NUCLEAR MAGRE'JEC RESONANQE swmss
O? CRYSTAL SYRUCTURE

Thais for the Dagm of Ph. D.
MICHEGAN STATE UNEE’AERSITY
Jchn Hubert Muilsr
1958

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This thesis is a denunstratlon of tee uscfulnflsa of
nuclear ca netic resonance (n r) as a tool for tau study of
crystal and liquid crystal etructuro.

The method is illustrated by ayplication to 33x cec-
ponnds. Tue flannetlc dipole-dipole Interacticn is the only
structure sensitive perturbation considered. This is useful
because of its dependence upon Separation and relative
orientation of the perturbing ard raconuut nuclei. Tue
data is presented as angular depuadaut Splittinga and
broadenings of tne nuclear resonance spectrum. Stereo-
Cnfiflical arguments concoct tue conclusions about mu5netic
ingredients to two rust of the atructurc.

in barium cnlorate m nexycrato. the tutor protons
are located by splitting cue to nearest reimhbor protons,
and tne "ore remote neighbors broaden tno Rpllt lines in
agreecext with tgeory.

Tue water molecules are located in ciogtase (c0p or
silicate hyirate) and two effect of toe electron paratugnetinm
of the ceppers :3 diaCusaod. A cannotic transition is
found near 80 degrees kelvin, below usich the proton
Spectrum vanisnes.

The structure an" 53181.15“ch eliamiu .m sulfate geniu-

hydrate is outlined solely on tno basis of nuclear resonance

JZJVTE " '57}: H"? IiiTL-LLR

data, the apnea arouy, and two number of rolcculea in the
unit cell.

The x-ray oorlved structure of enclose (torgllium
aluminum silicate nylroxice) is rejocted on the heels of
the nuclear rcsonarce spectrum.

I

In two final adapter the magnetic ordering in the

can

a and p-azoxypqonotole is

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NUCLEAR MAGNETIC RESONANCE STUDIES
of
CRYSTAL STRUCTURE

by

JOHN HUBERT MULLER

A THESIS

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics
1958

ACKNOWLEDGEMENT

It is indeed a pleasure to thank Professor Robert D. Spence
not only for suggesting the problems studied in this thesis,
but also for his all-inclusive help. He has given more I
than generously of his time in helping with a-thousand-and-
one pedestrian details, and has at the same time aided and
advised on every general aspect of the subject. Perhaps my
greatest debt to Professor Spence is the constant single-
minded pursuit of the problem from a practical standpoint

that he has taught and in so many ways exemplified.

ii

VITA

John Hubert Muller
Candidate for the degree of
Doctor of Philosophy

 

Dissertation:
N39123: Magnetic Resonance Studies 2: Crystal
Structure

 

Outline of Studies:
Major Subject: Physics
Minor Subject: Mathematics

Biographical Items:
Born: September 27, 1926; New York, New York

Undergraduate Studies:
Princeton University l9h3-l9hh; 19h6-19h9

Graduate Studies:
Michigan State University 1951-1958

EXperience:

United States Navy
Electronic Technician 19Lh-19h6

Brookhaven National Laboratories
Laboratory Assistant l9h8-19h9
Research Associate 19h9-1950

Michigan State University
Graduate Teaching Assistant 1952-1953;

1953-19Sh: 195

Membership in Professional Societies:
American Physical Society
Sigma Xi
Sigma Pi Sigma
Pi Mu Epsilon

111

TABLE OF CONTENTS
CHAPTER PAGE

ACKNOWLEDGEMENT.............'.......ii
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1
I. BARIUM CHLORATE . . . . . . . . . . . . . . . . 10
II. DIOPTASE . . . . . . . . . . . . . . . . . . . . 25
III. GUANIDINE ALUMINUM SULPHATE . . . . . . . . . . #7
IV. EUCLASE . . . . . . . . . . . . . . . . . . . . 59
v. LIQUID CRYSTALS . . . . . . . . . . . . . . . . 69
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 83

APPENDIX I
A table of magnetic properties of nuclei . . . . . . 87

APPENDIX 11
A calculation of the apparent width of two merging
gaussian curves . . . . . . . . . . . . . . . . . . 89

APPENDIX III

The number of independant constants deriveable from
the dependance of line width upon orientation . . . 9h

iv

LIST OF FIGURES

FIGURE
1. The function 300329 -1 . . . . . . . . . . . . .
2. Projections of barium chlorate structure . . . .
3. Lock-in derivative curves of proton resonance of
barium chlorate monohydrate . . . . . . . . . .
h. The proton resonance spectrum of barium chlorgte
monohydrate compared with the function 3003 O -1
versus angle . . . . . . . . . . . . . . . . .
S. Arrangement of atoms in the plane containing the
waters of hydration . . . . . . . . . . . . . .
6. The proton sublattice in barium chlorate . . . .
7. The line width of the satellite lines in barium
chlorate as a function of angle . . . . . . . .
8. The structure of dioptase . . . . . . . . . . . .
9. Typical nuclear magnetic resonance data for
diOptase . . . . . . . . . . . . . . . . . . .
10. Proton magnetic resonance spectrum of diOptase as
a function of angle about the 5 axis . . . . .
11. Proton magnetic resonance spectrum of dioptase as
a function of angle about the §_axis . . . . .
12. Angular relationships of dipole-dipole axes in
dioptase O O O O O O O O O O O O O O O O O O C
13. Possible location of protons with respect to water

11+.
153.

16>
217'.

oxygens in diOptase . . . . . . . . . . . . . .
The arrangement of the water molecules in dioptase

Angular dependance of nuclear resonance spectrum
in guanidine aluminum sulfate about the 5 axis

Typical proton resonance data for GASH . . . . .

Angular dependance of proton resonance spectrum
in guanidine aluminum sulfate about the.§ axis

V

16
17

19
26

29
30
31
3t

3t
37

so
51

53

Y)

LIST OF FIGURES

FIGURE PAGE
18. Typical proton resonance spectra for euclase . . 60
19. Angular dependance of proton resonance in euclase
about the 5 axis . . . . . . . . . . . . . . . 61
20. Angular dependance of the proton resonance about
the 1_axis in euclase . . . . . . . . . . . . 62
21. The structure of euclase according to Biscoe and
Warren 0 O O O O O O O O O 0 O O O O O O O O O 66
22. The molecular structure of p-azoxyanisole . . . 7O
23. The molecular structure of p-azoxyphenetole . . 71
2h. The proton resonance derivative in p-azoxyanisole 73
25. The proton resonance derivative in p-azoxyphenetole 7h
26. The proton resonance derivative in bubbled p-
azoxyanisole . . . . . . . . . . . . . . . . . 77
27. The proton resonance derivative in bubbled p-
azoxyphenetole . . . . . . . . . . . . . . . . 78
28. Two overlapping identical gaussian distribution
functions . . . . . . . . . . . . . . . . . . 91
29. The apparent half-width of overlapping gaussian
curves 0 O O O 0 O O O O O O O O I O O O 0 O O 92

vi

INTRODUCTION

 

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INTRODUCTION

Nuclear resonance has been successfully studied in
bulk matter since late l9h5 when Purcell, Torrey, and Pound
(l9h6) first found the resonance of protons in paraffin.
Almost simultaneously, and independantly, Bloch, Hansen,
and Packard (l9hb) found the proton resonance in water by
a slightly different technique. Shortly thereafter Bloch
(1946) gave a classical macroscopic theory of nuclear
resonance which discussed the subject from a purely phenome-
nological standpoint.

This field has now become one of the very active
branches of physics. Although there has been application
both to nuclear theory and to technical developments such
as the accurate measurement of magnetic fields; the most
extensive application of the method has been to the study
of the structure of matter.

Solids, liquids, and gases have been studied, though
the low density of gases tends to confine the use of _
nuclear resonance to the other two states of matter. The
:first observation of nuclear resonance in gases was made by
IPurcell; Pound, and Bloembergen (19h6) in hydrogen gas.
{Though there have been a number of experiments in gases they
are, in the main, of rather special interest, and will not
be discussed here.

Nuclear resonance was first found in a liquid or

quasi-liquid (paraffin): and one of the earliest discussions

2

of the effect of molecular environment upon nuclear resonance
was the paper on the relaxation of nuclear magnetic moments
in liquids by Bloembergen, Purcell, and Pound (19h7). This
was followed by an extensive general article on nuclear
resonance by Bloembergen, Purcell, and Pound (19h8) which

also gave more detailed consideration to spin-lattice re-

laxation in liquids.
In solids an early paper of importance by Pake (19h8)

discussed the magnetic dipole-dipole interaction and its
effect upon the nuclear resonance spectrum. Another major
contribution was the theoretical calculation by Van Vleck
(l9h8) of the magnetic dipole-dipole broadening of lines in
solids; and the accompanying experimental paper by Pake and
Purcell (19h8). The relaxation mechanism that is most im-
portant in solids has been shown by Rollin and Hatton (19u8)
and Bloembergen (l9h9) to be due to minute amounts of para-

magnetic impurities.
Knight (l9h9) extended the study of environmental

effects upon nuclear resonance by finding a minute shift in
the resonance frequencies of nuclei from compound to com-
pound, and especially between metals and salts. This effect
has been ascribed to the paramagnetism and orbital dia-
lnagnetism of environmental electrons by a number of authors
lactably: Townes, Herring, and Knight (1950), Ramsey (1950
l and b), and Proctor and Yu (1950). Further studies re-
8ulting in a very extensive literature on fine structure
aspecially in liquids and gases have been initiated by
Gutowsky, McCall, Slichter, and IcNeil (1951).

 

Nuclei having spins greater than one-half also
possess quadrupole electric moments. The first study of
this property in solids was by Pound (19h7). Nearly pure
quadrupole electric resonance was studied in individual
molecules by molecular beam.methods by Nierenberg, Ramsey,
and Brody (19h7). It was therefore a natural development
by Dehmelt and Kruger (1950) to find pure electric quadrupole
resonance in solids. '

The numerous advances in technique since the first
demonstration of nuclear resonance, though of great interest
and consequence, will not be treated here. That is be-
cause this thesis is not concerned with methods.

The purpose of this thesis is to extend the study
of magnetic dipole-dipole interactions as a tool for the
investigation of crystal structure. we will include liquid
crystals in our designation of crystals, despite the usual
connotation of solid with the word crystal.

It so happens that all the substances will be studied
by means of the magnetic resonance spectrum of protons. This
is accidental except in so far as protons form a very common
constituent of matter and possess a large magnetic moment.
3N0 general conclusions will be reached that cOuld not be
adapted to other nuclei possessing magnetic moments.

