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0-169

THE TEMPERATURE DEFENDENCE OF THE
FERRUMAGNETIC RESONANCE
LINE WIDTH IN

ELECTROPLATED COBALT

by
David Lyman Kingston

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics
1955

1:1:0'. ILE (3.3117223:

The author wishes to extend his deepest gratitude to
Dr. Robert D. 5301108 who directed this research. His ad-

fl

vice, helpful suggestions, end continue nteres t made the
work very enjoyable.
The author also wishes to thank Walter 6. Meyer who

worked side by side with him for many months.

“350

<3)
£3
(a

7’“
Tim.

FEFBOI-IAGITE‘T IC

WIDT

Submitted to the
State Colleg

in partial

A 3roved

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0

mm 'DTT‘ID 'I T Y?“ Tfi—Z‘ 1" fi‘?”"‘
.L 1—13..“ LLB-51‘ DALI-L ELVDL‘IVAL

on 1‘":
..~ ‘ [J
.44.“. 4- in.“

FEES OI-TAITC E LII‘TE

H IN E ECTROPLATED COBALT

by

David Lyman Kingston

School of Graduate Studies of Hichigsn
of Agriculture and Applied Science
fulfillment of the requirements

for the degree of

PM?
Did

MASTER OF 80133

Department of Physics

1955

 

73%

D s;vid L. Kin5ston
Thesis Abstract

This thesis reports a study of the temperature dependence
of the ferroma5netic resonance line width in electroplated
cobalt. At a frequency of 9300 megacycles per seconds, the
resonance was found at fields between 400 and 500 gauss, depen
in5 on the temperature. The resonance line width in electro-
plated cob: lt is much broader than that found in electroplated

nickel and considerably broader than that found in electroplat-

*5

ed iron. The line width may be expressed in terms of a e ip-

c
an ~v-n [T1 '1 9
rocal relei tion time, l/T2° -he value of l/T is 31 X i0
1 2 9
seconds— at room temperature, decreases to 17 X 10 se econds-l
at 300 de5rees centi rede and re ins ne early const: nt up to
600 de5rees centi5rade which was the hi5hest temperature at
which data wel -e taken. This is to be constrasted with the
behavior of electroplated nickel and for electroplated iron.
- 9 -1

For nickel ,l/T2 is 3.1 x 10 seconds at room temperature
and remains fairly constant up to 250 degrees centigrade.
At this point it re. see very sharply and reaches a value of

q -1 '9 ‘ ‘I §
5.9 x 109 seconds at 370 de5rees centi5rade wnicn is tne
Curie tem erature for nickel. For electrOplated iron, l/T2,

l

C)- ..
at room t mperature, is 15 x 10’ seconds . It decreases

-l
nearly line Hrl to 72109 seconds at 600 de5rees centigrade.

TABLE OF CONTENTS

I. INTRODUCTION
II. THEORY
A. FERROMAGNETIC RESONANCE ABSORPTION
IN AN ISTROPIC MEDIUH
I B. THEORX OF THE USE OF THE RESONANT
CAVITY
III. EXPERIMENTAL APPARATUS AND PROCEDURE
A. MICROWAVE APPARATUS
B. HEATING THE SAMPLE
C. MAGNET
D. PREPARATION OF SAMPLE
IV. REDUCTION OF DATA
V. RESULTS
APPENDIX
BIBILOGRAPHY

CO

Figure l.

{‘0

Figure
Figure 3
Figure 4.

5

Figure

Ch

Figure

“I

Figure

Figure 8.

Figure 9.

LIST OF FIGURES

Motion of the Magnetization Vector in
Ferromagnetic Resonance

Equivalent Circuit of Resonant Cavity
Picture of Apparatus

Block Dia5ram of Apparatus

Cavity Assembly

Magnet Calibration

Method of Finding HO and H0 From the
Experimental Curve R(H)

Typical Resonance Curve for Electroplated
Cobalt

Temperature Dependence of l/T2 for

ElectrOplated_Cobalt

Page

THE TEMPERATURE DEPENDENCE OF THE FERROMAGNETIC
RESONANCE LINE WIDTH IN ELECTROPLATED COBALT

I. INTRODUCTION

Ferromagnetic resonance was discovered by Griffithsi
in 1946. In his eXperiment a thin film of ferromagnetic
material was placed on one end of a cylindrical microwave
resonant cavity. An external magnetic field was applied
parallel to the surface of the film. With a constant mi—
crowave magnetic field of fixed frequency parallel to the
film, but perpendicular to the applied field, it was
found that a maximum power absorption occured for a parf
ticular value of the external field.

