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Effect of the large magnetostriction of Terfenol -D on microwave transmission and reflection

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EFFECT OF THE LARGE MAGNETOSTRICTION OF TERFENOL-D
ON M ICR O W AVE TRANSMISSION AND REFLECTION
By
Phoumyphon Sourivong
Master of Sciences, Mankato State University, 1994
A Dissertation
Submitted to the Graduate Faculty
of the
University of North Dakota
In partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Grand Forks, North Dakota
July
1999
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UMI Number 9949642
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This dissertation, submitted by Phoumyphon Sourivong in partial fulfillment o f the
requirements for the Degree of Doctor o f Philosophy from the University o f North
Dakota, has been read by the Faculty Advisory Committee under whom the work has
been done and is hereby approved.
(Chairperson)
I
K_^xr
This dissertation meets the standards for appearance, conforms to the style and
format requirements of the Graduate School o f the University o f North D akota.
and is hereby approved.
Dean o f the Graduate School
Date
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PERMISSION
Title
Effect o f the large magnetostriction of Terfenol-D on microwave
transmission and reflection
Department
Physics
Degree
Doctor o f Philosophy
In presenting this dissertation in partial fulfillment of the requirements for graduate
degree from the University o f North Dakota, I agree that the library of this University
shall make it freely available for inspection. I further agree that the permission for
extensive copying for scholarly purposes may be granted by the professor who supervised
my dissertation work or, in his absence, by the chairperson o f the department or the dean
o f the Graduate School. It is understood that any copying or publication or other use of
this dissertation or part thereof for financial gain shall not be allowed without my written
permission. It is also understood that due recognition shall be given to me and to the
University o f North Dakota in any scholarly use which may be made of any material in
my dissertation.
Signature
Date
in
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TABLE OF CONTENTS
ACKO W LED G EM ENT....................................................................................................... vi
LIST OF TABLES................................................................................................................vii
LIST OF FIGURES....................................................................................................... viii, ix
ABSTRACT............................................................................................................................ *
CHAPTER
I.
Introduction...................................................................................................1
II.
The theory.................................................................................................................... 3
2.1
Features of the different magnetic field configurations............................................... 3
2.2
Maxwell equation for H uL geometry..........................................................................5
2.3
Equation o f motion for the magnetization....................................................................7
2.3.1 The damping terms..................................................................................................... 11
2.3.2 The Landau-Lifschitz damping term.........................................................................11
2.3.3 The Gilbert damping term......................................................................................... 12
2.3.4 The Bloch-Bloembergen damping term.....................................................................12
2.4
Equation of motion for the lattice............................................................................... 14
2.5
The secular equation.....................................................................................................15
2.6
The boundary value problem for the H // ± geometry..................................................17
iv
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TTT
E xperim ental apparatus and sa m p le s ................................................................26
3.1 Static magnetic field....................................................................................................... 31
3.2 AC modulating magnetic coils.......................................................................................31
3.3 Microwave cavities........................................................................................................ 34
3.4 Cavity tuning....................................................................................................................34
3.5 Temperature control and measurement.........................................................................36
3.6 The chemical composition o f Terfenol-D and its physical properties........................36
3.7 X-rays of the samples...................................................................................................... 36
3.8 Data acquisition.............................................................................................................. 38
I V . E xperim ental results and conclusions................................................................. 39
4.1
Reflection measurements...............................................................................................39
4.2a Typical reflection results...............................................................................................39
4.2b FM R with M parallel to the [001] axis.......................................................................44
4.2c Experimental results for hysteresis.............................................................................. 49
4.2d Results for second sample...........................................................................................49
4.3
Summary o f reflection experiment............................................................................... 51
4.4 Transmission measurements..........................................................................................54
4.5 The experimental results................................................................................................54
4.6 Summary o f transmission experiments......................................................................... 56
4.7 Conclusions.....................................................................................................................56
R eferences.......................................................................................................................... 58
V
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ACKNOWLEDGEMENTS
The author wishes to express his gratitude to Dr. Graeme Dewar, Chairperson of the
committee, for his intelligent guidance, encouragement and patience. I also would like
to thanks Dr. B. Seshagiri Rao for his support and advising, and Dr. Soonpaa for his
excellent contribution and helping me with the computer.
Finally I want to express my grateful to my Graduate Committee for giving so
generously o f their time and effort in the evaluation o f my dissertation.
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LIST OF TABLES
Table
Pa§e
1.
The constant values of the root mean square voltage.................................................. 33
2.
The physical properties o f the Terfenol-D.................................................................... 37
The anisotropy constant and Gilbert damping parameter.............................................51
vii
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LIST OF FIGURES
Figure................................................................................................................................ Page
1. Illustration o f the three common magnetic field configurations.................................... 4
2. Magnetization with cosines direction angles.................................................................. 9
3. Fields inside the cavity and waveguide section............................................................ 24
4. Block diagram o f the apparatus for the reflection and transmission
experiment....................................................................................................................... 29
5. Waveforms illustrating for the modulation spectrometer used in reflection
microwave experiments................................................................................................30
6. Audio-Amplifier..............................................................................................................32
7. Root mean square induced voltage of the ac modulation coils vs magnetic field
33
8. Arrangement o f the cavities and the sample holder plate.............................................35
9. Dendritic platelets in Tb027Dy072Fe2........................................................................... 38
10. Derivative of reflection amplitude vs magnetic field at20 C ..................................... 41
11. Derivative of reflection amplitude vs magnetic field at 25 C ................................... 42
12. Derivative o f reflection amplitude vs magnetic field at22 C and 28 C ....................43
13. Derivative o f reflection amplitude vs magnetic field at 20 C ...................................45
14. Derivative o f reflection amplitude vs magnetic field at 29 C ...................................46
15. Derivative o f reflection amplitude vs magnetic field at24 C ................................... 47
viii
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16. Derivative o f reflection amplitude vs magnetic field after the sample was left at
12.0 kOe for 6 hours....................................................................................................... 48
17. Derivative o f reflection amplitude vs magnetic field at 26' C by sweeping
magnetic fields up and dow n..................................
50
18. Derivative o f reflection amplitude vs magnetic field at T= 32' C by sweeping
magnetic fields up and down......................................................................................... 52
19. Derivative o f reflection amplitude vs magnetic field at T = 2 0 'C for the second
sample.............................................................................................................................53
20. Transmission vs magnetic field at T = 2 0 'C for the second sample......................... 55
ix
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ABSTRACT
The purpose o f this dissertation is to study certain physical properties o f Terfenol-D, a
magnetic material which is extremely magnetostrictrive. O f particular interest is the
magnetoelastic behavior o f Terfenol-D when interacting with the microwave radiation
in the presence o f a static magnetic field. A microwave transmission spectrometer was
used to measure transmission and I modified the spectrometer to perform reflection
measurements. Experiments were done over a small temperature interval near room
temperature using static magnetic fields from 0 kOe to 13 kOe and with microwaves of
frequency 16.95 GHz.
The results of this study followed from observing at what
magnetic field ferromagnetic resonance (FMR) occurs. The enormous anisotropy
of Terfenol-D should shift the field for ferromagnetic resonance; the experimental
results confirm this effect and give an experimental value for the anisotropy
constant of AT, = (-2.00 ±0.025) xlO 6erg/cm3 at room temperature. Observation of the
microwave transmission through Terfenol-D should confirm the predicted role
magnetostriction plays in its response to microwaves.
x
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CHAPTER I
INTRODUCTION
The primary objective o f the experimental research work presented in this dissertation
is to improve understanding o f the radio frequency (rf) magnetic properties of the
ternary alloy Dy\ 73Tba,7Fe, known as Terfenol-D [1] by studying its reflection and
transmission properties using microwave radiation. This material is well known for its
large magnetostriction [2] and is considered as smart material.
In these experiments electromagnetic power at microwave frequencies is incident, in the
presence of an applied static magnetic field, on one of the surfaces of a slab-shaped
sample which is part of the common wall between two microwave cavities. The
transmitted signal at the opposite surface of the sample is collected in a second
microwave cavity and subsequently it radiates into the detector. A static magnetic field
in the plane of the sample allowed the study of the magnetic field dependence of the
reflected and the transmitted signal. The signal reflected from the front surface of the
sample, which is only weakly affected by sound waves, allowed the material’s
electromagnetic properties to be characterized . As will be shown in Chapter II, we
expect the transmission to be dominated by phonon propagation. We hoped to see
aspects of the magnon-phonon interaction. Because Terfenol-D has a large
magnetostriction parameter, and phonons are only weakly attenuated, we expected a
1
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2
large phonon transmission to the rear surface where a reconversion process allows the
phonons to generate significant microwave power in a receiver cavity. At certain applied
magnetic fields the ferromagnetic resonance (FMR) occurs and especially large
intensities o f sound waves are generated. Hence, a transmission peak is expected at
FMR. The spin waves do not contribute to the transmission. The field dependence of
the transmitted signal is a function o f the orientation of the magnetic field with respect to
the microwave magnetic field as well as of material parameters and temperature.