We will not consider electric quadrupole interactions
beheen nuclei and their crystalline environment. Some of

t5111c constituents of the substances studied will possess

quadrupole moments; but since we will not study their
resonance spectra, the only effect we need consider will be
due to their magnetic dipole moment.

To exemplify our thesis that nuclear resonance is
a tool of considerable value in the study of crystal struc-
ture, we will examine: two hydrates whose structure has
previously been determined by x-ray analysis except for
the protons in the waters of hydration; next a hydrated
crystal of previously unknown structure; a hydroxide mineral
whose structure has been suggested by x-ray technique; and
finally two organic liquid crystals.

Throughout our work we will use the magnetic dipole-
dipole interaction as a probe for the presence of other
magnetic constituents in the materials examined. We will
therefore consider that interaction in some detail before
proceeding to the individual cases.

The phrase, "dipole-dipole interaction" refers to
the perturbing effect of one magnetic dipole upon the nuclear
:resonance spectrum of another. The change in the spectrum
of the perturbed nucleus depends both upon the separation
13f the dipoles and on the orientation of the magnetic field
‘Vith respect to the line joining the two magnetic dipoles.
As a result, measurement of this perturbation yields
Structural information of considerable value.

In most cases here considered, the perturbing dipole

18 another nucleus. Only a small prOportion of the nuclei

S
are in transition at any given time from one spin orienta-
tion to another. Therefore in calculating the perturbation
upon a nucleus which is undergoing transition, it is a safe
assumption to consider that the perturbing nucleus remains
in the same state for the period occupied by the transition.

In one case, dioptase, one of the perturbing dipoles
is the magnetic moment of an unpaired electron associated
with the copper ion in the lattice. At room temperatures
however, the electrons pass so rapidly from one state to
another, that in spite of their large magnetic moment, the
time average effect upon the nuclei is Only a little different
from zero.

The change in the magnetic field due to a nearby
magnetic dipole is classically calculated to be guVrB)

(3 Cos20 - 1). Where/utis the magnetic moment of the per-
turbing dipole, r is the length of the line joining the
dipoles, and O is the angle between that line and the ambient
magnetic field. This yields an experimentally correct

effect for dissimilar nuclei of spin 1/2; but for the case

of identical nuclei it is necessary to consider one additional
.factor.

In this last instance one has to allow for the quantum-
:nechanical phenomenon of "spin exchange". Because the
Perturbing and the perturbed nuclei are of the same variety,
an interchange of their spin orientations, when they are

lFing in the same magnetic field, will not change the total

 

6

energy of the system. And because they are identical, the
coupling through precessional dipole magnetic radiation will
be large. As a result, the splitting of the levels due to
classically calculated fields of a magnetic dipole located
in the position and orientation of the perturbing nucleus
has to be augmented.

Pake (l9h8) has shown by rigorous quantum-mechanical
calculation that the total splitting then becomes
3/2Wr3)(300s20 - 1) where/u, r, and 0 are defined as
before. This formula is that of a surface of revolution
whose distance from the origin is prOportional to the
magnitude of the splitting, and whose long axis, which is
the axis of revolution, is parallel to the line joining
the nuclei. The median section of this is illustrated in
Figure 1. This type of dependance will be considered so
often in this thesis that it will simply be referred to
as "(300320 - 1) dependance".

Usually the data leading to the perception of these
interactions is taken as a series of line splittings measured
as a function of rotation angle about an axis perpendicular
‘to the external magnetic field. This will produce a graph
<1f splitting versus angle that will correspond to a planar
section through this three-dimensional surface of revolution.
Tmhe resulting data may be presented in polar form, or more
cOmonly, as splitting magnitude the ordinate against an

abecissa of angle. As can be seen at once from the formula

 

Figure 1. The function R = 3Cos20 - l plotted in polar
Coordinates. O is measured from the long axis of the figure.
An isolated magnetic dipole-dipole interaction causes a
Splitting proportional to R if 0 is the angle between the
dipole-dipole axis and the applied magnetic field. One should
think of this pattern as a section of a solid of revolution

about the long axis of the figure.

 

8
and Figure 1, this dependence is repeated every 180 degrees.
Therefore only data for a half circle will be presented.

Because the nuclear spin levels are so close together
at the magnetic fields usually used, there is at room
temperature only a small excess of nuclei in the ground
state over the upper state. The prOportion is about one
in ten to the fifth. This means that the signals in even
the best-relaxed substances are going to be small and not
easy to detect.

Furthermore, in most solids the relaxation mechanisms
are perhaps an order of’magnitude less effective than in
liquids. In some solids such as barium chlorate the re-
laxation time will be even longer, amounting to more than
one second. Extreme cases not studied here can only be
detected by methods using pulses separated by minutes.

Long relaxation times, besides being an experimental
nuisance, also give information; since they depend upon the
presence of certain factors. Close coupling to paramagnetic
electrons, for instance, or to nuclei with electric quad-
rupole moments may give strong spin-lattice coupling or
short relaxation times; while the isolation of nuclei from
thmse factors, or the location of nuclei with electric
(Iuadrupole moments in a lattice having a high degree of
Symmetry and therefore a low electric field derivative,

Vould tend towards long relaxation times.

In this study, we have used both the rf bridge
method of detection and the sensitive oscillator. The
work has been carried out in a temperature range from h..2
to h23 degrees kelvin. Because of the dependance upon
temperature of the relative population of the ground state

to the upper levels very low temperatures have been used

in.some cases.

BAR IUM CHLORA TE MONOHYDRA TE

 

1
.! . -
I . . ‘I

 

10
BARIUM CHLORATE MONOHYDRATE

This substance was studied in the form of single
crystals grown from an evaporating supersaturated water
solution. Crystals about 25 by 7 by 7 millimeters were
formed in about a month. The chemical was obtained in c.p.
form, but no particular precautions were taken to insure
that the material remained absolutely pure.

Crystals were grown both from distilled water so-
lutions, and from tap water solutions. No significant
differences were ever noticed nor shOuld they be expected,
since the nuclear resonance is an effect depending on the
gross number of nuclei of a partiular species and the
relaxation mechanism. Crystallization is a purifying
process that tends to limit the quantity of impurities. The
only possible effect of such impurities as would escape
this selective process would be to alter the relaxation
time and hence the signal amplitude. It would not affect
the observed spectrum or its angular variation.

The crystals grew in two habits. Those grown at
large supersaturations and high rates of growth were rela-
tively needlelike with the long axis being the g.axis.
Others grown more slowly tended to be more tabular, g_re-
maining the long axis though relatively shorter. In the
first case the dominant faces were the (110), (lIO) and
(lOD‘but not (lOI) faces. In the second case the dominant

11
faces were (110) and (ITO) as before but the remaining face
was replaced by (011) and (0T1) faces. It can be seen then,
that these crystals possess a distinct enantiomorphism in
the first modification; and indeed in all modifications
there are vestiges of this in the emphasis upon certain m:

small faces. No crystals were observed with opposite

handednes s .

746””-.. :31 r.» 9.9
‘._.

The structure of barium chlorate has been studied
by means of x-ray diffraction by Kartha (1952); and has y
been specified except as to the location of the protons in
the waters of hydration as in Figure 2. The crystal is
monoclinic of space group Cgh. The nuclear resonance
spectrum has been studied already by Spence (1955).
He found three lines, a doublet showing BCos20 - 1
dependance of splitting upon orientation and a weak sharp
central line. The maxima in the splitting of the doublet
occurred in the .a_¢_:. plane 30.50 from the positive _a_ axis
taverds the negative 3 axis. See Figures 3 and 14.. This
Spectrum indicates that most Of the waters in the crystal
have their proton-proton axes aligned in one direction only.
The spacing of the protons was determined from the magnitude
of the splitting to be 1.56 angstroms. Spence also noticed
that the widths of the lines in the doublet varied in such
8 manner as to seem due to the next nearest neighbor protons.
The central line was ascribed to minute amounts of water

trapped in the crystal during growth.

 

12

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15

Assuming that the protons are located approximately
along the lines joining the water oxygens to the chlorate
oxygens as in Figure 5, one gets a sublattice of protons
as indicated in Figure 6. One should note that adjacent
sublattice layers are translated not only one-half unit cell
distance in the )3. direction from each other, but also one-
half unit cell distance in the .a_ direction too. The location
of water oxygens is determined by x-ray analysis and their
specification as belonging to the water molecules is made
by valence and stereo-chemical arguments. The choice is
completely unique in this case.

In the present work we have re-examined the structure
0!” barium chlorate with a view to determining whether the
original hypothesis as to the source of the line width is
correct. And in particular, whether it is possible to
infer the positions of the next nearest neighbors from the
line width data, rather than from the actual spectrum. The
Present; experiment has been accomplished with considerably
“01‘. resolution then that reported by Spence. See Figure 3-

Measurements of line width were made from such re-
cordings. In this case, line width means the separation
between points of steepest slope of the absorption line.
Since these recordings are derivative curves, line width is

manured by the lateral separation between adjacent peaks

or Opposite sign.

 

16

.Borium

3 Water oxygen
0 Chlorate oxygen

- Proton

 

O

_> 8 axis

P 1 1 I
0 l 2 3
‘ Angstrorns

F 18111-e 5. The arrangement of atoms in the plane containing
the water of hydration. This plane contains the _b_ axis and
18 inclined 30.5 degrees below the positive 3 axis. The
cblarate oxygens are not quite in the plane.

—__ ___ 17
r ~~x

 

 

 

 

 

I
' l
O .|
CD
' l
1
‘l
I | .p I)
C I '.
L§~~§~
_ a + ~~~.-__J
CD
‘. (l
C)

Figure 6. The proton sublattice planes in barium chlorate
monohydrate. These layers of protons are located at

y/b = 0 and y/b 8 1/2; and are laterally displaced by an
amount a/2. The separation between the planes is 3.90
angstroms.

18

If one assumes that these curves are gaussian in
shape, this distance is just twice the second moment of the
line. Such an assumption is not so arbitrary as it might
seem. The shape of the line can be seen on the oscillisOOpe
to be not very different from.a gaussian and the error intro-
duced by small departures from this shape will be small
compared to line width.

The measured line width has been corrected for
modulation broadening by the following formula:

0-2 8 042 - lime/14.. Where «is the true second moment; a"
is the observed second moment; and Hm is the peak-to-peak
modulation amplitude. This correction has been shown to
be theoretically correct by Andrew (1953).