In l9u8 Kip and Arnold2 investigated the ferromagnetic
resonance in an iron single crystal as a function of the
orientation of the crystal in the applied field. The tem-
perature dependence of the width was first examined by

3 who worked Vvith polycrystalline nickel and

Bloembergen
supermalloy. His work on the temperature dependence of
the line width was extended by Healyh who studied ferro-
magnetic resonance of nickel ferrite as a function of

5

temperature. In 1953 Spence and Cowen6 examined the
temperature dependence of the resonance line width in
single crystals of iron and nickel, and electrOplated iron

and nickel.

A ferromagnetic may be visualized as consisting of a
great number of electron spin magnetic moments coupled to-
gether by exchange forces to give a resultant magnetic
moment. If the applied magnetic field and the r-f field
have the correct magnitude and orientation, the ferromagnetic
will absorb energy from the r-f field. Quantum mechanically
the applied magnetic field splits the degenerate energy
levels into a set of Zeeman levels designated by the magnetic
quantum numbers ms. If the selection rule Zlnk==il is
satisfied, the r-f field can induce transitions between these
levels. Since the most papulated level is the lowest energy
level the result is an absorption of energy from the r-f
field. The absorption continues because the spins give up
energy to the crystalline lattice tending to maintain an
excess of electrons in the lower state.

At.the present time it appears that a satisfactory the-
oretical explanation of the temperature dependence of the
line width has not been given. Bloembergen presented a
means of representing the line width by using spinspin re-
laxation times and spin-lattice relaxation times. Kittel

7

and Abrahams calculated the temperature dependence of the
Spin-lattice relaxation. No calculation of the dependence
of Spin-Spin relaxation times have been made. It is a com—
mon belief that the Spin-Spin terms make the major contribut—

ion to the line width. Kittel and Abrahams8 have suggested

that at temperatures below one-half the Curie temperature
the Spin-spin terms predominate, and above one-half the

Curie temperature the spin-lattice terms predominate.

II. THEORY

A. FERROMAGNETIC RESONANCE ABSORPTION

IN AN ISOTRUPIC MEDIUM

A thin sheet of ferromagnetic material, lying in the
xy plane, is subjected to a static magnetic field in the
z direcxion and a r-f field in the x direction. The mag—
netization will consist of a large constant component, M2
and a varying “X and My. The magnetization E and the total

angular momentum 3 are related by

=7" ’ (1)

where )z, the magneto-mechanical ratio is

_ 99
Warm-5,
g is the spectroscopic splitting factor, e is the charge of
the electron, m is the mass of electron, c is the velocity

of light. The classical equations of motion are,

d_Mxy=7[MXHeff]x _ MU. , (2a)
xy T2

dM — .. M2. M0

——2 = _. _—

d' Y[MXHEff]Z TI (2b)

In the case of an isotrOpic ferromagnetic material

M X Heff is the total torque. Hei‘f is the effective field

inside the material, and is given by

neff =H ’Nx 2M+fionis+ ova-M. :

ext y

(3)

where Nx is the demagnetization factor. The third term

yz
on the right of equation (3) is due to ferromagnetic anis-
tropy. Since, in this case, the medium is isotrOpic (see
appendix) this term will be dropped. The last term on the
right is due to exchange interactions. The exchange effects9
will be small for pure metals such as cobalt in the temper-
ature range in question. Since this term will have little
effect on the effective field, it will be drOpped.

The damping terms in equations (2) involve T1 and T2.
T1 is the spin-lattice relaxation time, and T2 is the rel-
axation time for all effects disturbing the epin system from
within. T1 and T2 may depend on the frequency of the applied
field and on the temperature. In general T1 and T2 are not
equal.

3

Following Bloembergen we obtain expressions for the

ferromagnetic frequency permeability and resonant frequency.