The dissertation is divided into five chapters. Chapter I is the introduction. Chapter II
is the theoretical framework for the experiments. It begins with a description o f the
features of the different magnetic field configurations, and continues on to show how
Maxwell's equations, the equation o f motion for the magnetization with the Gilbert
damping term, the equation of motion for the lattice, the secular equation and finally the
boundary value conditions can be used to calculate the expected transmission and
reflection coefficients.
Chapter EH contains a short description of the experimental
apparatus. Chapter IV presents the experimental results and conclusions.
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CHAPTER EE
THEORY
A theoretical expression for the amplitude of the microwave magnetic field transmitted
through, or reflected from, a thin ferromagnetic metal slab as a function o f magnetic
field is derived below. This expression includes the effects of the exchange energy, the
equation o f motion for the magnetization together with the Gilbert damping, the equation
of motion for the lattice, and Maxwell’s equations. Finally the boundary value problem is
solved. All o f these expressions are very important and necessary to explain the
ferromagnetic resonance effect. I begin with the magnetic field configurations.
2.1 Features of the different magnetic field configurations
There are three types of configuration for the orientation of applied and microwave
magnetic fields conventionally used in this sort o f experiment. When the static magnetic
field is perpendicular to the sample’s surface and to the microwave magnetic field
(Fig. 1), the geometry is called the perpendicular-perpendicular configuration, H _ . For
the static and microwave magnetic fields perpendicular to each other, but parallel to the
plane of the sample, the geometry is called the parallel-perpendicular configuration,
H _ . The case in which the static and microwave magnetic fields are parallel and in the
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4
z
H,
//.// >
H
x
Fig. 1. Illustration of the three common magnetic field configurations:
top
; center H l/±; bottom H nJ, . H a is the static magnetic field and h is
the microwave magnetic field, and k is the wave propagation vector.
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plane o f the sample is called the parallel-parallel configuration H 7 . In Fig. 1., which
—
►
depicts all three configurations, H a is the static magnetic field, h is the microwave
—
►
magnetic field, and k is the propagation vector for the waves. In our experiments we
used the parallel-perpendicular geometry. For a ferromagnetic sample, with no magnetic
anisotropy, the transmission through a slab in the H nj, configuration is dominated by the
ordinary electromagnetic wave and an electromagnetically excited longitudinal sound­
wave; in this configuration there is almost no magnetoelastic coupling. For the H
geometry, the transmission is dominated by a transverse sound wave, a spin wave, and by
the extraordinary electromagnetic wave.
Our magnet could not go to high enough field
to magnetically saturate the sample in this configuration. Even with a bigger magnet our
microwave frequency is too low for interesting effects to be observed. In the H
geometry the transmission is dominated by the extraordinary electromagnetic wave and a
transverse magnetoelastic sound wave. The spin wave makes a negligible contribution to
the transmission; therefore, to study the magnetoelastic coupling it is convenient for us
to choose the
geometry.
2.2 Maxwell equations for the H,l± geometry
The electromagnetic field inside and outside the ferromagnetic must obey Maxwell’s
equations.
V * 5 = if£ ;/+ M £ £ >
c
c dt
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(2.2.1)
and
VxE = - - — .
c dt
(2.2.2)
The metal is cubic and, neglecting the Hall effect, is adequately described by a scalar
conductivity a. Chm’s law becomes J = a E .
The magnetic induction is B = H + 4tiM , where M is the magnetization and H is
magnetic field, including the demagnetizing field. For the H nL configuration we
choose a coordinate system where the x -direction is normal to the surface of the sample,
the incident microwave magnetic field h is chosen in the ^-direction, and the applied
static magnetic field it is in the z -direction.
From Eqs. 2.2.1 and 2.2.2 we can write:
4n
„ 1 d(se. ) «
z — —=
cre.z H
—z
dx
c
'
c dt
- y ^ - = - - ^ ( h yy + 47anyy )
&x
c ot
(2.2.j )
(2.2.4)
I f we take for the field a space and time dependence of the conventional form e ‘(fo: fl*),
then Eqs. 2.2.3 and 2.2.4 become:
... , 4k
„ .
ikh z - — ae z - i
c
- ike.y = /' —
\ cJ
se.z
- 4 nmy)y .
Finally, we can write these two equations in the form:
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(2.2.5)
(2.2.6)
( s
. <r)
hy = - -----4-I — e .
KAn(Qj
\^Ak
coj
r kc "i
kc \
e. =
\4 kc o J '
1 .
h -w
Arc
(2.2.7)
( 2 .2 . 8 )
2.3 Equation of motion for the magnetization
The equation o f motion for the magnetization M is given by [3,4,5,6]
dM
a
= —y ( M x H eff) + Damping Term
(2.3.1)
where y = ge is the magnetomechanicai ratio, g is the spectroscopic splitting factor,
2me
and H ep is the effective magnetic field inside the ferromagnetic material. In general,
H eg- can be written as:
H eff = H a + h d + han + hrf + h ex+ h me
(2.3.2)
H a = applied magnetic field
hd = demagnetizing field
han = anisotropy field
= magnetic field associated with waves
ha = exchange effective field
hme = magnetoelastic field.
We approximate our sample as an infinite slab o f finite thickness, so the demagnetizing
field is in the x -direction is hd = —AjanJc. The demagnetizing factors in they and
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8
z -directions are equal to zero [7],
The anisotropy fields, given in reference [8], are;
han.x = ~ (Cimx + C2my + C3m.) / M 0
(2.3.3 a)
Kn.y =
+ C5my +C 6m .)/ M 0
(2.3.3b)
han: = - ( C 7mr + C%
my + C9m; ) / M 0
(2.3.3c)
C, = (2 K x/ M 0) { l - a ? ) + { 2K 2 / M 0) cc;a;
(2.3.4a)
where
C2 = C4 = 2 a ,a 2[2JS:i / M 0 + ( 2 £ 2 /M 0) a ; ]
(2.3.4b)
C3 = C7 = 2a, a 3[2£, /M 0 + (2K Z/M 0 ) a ; ]
(2.3 4c)
C5 = (2K x/ M 0)(l- a 2)+ {2K 2 / M 0) a,2a 2
(2.3.4d)
C6 = C S = 2a 2a 3 [2K x/ M 0 + (2K 2 / M 0)a,2]
(2.3.4e)
C9 = (2K x/ M 0) ( l- a \ )+ (2K 2 / M 0) a ~ a ;
(2.3.4f)
In Eqs. 2.3.4a-f, AT, and K 2 are the first and second order anisotropy constants and
a,,
1
. . .
.
,
m_
m
a~ are the direction cosines, where a, = — — = 0 ,a , = ------ --0
“ J
' M0
M0
and
a 3 = — ^ = 1. See Fig .2. In the H nL configuration the static magnetizationM0 is
M 0
parallel to the z -axis. We inserted these values into Eqs. 2.3.4a-2.3.4f, and obtained
h
_ 2-^1 m x -
M0
^ 2 K xm y^
^
'
"’
M, y '
We also know that the exchange field is of the form [9]
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(2 3 5 )
1
'
9
h
“
(2.3.6)
A /' 6tr:
where A is the exchange stiffness parameter.
J'COIO]
> x [100]
-
[001]
Fig. 2. Magnetization orientation with the direction cosinesa x, « , , a 3indicated,
a, = cos#, = 0, a , = cos#, = 0, a . = cos#5 = 1. Note that the crystal axes are assumed
to be aligned with thex, y, z coordinated system.
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10
From symmetry considerations alone the magnetoelastic energy [10,11] can be expressed
as a power series involving the a ,
U“
= Bx£ e„a,2 +
I=t
.
KJ
Keeping only the two lowest order terms yields
U me = Bx(eua f + ea a ; + e33a 3) + B 2(el2a xa 2 + e 23a 2a 3 + e3la 3a x)
(2.3.7)
where the et] are the strains and the Bt ’s are magnetoelastic coupling constants. The
Bt7s are related to Axxl and A100 by [6];
Here
B , = ~ ( C u + C a )Zm
( 2 .3 .8 )
S, = - 3 Q A , „ .
(2.3.9)
A = ~ is the saturation magnetostrictionor strain when the crystalis magnetized
and the strain is
measured along the direction o f themagnetization.The twoA 's in
Eqs. 2.3.8 and 2.3.9 refer to the directions [lOO] and [ i l l ] , respectively.
The CtJ are the conventional cubic elastic constants. The et] are strain components
defined by Kittel as [10]
g„ = ^ U ; for i=j and
(3c,
J
e
"
ac:
-i--------for i ^ j .