We have calculated the second moment of the lines
using the results given by Van Vleck (l9h8). Our calculation
is carried out to three degrees of approximation. See
Figure 7. The first includes only the four protons in the
waters of hydration immediately above and below the protons
of the water molecule under consideration. The second
refinement includes a considerably larger number of protons
and accounts for all these up to a lattice distance in
every direction. In addition this refinement includes a
factor of 1.02 to account for all similar nuclei in an
infinite linear lattice in the same direction as the nucleus
considered in the first lattice distance. This does not

include those nuclei occurring at new directions from the

19

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2O
perturbed nucleus and so accounts for only an indefinitely
small proportion of the more distant nuclei. However,
further corrections must be relatively small since the dipole
field falls off as l/r3. The last refinement includes the ,
magnetic moments of the barium nuclei, of which isotopic
fractions (amounting to eighteen percent) have small
magnetic moments.

The magnetic moment of the barium nuclei might be
thought to be too small, or the isotopic percentage of
magnetic nuclei among the non-magnetic might be considered
negligible to contribute to the line width in comparison
with the contribution of the protons. But, at the orienta-
tion in the magnetic field where the line width due to the
protons alone is at a minimum, the contribution of the
barium nuclei is at its maximum. This is because of the
great proximity of the nearest barium nuclei as compared
to the second nearest prohxxneighbors; and due to the
particular relative orientation of the barium nuclei which
causes this contribution to be near a.maximum at just this
point. See Figure 7. It is interesting to note that the
result of Van Vleck (l9h8) includes the case of isotopic
percentages directly without modification.

In using this formula for the calculation of split
line widths, we have neglected the fact that it is meant to
apply to the line width of single unsplit lines. However,

these lines are so completely separated over most of the

21
orientations of the sample in the magnetic field, that it
was thought a good approximation to do so. But we do not expect
this to be valid at small splittings. Here we are omitting
the first term in the formula and are considering it to
apply to the splitting separately.
This may be justified as follows: Consider the

second moment of the entire pattern

4-1::
s = [n2 P(h) an
-09

 

where h is the magnetic field measured from the center of
the resonance pattern and P(h) is the spectrum as an even
function of h. Let P(h) I W(h§8h) +W(h-bh) That is,P is
composed of two identical lines whose spectral functions
are W and which are centered at -5h and +3h respectively.
e-O *‘9
.Assuming that hW(h) dh = 0 and ‘er(h) dh = l
1’” -ee «we
then s . / h2(W(h¥8h) + wm-sn) )dh
@

 

/ When) . w(h-8h)dn

% fnzwmunmh + g h2W(h-8h)dh

Substituting h = h' -8h and h = h" 4. 8h in the two integrals

I‘espectively one gets the result
9-
S 8 /h2W(h)dh 4} 8h2. This says that the second moment,

-ee
3. of the entire pattern is equal to the second moment of

one satellite line about its center, plus 8h2, the square

22

of the separation of the satellite from the center of the
entire pattern. This is analogous to Steiner's theorem in

mechanics .

From Van Vleck's result one can get

" 2
where hi 3 Bjk Vs(s+1)73
and Bjk = -[g’8( BCosZOJk-l) /rjk3] [li- sen]

In these formulas s is the spin of the perturbing

dipole; g is its splitting factorz/G is the magneton (such
that gee is the magnetic moment of the perturbing dipole);

:- is the length of the line joining the perturbed nucleus

k
t: the perturbing dipole; and °jk is the angle between that
line and the external magnetic field; 8.1.31! is a specialised
kronecker symbol which is zero except when the perturbing
dipole, k, is the same variety as the perturbed nucleus,
J. h1 is the same as the splitting that one would get
from calculating a single isolated dipole-dipole inter-
action separately. See Pake (1914.8). Separating off the
largest term which we will call h: we get 8 8 hlz # ghiz.

We assume that for large splitting of narrow lines we can

identify n12 with She.

This may be rationalized by noting that for a nearly
18 olated pair of nuclei the first term hla is 1/3 of the
acIllare of that calculated by Pake for a completely isolated
Pair. This pair by itself would give two sharp lines. As

23
perturbing nuclei are brought nearer to the pair we can see
that the first effect must be a broadening caused by a
spread in magnetic field about the sharp value. This is
due to the permuted orientations of the perturbing nuclei.
When the perturbing nuclei come so close that the broadening
is of the order of the splitting, it is clear that we cannot
allow the approximation. The region in which the assumption
starts to fail is very narrow since the dipole magnetic
fields fall off as l/r3.

In Ba(ClO3)2.H20 this condition of widely split
narrow lines is generously satisfied as the line shape
shows. If we can identify hl with an then S = She + ghiz'
As a result, it is possible to assign the broadening of the
lines to the remaining terms ghiz. Our calculation of
these terms depends upon Kartha's structure and on the
single fixed orientation of the waters. It is found to
agree very well with the variation in the experimental line
width.

The most significantly incorrect feature in this
calculation is that the protons have been assumed to lie
.1n the‘gg plane at y/b=0 and y/b=§. The protons in fact
<10 not quite lie in these planes. Both members of any pair
<Jf protons are on the same side of the plane. But nearest
Imeighbor pairs near any given plane are alternately on
Opposite sides of the plane. The distance from the plane

to the protons is plus or minus .11 angstroms. However,

I16

t?

2h
neglecting this feature changes the largest three terms in
the broadening only slightly.

Thus the second order interactions that we have
measured form an integrated picture in relation to the
structure of the proton sublattice and the remaining magnetic
ingredient, the barium molei. Though we have not carried
out a calculation of such structure from the data, but rather
have worked in the reverse direction; the confirmation shows
the possibility of locating next nearest neighbors by line

width measurements.

DIOPTASE

25
DIOPTASE

Dioptase is a handsome green mineral possessing
iridescent reflecting planes in the interior of the slightly
translucent crystal. It is comparatively rare, and single
crystals large enough for nuclear resonance study are ex-
tremely rare. The lack of perfection in large specimens
accounts for the fact that diOptase is not ordinarily more
than semi-precious.

The formula is CuSiO3-H20. There are eighteen
molecules to the unit cell and the space group is R}. The
mineral has been examined by x-ray diffraction by Heide
and Boll-Dornberger (1955) who give a complete structure
except for the location of the hydrogen atoms.

The crystal consists of hexagonal rings of six
8 ilicate tetrahedra in the 353 plane while cOpper ions are
disposed around the rings. The water oxygens lie in
puckered rings of six between and coaxial with the silicate
1‘ings. All oxygen atoms except those chosen for the waters
or hydration are in tetrahedral inter-relationship about the
Silicon atoms, as centers. See Figure 8. Heide and Boll-
Dornberger speculated that the hydrogens are somehow located
in the interior of the oxygen rings where there is certainly

considerable space otherwise vacant.
Our problem in this section of the thesis is to

show: a) how nuclear magnetic resonance can locate the

26

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.enpossapeu
nooaflan c» wcfiwuoaon cowhmo "menu .oasoeacfi meant a Mo anon nu scans newhno
“mocha .acouaam "com .aomaoo newceao umBOHHcm we macaw ho ecoaueczm no seasons
uceaeuhfip uneneaaoa maoaoo pcosemuav one

.Q easwfim

 

27
protons in the waters of hydration in a more complicated
crystal than barium chlorate monohydrate; and b) the effect
of the paramagnetic OOpper ions upon the proton resonance
as a function of temperature. This will further demonstrate
the utility of nuclear resonance in structural analysis,
and its application to the gross magnetic properties of a
strongly paramagnetic crystal.

This work has been made possible by the extensive
x-ray analysis. Not only have the locations of all the
atoms heavier than hydrogen been determined, but also the
selection of those oxygens belonging to the waters of
hydration has been set by valence bond saturation arguments.
These oxygens are the only ones with bonds unsaturated by
attachment to other atoms. Each one is part of the sixfold
coordination of oxygen atoms around two copper atoms, but
is not otherwise attached to any of the atoms besides
hydrogen.

Furthermore, we know that the waters of hydration
are difficult to remove from the lattice by heating.* There-
fore the bonds holding the waters in the crystal cannot be
extremely weak, as they are in the case of barium chlorate
monohydrate.

The nuclear resonance data was readily obtained

with the strong signals that might be expected in a crystal

*Heide and Boll-Dornberger, (1955).

28
containing a large number of paramagnetic ions. Figure 9
shows some typical data. Both the apparent number and
splitting of the lines varies with orientation of the sample
in the magnetic field. At some orientations there is so
much overlapping that only one broad line showing little
detail is seen.

Figures 10 and 11 show the resonance spectrum as a
function of angle about two mutually perpendicular axes.
These are the g or trigonal axis, and the 5 axis perpendic-
ular to it.

Rotation about the 5 axis shows hexagonal symmetry
which is the result of the inherent 180 degree symmetry of
nuclear resonance dipole—dipole interactions superimposed
‘upon the trigonal symmetry of the crystal.

An instructive detail of the angular splitting de-
pendance is the maximum splitting of 10 gauss. Knowing
that waters of hydration reside in the crystal; and knowing
that the smallest maximum splitting so far reported for
firmly bound waters of hydration is about 19 genes; we can
be assured that the proton-proton axes are not in a plane
perpendicular to the 5 axis. If such a plane were perpendic-
‘ular to the 5 axis we would observe a much greater splitting
<3f the resonance pattern at some angle in this rotation.

Furthermore, the maxima of the splitting occur
tapproximately at an angle of thirty degrees from the 5 axes

<>r parallel to the‘g axes. This is also perpendicular to
the _x axes. ‘

29

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30

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32

Figure 11 showing the splitting of the proton re-
sonance pattern as a function of orientations in the magnetic
field perpendicular to the 5 axis, shows the broadest split-
ting observed in the proton resonance spectrum of dioptase.
The proton-proton axis is evidently perpendicular to the 5
axis and tipped about h7.5 degrees from theIE axis. There
is a great deal of additional evidence for such a conclusion
in this figure.

To appreciate this evidence prOperly we must con-
sider both the symmetry operations for the space group of
the crystal and the effect of the geometric detail upon the
nuclear resonance pattern. Now all the waters of hydration
located regularly in the crystal must be subject to the
symmetry operations of the space group R3. All possible
permutations of the proton-proton axes in a unit cell can
be generated from the orientation of only one water molecule.
This is because there is only one water molecule in the
chemical formula; and there are eighteen molecules in the
unit cell; and eighteen different equivalent locations are
generated in the unit cell by the symmetry operations.

A set of symmetry operations sufficient to generate
all eighteen are: one threefold rotation axis, a glide
plane with three equivalent positions, and one space in-
version. The translation along the glide plane cannot be
detected by first order measurements of nuclear resonance

splittings. Furthermore, a space inversion leaves

33
orientations altered by exactly 180 degrees which cannot

affect the nuclear resonance angular dependance which already
has this two fold symmetry. As a-result one is able to.
detect three, and only three, dipole-dipole axes by first
order nuclear resonance measurements.