A solution of equations (2) can be obtained by letting

M =Mx e'wt . Combining equations (2), (3) and

xy Y

ext _ iw'r ext_ ext__

we get expressions for Mx’ My, and M2. In these calculations
terms involving products such as MxMy, which are small (see
figure 1) are dropped. Letting M3 3 M0, the last term of

5

 

 

 

 

 

FIGURE l MOTION OF THE MAGNETIZATION VECTOR
IN FERROMAGNETIC RESONANCE

6

equation (2b) becomes zero. Hence Tl drops out of the final
equations. Using the relation for the susceptibility,

Il'L—|=ll'Ll—l—'/L2=C4’—HMX ' (5)

where H1 is the r—f magnetic field, we get

47ry2Mo£rh+(Ny—N2)Mo](w§— mg) l
2 Z 2 4w?! , ,
(0)0 -a)) +- ’/EE (6)

__ 47f] 2M0[Ho +( Ny- N ZIMO] Zea/T?

’U'Z- (w:— w 7) + 4071-2 ’ (7)

where (u: , the resonant frequency is

w: = y2[Ho + (Nx- NZ)M0] [Ho+ (Ny - NZ)MO]+% ' ,
2 I8)

T2 is the quantity to be evaluated in this experiment.

B. THEORY OF THE USE
OF THE RESONaNT CAVITY

A microwave resonant Cavity is a dielectric filled re—
gion completely surrounded by conduction walls, except for
an iris which couples the cavity to the rest of the micro-
wave system. The boundary conditions are satisfied by par-
ticular configurations and frequencies of electromagnetic
field solutions. These normal modes correspond to the res-
onant frequencies of the cavity. In this experiment we are
interested only in those which have linear dimensions of the
cavity of the order of the wave length in the wave guide.

For convenience, the cavity may be represented by an
equivalent circuit made up of lumped elements. The cavity
and coupling iris may be replaced by parallel resonant cir-
cuit which is coupled to the transmission line by a trans-

former (see figure 2)

 

 

R nTV R ' :1J__ C

I I

FIGURE 2. EQUIVALENT CIRCUIT OF RESONANT CAVITY

Zo

(_—

T
v
I

 

It is assumed that the transmission line is terminated by
its characteristic impedance Z0 and the line sees an effect-
ive resistance R. The turns ratio of the transformer is n.
It is assumed that no energy is lost in the transformer.

With the Cavity at resonance, conservation of energy

 

 

gives
_\_/_2=n2V2
R R' (9)
hence
R'=n2R ,
and (I0)
.- 2
20— n ZO
The absorbed power is then
2 2 2
P =_v_=_<_!-I:_V_ri=Vu2<'+‘"),
O R I? R . (ID
where ,r', the voltage reflection coefficient, is
_. VF
F Vi (l2)
At cavity resonance
F=Jiii
R+IZo ’ (I3)
and
£9: I—F ,
R |-+1" (I4)

V2 2 V2 2 V2 V-2
pO-—|-R(I+T )..-Z—'O(I ELF} 20,8 ZOI",

(I5)
where P1 and Pr are the incident power and the reflected
power. They are related to the absorbed power by

a=P—e

(I6)

At resonance, the reflection coefficient I1 is real
and may be positive or negative since R and 20 are real.
l/n is a measure of the coupling between the guide and the
cavity. In this experiment all the data was taken with an
undercoupled cavity. Therefore only the case of the under-
coupled cavity will be considered.

For the undercoupled case, l/n is small compared to

Zo/R', I. is negative and Vr is negative. Thus

 

,Vrnax :: Vi —vr ’
(I7)
Vmin =Vi+vr
The voltage standing wave ratio is
___Vmcm= Vi-Vr= I_P__Zp_
P me va+V. I+F R ' (IS)

The Q of the cavity will now be defined and related
‘to quantities which can be measured, namely p, Irw , PT.

The total Q is

10

energy stored in cavity

 

:: 277' '
Q TOTOI energy lost per perIOd ('9)

The total losses of the cavity are made up of the losses
of the unloaded cavity and the external losses. These
losses are prOportional to the reciprocals of the total Qt,

unloaded Q,L1 and the external Q8, reSpectively.

I I I
= — + -— .
Qt Q" Qe (20)

The unloaded Qu is made up of all of the losses in the
cavity proper, QC, the losses in the c0pper walls of the
cavity and Qfer' losses in the ferromagnetic sample. The

external Q6 is the energy lost by the cavity to the guide.

Qfer . (2I)

The Qfer' is due to the magnetic losses in the sample, and
losses due to eddy current and dielectric losses. However,
the magnetic losses in the sample are predominate.

The fundamental quantity measured in a ferromagnetic re-
sonance experiment is the ratio of the external Qe to the

unloaded Qu. This is defined as

a. =.___.'+T‘
Q" ’0 I-F' (22)

.93- Qe(SB_ [[sz 0-)somple+( 17H 0mm .
0., (17qu d1, m of “my (23)

ll

where

we ewe-

Therefore we can express the ratio as to Qu as

Qe

'6} = A fl-tEB .

