NUCLEAR RESONANCE STUDEES OF
ANTIFfiRRO’MAGNEHC CRYSTALS

{N ZERO-FIELD

Thesis {or “19 Degree 0‘ M. 5.
MICHIGAN STATE UNIVERSITY

Doohee Kim
1962

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M iChigan S tare
Univ ersity

 

NUCLEAR RESONANCE STUDIES OF ANTIFERROMAGNETIC
CRYSTALS IN ZERO-FIELD

BY

Doohee Kim

A THESIS

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

MASTER OF SCIENC E

Department of Physic s

1962

ACKNOWLEDGMENT

I wish to express my gratitude to
Professor R. D. Spence for suggesting the
topic and for his kind guidance throughout
this work.

************

ii

TABLE .OF CONTENTS

Page

I. INTRODUCTION ....................... 1

II. THE THEORY AND THE TECHNIQUE OF THE METHOD . 2
III. APPLICATIONS ........ ' ............... 9
1. CuClz-ZHZO ...................... 10

‘2. NiClz-6HZO ......... . ............ 10

3. COBrzo 6Hzo ...................... 13

4. MnClz-4HZO. . . . . . . ..~ ............. 13

LIST OF REFERENCES .......... . ........ 16

. iii

I. INTRODUCTION

In the study of nuclear magnetic resonance of antiferromagnetic
crystals, it is customary to find the direction of the local magnetic
fields at probe nuclei sites in the presence of a large external magnetic
field. This method is rather tedious, and a number of helium runs are
required to complete the study of a single crystal. In addition to this,
it has been shown that the external magnetic field perturbs the spin
arrangement of the magnetic ions, and the results are not completely
characteristic of the crystal itself. In view of these difficulties, we
have examined an alternate method of finding the local magnetic field.
It is the purpose of this thesis to show that nuclear magnetic resonance
experiment in zero field (no external magnetic field) removes these

difficulties.

II. THE THEORY AND THE TECHNIQUE OF THE METHOD

The apparatus used for the zero field method is the same as that
for the applied field method except there is no magnet. In place of the
magnet, one uses a modulation coil, by which a time varying magnetic
field can be oriented in all directions. The set-up is shown in Figures
1 and 2.

The nuclear magnetic resonance occurs if the total magnetic field
Ht; which the nucleus experiences, and the frequency of the detecting

device satisfy the following relation.

(1) IN = gHHt
where g: the g factor for the nucleus
II: the nuclear magneton
Ht: the total magnetic field at the nuclear sites
h: Planck's constant
v: the detector frequency

Here Ht consists of the internal field of the crystal at the nucleus
sites and the modulation field. The total field is given by:

—-> —9
(2) th=le+Hrzn +ZH1-Hm

where H1 is the local field and Hm is the modulation field.

If in an antiferromagnetic crystal there exists a set of proton
sites which experience a given local magnetic field, there exist an
equal number of proton sites which experience a local magnetic field
of the same magnitude but of opposite direction. Thus the total field

for these two cases is given by:

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(2') th =H12+Hr7h+ZIHll lel sin6cos¢
(3) th=le+Han-ZIH1Ilelsinficosq)

where ‘Hm is along x-axis, and 6, (b the polar and azimuthal angle of
one of the local field vectors respectively.

As the result of the time variation of the modulation field, the
total magnetic field Ht fluctuates within the limits determined by the
amplitude of the modulation. If, within these limits, the total magnetic
field satisfies equation (1), the nuclei will absorb the power from the
detector. Except for large modulation field, the absorption will occur
at two places in the period of the modulation cycle.

If the absorption signals are viewed on the oscilloscope, which is
synchronized to the modulation, resonance lines are observed when

the following conditions are satisfied.

 

(4) Hmosin(wrntl) = -IH1 sin 9 cos (b | + I\/(Hlsin 9 cos ¢)Z+(H§ - Hf)

 

(5) Hmosin(comtz) = +|H1 sin 6 cos 4; I - ~/(Hlsin 9 cos OFT-(H7 - H?)

where w is the modulation frequency, and H is the modulation
m mo
amplitude. With this definition of t1 and t2, the separation between the

two signals is given by:

 

 

(6) tl-tz = éarc sin (-- IHI sin 6 COS ¢I + figsin 6 COS ¢F+(H:'HI) )
Figure 4 shows the separation as a function of the orientation of
the modulation field for CoClz' OHZO, where the solid line represents the
calculated values and the circles the experimentally observed values.
The smallest separation occurs when the modulation field and the
local field vectors are parallel; therefore, to determine the direction
of the local magnetic field, it is sufficient to measure the separation as
a function of the orientation of the modulation coil with respect to the

sample-crystal.

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This can be done very rapidly by watching the separation of the two
resonance lines on the oscilloscope, and this speed is one of the
advantages of the method.

The magnitude of the local field is determined by setting the
frequency of the detector so that the two signals coalesce at all orien-
tations of the modulation field. The value of this frequency inserted to
the equation (1) gives the magnitude of the local field.

The detecting system consists of the marginal oscillator, modu-
lation coil, oscilloscope, and the frequency counter. The circuit
diagram of the marginal oscillatorris shown‘in Figure 3. -In order that
the Hewlett Packard frequency meter respond to the small power output
of the oscillator it is necessary first to amplify it through a broad
band amplifier.

To achieve all possible orientations of the modulation field, the
field must be generated by a single coil with the crystal displaced from
its center. The modulation field this arrangement produces at the
sample is not very uniform in magnitude or direction.