3ct
J
(2.3.10)
Here the ir ’s are the displacements o f the lattice along the i 1/1 direction. These
quantities are functions of space and time; e.g., ui =
.
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11
To get the hme, we assumed that the magetoelastic energy U me is o f the form
U me = - M - hme. We compare this to Eq. 2.3.7 and, after some manipulation, we
get
h.m e
M0 d x
x .
(2.3.11)
2.3.1 The damping terms
Now we introduce the damping term in the equation o f motion for the magnetization.
We know from experiments that when the microwave field is switched off the precession
o f M about the direction o f H eff dies out rather rapidly. This means there is some
process or processes which damp out the precession [12,13]. We consider first the
Landau-Lifschtz form.
2.3.2 The Landau-Lifschitz damping term
(2.3.12)
Landau and Lifschitz [9] introduced this term in 1935, where A has the dimension of
frequency and is an adjustable parameter indicating the degree o f damping in a magnetic
system. The damping term represents a torque exerted on the magnetization, which
tends to align M along H a.
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12
2.3.3 The Gilbert damping term
The Gilbert [14] damping term was introduced in 1955 and it is equivalent to adding
a fie ld
-(M x
) to the effective field H eff ; for a small departure from
M l
&
equilibrium the Landau-Lifschitz and Gilbert damping terms are equivalent.
2.3.4 The Bloch- Bloembergen damping term
_^y_
r,
~ ¥ -- ~ M o
Tx
(2.3.13)
This form of damping was introduced by Bloembergen [15] in 1966 as a
modification o f the Bloch equation for nuclear magnetic resonance. Here Tx is the
relaxation time for the component o f the magnetization along H eff, and Tz is the
transverse relaxation time, Tx is very short and irrelevant to our experiments. Tz is a
measure of the relaxation rate o f the transverse components o f M .
In this dissertation we use the Gilbert damping term to describe the relaxation of the
magnetization to its equilibrium position because the magnitude o f the magnetization is
conserved and the damping term is always aligned perpendicular to ^
< ~ . Using the
dt
Gilbert damping term for the H „ L configuration, the equation of motion (2.3.1) can be
written as:
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13
8M
- - y M
dt
2K, ( »
.
2A d 2M
,Wxr + /ny.yJ+A<fX + Ay.y + ---M l dx2
M:
x
G 8 M '1
B, r?z/z ,
?---- — X M 0 8x
y 'M l d t J
(2.3.13)
Taking the cross product and keeping only terms to first order in the small, time
dependent quantities, equations for the two transverse components of M result. Having
the applied magnetic field pointing along the z -axis, and the oscillating magnetic field
along the _y-axis, and having hd - -A7anx, these equations become:
a
y dt
y
y zM 0 d t
M 0 dx2
Mn
1 dm
a- x * \
- T >r- = - \(Hli a +^7tM0)mx
- 2K^m „ ,G
M0
y 'M 0 d t
y dt
(2.3.14)
oy
(2 .3.15)
M 0 d x~
dx
As usual we use the time and space dependence of mx, m and u. o f the form
e,{kx a<). These two equations reduce to:
IC O
• m ,
-
r
1(0
my =
h
-
Gico
2Ak2
• +
y -M o
2 K.
-
M ,0
M0
H „ +4n M n-
Y
Gico
my - h y M 0 and
2Ak2
■+
y -M 0
2K,
-
Mq
M0
mr + ikB^u.
(2.3.16)
(2.3.17)
In the absence o f any magnetoelastic coupling, this is the form o f an ellipse for
coordinates m and
.
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14
2.4 Equation o f motion for the lattice
We consider an elastic continuum, which is assumed to be anisotropic and the
crystal structure is cubic. Then the equation o f motion for the lattice, which is merely a
restatement o f Newton’s second law, is of the form[16].
d t~
dxk
i, k = 1,2,3
(2.4.1)
Where p is the mass density o f material and the Tlk are the components o f the total
stress tensor due to the forces acting on the material. The total stress tensor [16] can be
written as:
(2.4.2)
The magnetoelastic contribution to the stress tensor Tike is given for a ferromagnetic
material [17] by
(2.4.3)
where U meis from Eq. 2.3.7. The
are the elastic stresses in the cubic crystal
lattice [18], For our assumed configuration with the applied the magnetic field in
thez -direction, the [001 ] axis parallel z , and the propagation of waves in the x-direction
ox
(2.4.4)
The magnetic components mx and my are coupled with u. ; also M 0and H a do not
change. Further, ux = hy = uy = 0 . Then equation 2.4.1 reduces to
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15
p
d~u_
d 2u.
B, d m x
t~ —C A
- J
z?x 2 M 0
x
d t2
(2.4.5)
w. and mx have the space and time dependence as stated earlier. This equation then
reduces to:
( - t y 2p + C Mk z}u_
Mn
ikmx = 0
(2.4.6)
2.5. The secular equation
For convenience, we now gather together all the equations that lead us to secular
equation. These are:
H
a+
4 k
ICO
------- m x
r
^4kco j
-h ..
4 k
M
q
0
r rr
G .
2A .
2K X^
H „ -----;----- ico + ------ k +
my -h yM ^ = 0
M,o y
a r 2M 0
hy +
f e
. c rA
h i —
4 k
+
2 A k2 2K.
1(0 m v + i-kid
n
Bm . = 0
+ — L mr h
2; i / ico + ---------M
0
M
0
j
Y
Y M o
oo
e=0
r kc ^
e. + my = 0
(2.5.1)
(2.5.2)
(2.5.3)
(2.5.4)
\_ 4 k c o j
i-co2p +Cj,Ar2W. + ik - ^ - m r = 0 .
V
H
44 ' -
(2.5.5)
From these equations we can set up in the matrix o f the coefficient o f the independent
variables mx, my, ex, hy and u. . This system of equations has non-trivial solutions
only if the determinant o f the matrix o f coefficients is zero, that is,
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16
ico
V+2M 2
K
ico
7
2A k 2
Z + --------
0
0
-M 0
ikB2
0
M0
r
0
0
1
ikY
kc
X
0
where
0
Y
0
kc
4 71CO
1
4 KCO
0
4k
0
0
0
- a)2p + CMk 2
V - H -\- 4;zM0 -
(2.5 .7)
r K
0
M>
X =— +— .
4;r oj
(2.5.9)
This leads to a secular equation which is quartic in&2, namely
r i ^ + r ^ 2)3 +<d(*2)2 + 'p (* 2) + n = o
where the constant coefficients n , T,d>,
r _
A 2c2p t
4M 0k 2
A B ;c 2
8M 2k 2c o 2
A c C
and Q are: n = ------ '44, ,
A c -C J f ^ A 2C ^ X
8M 0k 2c o 2
k ~
8M 0k ~
2M q T C
(2.5.11)
An ~M Qco"
M
A c2C ^ Z
(2 5 12)
8k 2a ) 2
0k
0 = _ £ ^ ^ + A c2 p} ^ _ A B £ ^ + 2 a M qc ^ x +
16/ ~
( 2 .5 . 10)
^
^
2M
qk
+^ ^ ^ - - ^ - ^ M 0k
8k ~
(2.5.13)
B \ c2Z
\
6M
qk 2c o 2
c - C jn
16k 2c o 2
A C ^X Z
2k
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17
...
c2pea2
¥ =—
16—
/ 7i—:—
1
-
,
■>„ A pV co rX C^co2X
X + M aC
= -;----0 „44V X + 2A M 0p
OH co'X + — ----------------- Ay
2K
(2.5.14)
c 2p VZ
16 k
and
B \X Z ; CJVXZ | Ap co2XZ
Ak M 0
4t t M q
Q. = M 0pVco2X - pO} * + ? V a
4 y 'K
4k
2k
.
(2.5.15)
Solving this equation leads to four different pairs o f values for k which correspond to
an electromagnetic wave (kem), two spin waves iksX,ksl), and a sound wave {ksd); each
pair consists of one wave propagating in the positive x-direction, and one in the negative
x-direction. Equation (2.5.10) and certain generalizations of this configuration are solved
by a Fortran program called TRANS_7 written by Dr. Graeme Dewar.
2.6 The boundary value problem for the
H UL
geometry
In this section we set and solve the boundary value problem for the H„^ geometry.