Let us visualize the pertinent angular relation-
ships by considering the dipole-dipole axes to lie along
the inclined edges of a three sided pyramid as in F1gure
12. The base of the pyramid is equilateral. The angle of
the inclined edges to the altitude of the pyramid is equal
to the angle between the dipole-dipole axes and the 5 axis
in the crystal. Call that angle A0. In this case it will
be h7.5 degrees. The axis about which our particular rota-
tion is taking place is parallel to one edge of the base of
the pyramid.

We will get two splittings. One will be from the
proton-proton axis perpendicular to the axis of rotation
and the other resonance pattern will be due to the other
two hydrogen-hydrogen axes which will have identical
resonance patterns. The identity of the last two resonance
patterns arises because the dipole-dipole axes are coplanar
with, and inclined symmetrically to, the axis of rotation.
The angle of these axes to the plane of rotation is
co : Sin-1(V37E Sin A0).

The angle between the 5 axis and the side of the

pyramid containing these two dipole-dipole axes and the

31+

UV

 

K:

 

X

Figure 12. Angular relationghip of dipole-dipole axes in
dioptase.

~\\

  

Figure 13. Possible locations of protons with respect to
water oxygens.

35

axis of rotation will be B0 = Tan'1(1/2 Tan A0). The sum
A0 + Bo has to be the separation between the maxima of the
two angular dependance curves in Figure 11. The'g axis

has to lie between these two maxima so that we must measure
this sum in Figure 11 backwards from the largest maximum to
the near 3 axis and continuing in the same sense from the
5 axis at the other end of the diagram to the smallest
maximum. This comes out to be 76.5 degeees as can be seen
in the figure. The curves are drawn according to the
values calculated for the geometric situation outlined
above, while the data are shown by the circles.

We thus confirm by nuclear resonance methods our
selection of the 5 axis by external morphology. Consequently
‘we assure ourselves of the correct choice of the §g_plane.
Our data in Figure 11 indicates a slightly incorrect
selection since the errors are systematic in their angular
(dependance, but not merely shifted in angle. What we had
(shosen as the §,axis is therefore a little out of the true
253 plane.

We will now consider what the proton-proton axes
'that we have found mean in terms of the structure of the
mnineral. As noted before, the water oxygens lie in puckered
IPings of six between and coaxial with the silicate rings.
littached to each oxygen within a radius of approximately
~198 angstroms must lie two hydrogen nuclei. Approximately

this distance has always been found for the 0-H bond

36
length when measured by neutron diffraction or infra-red
absorption spectra in a great variety of compounds.

By specifying the orientation of the proton-proton
axes, and the radii of the protons about the oxygen atoms
we restrict the spatial location of the protons to Opposite‘
ends of lines parallel to the specified direction and lying
on a paraxial circular cylinder as in Figure 13. The only
atoms near enough to share these protons with the water
oxygens are other water oxygens or certain oxygens in the
silicate rings. The particular inclination of the proton-
proton axes suggests a unique assignment of the locations
of the protons.

To consider possible locations for the protons we
nmy'imagine a circular cylinder coaxial with the‘g axis
(and the silicate rings and the water oxygen rings) and
passing through the water oxygen atoms. Such a cylinder

‘V1ll have a radius of 2.h1 angstroms. We may then unwrap
that cylinder and spread it out in a plane as in Figure 11;.

We can now see that the only rational location for

,the protons is approximately in the surface of this cylinder
and with locations specified by the proton-proton axis
<Irientation and the standard separation from the oxygen
Eltoms.

This picture explains why the waters are so firmly

bound in the mineral. The waters are tied to each other

111 a two-dimensional ice structure and the alternate waters

37

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38

are attached to silicate oxygens in the silicate rings above
and below the waters of hydration. As a result each water
molecule is tied to the rest of the structure by three

hydrogen bonds as well as being part of the coordinations

about the capper atoms.

The 0-H-0 bonds within the ring itself are probably
bent somewhat with the proton slightly shared with the
second nearest neighbor oxygen atom. Therefore these pro-

tons probably lie slightly towards the inside of the

c ylinder.

Considering all these factors, we assign the.follow-

1mg relative coordinates to the protons.

x1e 1‘8 2‘0

P1 .l7h .067 .29h

P .12h .966 .hlh

2

Where a I 111.61 angstroms and c = 7.80 angstroms. These
Values are Operated upon by the symmetry operations of the

Space group R3 to produce all eighteen sets of two protons

each in the unit cell of dioptase.

We will now discuss the magnetic moment of the '
unpaired valence electron on the capper ions. Because
nuclear resonance. is the matter under discussion we will
Cons ider only the effect of this component upon the proton

resonance and the uses of the proton resonance in studying

the electronic paramagne tism.

39

The first matter to be considered is the effect of
the electrons upon the proton resonance spectrum at room
temperature. To do this, we must estimate the magnetic
moment associated with each copper ion. This will be done
in two ways. In the first the magnetic moment of the copper
ion is calculated assuming a magnetic moment equal to that
from exactly one electron spin per ion. In the second we
make an estimate of the magnetic moment of each OOpper ion,
starting only from the bulk paramagnetic susceptibility of
the crystal. The net effect of such a moment upon the
Spectrum is then calculated.

The susceptibility of a single unpaired electron
is given by 3362 S(S+l)/3kT where g is the Lande splitting
factor of the electron, b is the Bohr magneton, S is the
Spin quantum number, k is Boltzmann's constant, and T is
the absolute temperature. If this is multiplied by the
magnetic field, 7000 gauss, that we have used throughout
our experiments one gets the average magnetic moment of a
aingle electron in this field.

This average moment is what has to be used in
Computing the effect of the electronic paramagnetism upon
the nuclear resonance because the electronic relaxation
tfilms is short compared to the nuclear resonance period. As
a result, the nucleus sees a field which fluctuates so

I‘Qpidly that the only effect of the electronic moment is

an average one .

to

Inserting the values of 2.00 for g; .927 x 10"20
erg/gauss for b; 1/2 for S; 1.38 x 10‘16 erg/degree for k;
and 289 degrees for T; one gets 2.33 x 10'“27 erg/gaussz
for the susceptibility. Multiplying this by the magnetic
field of 7000 gauss yields a moment of 16.25 x lO'zh
erg/gauss for the average magnetic moment of the electron.

We can arrive at a value for this from bulk measure-
ments upon the crystal. Foex (1921) gives the specific
susceptibility of powdered dioptase at this temperature as
8 -8 x 10"6 erg/gaussz gm. This is an average over the
Various orientations of the powder particles so that we can
only consider the isotropic average paramagnetism. The
specific susceptibilityxa is simply converted to the
atomic susceptibilityxa bythe following formula:
Hxs/L =Xa Where L is Avogadro's number,6.03 x 1023, and
II, the molecular weight, is 157 grams, for dioptase. X a
comes out to be 2.30 x 10"?7 erg/gaussz. The resulting
value for the magnetic moment of the copper ion willbe
16.1 x 10'2“ erg/gauss. Thus at room temperature the gross
paramagnetism of the material is in excellent agreement
with the supposition of one unpaired electron per capper
ion.

Let us calculate the effect of this moment upon
tPhe proton resonance. There are two capper ions near each
Proton pair. More remote coppers may be neglected due to

the rapid falling off of the field of magnetic dipoles with

1+1
distance. The distances of the near copper ions from one
proton of the pair are 3.0 angstroms and 3.2 angstroms. The
distances of the coppers from the other proton in the pair
are 2.9 and 3.2 angstroms. Both coppers are almost in the

same direction from the proton pair.

For simplicity let us assume that both cOppers are
as close as 2.9 angstroms and in the same direction from
the protons. To begin with we will treat the case of one
copper ion only.

We are here considering the magnetic effect of an
electron which passes so rapidly from one state- to the 8
<>ther that the only net effect is due to a slight preponder-
ance of time spent in one state. Thus we will use the
Elverage magnetic moment instead of the total magnetic
moment of the electron. Let us call this/E. Then WP3)

( BCoszo - l) is equal to the magnetic field at the proton
due to the electron. Here r is the length of the line
from the ion to the proton, and 0 is the angle between this
line and the external magnetic field. We can find the
upper limit of the field due to the electron by picking the
maximum value for (300320 - 1) which is 2; and simply
nIultiplying the magnetic moment divided by the cube of the
distance by this same factor of two. 2x 16.13 x lO'Zh/
alph x lO’zh = 1.32 gauss at room temperature. Now because
this is due to an average moment we will not see both a

plus and minus effect upon the resonance, but rather a

1p?

shift of the resonance line of approximately one-and-one-
third gauss. This will be anisotrOpic for a particular
pair of protons. However, the lines are so broad that the
observation of merely the presence or absence of such a
shift is uncertain.

At 87 degrees kelvin the susceptibility and the
average magnetic moment get be calculated in the same way
as at room temperature. We can an atomic susceptibility
of 7.18 x 10'27 erg/gau332 and a magnetic moment at a field
of seven thousand gauss of 50.3 x lO'Zh erg/gauss. From
the specific paramagnetic susceptibility of 21.1 x 10'6
erg/gauss2 gm measured at this temperature by Foex (1921)
we calculate an atomic susceptibility of 5.53 x 10'27
erg/gauss2 and a resulting magnetic moment per electron of
38.7 x lO'zh'erg/gauss. Using the experimental value for
the calculation of the maximum possible shifting of the
resonance lines as before, one arrives at 3.2 gauss. Since
the lines are considerably broader at liquid air temperatures
than at room temperatures we cannot report observing this
{anisotropic shift. No attempt was made to observe the
static shift due to the bulk paramagnetism of the sample.

Actually this is much less than the anisotropic component

at the protons, amounting only to .1117 gauss at 87 degrees

kalvin.

So far we have ignored the fact that there are

a<=t:ually two copper ions instead of one as near neighbors

1+3

to the protons. We can justify this as follows: Consider

the effect of the addition of a second unpaired electron to
the first. We assume that each electron produces the same
effect at the location of a particular proton. This assump-
tion is very nearly fulfilled in practice, because of the
proximity of the coppers to each other and the same general
(iirection between the proton pair and each copper.

Furthermore we assume that the orientations of the
electrons fluctuate in an uncorrelated fashion from
alignment with the field to alignment against the field.
The first electron spends only a little more than half of
1.ts time oriented with the applied magnetic field, and only
EL little less time aligned oppositely. Accordingly, the
two electrons considered together spend only about one
cxuarter of the time both aligned against the field and one
cruarter of the time aligned c00peratively with the magnetic
1?ie1d. They are aligned antiparallel to each other for half
the time.

