

 

 

Fig. 5.--Zero—Splitting Data for NaClO, With
Theoretical Plot for Fig. 3 Superimposed.

2O

experimentally. In some cases broadening of the Spectral
line made it impossible to separate signal from noise in
these regions. The departure of some of the experimental
points from the expected curves is evidence, perhaps, of
imperfections in the crystal lattice. In some of the
Specimens studied later this effect was quite pronounced.
Solid dots represent points projected from the upper
hemiSphere onto the projecting plane, and open dots
represent points projected from the lower hemiSphere
upward onto the plane.

Figure 6 is a stereographic projection of the zero-
Splitting data for a crystal of paradichlorobenzene
06H4Cl2’ which was freshly grown from the melt. The
splitting was carried out at a temperature of about 25°C.
Since this crystal was grown in a piece of glass tubing
which had been pulled out to a fine tip and lowered slowly
into cold water, there were no observable faces and no
crystalline orientations were known. Therefore it was
impossible to construct a theoretical plot of the
splitting. However, the plot Shows quite clearly that
there was only one molecular orientation in sufficient
abundance to produce strong Signals. The presence of
points some distance from the curves is evidence that
phase transitions had taken place. In this Specimen the
field gradient axis AA' is inclined at an angle of about

72° 30' with the axis about which the crystal was rotated,

21

 

 

Fig. 6ru-Stereographic Projection of Zero—Splitting
Data for Paradichlorobenzene Crystal, High
Temperature Phase.

22

according to measurement on the stereographic net. This
measurement was predicated on the assumption that, since

(4)

Dean and Pound had found the value of the asymmetry

parameter for C135 in paradichlorobenzene to be 0.081
0.03, then the predictions of theory as discussed
previously here apply, within the limits of experimental
methods used here, and so the zero—splitting angle is

54° 44‘. This crystal evidently was in the high tempera-
ture (P) phase and this is consistent with the results
reported by Dean and Poundqu in which they report that
the backward transition from )3 tofiphase occurred slowly
on cooling from the melt, and could not be accelerated by
cooling below 22°C.

Figure 7. is for a crystal of paradichlorobenzene
which had been grown from melt some years before. Here
there are two molecular orientations, AA' and BB', making
an angle between them, as measured from A to B on the
stereOgraphic net, er about 75°. This agrees with
results given by Dean and Pound(°) , which gives the angle
between the 01-01 direction (Z axis) and the symmetry
axis of the monoclinic crystal (b axis) the value 52° 30'.
The two molecules in the unit cell are related by rota-
tion through 180° about the b axis. By locating a point
on the stereographic net at an angular distance of 52° 30’

from both A and B' the b axis of the crystal was located

23

 

Fig. 7.__Stereographic Projection of Zero—Splitting
Data for Paradichlorobenzene Crystal, Low
Temperature Phase.

24

at the point marked ‘bi, at an angle of about 680 from
the axis about which the crystal was rotated.

Resonant frequency in the“ (low temperature) phase
was measured at 34.16 megacycles, at about 25°C. Duchesne
and Nonfils (5)have measured the resonant frequency at
34.28 megacycles at 21°C and Buyl-Bodin and Dantreppe(6)
obtained a value of 34.16 mc.

It would be instructive to carry out splitting
measurements on the specimen of figure 5 at intervals over
a period of time, in an effort to observe the transformation
from the )3 phase to theofiphase.

Figure 8 is a theoretical plot of the splitting
pattern of a single crystal of cuprite (Cu20), rotated
about a line parallel to the edge of the unit cell. X-
ray data(7)indicates that the unit cell of cuprite is
cubic, containing two molecules, with each oxygen atom at
the center of a regular tetrahedron having a COpper atom
lying midway between the two oxygen atoms. Since the
oxygen-COpper—oxygen axis is the symmetry axis of the
electric field gradient present at the COpper nucleus,
there are four physically inequivalent copper sites, with
the field gradients parrallel to the body diagonals of the
cube.

Since the angle between a body diagonal and an edge
of the cube is 54° 44', the same as the angle of zero-

splitting all the cones of zero Splitting pass through the

25

 

Fig. 8.--Theoretica1 Plot of Zero-Splitting of Cuprite
Crystal, about Axis Parallel to Edge of Unit
Cell.

26

center point of the diagram, the axis about which the
crystal was rotated. In this case elementary spherical
trigonometry shows that the smaller arc of the circle of
intersection of the cone with the Sphere projects onto the
plane as a straight line. Also, the projection of the
second nappe of the conesflxnflaAcoincides with the projection
of the first nappe about C, and likewise for each of the
other cones.

Figure 9 represents the theoretical plot of figure
(7) superimposed on the stereographic projection of the
Splitting data for a cuprite Specimen. This Specimen was
cut from a matrix containing several octahedral cuprite
crystals and was by no means a single crystal. In fact,
one of the principal difficulties in nuclear quadrupole
resonance work is that of obtaining single crystals of
sufficient volume to provide a strong absorption signal.
The Specimen of figure 8 appears to consist of a single
large crystal with parts of two or three other crystals
adhering to it. Since the Specimen did exhibit several
well defined faces, it was possible, by goniometric
measurements, to determine a probable symmetry axis about
which to rotate it. The fit of figure 9 is evidence for
the above statement.

The resonant frequency of the Specimen was measured at
25.94 megacycles. Williams(8)gives the frequency at about

63 65.

26 megacycles for Cu and about 24 megacycles for Cu

27

 

Fig. 9.-—Splitting Data for Cuprite Crystal, with
Theoretical Plot of Fig. 7 Superimposed.

28

It is evident that quadrupole resonance can be a
very effective tool for dealing with problems of crystal
structure. In particular, one can use it-—

a. To locate crystallographic axes. This application is
especially useful when the specimen exhibits no well-
defined faces, so that goniometric measurements are

not possible, as in the case of the paradichlorobenzene
crystal which was studied.

b. To determine the number of nonequivalent mole-

cules in the unit cell. Chemically nonequivalent
molecular Sites exhibit nultiple lines in the pure
quadrupole Spectrum while physically nonequivalent mol-
ecules can be distinguished by a study of zero-splitting
data.

c. To determine the order of the crystallographic axes.
d. To study phase transitions. AS indicated previously,
day to day observation of the transition of paradichloro-
benzene from the high temperature, p phase to the low

temperature,cx;phase is an intriguing possibility.

APPENDIX

3O

.hpam
. cops
H paw
Hm cm
EooN
mo Q
OHpocsm m m
m on
ad o
Hmca
.m e
no m:
prH
Hem-
n.0H

.wam

o A V

mm
zoo 9 A3 _
I

31

 

   

31
(CI (D

Fig. 11.--Splitting of Resonance Lines Resulting From
Two or More Physically Inequivalent Nuclear
Sites in the Unit Cell of the Crystal of
Sodium Chlorate.

BIBLIOGRAPHY

J. H. Smith, E. M. Purcell, and N. F. Ramsey.
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C. Dean. Phys. Rev. Vol. 96, No. 4, 1053-1059,
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T. P. Das and E. L. Hahn. Solid State Physics,
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(3

Dean and R. V. Pound. J. Chem. Physics 20,
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Duchesne and Nonfils. Comptes Rendus 238, 1801,
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Buyl, Boden and Dantreppe. Comp. Ren. 233, 1101
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