Applying the standard procedure of integrating equations (2.5.3), (2.5.4) around a
surface enclosed by a loop traversing the two different media [19], it is found that the
tangential components o f e and h must be continuous across the interface. At the
surface o f the sample the spins are assumed to be [4,20,21,22] unpinned, i.e., the
derivative of the magnetization is continuous at the surface,
dx
- 0 . Further, the surfaces
are traction free [ 20 ,21 , 22 ]; in essence, there is no surface stress originating in the
vacuum. In general, the boundary conditions are:
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18
(i) the continuity o f tangential e :
em = ef"
at x =0 and x=d ,
(2 .6 . 1)
(ii) the continuity o f the tangential h
h‘vn = h°ut
( 2 .6 . 2 )
at x =0 and x=d ,
(iii) boundary conditions for the magnetization
cmx
=
dx
dmx
=
dx
0
0
dmy
dx
r= 0
dmy
=
(2.6.3)
0 ,
(2.6.4)
= 0.
dx
(iv) traction free surfaces,
dii.
B.,
Cj, — —h— —m. lr=0 = o,
^
dx
M0
j
du_
B-,
Cj, — -h ---- —m.
v
dx M 0
j
=
0
(2.6.5)
.
( 2 . 6 .6 )
We can use these 10 boundary conditions to solve for all waves in terms o f the
incident wave, since we have four pairs of waves within the sample slab and reflected
and transmitted waves. Outside the slab, at the front surface, there are two waves: k() for
the incident microwave wave vector and kref for the reflected microwaves wave
vector. At the back surface kt is the wave vector o f the transmitted wave. By applying
the boundary conditions to the front surface o f the sample where x= 0 , the tangential
components of e and h fields are:
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19
(e 0 + e r e f )
fa
+ h r e f)
L
= C
Uq
=
+ < , -h e ' + e;, + e ' + e;2 + e s\ ^
+
K m
+
K d
K d
+
+K
l
+
K
+
K
l
+
K
l
(2.6.7)
Lq
( 2 .6 .8)
Similarly, the boundary conditions for the magnetization at the front surface are:
0=' W
-
+ ‘ks<imx.sd ~ ik,dml.sd
(2.6.9)
+
0
‘ k s 2 m x .s 2 -
i k s2m l. s 2
+ i k s d m l. s d ~ i k , d m y .s d
=
( 2 .6 . 10)
+ l ’K
K
. s l
~
i k s l m ~y.sl + i k s 2 m y . s 2 ~
i k : 2 m ~y.s2
and the boundary condition for the traction free surface is:
o = c M [ kemii;m- i k emu;m+ ik sdu;d - i k sdu;d + /*„ » ;,
+ / * , 2«;2
(2.6 . 11)
B-, r ^
,.
+
+ w— +/wx.^ + OTr.*/ +™ „i
-
-1
+mxs2 +mxs2J.
For the rear surface o f the sample, at x = d, the tangential components of the e and h
fields are:
- ik . , d
e~
e~,k,md +e~em e~‘ktmd + es\ae " ktdd + e~de~lkaid
+e~e~,k,,d
+e~.e~,k,'d
em
sa
ji
(2.6 . 12)
+ ex2e rdc,zd + e~2e ~,k’zd =
et
nkmd , u- „-<k^,d ,
^ik^d , l.~ ~~>k*id , k
- , U~^-'k,id
K y ' kmd ^ K f i " ^ +h;de ^ d +h;de -k«d +h;xe ^ ' d + K K "
(2.6.13)
h ; 2e ^ - d + h ; 2e - ^ d =
h, .
For the magnetization we have
iK jK ,^ -iK jn -^ e ^
-ik jr i^ ^
(2.6.14)
+ * , ! < , . e**'1' - i K ^ e - ^ d +i ks2m l 2e ^ d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
=0
and for the traction free surface we have;
M
n
+ ™ : ,^ ',k!ld + ^ :.s ^ 'JC’>d + k , i e ~ ,k,'d +™:.sie~*'i d
)
(2.6.16)
For convenience, we write the fields inside the medium in terms o f generalized
impedances Z as follows:
f e { e . , mx , mv , u. }
= Z~r (± k)h'
From Eq. 2.5.3,
kc
e. = — Anco -h„
—
Ak
+ /
(2.6.17)
—
co
S
O
For good conductors we have — « — ; hence
An
co
ick ,
e-- =~Ana
a
hy
(2.6.18)
Solving Eq. 2.5.4 by using Eq. 2.6.18, we then get
f
- r 2
'
ik
c2 \
1 h---------my = ~ An
l T v Anaco y
Also, by using the expression for m from Eq. 2.6.19 in Eq. 2.5.2, we get
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.6.19)
21
m, =
iy
4KCO
4k M 0 — 1+
i k 2c 2 V
4Tract)
H -
icoG
y 2M 0
2A , ,
2K x
k~ + - 1
M0
M oy
( 2 .6 .20 )
Finally the expression for the u. from equation (2.5.5) is
ykB z
u. =
4k M 0 -
'
1+
ik 2c2 V
-
4KcaMQ( C 44k 2 —<o2p )
4KCJCOJ
H ,a -
icoG
*>
yM 0
2Ak2
2 K.
M0
Mn
( 2 . 6 .21 )
From these four equations, we can exhibit the impedances for forward propagating
waves:
z ; (k) =
ick
( 2 .6.22 )
4kct
iy
z ; (k) =
'
ik 2c2 Y TT
4 k M 0 — 1+
\H a
4 KCO
4 k <jo.)
Z ’ {k)=-
■t 2
ik
c2
4k ^
(2.6.23)
(2.6.24)
1 + ---------
4K<TCOJ
kyBz
z ;, (*)=
icoG 2A . ,
2K x
:----- + ------k - + --M,
y~ M 0 M 0
(co2p - C Mk 2)4 K 6 )M Q
4k M 0 -
'
ik 2c 2 N
I + -------v
4KOCO Ha
icoG
2A .
2K
+ — k-+y 2M 0 M
M, o y
(2.6.25)
Now we can apply these generalized impedances to the boundary conditions at the front
surface. The wave propagating in the positive direction is related to the reflected wave
propagating in the negative directions by k~ = - k + . Thus we see that the general
impedances for forward and backward propagating waves are related by:
z ; ( - * ) = - z ; (k ), z ; ( - k ) = z ; ( * ) , z_
( - k) = z ;
(k).
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( 2 .6 .26 )
22
Hence Eqs. 2.6.1 through 2.6.6 become:
*«+v=Z k - K )
(2.6.27)
a
(2.6.28)
- k „ = y . z . ( k )( k - k )
k
a
(2.6.29)
° = T . ‘^ z . . k ) k - K )
a
0=Z.'KZ-(K)k-K)
(2.6.30)
° =Z
(2.6.31)
c j k . z , ( K ) { h : - K ) + j j - z mi( K ) k - k )
Where a e {em,sd,si ,sz } andka is the wave propagation constant which corresponds
to the index a .
At the rear surface
(2.6.32)
+ K ^ - d)= > ’,
(2.6.33)
' L z . k ) k ‘ ”, -‘‘ + K e - * - d) = K
a
o
(2.6.34)
) k ^ ‘ d - K ^ ' d) = o
(2.6.35)
Y . ik. z „ k ) k e * - d
a
Z * . z ., k
Z
B,
( C J k ,Z ,( .K J + - ^ - Z , ^ ) W ^ ‘k-d - K e ~ “-d)
M 0
=
0
By using these equations we can calculate numerically the transmission and
reflection coefficients
—
and —— .
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(2.6.36)
23
Now we are going to relate these coefficients to the ratio of the microwave power
entering the receiver waveguide to the input power to the transmitter cavity. The fields in
the waveguide and the cavities are different. In particular, fields in the cavity
are ~
time fields in the waveguide, where Q is the cavity quality factor defined
below. We will now identify all fields in the waveguide in the cavities with help of
Fig. 3.
Consider first the waveguide-cavity transmitter section. In the transmitter waveguide
ht and et are the magnetic and electric field amplitudes, respectively, o f microwaves
coming from the dielectric resonant oscillator (DRO) as they propagate in the waveguide.
In the transmitter cavity ha is the maximum microwave magnetic field, h; , e, are the
magnetic and electric field amplitudes o f microwaves incident on the sample,
respectively, andhR,eRare the reflected microwave magnetic and electric field
amplitudes. Next, consider the receiving cavity-waveguide section. Here ht,et are the
magnetic and electric fields amplitudes of microwave in the receiver cavity and, hr,er
are the magnetic and electric fields amplitudes of microwaves in the receiver waveguide.
According to Cochran et al [23] the fields in the respective waveguides satisfy the
following equation.
2
(V
UJ
f 1 1 (V |
U z oJ A ,
where /? is defined as
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24
Transmitter cavity
Receiving cavity
Waveguide
Coupling hole
Sample
Fig. 3. Fields inside the cavity and waveguide section.
P =
K Ac a v ity
Q
(2.6.38)
^ s a m p le
and where ko and k are microwave propagation constants in free space and in the
waveguides, respectively. Z 0 is the impedance of free space.