For the modulation coil geometry and crystal size used in these
experiments, the modulation amplitude varies by 5% and the direction
of the modulation field varies by :h 30 over the sample. The amplitude
of modulation was about 50 gauss with the variac setting at 70 v.

Since the natural line width is the order of 5 gauss, the broadening
effect of the resonance line due to the modulation amplitude spread
can be reduced by using a small modulation amplitude.

Thus the zero-field method has two advantages over the applied
field method; namely, the rapidity in finding the local field vectors

and the negligible distortion of the magnetic moments.

III. APPLICA TIONS

In this section, we examine some of the information Obtained from
antiferromagnetic crystals by using the zero field method. Oscilloscope
pictures of the proton resonance of typical antiferromagnetic crystals in
zero-field are shown in Figure A to Figure 0..

Analysis of our zero field study of CuClz- 21-120 shows that there
are eight local fields, all having the same magnitude but different
directions. Each signal arising from these fields is split into two com-
ponents by the dipole-dipole interaction of protons in the same water
molecule. Within experimental error, the results are in agreement
with those of the Leiden group [2].*

The interaction potential V12 of two magnetic dipoles is given by:

 

—> —> 3(—-> —>) (—-> T?)
o . r o
(7) V12 .: Eirlzflz _ H1 1%?sz 12

where :2 is the vector connecting two dipoles of moment Wand I:

In hydrated crystals, the distance between the protons in the same
water molecule is shorter than that between protons in different water
molecules. In most of the cases the distance is so much shorter that only
the dipole-dipole interaction of protons in the same water is important,
and in evaluating V12 one has only to consider such protons.

As a result of V12 the energy levels are distorted, with a conse-
quent shift of the resonance frequencies, which is a function of the angle
between the local magnetic field at the proton sites and the proton-proton

direction.

 

>4:
Numbers,1n brackets are reference numbers.

10

1. CUC120 21-120

The resonance frequencies and the orientation of the local fields
in CuClzo ZH'ZO are shown in Table 1. Here 9 is the angle measured from
the crystallographic c-axis and (b is measured in the a-b plane from the

a-axis. The orientations are shown graphically in the stereogram of

 

 

 

Figure 5.
Table 1.
W
The proton The direction of the
resonance frequency local field Temperature
(MHz) 9 ¢
vA*- 3.12 65° 37° 2.15°K
* o o
. ,0
VB 3 4 65 37

 

2. ~ NiC12.6HzO

There are four local magnetic fields at the proton sites in this
crystal below the Néel temperature (6. 20K). The highest and the lowest
frequency resonance lines Show the doublet -splitting but the intermediate

frequency lines did not. *Figure 6 and Table 2 show the results Obtained

 

 

 

 

at 4. 2°K.
Table 2.
=1 _=: L
The local magnetic field The resonance frequency Tempgra-
vector at proton sites (MHZ) ture ( K)
A 6. 39
'B 5. 17 o
C 4. 67 4° 2 K
D 3. 18

 

11

 

 

 

 

 

1: MSU's
Z: Leiden's
I
001
G . . b
F
- 0
Go?- 2 0
' . I:
' t

Figure 5. Stereographic projection of the local magnetic fields at
the proton sites in CuClz- ZHZO (ours at 2. 150K).

lZ

 

 

 

 

 

 

 

a!

Figure 6. Stereographic projection of the local magnetic fields at
the proton sites in NiClz- 6HzO at 4. 20K.

l3

3 . COBrz° 61-120

The numerical results of the local magnetic fields are given in

Table 3.
.Despite the fact that the crystals CoClz. 6HZO and NiClz- 6HZO are
isostructural [3, 4], the COBrz- 6HZO and NiClz- 6HZO are quite different

magnetically.

Table 3.

 

 

The local magnetic

 

field vector at proton The; resonance, ‘Ifmperature
sites frequency (MHz) ( K)

A 6. 99 o

B 5 . 80 2. 5 K

C 3 . 7O

 

4. MnC12‘4HZO

Three groups of proton resonance lines were observed for this
crystal below the Néel temperature (1. 620K). The local fields seem to
lie all in the a'-b plane of the crystal as shown in Figure 7. The

numerical results are given in Table 4.

Table 4.

W

The local magnetic

 

field vector at proton The resonance %emperature
sites frequency (MHz) ( K)

A 8. 1

B1 7 . 0 0

B2 7. 0 1. 17 K

C 5 . 7

 

l4

 

 

 

Figure 7. Stereographic projection of the locac.)l magnetic fields at the
proton sites in MnClzy4I-IZO at 1. 17 K.

-15-

Fig.B F1g.C

 

Fig.D

 

Fig.F rig.G

 

Fig.H

 

Fig.3 to Fig.G : Zero-field signals of proton(s) in crystals
in antiferromagnetic state.

F1g.A; A coalesced proton signal in 00012.68é0.
Fig.8 and C:Sdgnals in N1C12.6H20.

Fig.D: or CoBr2.6H20.

Fig.3: of CuClg.2H20.

Fig.F and G: of mn012.4H20.

Fig.H: Zero-field proton Signal in Mh(CHSCOO)2.4H20 at 2.4.K.

LIST OF REFERENCES

. Andrew, E. R. Nuclear Magnetic Resonance. .Cambridge
University Press (1956).

. Poulis, N.IJ., Hardeman, G. E. G., Van der Lugt, W., and
Hass, W.~P. A. Physica, XXIV, 280 (1958).

. Mizuno, Joji. Jour. of the Phys. Soc..of Japan, Vol. 15, No. 8,
1412 (1959).

. Mizuno, Joji. Jour. of the Phys. Soc. of Japan, Vol. 16, No. 8,
1574 (1960).

16

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