(24)
EXperimentally a quantity p, which is prOportional to

reflected power, is measured.

2
p = CB ==CHW R

(25)
This equation assumes that the amplifier used is linear.
Therefore
I
{1:9},
(I
where

a = w/cPi

(I may be determined by measuring the standing wave ratio

f) at one value of p. For the undercoupled case

I + NW .
I - NW (26)

In the case of an overcoupled cavity f) is the same as above.

. —I
LF| 73777 ' (27)

Then

(28)

12

Hence

Afi+s=p=l_|fl. (29)

I “ DZ, (30)

" 95 (am
where A and B involve Q,c and Qu and are constants for a
given cavity at a given temperature. Thus, a quantity pro-
portional to the effective permeability can be obtained from

measurments made on the cavity.

13

III. EXPERIMENTAL APPARATUS AND PROCEDURE
A. MICROWAVE APPARATUS

The apparatus is the conventional microwave apparatus
as shown in the block diagram (Figure 4). The klystron is
a 723AB low power oscillator operating at about 9000 mega-
cycles per second. It is modulated with a 1000 cycles per
second square wave which, along with necessary DC voltages,
is isolated from the lOad by a flap attenuator. The klystron
and attenuator are matched with a Hewlett-Packard E-H tuner
to the H-arm of a magic T. The E-arm is matched with another
E-H tuner. The output of the matched detector is read on a
Browning TAA-l6 twin-tee tuned amplifier peaked to match
the frequency of the square wave. The two side arm of the
T feed the energy to a matched load coupled to one arm and
through a standing wave detector to the resonant cavity,
coupled to the other arm. Energy from the klystron is coup-
led into the two side arms but not into the E-arm of the T.
That portion of the energy which goes into the arm with the
matched 10ad is completely absorbed. The portion which goes
in to the arm with the cavity is reflected from the cavity
and partially coupled into the E-arm. Only power that is re-
flected from the cavity goes into the E-arm. This is the
power that is measured by the detector, and the meter read-
ing is pr0portional to this reflected power. The matching
devices prevent any multiple reflections in any of the arms

of the T. The overall matching of the system is to a volt-
1h

 

FIGURE 3 APPARATUS

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age standing wave ratio of about 1.05 except in the arm con-
taining the resonant cavity. In this arm, the standing wave
ratio is determined by the type of cavity used. A mica win-
dow was cemented in the guide to prevent leakage of the hy—
drogen gas which was used as a reducing atmosphere. The
mica introduced so little reflection that it was not necess-
ary to match these reflections out.

The resonant cavity used was three half wave lengths
long. It was coupled to the guide with a symmetrically
placed circular iris. An iris with a diameter less than
5/16 inch resulted in an undercoupled cavity. An undercoup-
led cavity was used throughout the experiment.‘ The cavity
was assembled with silver solder. Silver solder was used
because of its high melting point. Oxidation was removed
from inside the guide by polishing with a fine grade of
carborundum and then washed with a 10% solution of nitric
acid.

The sample was plated on a brass disks(copper disks were
also used). The disk was then silver soldered on the end
of the cavity, thus completing the cavity. The preparation

of the sample will be discussed later.

17

 

F LANGE—<

I

X-BAND
' WAVEGUIDE

 

TI MICA WINDOW

 

 

 

 

 

 

 

FLANGE 4 F

 

 

T

3:34—H2 INLET

4—WATER INLET

 

_[::I*—WATER

OUTLET

--— IRIS

RE SONANT
CAVITY

L—

 

/

 

 

 

SAMPLE

 

mfg.— H2 OUTLET
DISK —- I:_l__m_

 

FIGURE 5 CAVITY ASSEMBLY

l 8

 

B. HEATING THE SAMPLE

The heating coil had resistance of approximately 9

ohms and was fabricated of .112 inch by .005 inch Nichrome
ribbon wound on the cavity in a non-inductive manner. The
coil was Operated on AC supplied by a Variac. The coil was
wound on a layer of asbestos on the cavity and was held in
place by a layer of baked potters clay. An insulating layer
of glass wool was wrapped on the clay. This oven was cap-
able of temperatures up to 800 degrees centigrade with a
power input of 500 watts. The temperature was measured with
a calibrated Chromel-Alumel thermocoupheinserted in disk on
which the sample was plated. Thus making it possible to
read the temperature at a point adjacent to the sample. The

5

thermocouple was checked by Cowen and was found to be accur-
ate to 2 degrees centigrade from room temperature up through
the range needed. The output of the thermocouple was read
on a Leeds and Northrup type K potentiometer. The hydrogen
gas in the cavity prevented oxidation of the sample at high
temperatures. The gas entered above the cavity and paSsed
out the bottom where it was burned. A water cooled section

of the guide was placed above the cavity to prevent excessive

heating of the rest of the microwave apparatus.