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25
The power ratio for the cavities PRcm is defined as;
PR~ =
f O
2_ Q2
A tc2
j
U
( a s a m p le ^ ( K \
A
^
c a v ity
j
l O
6
f A- l
U J
(2.6.39)
where Asample is the area of the sample exposed to the microwaves, Acavt[y is the area of
cavity end wall. The inverse impedance is
=
Z 02
1- ' A '
(2.6.40)
KkoJ
where k is the microwave wavelength in free space and kc is the waveguide cutoff
wavelength.
We define the Q as
oL = — ,
“L
A /’
OuL = 20 L
(2.6.41 )
where 0 L, 0 uL are loaded and unloaded Q ’s, respectively.
The cavity quality factors, Q, were determined from the frequency shifts required to
detune the critically coupled cavities so that half the incident power was reflected, the
Af is the full width at half maximum o f the frequency resonance curve.
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CHAPTER IE
E XPER IM EN TA L APPARATUS AND SAMPLES
In this chapter we present a short description o f the experimental apparatus used for the
reflection and transmission studies of Terfenol-D and describe the physical properties of
the samples.
The microwave spectrometer used in these experiments can be divided in five different
sections:
A: Microwave generation
B: Calibration line
C: The signal line with the cavities
D: Reference line
E: Recording system
A block diagram o f the microwave circuitry [24] is given in Fig. 4. The
microwave generator (shown in the lower left hand comer o f the figure) is a MITEQ [25]
DRO-J-17000-SP Dielectric Resonant Oscillator which delivered approximately lOOmW
and was energized by a power supply built by Chris Reese [26]. This power supply was
also used for the M ITE Q amplifier in the recording system. The microwaves are
generated at a frequency o f 16.95GHz . Following the line from the DRO on the
diagram, we see that this power passed through a 24dB isolator ( necessary to protect the
microwave source against reflected microwave power) and then was split into two parts.
26
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27
The major portion o f the power continued to the transmitter cavity while the remaining
10% of the power continued through the reference line. The microwaves were fed into a
slightly under-coupled transmitter cavity. The signal transmitted through the sample
passed on through the receiving cavity to an ARRA [27] 62-115-LD chopper operating at
2.304 M H z and was fed, via a M ITEQ AMF-65-1821-30 Amplifier, into an M 1216M®
mixer where it was mixed with the signal from the reference line.
The difference signal from the mixer was amplified as it passed through an IF-preamplifier, operating at 2.3MHz, and an EF-amplifier before it reached the PAR [28]
phase sensitive detector [29], Two CP 6282-OT 35 dB isolators between the mixer and
amplifier, as well as the two isolators on either side of the chopper, buffered the amplifier
and the chopper.
The reference signal (local oscillator) for the balanced mixer was tapped directly from
the DRO output by a 10 dB directional coupler. Its phase could be altered through 360"
by a Hewlett-Packard [30] P 885A precision phase shifter that was buffered by a
22dB U T D [31] isolator CT-6437-OT. The output o f the mixer is proportional to
(3.0.1)
where the E^ o f interest is very small, hence
(3.0.2)
Here E hc is the reference signal (local oscillator), E ag is the microwave signal,
and 9 is the phase angle between the local oscillator and microwave field.
It is E ag
which is modulated at the 2.3 M H z EF while E loc is fixed. The phase shifter allowed
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28
alteration o f & . The purpose of the calibration line was to calibrate the gain of the
receiver. It consisted o f a phase shifter, a Hewlett packard [32] P 382A variable
attenuator (0 to 70dB), a modulator, and 30dB fixed attenuator.
The two switches in the signal line were used to tune the receiver cavity. The reflected
signal from either the transmitter or receiver cavity were tapped by a 20dB directional
coupler into a microwave diode which was connected to an oscilloscope. A variable
attenuator between the two switches provided additional isolation between the transmitter
and receiver portions of the apparatus when the switches were in the normal position.
For the reflection experiment a small oscillating magnetic field was added to the static
field by a pair o f modulating coils attached to the pole pieces of the magnet. These coils
were driven at 90 Hz from the reference channel o f a phase sensitive detector ( PSD).
See Fig. 4. The 90 Hz reference signal was fed through an audio-amplifier to the
modulating coils. The impedance of the output stage of this audio-amplifier was matched
to the impedance o f the modulating coils. The strength of the actual magnetic field
produced at this frequency at the site o f the specimen is relatively small and hence the
magnetic field modulation was used to sample the gradient of the absorption line rather
than sweep through it. The way in which this 90Hz modulation is used in measuring the
ferromagnetic resonance absorption as indicated in Fig. 5. The signal obtained from
the spectrometer is proportional to the first derivative o f the absorption line with respect
to the magnetic field. The absorption spectrum appears at the output of the microwave
detector in the form o f a periodic 90 Hz waveform on a dc offset. The periodic signal is
amplified by the narrow band amplifier of the phase sensitive detector. The waveforms
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29
obtained in this experiment are shown in Fig. 5.
Isolator 30 dB
Isolator >45 dB
20 dB
I
Amplifier 35dB Gain
Isolator 35dB
1
Chopper
Attenuator
30 dB
S ig n a l lin e
Isolator 35dB
Mixer H IF Pre-amp
Chopper
to
IF Amplifier
j
Phase
shifter
f
Y' I
Calibration line
Isolater 22dB
Variable
Attenuator
R e fe re n c e lin e
Variable
Attenuator
0-70 dB
Phase
shifter
Microwave Source
Magnet
Magnet
Isolator 24 dB
10 dB
Frequency
Meter
i
20 dB
Modulating
Digitizer
and
Recorder
Audio
Amplifier
Fig. 4. Block diagram o f the apparatus for the reflection and transmission
experiment
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Time/IVbgTeticfiekJ
IVfegneticField
*
(b)
(a)
▲
i
rvfegteticfield
(C)
Fig. 5. Waveforms illustrating operation of the modulation spectrometer used in
reflection microwave experiments
(a) The way in which the first derivative output is produced by field modulation,
(b) The first derivative before phase sensitive detector as the magnetic field is swept,
(c) The pen recorder trace after phase sensitive detection.
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31
3.1 Static magnetic field
The static magnetic field was produced by a water cooled Varian [26] V-H F 3401
electromagnet, having a VFR 2601 M K I I Fieldial power supply. The DC magnetic
field can be swept over ranges up to 0 to 13 kOe (or from 13 to 0 kOe) over periods
from a few seconds to two hours. To measure the magnetic field we used an
F.W.Bell 9550 gaussmeter [33], We put this gaussmeter’s Hall probe between the
magnetic poles and near the sample.
3.2 AC modulating magnetic coils
The ac magnetic field was produced by modulating coils using a sinusoidal current of
frequency 90 H z from the reference signal o f the phase sensitive detector. The sinusoidal
current was amplified by a 5W audio-amplifier before its reached the modulating coils
attached to magnet poles. Because the current drive capability o f the PSD was very small
we amplified it before it reached the modulating coils, otherwise our sample would not
experience sufficient modulation of the internal magnetic field.
We used an LM384 5W [34] amplifier and connected as shown in Fig. 6 . Vm was
connected to the PSD and the output was connected to modulating coils.
The magnitude of the modulation at the position o f the sample was calibrated in the
following manner. We connected a pickup coil to the digital voltmeter and then placed it
between the two magnet poles. The pickup coil was used to measure the induced voltage
which was produced by the modulation coils. We took data by increasing the dc
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32
magnetic field from 0.1 to 12.7 kOe while recording the root mean square induced
voltage produced by the modulation coils.
+22V
0.1 pF
Modulation
coils
LM384
I Ok
3,4,5
7,10
+
11,12 Z Z I 5p.F
2.7
0.1 pF
Figure 6 . Audio-Amplifier.
By using a curve fitting calculation we saw that the induced voltage could be adequately
described by a polynomial function of fourth order in magnetic field. See Fig. 7.
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33
VRMS= l + a H - b H 2 + c H 3 + d H \
(3.2.1)
Where a, b, c and d are constants. See Table I.
Table I
a
7.35 x 10”6 V/Oe
±0.0066 V/Oe
b
- 6.84 xlO ' 9 V/Oe 2
±0.0617 V/O q1
c
1.698 xlO "13 V/Oe 3
± 0.0142 V/Oe 3
d
2.889 x l 0 ~18V/Oe 4
± 0.0254 V/Oe 4
where Vms stands for the root mean square voltage produced by the modulation coils
and H is the static magnetic field produce by the electromagnet. Because the induced
voltage produced by the modulation coils was not constant we had to divide the
experiment data obtained in reflection by equation (3.2.1).
1.1
•o
a
1—
UJ
S
^
&
i
i
E
E
0-9 j_
h
nJ.tJ
o r __
V
,
:
A.
. ___
__
.............................................
J
..................................................
.................
n 7
t-..............
U.
1 tr
F
Polynomi ai fit ............... .................. X .............. ...........