The only contribution to the net magnetic moment
will be the surplus of time spent cooperatively parallel

'1th the field over the time spent both parallel to each

other but against the field. Although the magnetic moment

or these states is twice as great as in the case of the

single electron, the surplus time spent in one over the

other, is only half as great. As a result the average

magnetic moment is unchanged.

ML

Exchange coupling between the electrons can destroy
the assumption that the electrons fluctuate independently.
However so long as the substance remains paramagnetic as
opposed to ferromagnetic or antiferromagnetic this assumpt-
tion will probably hold.

When dioptase was cooled further the proton resonance
.suddenly disappeared in the neighborhood of 20 degrees
absolute. It was not observed below this temperature.

In a purely paramagnetic substance one would still
OXpOOt to see nuclear resonance at such temperatures since
the relaxation mechanism through the spin-lattice inter-

action should not be quenched yet.
As a result we have concluded that the material has

undergone a magnetic transition which has two effects. The
first is that the spread in the values of the magnetic
1?1eld at the protons has been greatly increased by the strong
alignment of the electrons by cooperative interaction.

The second is that a definite alignment may effec-
tively diminish the spin-lattice interaction of the electrons.
The consequence of this is the reduction of the spin-

1‘1 ipping frequency of the electrons to the point where the

alain-lattice mechanism for the relaxation of the proton

Bpins may be interfered with.

We have sought to distinguish whether this trans-

it ion is to a ferromagnetic or antiferromagnetic state. It

18 possible that the transition is to some intermediate

hs
state, such as ferrimagnetic. In order to make such a dis-
tinction we have measured the location of the resonance line
as a function of temperature just above the transition
point. This would furnish an estimate of the bulk suscep—
tibility of the material.

The suceptibility of ferromagnets in the region
above the curie temperature is a monotonically decreasing
function of temperature. The susceptibility of anti-
ferromagnets tends to have a knee in the dependance on
temperature such that there is either a maximum in the sus-
ceptibility above the Curie temperature or at least a point
where the slope of the paramagnetism with temperature
suddenly becomes more greatly negative.

We have not been able to measure any _such change
in the susceptibility by comparing the proton resonance
against a reference sample. However, the very lack of slaps
to the paramagnetic susceptibility curve near the trans-
ition is an indication against ferromagnetism. Thus there
is a strong likelihood that what is observed in dioptase
at 20 degrees is an antiferromagnetic, or at most a ferri-
Inagnetic transition.

The accuracy of measurement of temperature in this
region is unfortunately not very great as we had only an
erratic carbon resistance thermometer available. We there-
fore set limits of plus or minus six degrees on the

temperature measurement.

to
The conclusion that is to be drawn from this study
in the proton resonance of dioptase is that nuclear resonance
has proven to be not only useful in locating the protons
in the waters of hydration; but also has demonstrated for

the first time in this material the existence of a magnetic

transition. Something about the nature of that transition

laas also been shown, and it is perfectly possible that

further resonance experiments would show conclusively the

zlature of this transition. The effect of the electron

Isaramagnetism above the transition has been estimated, but

tsecause of its relatively small size has not been definitely

dietected.
We believe therefore that the utility of nuclear

xresonance as a tool for the study of matter has been force-

1?ully demonstrated once more.

GUANIDINE ALUMINUM SULFATE

in

GUANIDINE ALUMINUM SULFATE

We next come to a material whose structure was
unknown prior to our nuclear resonance observation. It
represents perhaps the first compound whose main structural
features have been deduced from nuclear resonance data and
stereo-chemical arguments. As such, it is a very favorable
case for the application of nuclear resonance to structure
determination.

Guanidine aluminum sulfate hexahydrate, or as is
usually abbreviated GASH, is a ferroelectric trigonal
crystal of space group 03v according to Wood (1956). It is
grown from a water solution of guanidine sulfate and
aluminum sulfate as apparently perfect clear crystals. The
formula is C(NH2)3A1(SOu)2'6H20.

There are three types of ferroelectric regions,
each of which can have two opposite orientations in a
single crystal. The type of region depends upon the direc-
tion of growth in the original crystal. These properties
are a permanent part of the crystal, since the Curie
temperature is above the melting point. See Holden,

Merz, Remeika and Matthias(1956).

The three regions differ from each other in the
amount of bias that the ferroelectric hysteresis loops of
the regions exhibit. Growth along the §.or trigonal axis

produces the greatest bias, while there are two types of

1+8
region with lesser bias produced by lateral growth. Pre-
ferred growth is in only one direction along the 5 axis.

Since a complete structure analysis has not been
made separately for the three ferroelectric regions of GASH,
it is not certain that the three regions are of identical
structure. The permanent nature of the different regions in
the crystal suggests that there may actually be a structural
difference between them.

Etch pits made in the (0001) planes of the crystal
with water are triangular pryamids. The pyramids are
symmetric if growth in that region has taken place along
that axis. Where the crystal has grown laterally the
etch pits are tilted. See Wood (1956).

Our study has been made upon two crystals which
have been very kindly supplied by the Bell Tblephone
laboratories. The samples were cylinders of rotation about
the g and 5 axes. The crystal cylinder concentric with
the 5 axis showed both symmetric and tilted etch pits. We
therefore cannot be sure whether we have studied a crystal
that was "single", or twinned, or even more radically
different, from portion to portion. Rotations of these
crystals in the magnetic field were made, and the nuclear
resonance derivative curves were recorded for every ten
degrees.

The nuclear resonance data as a function of orienta-

tions of the crystal in the magnetic field perpendicular to

(4-9

thqu axis showed hexagonal symmetry. See Figure 15. This
is to be expected from a trigonal crystal. The large

nmjor splitting of 20 gauss is characteristic of the maximum
splitting of waters of hydration. It therefore seemed
evident from the first that three sets of proton-proton axes
belonging to the waters of hydration lie in the (0001)
jplane. The relative orientation of the maxima about the

_z_ axis showed that the p-p' axes were perpendicular to the

35 axes or parallel to the‘g axes.

The orientation of the crystals was marked on them
at the Bell Laboratories. The choice of axes was checked
and confirmed by the alignment of the sides of the etch
laits in the cleavage planes at the ends of the crystal.

The supposition that the most widely split lines
swore due to the waters of hydration was strengthened by
rneasurement of the relative area under these lines. 0n
lihotographs of the absorption spectrum these seemed to
kuave one-ninth of the total area. See Figure 16. As there
HJFe two such lines symmetrically displaced about the center
<31? the pattern, the combined area of the two is two-ninths
<31? the total area under the absorption spectrum.

Since there are three such sets of maxima, the total
Eixrea of absorption belonging to the most widely split lines
InEast be three, times two-ninths. This is Just two-thirds.
133st is the stoichometric proportion of protons associated
with the water of hydration in C(NH2)3A1(SOh)206H20. There

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51

 

L l

0 IO 20
Gauss

Figure 16a. A schematic representation of the proton re-
sonance absorption in GASH. This is taken from photographs
of an oscillosc0pe trace of the absorption at the angle of
maximum splitting. This occurs'for Ho parallel to the

2 axis. The area under the two humps is approximately
one-ninth of the total.

 

L l

0 I0 20
Gauss

Figure 16b. The absorption spectrum of GASH for Ho parallel
to the 5 axis. This pattern is one found in the rotation
of the crystal about the x axis.

52
is no guarantee that this reasoning is correct, but it
does lend an air of probability to the conclusion that the
most widely split lines belong to the waters of hydration.

For such reasoning, a necessary condition is that the
protons should all contribute equally to the total absorp-
tion in this region. This condition was insured by the
following. The lines were very difficult to saturate; the
signal-to-noise ratio of the absorption spectrum.was high;
and the area measurements were made with the applied radio-
frequsncy power sufficiently low to ensure no saturation
of the resonance absorption.

The central portion of the spectrum contains not
only a large number of lines which are widely split at
some orientations as we have just discussed, but also
additional absorption lines which are not distinguishable
in this rotation of the crystal.

A rotation about the x axis was therefore also made
yielding the data shown as circles in Figure 17. Based
on the conclusions from the previous rotation, both the
position and the relative area under two sets of lines
could be predicted. The most widely split line shows Just
one ninth of the total area and as before is due to a
proton pair in the (0001) plane perpendicular to the 5
axis which is the axis of rotation for the crystal. In
addition, there is one pair of lines which is due to the

two proton pairs which are perpendicular to the other);
axes. These are at 120 degrees to the axis of rotation.

 

53

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Sh

The remaining data suggests a set of proton pairs
which are parallel to the g_axis. The splitting of this
pair of lines is less than what one would expect if it were
due to the interproton distance in the hydrogen pairs in
guanidine. Furthermore it would seem from the data that
all of the guanidine proton-proton axes are parallel to
each other. The guanidine radical is usually supposed to
be planar in structure and trigonal in symmetry. The carbon
is supposedly' surrounded by three nitrogen atoms and to
each of these are attached two hydrogen atoms. If the
proton-proton axes are parallel to each other in such a
model, either the trigonal symmetry or the planar structure
would have to be violated.

Furthermore, the wide spacing of the hydrogen atoms
is contrary to previous knowledge of the structure.

These difficulties can be simultaneously resolved
by the supposition that the guanidine radical is in rota-
tion about its axis of symmetry. This axis is assumed to
coincide with the g or trigonal axis in the crystal. The
rotation is supposed to take place with a frequency that
is of the order of the nuclear resonance frequency. The
effect of such rotation upon the spectrum is to yield a
pattern equivalent to a pair of resonant dipoles whose
connecting axis is parallel to the axis of rotation. See

Gutowsky and Pake (1950). The line splitting of such a

55

case is just half of what it would be for the same separa-
tion of similar static non-rotating dipoles.

The actual dipoles could then lie at angles of 120
degrees to each other in the plane of the radical as is
commonly supposed to be the case. Each dipole-dipole axis
would then be rotating about the carbon at the center of
the molecule and there would be no distinction between
them in the spectrum. ~

Support for such a view of the structure of guanidine
aluminum sulfate hexahydrats is to be found in the classic
demonstration by Andrew and Eades (1953 a and b). They
showed that some entire benzenemoleculss rotate about
their axes of symmetry in solid benzene at temperatures
above 90 degrees kelvin. Above 120 degrees kslvin almost
all the benzene molecules are in rotation about their
hexagonal axes. The melting point of benzene is 278.5
degrees kelvin. It is not surprising therefore if a
radical of smaller moment of inertia and similar size
should be in rotation at temperatures of the order of 293
degrees kelvin. .