19

C. MAGNET

The magnet was constructed on SAE 1020 low carbon steel
in the form of a square box 22 inches by 22 inches by 9 inches.
The sides of the box which are 3% inches thick were machined
and bolted together with 2 inch bolts. The pole pieces are
7 inches in diameter. They were machined to slip fit holes
bored in the box, and were threaded and held in place with
retaining rings 1 inch thick. Because the ferromagnetic res-
onance line is Very wide no special treatment of pole faces
to insure a homogeneous field was undertaken. The copper
wire used in the winding was .072 inches square formvar in-
sulated wire which was wound on four cOpper bobbins. The
bobbins were made of 3/16 inch thick capper sheet and were
designed so that the finished spool was 15 inches in diameter
and 1 3/A inches thick. Each coil had a resistance of approx-
imately 5 ohms. The wire was insulated from the bobbin with
a layer of .010 inch asbestos paper. After winding,the coils
were dipped in formVar and wrapped in linen. The two coils
on each pole were connected in series. The windings on one
pole were connected on parallel with those on the other pole.
The magnet was cooled with 6 water coils made of t inch square
cOpper tubing. It was found that the magnet itself did not
heat up to any great extent. However, the cooling was nece
essary because of the high temperature to which the cavity
was heated. The power source was a bank of 12 marine batter-

ies with a 300 ampere hour capacity. The magnet was control-

20

led with a water cooled rheostat. The current was read on
a Westinghouse Type PX—h ammeter to an accuracy of approx-
imately 25 milliamperes in the range 0—5 amperes. Since the
hysterysis of the magnet is approximately 65 gauss, care
was taken in always taking readings with increasing current.
The magnet was calibrated with a Sensitive Research Company

model F. M. fluxmeter.

D. PREPARATION OF SAMPLE

The cobalt was plated on brass and copper disks. The
disks measured 1 l/h inches in diameter by 1/8 inch thick.
The disks were polished with various grades of emery paper.
The final polish.was done with.Jewdu$ rouge until a near
mirror finish was obtained. After polishing, the disks were
thoroughly cleaned, first by washing with a detergent and
then electrolytically in a Special cleaning solution. The
cleaning solution consisted of 23 grams of sodium carbonate,
23 grams of sodium triphosphate, 15 grams of sodium hydroxide,
and 8 grams sodium meta silicate dissolved in one liter of
water. The sample to be cleaned was used as cathode. The
current density used was of the order of 5 amperes per square
inch. The disks were then plated in a solution containing
cobalt sulphate, magnesium sulphate and ammonium sulphate.
The current density used was .012 amperes per square inch.
Oxidation at room temperature was unimportant with cobalt

plates. For this reason it was possible to plate several

21

20_._.<mm_._<o ._.u204_2 m mmDoE

 

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9m O.m ON ON m._ 0.. m.

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H

00m _
9
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22

samples at one time and then store them. It was not neces-
sary to polish the sample after plating. A bright plate
was relatively easy to obtain. The samples were then ready
to be silver soldered to the guide to form the resonant
cavity. Hydrogen gas was used as a reducing atmOSphere in

the soldering process.

23

IV. REDUCTION OF DATA

It is desirable to construct a quantity which deperds
on fL but does not involve the unknown constants of equation
(31), namely A and B. By so doing we can eliminate the

evaluation of these constants. Rewriting equation (31)

P2 (31)
Let ,LLres be the value of ILL at ferromagnetic resonance.

Equation (31) becomes
I

(I-+ p3.
A +s= ——EP -
#res 0—D? (32)

res
At very large values of H0, ,u, should equal unity.
Hence

.i
a + 93
—__—_—TL . A
c1- g3 (93)

Combining equations (3|), (32), and (33) we get,
u n

_. .L ~
L" = (inc—pf”)
Jive-s. ' 9,;— pg (31+)

R involves only equantities which can be measured directly.

A -+ E3 =

(1" p%

It can easily be determined as a function of H0‘ Next it
must be shown that l/T2 can be found from R (H0). The me-
thod employed to find l/T2 is approximate. The exact method
is considerable more tedious. Although the method is approx-
imate its accuracy may well be within the limits imposed by

the fundamental data.