0.6 t
----* ---- Experime nt
*X
0.5 ^
...........
............................... - - V F
i
...
* *
0.4 r - ; : - 1
0
20 DO
40 DO
6000
8 0 00
10000
12000
i
H[Oe]
:
•;
:
^
14000
Fig. 7. Root mean square induced voltage o f the ac modulation coils vs magnetic field.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
3.3. Microwave cavities
The transmitting and receiving cavities were constructed to be as nearly identical as
possible. They had the function o f matching the impedance o f the sample to the
waveguide. The cavities was designed to operate at 16.95GHz and supported
the TEl02 mode. The sample was sandwiched between the two cavities and, to avoid
microwave leakage around the sample, we used an indium ring to form an electrical seal
of the sample to the sample holder and o f the sample holder to the cavities. The screws
securing the sample holder to the cavities were tightened very hard. The indium ring
sealing the sample to the sample holder was tightened gently to make sure that we did not
shatter the sample. I repeated the tightening procedure several times since the indium
flowed around the sample and as it stopped the microwaves from leaking around the
sample or around the sample holder. See Fig. 8 .
3.4. Cavity tuning
The cavities must resonate at the frequency o f 16.95 GHz. To obtain this resonant
frequency we had to tune the cavities by extending a Teflon rod into the cavities to
increase their dielectric volumes. The diameter of the tuning hole is 0.63 cm and the
length o f the Teflon tuning rod is 15 cm, of which only «0.63 cm actually
protrudes into the cavity.
The Teflon screws could alter the resonance frequency by 200MHz per turn [35]
which is equivalent to pushing the Teflon rod into a cavity by 0.80 mm [35], Tuning was
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35
accomplished by turning the Teflon tuning rod until we obtained the correct resonance
frequency.
Cavity
Sealing ring
Aperture
Sample
Sample holder
plate
Cavity
block
Tuning hole with
extension pipe —
Fig. 8 . Arrangement o f the cavities and the sample holder plates
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36
3.5. Temperature control and measurement
The cavities were heated by an 5W resistor. The temperature of the cavities was
controlled by varying the power supplied to the heater by a Varitec [36] model WP-710
power supply. The maximum temperature I heated sample number one to was 33* C and
for sample number two the maximum was 50* C .
I drilled a hole 0.8cm in diameter and
1.5 cm deep into the cavity wall. This hole was used to support the thermometer with
which I read the temperature of the sample. The minimum temperature that I
operated at was room temperature.
3.6. The chemical composition o f Terfenol-D and its physical properties
The chemical composition o f Terfenol-D is Tb027DyOTiFe2. Other physical
properties use in the calculation are given in the reference [ 6 ] are shown in Table II.
3.7. X-rays of the samples
The samples were prepared by the Czociiralski method. This produced
polycrystalline boules with individual crystals for which 0.5mm X 5mm X 3mm was a
typical size. The crystals grew [37] preferentially along the [l 12] direction
in twinned dendrite sheets with the[l 11] direction normal to the sheets. See Fig. 9. We
took X-rays of the samples using the back-reflection Laue method by placing a polaroid
camera at several sample-film distances. The Laue pattern for a single crystal consists of
a set o f diffraction spots on the film and the positions o f these spots depend on the
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37
orientation o f the crystal plane. By using a Greninger net we were able to identify the
orientation of the crystal planes.
Table II
The exchange stiffness
X = 9 . 0 x l 0 -7 erg/cm
AtcM = 10116G
The conductivity
cr= 1.497 x lO 16 1/s
The anisotropy constant
K x = - 6.00 x 103 erg/cm3
The magnetostriction in [ 1 11] direction
A,,, = 1.64 x 10”3
The magnetostriction in [100] direction
= 1.0 x 10-*
The anisotropy constant
K-, = -2 .0 0 xlO 6erg/cm3
The cubic elastic constant
C u = 1.41 x 1012erg/cm3
The magnetoelastic coupling constant
Bx = —1.4 x 108 erg/cm3
The cubic elastic constant
C ,2 = 6.48 x 10n erg/cm3
The magnetoelastic coupling constant
B2 = -2.3 x 109 erg/cm3
The cubic elastic constant
C A4 = 4.87 x 10u erg/cm3
g factor = 2.2
The mass density
p - 9250 kg/cm3
For sample # 1 the direction in the x-direction was [120], the ^-direction was [111],
and the z -direction was [1 10], For sample # 2 the direction o f the z-axis was in the
magnetically easy direction [ 111].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
[i 12 ]
M
a
[ 111 ]
[n o ]
Figure 9. Dendritic platelets in TbQ21DyoriFe2
3.8. Data acquisition
The output o f the phase sensitive detector was applied to the _y-axis o f an x - y chart
recorder which corresponded to the voltage of the reflected or transmitted signals. The x axis was driven by the output voltage of the Fieldial magnet controller which
corresponded to the magnetic field.
Simultaneously, the output o f the phase sensitive
detector and gaussmeter were fed to a computer through an EEEE -488 [38] interface
board. The data were recorded and stored in the computer for further analysis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER IV
EXPER IM EN TA L RESULTS AND CONCLUSIONS
In this chapter we present and analyze experimental results for the reflection and
transmission measurements on Terfenol-D. Where appropriate, comparisons are made to
the theoretical model presented in Chapter H.
4.1 Reflection measurements
We performed experiments by recording the signal reflected from the front surface of
the sample. The instrument offsets were subtracted from the experimental data and then
the data were divided by the Eq. 3.2.1.
4.2a Typical reflection results
For the first sample we performed more than 100 reflection experiments. Most of
the data had poor reproducibility; some o f these data are presented here to illustrate
typical measurements. See Figs. 10-12. The data shown here are divided into three
groups, each group represents data taken over a short period of time, i.e., on the same
day.
These data show that the run to run reproducibility is fair to good. However, over
long periods of time during which the sample was cycled in temperature and magnetic
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
field, the reproducibility is quite poor.
I speculate that this evolution o f the shape of the reflected signal is due to the large,
random internal magnetic and strain fields which are know to exist in this material [38],
As the sample is cycled in magnetic field the magnetization o f the sample does not
evolve through the same set of microstates as it becomes magnetized with increasing H
and then returns to a nearly demagnetized state when H decreases. The large
magnetostriction o f Terfenol-D makes the situation worse by causing the varying state of
strain of the several crystallites in the sample to influence the magnetization.
Cycling the temperature further complicates the situation by causing the
magnetocrystalline anisotropy to vary: Terfenol-D is designed to have nearly zero
anisotropy near room temperature and relative changes in the anisotropy constants are
large even for small temperature changes. These anisotropy constants play a large role in
determining the direction of the magnetization within each crystal o f the multicrystal
sample.
It is clear from Figs. 10-12, that the FM R absorption, if present, is enormously
broadened. This is to be expected for our multicrystal sample because each crystal has
its own FM R field; the data represent the sum o f the absorptions for several crystals.
Also, the magnetization in each crystal is generally not parallel to the external magnetic
field and, as the magnetic field is swept, the reorientation of the magnetization broadens
the FM R absorption line for each crystal.
The evolution o f the reflection signal in Figs. 10-12 suggests that the sample must be
"trained" with many cycles of magnetic field sweeps before the reflection results become
Reproduced with permission of the copyright owner. Further re p ro d u c tio n prohibited without permission.
41
somewhat reproducible. Indeed, the results described below were obtained after just such
a training period.
8
CD
TJ
6
Run#1
Run#3
Run#5
o.
E
CD
4
C
o
a
2
jd
£
o
0
<13
•2
>
CD
>
ai
-4 hV--
Q
-6
0
2000
4000 Hj-Qe] 6000
8000
10000
Fig. 10. Derivative o f reflection amplitude vs magnetic field at T= 20' C for sample#!.
Every 30th data point is represented by a symbol. The sensitivity o f the PSD w as0.2//F
and the time constant r = 3s for run # 1 and # 3 and r = 10s for run # 5. The sweep rate
was 0.4 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
0)
T3
13
"q.
E
CO
c
o
o
JD
'S
>«—
0
4
'— Run#7
2
s— Run#8
'— Run#9
0
1
IL .
•2
0
Q
-4
0
2000
4000 Hj-0 e j 6000
8000
10000
Fig. 11. Derivative o f reflection amplitude vs magnetic field at T=25" C for sample # I
Every 30th data point is represented by a symbol. The sensitivity o f the PSD was 0.2 fiV,
the time constant x = 10s, and the sweep rates were 0.2 kG/ min, 0.4 kG/min and
0.1 kG/min for runs # 7, 8, 9 respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UJ
<D
>
CO
>
'l .
CD
-10 r
Run#12 at T=22 C
Run#14 at T=28 C
Run#15 at T=28 C
N>\
.