An attempt was made to study the resonance at low
temperatures by cooling a sample of GASH to liquid air
temperatures. The large internal stresses induced by the

temperature change however destroyed the crystal. At some

56
sufficiently low temperature the rotation of the radical
should cease, and a corresponding broadening of the reson-
ance should be observed.

Finally, the second moment of the line shape of a
powdered sample has been observed by McCall (1957) at low
temperatures. The second moment increased from 21.7 gauss
at 296 degrees kelvin to 26.9 gauss at 77 degrees kelvin.
He concluded that some of the proton groups in the material
were in motion at room temperatures, and hypothesized that
these were in fact,in the guanidine radical, which rotated
about its trigonal axis.

Knowing the orientation of the proton-proton axes
in GASH, and assuming that we can assign the different
resonances to the different constituents of the crystal,it
is possible solely through the addition of stereo-chemical
arguments to suggest an approximate atomic structure for
the crystal.

Let us first note some pertinent matters. The formula
of GASH is very similar to theformula of the slums. The
only change is the substitution of the guanidine radical
for a cation such as sodium or potassium. Waters of hydra-
tion in most hydrates generally tend to cluster around the
highest valence cations. This suggests that the waters
might be clustered around the aluminum. Aluminum has a
valence of three, and in the alums has a coordination of

six oxygen atoms around it. These are arranged like the

S7
corners of a regular octahedron. Such coordination has an
axis of three-fold symmetry. The nuclear resonance patterns
imply that the dipole-dipole axes associated with the waters
are arranged in three pairs. All this suggests that the
aluminum is surrounded by six waters of hydration in trigonal
symmetry with respect to the 5 axis.

Let us consider how an alum structure which is cubic
could be distorted into trigonal form. The trigonal axis
associated with a cube is along the body diagonal. In an
alum the cations are located at the corners of the unit
cube. The two species of cation alternate along all three
axes. The unit cell has twice the dimension of the unit
cube and contains four of each cation, and eight anion
complexes such as sulfate radicals.

Along the body diagonal therefore, the cations
alternate also. Thus distortion of this structure into a
trigonal form.must have both the aluminum and the guanidine
lying on trigonal axes and alternating with each other.

From valence considerations sulfate radicals have to be
interspersed between the aluminums and the guanidines. The
waters of hydration must lie around the aluminums with
their axes of symmetry along what would otherwise be the
cubic axes but now are diagonals in the trigonal crystal.

This is as far as one can go in the structure
without x-ray data. If one adds the additional information

from x-rays that there are three molecules in the unit cell,

58
one can tell that the three aluminums have to be coplanar
(in the (0001) plans) because the space group allows only
coplanar locations to have a multiplicity of three. Like-
wise the six sulfate radicals have to lie in general
positions in the molecule in order to allow for their
number.

A complete structure will require a thorough x-ray
analysis. Our tentative arrangement has been supported in
part by Geller (1957) who stated that a partial x-ray
analysis showed that the aluminums were indeed on the three-
fold axes of symmetry. Because of the number of possible
arrangements of the atoms in the unit cell included in our
general arrangement of the structure, and because of the
unresolved question of whether we are dealing with one or
with several structures we offer no definite structure.

Any possible structure however, must agree in major features
with our conclusions.

We believe that this example is outstanding as a
demonstration of the power of nuclear resonance as a crystal
structure tool. It is accidentally very much more fortunate
than most cases, but it is not by any means negligible or

trivially simple.

EUCLASE

A”.
I

 

59

EUCLASE

We have attempted to extend the analysis of hydrogen
compounds by proton resonance to a hydroxide, as well as
to the more favorable cases usually presented by the hydrates.
Other workers have also studied hydroxides. For example see
Ellsman and Williams (1956) for a study of brucite. The
substance chosen for this study is the mineral euclase. The
nuclear resonance spectrum of aluminum in euclase has
already been carried out by Eades (1955), but no study of
the proton resonance spectrum has appeared.

Euclase is a basic orthosilicate of aluminum and
beryllium, occurring as prismatic crystals. The (010)
plane shows perfect cleavage. It is an attractive trans-
parent vitreous mineral that varies from colorless to pale
green or blue.* Our specimen came from Minas Geraes in
Brazil, and exhibited only small imperfections.

Rotations of the crystal in the magnetic field about
the g.axis and the b axis were made. Only one proton
resonance line was found, which varied in width as a function
of orientation. See Figures 18, 19 and 20.

The lock-in amplifier records of the line at
orientations near the greatest width of the line suggested

that the line was made up of two overlapping components.

 

*Sse Dana (1932). pp. 618, 619.

60

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63
This is shown by the decrease of the slope of the derivative
curve near the very center of the line in Figure 18 a).

If one assumes that the overlapping components are
identical and are gaussian in shape, one can derive a
relation between the separation of the centers of the in-
dividual lines and the separation between the maximum and
minimum derivative points obtained from the lock-in amplifier
records. See Appendix II.

The use of such a relation implies that one knows
the width of the individual components. Tb avoid this
difficulty, one may assume that at the point of narrowest
line width the components have coalesced, and hence the
observed line width at that point is the width of the com-
ponents. This is a rather dangerous assumption since there
is no reason to assume that the line widths of the components
are independant of orientation. Nevertheless we have deduced
component separation by this method. The general conclusions
that we reach are not dependant upon any assumption of the
width of the components, or, in fact, upon the number of
components. The justification for such treatment of the
data is rather in the form of the experimental curves them-
selves.

0ur nuclear resonance data show maxima in the width
of the line approximately along the g,‘b, and g axes of the
crystal. Since one observes only a single broad line one

can identify only the main sources of broadening. As the

6h
greatest width of line occurs for Ho parallel to the 3
axis, one must suppose either a magnetic dipole along the
.2 axis or a confluence of magnetic dipoles in the 22
plane about the protons in the compound.

The variation in the 2Q,plane requires that the broad-
ening cannot be solely due to magnetic dipoles displaced in
the‘g direction from the protons. Furthermore the broaden-
ing is larger in the g orientation of the crystal in the
magnetic field than in the b, As a result, one would
suppose that any predominant magnetic dipoles in the 2b
plane about the protons must lie in the 3 direction.

The space group for euclase according to Biscoe and
Warren (1933) is Cgh' P21 /c. There are four non-equivalent
molecules in the unit cell. In this space group any one
molecule can be transformed into any other by glide planes
and twofold screw axes. It is impossible to distinguish
among molecules related by these symmetry operations by
means of nuclear resonance. The reason for this is that
nuclear resonance patterns arising out of magnetic dipole-
dipole interactions have 180 degree symmetry. We can
therefore consider the orientation dependance of the resonance
spectrum of only one proton and can be assured that any
result will apply to the others as well.

Biscoe and Warren (ibid.) have made a selection of
the oxygens associated with the hydrogens. This selection

is made upon the basis of the location of all the atoms in

65
the crystal, and upon valence grounds. If one accepts the
rest of the structure, one has to accept the selection of
the hydroxyl oxygens since valence arguments leave only one
oxygen unsaturated.

We are therefore obliged to consider where the pro-
tons could be located within about one angstrom radius
from the oxygen atoms. This is about the upper limit on
0-H bonds. Within this radius there is no feasible loca-
tion that can satisfy the requirements of the nuclear
resonance spectrum. It is easy to satisfy the requirement
that there be minimal splitting due to dipoles in the 2
direction. However as one can see from Figure 21 it is
hard to satisfy the requirement for a largest splitting
in the 3 direction.

We cannot achieve even a fraction of the necessary
broadening from the interaction with magnetic moments of
other protons. See Appendix I. These are too far away.
It is not possible under the symmetry Operations allowed
to bring the protons sufficiently close to each other to
provide the necessary broadening.

If we examine possible splitting from other nuclei
we come to the conclusion that only the aluminum has
sufficient magnetic moment that a reasonably large separa-
tion between it and the proton would not rule out the
observed broadening. However, even in this case such a

location of the proton in relation to the hydroxyl oxygen

66

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67
seems extremely unlikely. This is because this oxygen is
a part of the sixfold octahedral coordination of oxygen
atoms about the aluminum. Since the proton and the aluminum
and the beryllium are all cations, it would be strange if
any two of these were closer together than their common
anion. This reasoning leads to an upper limit for the
broadening due to the aluminum of h.8 gauss. This is too
small since there is no other interaction sufficiently
large to warrent consideration in such a location. Further-
more the only possible location so near to the aluminum
would give the wrong angle dependence of line width. There-
fore we must reject the tentative structure proposed by
Biscoe and Warren.

Eades' study of the aluminum resonance agreed with
the symmetry of the crystal as prOposed by Biscoe and
Warren. The proton resonance results also agree with the
symmetry of the crystal. This is not surprising since the
space group of crystals is generally known with greater
certainty than their structure. The aluminum resonance,
however did not yield any specific disagreement with the
complete structure according to Eades.

In their paper on euclase, Biscoe and Warren stated
that there were other possible structures. Had all the
alternatives been published, one of the structures might

have fitted our results. We were not able to supply an

68

alternative structures on the basis of the nuclear resonance
data alone. This shows the interdependence of the two
methods of structure study.

We conclude that nuclear resonance has shown itself
to be a useful tool in this case not only by rejecting a
tentative x-ray derived structure, but also by furnishing

a basis for choice among other possible structures.

L IQU ID CRYS TALS

69
LIQUID CRYSTALS

In this last section we demonstrate a useful appli-
cation of nuclear magnetic resonance to the study of the
structure of liquid crystals. Liquid crystals are materials
possessing a separate phase between the solid and liquid
states. This is a true phase separated from the neighbor-
ing states of the substances by heats of transition. The
macroscopic appearance and behaviour of liquid crystals is
that of a cloudy liquid, which clears completely when the
temperature rises above the transition to the liquid state.

It appears that there are at least three distinguish-
able types of liquid crystals, nematic, cholestsrine, and
smectic.* We will not discuss the particular features of
the last two types, as the liquid crystals under discussion
here are nematic.

Nematic liquid crystals are to be found only among
compounds whose molecules are long and rod-like in general
shape. We will be concerned with just two: p-azoxyanisole
and p-azoxyphenetole whose chemdcal structures are shown in
Figures 22 and 23.

It has been established that the molecules in liquid

crystals are oriented parallel to each other over regions

 

'—

*Friede1 (1922). Annalee d2 Physigue.