V *
\y
<C-
a
-12
0
2000
4000 H[Oe] 60 00
8000
Fig. 12. Derivative of reflection amplitude vs magnetic field at T =22*C and
T = 28' C for sample # I. Every 30th data point is represented by a symbol. The
sensitivity o f the PSD was 2/uV, the time constant r = 10s, and the sweep rates were
1 kG/min, 0.2 kG/min and 0.4 kG/min for runs # 12,14,15 respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10000
44
4.2b FMR with M parallel to the [001] axis
The sample used to perform the measurements shown in Figs. 10-12 was found from
X-ray measurements to have one relatively large crystallite orientated with the [001]
direction within 1* o f the external magnetic field.
After performing many experiments, we were able to obtain data with a good FMR
absorption profile with M 0 accurately aligned with H 0 and both nearly parallel to the
[001] direction. This is a magnetically hard direction and accounts for FM R at such a
high magnetic field.
X-rays indicate that the [120] direction o f the crystallite was perpendicular to the
sample plane. Below we analyze our FM R data using a computer calculation based on the
theory presented in Chapter II; that theory assumed that the [100] axis was perpendicular
to the sample plane. Ferromagnetic resonance is dominated by the electromagnetic wave
which propagates into the sample; this wave is only weakly influenced by the crystal
orientation. On the other hand, the total field the magnetization experiences includes the
anisotropy field which is critically dependent on the orientation o f the crystal axes. Thus
the FM R absorption calculation is nearly the same for any direction o f the wave
propagation provided that the magnetization is along a [001] axis and the comparisons o f
the theory to experiment below are valid.
The data shown in Figs. 13-15 were taken over a small range o f temperature near
room temperature. Basic features o f the experimental data are well described by the
calculation using fitting two parameters: the anisotropy constant and the Gilbert damping
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
parameter. Note, however, that there are some secondary maxima and minima in the
experimental data. Since the crystals are twinned this could represent FM R absorption in
the parent that is somewhat different than in the daughter. Another possibility is that there
exist different states o f strain in the parent-daughter twins. Another possibility is that
multiple reflections o f the electromagnetic wave at twin boundaries give rise to
dimensional resonances, i.e., standing waves at certain wavelengths. These secondary
maxima and minima are not present in the calculation.(Some numerical noise, due to
numerical differentiation o f the absorption lineshape , is apparent in the calculation).
15
<d
3
I
1
'
'
j
i
i
:
i
1
i
'
;
1
:
1
r
-jo
t- —
3
— Experiment, R u n # 21
Theory
Q_
£■
E
CO
5 F ....................->....................... ...............
.2
o
a?
P
<D
f-
■
I /
b
J ....." ' 7' —
0
-5
>
-I
<D
Q
-10
\ -A i- ■■
li3
-15
4000
6000
8000 .
,
H[Oe]
10000
12000
Fig. 13. Derivative o f the reflection amplitude vs magnetic fields at T= 20* C
for sample # 1. The unit o f the derivative o f reflection amplitude is arbitrary.
Every 30th data /calculation point is represented by a symbol. The calculation has
been normalized to the experimental data . The sensitivity o f the PSD was 10 juV,
the time constant r = 10s, and the sweep rate was 0.5 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14000
46
o
T3
_2
Q.
E
(a
c
o
"o
J32
‘0
6
4
Experiment. Run# 23
Theory
2
0
r ii
-2
P / -
*v
-4
I
»
/
■r /j ‘
-
j '
•"w i
5 ^ -
>
0
Q
■
-6 h- -8 L .
4000
6000
8000 ,,r~ . 10000
H[0e]
12000
Fig. 14. Derivative of reflection amplitude vs magnetic field at T=29*C
for sample # 1. The unit o f the derivative o f reflection amplitude is arbitrary.
Every 30th data /calculation point is represented by a symbol. The calculation has
been normalized to the experimental data. The sensitivity of the PSD was \0juV,
the time constant r = 105, and the sweep rate was 0.5 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14000
47
10
a>
3
~
Q.
E
to
c
o
*=
o
<D
>♦—
9)
•a
5 b
Experiment. Run# 26
— Theory
0
-5 <D
a
-10
4000
6000
8000 H [0 e ]10000
12000
Fig. 15. Derivative o f reflection amplitude vs magnetic field at T = 2 4 *C
for sample # 1. The unit o f the derivative of reflection amplitude is arbitrary.
Every 30th data /calculation point is represented by a symbol. The calculation has
been normalized to the experimental data. The sensitivity of the PSD was 10jjV ,
the time constant r = IO j, and the sweep rate was 0.5 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14000
48
Finally we present the data for the case in which the sample interacted with a large
magnetic field for 6 hours before measuring the microwave reflection. See Fig. 16. The
magnetic field was swept from 0.1 kOe to 12.7 kOe for these measurements. Note that
the magnetization M 0 is not quite aligned with H 0 and both are not quite parallel to
the [001] direction.
0
0
•o
rj
o.
-2
E
to
c
o
CJ
_0
v*—
0
-6 r
0
>
Run# 48 at T=26 C
Run# 50 at T=32 C
Run# 52 at T=19 C
1l _
0
Q
-10
0
2000
4000
6000
8000
H[Oe]
10000
12000
14000
Fig. 16. Derivative o f reflection amplitude vs magnetic field at T = 1 9 'C ,2 6 ‘ C and
T=32*C for sample # 1 after the sample was left at 12.0 kOe for 6 hours. Every 30thdata
point is represented by a symbol. The sensitivity of the PSD was 20ju F , the time
constant r = 10s, and the sweep rate was 1 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
4.2c Experimental results for hysteresis
We also performed experiments to determine the hysteresis of Terfenol-D by
taking data while sweeping the magnetic fields either up or down at different
temperatures. See Figs. 17-18. From these experiments we observed hysteresis in the
field for FMR. By measuring the difference in magnetic field for several points on
identifiable features o f the experimental data, we obtained the average field within the
sample which opposed changes caused by sweeping the external magnetic field. This
hysteresis field was 1950± 5 0 0 e a t 26’ C and 2050± 50 0 e a t 32’ C.These
results agree very well with the hysteresis field of 2000 Oe given by reference [38],
4.2d Results for second sample
For the second sample we also performed reflection experiments. Because the
magnetically easy direction was nearly parallel to the magnetization we did not see
FMR. In this case FM R is expected at negative magnetic field for the anisotropy
constants determined below. Such fields are experimentally inaccessible. See Fig. 19.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
— Sweeping up, Run# 48
-a — Sweeping down, Run# 49
0
a)
"O
"5.
E
co
c
o
o
i
x
-4
>4—
0)
M—
-6
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1
4
:1
‘
: V
'
*
'
.
<
!
\
■ \
\ x *v '
v
-8 h
'l—
0
Q
-10
0
2000
4000
6000
8000
H[0e]
10000
12000
14000
Fig. 17. Derivative of reflection amplitude vs magnetic field at T = 26*C for sample # 1,
obtained by sweeping magnetic field up and down. Every 30th data point is represented
by a symbol. The sensitivity o f the PSD was 20juV, the time constant r = 10s, and
the sweep rate was 1 kG/min.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
51
4.3 Summary o f reflection experiments
From these experiments the FMR field, once the hysteresis field o f 2000 Oe is
subtracted is given in Table H I . These fields are derived from the best fitting
calculations shown in Figs. 13-15. From these we determine the anisotropy constant
and the Gilbert damping parameter .
Table EH
Gilbert damping
constant G
Temperature
Ferromagnetic
resonance (FMR)
Anisotropy constant
K x[erg/cm3]
21’ C
7425 ± 6 5 Oe
(-2.00 ± 0.025) x lO 6 9.375 x 108s~‘
24* C
7O5O±650e
(—1.96 ±0.025) x 106 1 .1 2 5 x 1 0 V 1
29* C
6690 ±65 Oe
(-1.71 ±0.025) xlO 6
1.50 x 1 0 V
The Gilbert damping parameters are 4 to 6 time larger than the largest known intrinsic
damping parameters. It is most likely that our damping parameter does not describe the
intrinsic magnetic viscosity o f Terfenol-D but instead indicates that the sample changes
in someway, perhaps with changing strain, as the magnetic fields is swept through FMR.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
-*— Sweeping up. Run# 50
* — Sweeping down, Run# 51
0
r
0
2000
4000
6000
8000
H[Oe]
10000
12000
14000
Fig. 18. Derivative o f reflection amplitude vs magnetic field at T = 32 *C for sample # 1
obtained by sweeping magnetic field up and down. Every 30th data point is represented
by a symbol. The sensitivity o f the PSD was 20 juV, the time constant r = 10.s\ and
sweep rate was I kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
0.8
TJ
0.6
CL
0.4 h
Run#1
Run#2
0.2
-
0.2
-0.4
-
0.6
0
2000
4000
6000
8000
H[Oe]
10000
12000
Fig. 19. Derivative o f reflection amplitude vs magnetic field at 21'C for sample # 2 .