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72
large in comparison with molecular dimensions. See Frisdel
(1922). On the average these regions contain approximately
105 to 106 molecules. Perfect alignment within these regions
has not been demonstrated, but the average alignment of
the molecules cannot be disputed. The boundaries of these
regions are probably extremely vague and change rapidly in
response to electric, magnetic, or mechanical forces.

The effect of magnetic fields upon liquid crystals
is to introduce a common alignment to the various regions
and hence provide a general ordering throughout the liquid.
This has been discussed by Chatelaine in an extensive review
(l95h). Without the local self-orientation, the diamagnetic
anisotropy of the molecules would be insufficient to cause
alignment against the disordering effect of thermal
vibration.

The nuclear magnetic resonance spectrum of
p-azoxyanisole has been reported by Spence, Moses and Jain
(1953); and the spectrum of p-azoxyphsnetole has been re-
ported by Jain, Moses, Lee, and Spence (1953). These
spectra are shown as lock-in derivative curves in Figures
2k and 25.

These are the broadest nuclear resonance spectra to
be found in fluids. The explanation for this splitting
and broadening has been given by Spence, Gutowsky, and

Holm (1953) and Jain, Lee, and Spence (1955).

73

 

 

 

 

 

 

 

 

I I

C) 5 K3
scale in gauss

P

Figure 2h. The lock-in derivative curve of the proton
magnetic resonance absorption in p-azoxyanisole.

 

 

 

 

 

 

l_

0 . 5 l0
scale In gauss

J

Figure 25. The lock-in derivative of the proton magnetic
resonance absorption in p-azoxyphenetole.

7h

 

75

According to these authors, the molecules of both
compounds are aligned approximately along the magnetic
field. p-azoxyanisole is aligned not nearly so parallel
to the magnetic field as p-azoxyphenetole. The former is
strongly oriented approximately to degrees away from the
magnetic field direction. This alignment causes adjacent
protons on the benzene rings to be similarly oriented. The
magnetic dipole-dipole interaction between these protons
results in a pattern consisting of two split lines. Due
to the angle between the average orientation of the mole-
cular axes of p-azoxyanisole and the applied magnetic
field, that compound has a smaller splitting than
p-azoxyphenstole has.

The end methyl or ethyl groups in the two compounds
are relatively free to reorient about the molecular axis,
and so provide a central broad line. This was confirmed
by using deuterated methyl groups in the case of p-azoxyanisole.
The central line was nearly absent from the spectrum of the
deuterated compound as shown by Jain, Lee, and Spence (1955).

In this research we have sought to reinforce that
conclusion by destroying the ordering of the liquid crystal
by mechanical means while simultaneously observing the
nuclear magnetic resonance signal. In order to destroy
the magnetic ordering it was thought sufficient to stir
the liquid on a macroscopic scale. The ordering in a

magnetic field is due only to the fact that the molecules

in...

‘Efig

 

 

76
are already in c00perative local alignment. As a result,
the only restraint that the magnetic field ordinarily has to
overcome is that of the gross visCosity of the material
against the rotation of the oriented liquid crystal. Without
the local self-alignment, the magnetic field would be power-
less to maintain orientation against thermal disordering.
The effect of stirring upon the liquid crystal should be to
destroy the over-all alignment of the fluid.

In order to avoid large microphonically induced
voltages in the nuclear resonance sample coil that would
prevent observation of the resenance signal, some method
was sought to provide smooth and continuous stirring of the
sample.

The first method tried was bubbling gas through the
liquid. Helium was used because of its chemical inertness.
The gas was preheated by passage through a glass loop which
was immersed in the same air bath used to maintain the
sample at the high temperature at which it is a liquid
crystal. A glass nozzle projecting into the test tube con-
taining the sample was an integral part of the leap. Re-
markably, this arrangement worked on the first try.

Though microphonics occurred, they were of entirely
tolerable magnitude. Both photographic records and lock-in
derivative curves of the spectrum were obtained. See
Figures 26 and 27 for lock—in curves. A powerful narrowing

of the lines is immediately evident.

 

 

 

 

 

L

C) 5 K)
scale in gauss

L

Figure 26. The lock-in derivative curve of the proton
magnetic resonance absorption in bubbled p-azoxyanisole.

77

78

 

 

 

 

0 5 I0
scale in gauss

Figure 27. The lock-in derivative curve for the proton
magnetic resonance absorption of bubbled p-azoxyphsnetole.

79
Though perfect alignment in the undisturbed state is

not to be expected, the great difference in the spectra
between the stirred ani undisturbed samples is convincing
proof of a high degree of ordering in the undisturbed samples.
The effect is found in both liquid crystals. Curiously,
both have about the same line width in the stirred state,
whereas they have very different splittings in the quiescent
condition. This greatly strengthens the supposition pre-
viously postulated by Jain, Lee, and Spence (1955) that the
narrower splitting in p—azoxyanisole is due to a definite
orientation of the proton-proton axis at an angle to the
magnetic field.

The spectrum remaining during stirring is to be
interpreted as a random orientation pattern. Near equality
of width of the spectra of the two compounds when stirred
shows that the interproton distance causing the major
splitting is the same for the two compounds.

One feature of interest is the time required after
stirring stops for the liquid crystal to return to its
ordered condition in the magnetic field. We have determined
this time by using a moving film oscillograph to measure
the rate at which the resonance spectrum returns to that
found for the undisturbed state of the liquid crystal. It
turns out to be approximately .03 second for p-azoxyanisole.

This can be compared with the time constant required

for the reorientation of a diamagnetically anisotropic

80
rigid sphere immersed in a viscous liquid. It is acted
upon by a magnetic field and by the viscous drag of the
liquid. The differential equation.for this situation is:

I 9.2% 4 8rrro§1 9.9 l’ HOZXs %rrr03/o Sine CosO = 0

dt dt

Where I is the moment of inertia of the sphere; 0 is the

angle of the sphere with respect to its equilibrium orienta-

tion; t is the time; ro is the radius of the sphere;‘n is

the viscosity of the liquid; H0 is the applied magnetic

field;)(a is the surplus diamagnetic moment per unit mass

of the sphere; and P is the density of the sphere.
Assuming that we can neglect acceleration torques

in comparison with viscous and magnetic torques, we drOp

the first term of this equation. The resulting equation

integrates directly to

Tan 0 3 e't/T TanOo where T I 651/ Hoax ale
and TanOo is the tangent of the angle of orientation of
the sphere at time t-O with respect to the orientation of
the sphere in equilibrium.

Inserting the values ofrl ,X, and p for para-
azoxyanisole and 7000 gauss for the value of 30' the magnetic
field that we have been using, into the expression for T;
one arrives at a range for T of .03 to .1 seconds. This
range occurs principally because of the spread in the value

for the viscosity of the liquid crystal. This depends not

81
only on temperature; but more importantly on whether the
shear in the fluid crystal is parallel or perpendicular to
the magnetic field. According to Miesowicz (1936) the
viscosity parallel to the magnetic field is about 250 x
lO'h; while the viscosity perpendicular to the magnetic
field ranges from 550 x IO'h at 135 degrees centigrade to
950 x 10-h at 120 degrees centigrade which is nearly the
liquid crystalline range of the compound.

The agreement between such a very crude model and
the measured value is probably accidental. It does suggest
some additional justification for supposing the existence
of some sort of swarms. Furthermore the work of Heat (1931)
on the dependance of the x-ray diffraction pattern of liquid
crystalline p-azoxyanisole upon the frequency of an applied
electric field suggests that the assumption of negligible
acceleration forces is valid. Kast found that up to a
frequency of 25,000 cycles there was no change in the
diffraction pattern indicating that the permanent electric
moments of the molecules and hence the molecules themselves,
were following the field up to that frequency. Significant
inertial forces could be expected to have altered the
pattern by preventing the following of the field by the
permanent dipole moments of the molecules. We can therefore
be confident that there will be no inertial effects worth
consideration at frequencies as low as 33 cycles per

second.

82

Finally we may compare our measurement of the re-
orientation period in a magnetic field with that of Carr
and Spence (l95h). They found that a period of the order
of one minute was necessary for realignment at the
field of 200 gauss that they employed. In this case align-
ment was studied by microwave dielectric measurements.
According to our model T depends inversely on H02. There-
fore their value reduces to .05 seconds at our magnitude of
the magnetic field. Considering the crudity of our measure-
ment which depended upon subjective estimation of the
relative appearance of the nuclear resonance pattern, this
is good agreement. We thus may be confident not only of
the order of magnitude of this measurement, but also of
the general picture of the reorientation of local swarms.

We believe that this study of liquid crystals has
demonstrated the ability of nuclear magnetic resonance to
show the state of a material undergoing changes between an
ordered state and a disordered state. Furthermore, it has
been able to measure roughly the rate of change of the
state. This method is moreover capable of considerable
refinement. Thus the utility of nuclear resonance for
structure study even in such a borderline case as liquid

crystals has been demonstrated.

B IBLIOGRA PHY

BIBLIOGRAPHY 83

Andrew, E. R. (1953). Phys. Rev. 91, h25.

Andrew, E. R. and Eades, R. G. (1953a). Proc. Phys. Soc.
London. A, 66, hlS.

 

Andrew, E. R. and Eades, R. G. (1953b). Proc. 321. §2_
A, 218! 5370 r

O

 

Biscoe, J. and Warren, E. E. (1933). Zeit. f. Krist.
86, 292.

 

Bloch, F. (19u6). th . 523. 70, ueo.

Bloch F. Hansen, W. W. and Packard M. E. (l9h6). Phy .
' a.;. 69, 127. '

Bloembergen, N. (19h9). Physics. 15, 386.

Bloembergen, N., Purcell, E. M. and Pound, R. V. (l9h7).
Nature. 160, h75.

Bloembergen, N., Purcell, E. M. and Pound, R. V. (19h8).
Phys. B. . 73. 679.

Carr, E. F. and Spence, R. D. (195k). ‘g. Chem. Phy .
22, 1&81.

Chatelaine, P. (l95h). Bull. Soc. Franc. Mineral Crist.
77. 323.

Dana, E. S. (l95h). ‘A Textbook 2; Mineralogy revised and
enlarged by Ford, W. E., John Wiley and Sons, New
York.

Dehmelt, H. G. and Kruger, H. (1950). Egturwiss. 37. 11.

Eades, R. a. (1955). Canad.'g. th . 33. 286.

8h

Elleman, D. D. and Williams, D. (1956). J. Chem. Phys.
25, 7&2.

Foex, G. (1921). Annalee g2 Physique. 16, 17h.
Friedel, G. (1922). Annalee g2 Physique. 18, 273.

Geller, S. (1957). Private Communication by courtesy of
Spence, R. D.

Gutowsky, H. S., McCall, D. W. and Slichter, C. P. (1951).
Phys. Rev. 8h, 589.

Gutowsky H. S. and Pake, G. E. (1950). ‘g. Chem. Phys.
16, 162.

Heide, Von H. G. and Boll-Dornberger, K. (1955). Acta.
Crysta. 8, h25.

Holden, A. N., Merz, W. J., Rsmeika, J. P. and Matthias, B.
T. (1956). Phys. Rev. 101, 962.

Jain, P. L., Lee, .I C. and Spence, R. D. (1955). g, Chem.
Ph:! 0 23' 6780

Jain, P. L., Moses H., Lee, J. C. and Spence, R. D. (1953).
Bull. Amer. Phys. Soc. 28, 8.

Kartha, G. (1952). Proc. Indian Acad. Sci. 36A, 501.

 

Kast, w. (1931). ‘g. Physik. 71. 39.
Knight, w. D. (19h9). Phy . 531. 76, 1259.

McCall, D. w. (1957). pg. Chem. Phy . 26, 706.

 

 

85

Miesowicz, M. (1936). Bull. Acad. Polon. g3 Sciences gt
Lettres. (Serie A), 228. Quoted by Carr, E. F.
. Phd. Thesis, Michigan State College.

Nierenberg, W. A., Ramsey N. F. and Brody, S. B. (19h7).
Phys. Rev. 71, h 6A.

Fake, 0. E. (19118). ,1. Chem. m. 16, 327.

Pake, G. 8h and Purcell, E. M. (19118). Phys. 33;. 711,
11 .

Pound, R. v. (19117). th . 39.1. 72, 1273.
Proctor, W. G. and Yu, P. C. (1950). Phy . Egg. 77. 717.

Purcell, E. H., Torrey, H. C. and Pound, R. V. (19h6).
Phy . .1333. 69. 37.

Purcell, E. M., Pound, R. V. and Bloembergen, N. (19h6).
Phy . Rev. 70, 988.

Ramsey, N. F. (1950a). _P_h_y_. m. 77, 567.

Ramsey, N. F. (1950b). Phys. 391. 78, 221.

R6111n, B. v. and Hatton, J. (19118). 11131;. 531. 71., 31,6.
Spence, R. D. (1955). .g. Egg . §hy_. 23, 1166.

31301106, R. De, GutOWSky He Se and 11011!“ CO H. (1953). :1.
Chem. Phy . 21, 1891.

Spence, R. D., Moses H. and Jain, P. L. (1953). .g. Chem.
P1128. 21’ 3800

 

86

Townes, C. H., Herring, C. and Knight, W. D. (1950). Phy .
RBV. 779 8510

Van Vleck, J. H. (191.8). Phy . 391. 711. 1168.

Wood, E. A. (1956). Acta. Crysta. 9, 619.

APPEND IX I

87

 

 

 

 

 

 

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APPENDIX II

89
APPENDIX II

In this appendix we calculate the apparent width
of a line composed of two overlapping gaussian curves.

Width of line is taken to mean the separation between peaks
of Opposite sign on a lock-in amplifier record. Such a
record is a graph of the derivative of the absorption
spectrum. Consequently this width is the separation between
the point of maximum positive slope and the point of maximum
negative slope of the spectrum.

In order to calculate this width for two overlapping
gaussian functions we must add them together, and find the
points of steepest slope of the sum.

A gaussian absorption curve can be expressed as
G 8 e‘hg/Zogwhere G(h) is the absorption; h is the magnetic

field measured from the center of the resonance line

1'-

such that th dh - O; and c- is the second moment of G

‘I’O
such that hZG dh
_;ag____ : c'

*0
'[G dh
-m
It so happens in the case of a gaussian function
that <r is also a measure of the distance from the center
of the gaussian to the point of steepest slope. It is

therefore the half-width of the gaussian function itself

by our criterion for line width. That is, + a- are the

9O

roots of dZG a O. This last may be more fully expressed, by
db2 2 2
saying that the equation (xz/a'h - 1/0'2) e'h /2 o- 8 O has

the root h ' i 0'. The second moment generally does not
equal the line width for other functions.

Let us designate the separation of the two gaussian
absorption curves center-to-center by (8h. We will suppose
that they are symmetrically displaced from h-O by an amount
8h/2. The total absorption will then be proportional to
pm) . e-(h +8h/2)/2 0’2 , e-(h - 8h/2)/2 0'2. The um-
ation is illustrated in Figure 28. Calculating the second
derivative, sz/dhg, and setting it equal to zero yields
the equation:

2(h/o')(5h/20-) Hyperbolic Tangent ((h/c')(8h/2q-)) :
(h/o' )2 + (8h/20')2 - 1

This transcendental equation can be solved numerical-
ly for g = h/o- as a function of at = Sh/ZG' . The solution
is displayed in Figure 29. The points of steepest slope
occur at those values of h satisfying this graph for partic-
ular values of 8h and a- .

Given an and a- we can find h directly in units of
(r. We can then find 2h, the width of the combined gaussian
absorption curves in terms of: 8h, the separation center-
to-center, andcr , the half-width of the components.

Experimentally, we measure 2h, make some assumption

about the value of 0'. (for instance we may let c'be the

91

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Figure 29. The apparent half width of two overlapping
gaussian curves. This shows the apparent half-width, h,
of two overlapping identical gaussian curves as a function
of their half-separation, h/2, in units of their individ-
1181 half-W1dth’ e

93
minimum value of the orientation dependant line width) and

then 8h.can be found with the aid of the graph.

APPENDIX III

9h

APPENDIX III

The purpose of this appendix is the calculation of
the total number of unadjustable constants that can be
obtained from measurements of the second moment of a line
in the most general case.

We will assume that the second moment of a line will
be generated entirely by magnetic dipole-dipole interactions.

The square root of the sum of the squares of such
interactions was shown by Van Vleck (19h8) to be equal to
the second moment.

Van Vleck's results can be cast into the form

n
0.2 = Z k12(PZCos 01) )‘2 where o- is the second moment

i=1
of a line; k1 is a constant for each particular dipole-
dipole interaction perturbing the resonant species of
nucleus; and Oi is the angle between the dipole-dipole axis
and the magnetic field. k1 depends upon the kinds of per-
turbing and perturbed dipoles and their separation. P2 is

the legendre function of the first kind, (l/2)(BCos20 - 1).

i
We are going to show the number of constants required

to express the square of such an interaction at an arbitrary

angle in an arbitrarily oriented coordinate system. The

sum of such interactions will then simply result in separate

sums of these constants.

95

Because these constants will be multipliers of
functions chosen so as to be linearly independant in the
arbitrary coordinate system, addition of the squares of
interactions can only result in sums of coefficients of the
orthogonal functions. Consequently no new kinds of functions
will be generated by the addition, and a smaller number of
functions can arise only in special cases. Usually all
constants will be non-zero.

Consider then (P2(Cosy) )2 where‘y is some particular
01 . Expressing this as first powers of legendre functions
of the first kind, one gets:
was...» >2 - (18/35) rumour) - (5/7)P2(Cos)’) + 9/20
Now consider this in an arbitrarily oriented spherical
coordinate system such that )'1s a great circle are from the
point (1,0o,¢o) to the point (1,04); where Go and p0
represent the angular orientation coordinates of a dipole-
dipole axis in the 91 direction; and 0 and ¢ are the angular
coordinates of the magnetic field direction.

In such a coordinate system

Cos)’= 00800 C039 4 SinOoSinOCos (¢ -¢o) ; and Pn(Cos7)
n
Pn(CosGo)(PnCosO) 1» 2 Z ((n-m)!/(n+m)1)

m=1

P1300300) P:(Cos0) Cos m(¢>o -¢) where P are legendre

n
functions of the first kind and P: are associated legendre

functions of the first kind.

96
Making use of these relations we can now transform

(P2(Cosy))2 into:

(18/35) [Ph(CosOO) Pu(COSO)
+ (l/lO)Cos (¢ - ¢O)Pfi(CosG°)Pfi(CosO)
+ (1/180)C082( ¢ - ¢O)Pfi(Cos00)Pfi(CosO)
+ (1/2520)Cos3(¢ - ¢O)Pi(CosO°)P&(Cos0)

f (l/20160)Cosh( ¢ - ¢O)Pllt(CosOO)Pk(CosO)]

-(5/7) [P2(CosOO)P2(CosO)
+ (1/3)c63(¢ - ¢O)P:(Cosoo)P;'(Cos0)

+ (1/12)C082(¢ - eo)p§(c°so°n§(coso)]

+ 9/20

The legendre functions in this expression may be defined as

follows:
P2(C089) = (l/h)(306320 4 1)
P%(COSO) = (3/2)(Sin20)
P§(Coso) = (3/2)(1 - C0829)

(l/6h)(35Cosh0 4 2000329 + 9)

Ph(CosO)

97

Pfi(COSO) = (5/16)(ZSin20 + 7Sinh0)
Pfi(CosO) = (15/16)(3 + hoosao - 7Cosh0)
Pi(Cos0) = (105/8)(ZSin20 - SinhO)
Pfi(6630) = (105/8)(3 - hCosZO + CoshO)

We have now expressed in an arbitrarily oriented
spherical coordinate system a single dipole-dipole inter-
action contribution to the square of the second moment. Not
all these terms are linearly independant in G and ¢»over

the unit sphere, so we must regroup them accordingly.

(18/35) [Ph(Cosoo)Ph(Cos9)
+ (1/180)CosZ( ¢ - ¢O)P:(CosOO)P:CosG)

+ (l/20160)CosL|.( 4’» - ¢O)Pt(CosOo)Pt(CosG)]

+ (18/3S)El/10)Cos(¢ - ¢O)Pfi(CosO)

«I» (l/2520)Cos( ¢ - ¢O)Pfi(CosOO)P’i(CosO)]

+ (5/7)[: P2(CosOO) P2(CosO)

)- (1/12)CosZ(¢-¢O)P:(Cosoo) P§(CosO)]

+ (5/7) [(1/3) Cos(¢-¢o) P%(Cos00) P;(CosO)]

+ 9/20

98

Each square bracket is linearly independant in G
from the others. Within each bracket there are a number of
terms which are independant in ¢ . In the first bracket
there are five; in the second, four; in the third, three;
in the fourth, two; and in the fifth is the remaining
linearly independant term.

Thus there are exactly fifteen constants generated
by (P2(Cos )y)2 which are orthogonal over the entire range
of orientations that can be assumed by the magnetic field.
Therefore from second moment measurements alone, one can

determine fifteen independant constants.