Every 30th data point is represented by a symbol. No FM R absorption is apparent. The
sensitivity o f the PSD was 2 juV, the time constant r = 10s, and sweep rate was
0.84 kG/min.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14000
54
4.4 Transmission measurements
Transmission measurements were performed on both samples. The first sample, 94 /jm
thick, transmitted no discernible signal, presumably because it was too thick. The second
sample, 77 jjm thick, did produce a measurable transmission signal. See Fig. 20. This is
the first observation of the transmission of microwave energy through Terfenol-D. For a
sample o f this thickness the only plausible modes of transmission are acoustic.
The transmission is proportional to — cos{6S+ 0 ^ ) where Qt is the phase of the
K
transmitted signal and 9 ^ is the phase angle of the receiver’s local oscillator. 0Lo was
fixed at a constant value for these measurements.
4.5 The experimental results
From the experiments we see that the transmitted amplitude increased from near
zero to a plateau at a magnetic field o f« 4 kOe. For this sample the magnetically easy
direction was nearly parallel to the magnetization. In this case ferromagnetic resonance
is calculated to occur at negative external fields which are experimentally inaccessible.
The hysteritic field of « 2 kOe determined above must be substantially exceeded by the
external magnetic field before the sample becomes magnetized. For H < 2 kOe, the
sample is broken up into domains and no calculation is available to compare to the
experiment. However, the increase o f the transmission from near zero to non-zero values
at high field is qualitatively the same as our experience with other materials such as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
nickel, iron and Invar. A demagnetized sample always displays diminished transmission.
The interesting effects to be observed near FM R were inaccessible with this sample.
The transmission at H > 4 kOe is nearly constant; this is in agreement with the calculation
which attributes transmission to sound waves.
7
6
5
4
3
2
Run# 18
Run# 19
Run# 27
1
0
1
0
2000
4000
6000
8000
10000
12000
14000
H[0e]
Fig. 20. Transmission vs magnetic field at T = 20 *C for sample # 2 obtained by sweeping
magnetic field up. The unit o f transmission is arbitrary. Every 30th data point is presented
with a symbol. The sensitivity o f the PSD was lm V, the time constant r = Is, the
sweep rate was 1 kG/min, and the gain o f the an amplifier was 0. ldB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
4.6 Summary of transmission experiments
The transmission experiments clearly show that there exists microwave transmission
through Terfenol-D. We attribute this transmission to sound waves traversing the sample.
From the experiment we also learn that the power ratio for the signal is
PR |
= -90.9 dB+201og ( V s,-g ) = -74.34 dB [35] where V Sjg = 6.0 V and the expected
power ratio based on the calculation is PR \calc = 20 log— =-113.32 dB. In terms o f the
transmission signal, we calculate that the phase sensitive detector output at H= 6.0 kOe
should be 76 mV whereas it is measured to be 6.0 V. The discrepency is unresolved.
4.7 Conclusions
We observed microwave transmission through Terfenol-D and performed FMR
reflection measurements. From these experiments we were able to measure the anisotropy
constant K xover a small temperature range near room temperature. At room temperature
the anisotropy constant K x = (—2.00 ± 0.025) x 106erg/cm3. The Gilbert damping
constant G which fitted the data is not the intrinsic magnetic viscosity. In addition, we
observed hysteresis in the field for which FM R occurred. We did observe transmission
through Terfenol-D. Our sample was so thick that only sound waves, copiously generated
by virtue of Terfenol-D’s huge magnetostriction, could plausibly account for this
transmission. The transmission amplitude was remarkably close to the expected
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
transmission — =2.18 x 10“6. We did not see any interference between sound and spin
K
waves as predicted by theory.
For future experiments I suggest using cavities which can allow rotation of the
magnetic field; it would be useful to change the orientation o f the magnetic field with
respect to the crystal. I also suggest performing experiments over a wider temperature
range to observe how the anisotropy constant behaves, because Terfenol-D’s physical
properties are very sensitive to temperature.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
[1]
J.P. Teter, K. Mahoney, M. Al-Jiboory, D. G. Lord and O.D. McMasters J. Appl.
Phys. 68, 8 (1991).
[2]
D. C. Jilies, S. Hariharan, J. Appl. Phys. 67, 5013 (1990).
[3]
A. G. Gurevich, Ferrites at Microwave Frequencies, (Boston Technical Publishers,
Cambrige, Massachusetts, 1965).
[4]
J. F. Cochran, B. Heinrich and G. Dewar, Can. J. Phys. 55, 787 (1977).
[5] G. Dewar, Phys. Rev B 5, 7805 (1987).
[6] G. Dewar, J. Appl. Phys. 81, 5713 (1997).
[7] B. D. Cullity, Intoduction to Magnetic Materials, (Addison-Wesley Publishing
Company, Reading, Massachusetts, 1972).
[8] C. Vittoria, G. C. Bailey, R. C. Barker and A. Yelon, Phys. Rev. B 5,
2112(1972).
[9]
[10]
L. D. Landau and E.M. Lifshitz, PhysikZ. Sowjiet Union 8, 153 (1935).
C. Kittel, Introduction to Solid State Physics, 4th ed. (John Wiley & Sons Inc.,
New York, 1971).
[11]
G. T. Rado and J. R. Weertman, J. Phys. Chem. Solids 11, 315 (1959).
[12]
G. Dewar, B. Heinrich and J. F. Cochran, Can J. Phys. 55, 787 (1977).
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
[13]
J. F. Cochran, B. Heinrich and G. Dewar, Can. J. Phys. 55, 787(1977).
[14] T. L. Gilbert, Phys. Rev. 100, 1243 (1955).
[15] N. Bloembergen, Phys. Rev. 78, 572 (1950).
[16] L. D. Landau and E. M . Lifshitz, Theory of Elasticity, 3rd ed. (Oxford, New York,
1986).
[17] G. C. Alexandrakis and G. Dewar, J. Appl. Phys. 55, 2467 (1987).
[18] Charles Kittel, Introduction to Solid State Physics, seventh ed. (John Wiley &
Sons, Inc, pp 80-90, 1996).
[19] J. D. Jackson, Classical Electrodynamics, 2nd. Ed. (Wiley, New York, 1975).
[20] B. Heinrich and J.F. Cochran, J. Appl. Phys. 50, 2440 (1979).
[21] B. Heinrich, J. F. Cochran and K. Myrtle, J. Appl. Phys. 53, 2092 (1982).
[22] K. Myrtle, B. Heinrich and J. F. Cochran, J. Appl. Phys. 52, 2250 (1981).
[23] J. F. Cochran, B. Heinrich and G. Dewar, Can. J. Phys. 55, 834 (1977).
[24] Carlo Waldffied, Scott Wadewitz, and G. Dewar, J. Appl. Phys. 75, 5919
(1994).
[25] Address: M IT E Q , 100 Davis Drive, Hauppauge, New York 11788.
[26] Chris Reese, MS Thesis, University o f North Dakota, 1993.
[27] Address: Antenna & Radone Research Association, Inc., 15 Harold Court,
Bayshore, Long Island, New York 11706.
[28] Address: Princeton Applied Research Corp., P.O. Box 565, Princeton
New Jersey 08540.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[29] T. H. Wilmshurst, Signal Recovery from Noise in Electronic Instrumentation,
(Adam Hilger Ltd, Bristol, Boston, 1985).
[30] Address: Hewlett-Packard Co., 5651 West Manchester Ave., Los Angeles,
CA 90045.
[31] Address: UTD Microwave Inc., 3500 Sunset Ave., Asbury Park,
New Jersey 07712.
[32] Address: Hewlett-Packard Co., 5651 West Manchester Ave., Los Angeles,
CA 90045.
[33] F.W.Bell Gaussmeter, Bell Technologie Inc., 6120 Hanging Moss Road, Orlando,
FL 32807.
[34] L M 384 5Watt Audio-Power Amplifier, National Semiconductor Corporation,
Santa Clara, CA 95051.
[35] Carlo Waldfried, MS Thesis, University o f North Dakota, 1993.
[36] D.C. Power supply W P-710 Vector-VIZ, Instrument division. 10101 Foothills
Blvd. P.O.Box 619011. Roseville, CA 95678.
[37] A.E. Clark, J. D. Verhoven, O. D. Macmasters, and E. D. Gibson, IEEE
Trans. Magn, Vol. 22, 973 (1986).
[38] Keithley EEEE-488 Interface Board, Keithley Instruments., Inc. Aurora Road
Cleveland, Ohio 44139.
[39] J.B. Restorff, H.T. Savage, A.E. Clark, and M . Wun-Fogle, J. Appl. Phys. 67,
5016 (1990).
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