2h

Let Ho% and Hué be two values of HU such that

2 2 _ 2w _ +

w° - w — T2 ; H° _ Hole (35)
2 2- _M o = _

(00" (U " T2 ’ Ho Ho% (36}

From equations (8) it can be seen that 33% and Hag are

solutions of

2 2 ._
Ho'3_ Hores.+ 277(Ho'3_ Hare) i 6 - O ’ (37)
where
2 = (N +N' —2N')M ,
77 x y z o (38)
and
6 = _2__
2
7 T2
Then
i-
+ 2
Hl+ =[(H + )+e],
02 77 ores 77 (39)
and
_ 2 %-
Hoé + 77 = [(H°'es+ 77) — E] ' (40)
We define the line width as
= + -- -
AH - H°$ H“: ' (1+1)

Expanding the right side of equations (39) and (40) by the

binomial theorem,

25

 

 

LOO ““"““”“"""

 

.75

R(Ho)

.50

.25 am.)

 

 

 

 

FIGURE 7 THE METHOD OF DETERMINING H:
AND H; FROM THE EXPERIMENTAL

CURVE R(H)

26

 

 

e e 2
AH: Hores+77 [I + %(Hores+ 7]) + m]

Upon examination it can be seen that the second term will

((+2)

be small compared to the first term and therefore may be

dropped.
Then
6 :— (Hores+ 77)AH ’ (1+3)
and
I If
— g H + AH
T2 2w( ores 77) ((41+)

Thus by measuring the resonance field and the line width we

can evaluate l/Tz. 13H can be obtained from a plot of R (H0).
+ .-

Let [(12% and [(1.2% be the values of [$2 at H0 equal to

H05 and Hot respectively. From equations (7), (35), (36)
+ nu

— N L
lu'z% — lug-é- — Zluz res (“5)

The superscripts and will now be dropped. From (6),

l

N 1
Ian-'5 "’ [“25 = ‘é'l‘zres + ' (#6)

From the expressions
l_ 2 2%
swede.) +#2I

and from (45), and (46)

. g l 2 J— 2 i
[(1% —- [(l + 2#2res)-+ (2#2res)] + z/J—zres (47)

Similarly,

,u'res 9- [l + (,LLzres)2]% + luares ' (48)

These expressions for [1,: and [u'res are inserted in equat-
2

ion (BH) giving the ratio,

= {[(I + 7%;qu +(%"‘2“’-")2]2L + alarm} - I 449)

{[l+ (Muff + pansy —- I

The intersections of Ra with R (H0) give approximate values

NI-

 

of H:% and H;%‘ as varies slowly with #2 ,es as shown
by Table I.
TABLE I
VALUES OF Ra FOR VARIOUS VaLU'aS OF [(1.2 res

 

(fiese _5a_
65 .759
60 . 758
55 .757
5D .75?
#5 .756
LID . 755
35 - 754
30 . 753
25 . 751

 

28

An approximate method is now needed for calculating

#2 res 0 From eQUationa (7) and (an),

ILL E 477M0(Ho res + 47%)
ares AH(Ho res + 77) (50)

We find l/T2 by proceeding as follows:

1. Data consists of a plot of p against applied field
HO' p is measured at a point near resonance. C! can be
calculated from equation (28). R(Ho) can be obtained from
equation (34).

.2. Assume Rais .75. Draw a line on the plot of R(Ho)
at .75 and read off the distance between intersection points.
This the first approximation of AH .

3. From equation (50) calculate/Lzfim. Find the cor—
responding value of Ra from Table I. Then from the plot of
R(HO) find second approximation of'llH.

u. Calculate l/T2 from equation (an).

Usually two approximations are sufficient to give all
the available accuracy in this method. The advantage of us-
ing the above method is eliminating the necessity of evaluat-
ing the constants A and B of equation (31). These constants
involve the uncertain conductivities of the Cavity walls. If
they are introduced they increase the possibility of erroneous
results.

Values of the demagnetization factors and the saturat-
ion magnetization had to be inserted in reducing the data.

The totad.demagnetization factors consist of the sum of the

29

shape demagnetization factors and the anisotropy factors.

0
N, =N,+N,

N =N +N° . (51)

The Na's are anisotrOpy factors and the N's are shape demag-
netization factors. Since the sample used in this experiment
is isotropic, the anisotropy terms can be drOpped. The shape

demagnetization factors for thin oblate spheroids are,

4.
N. = "$0 '— 7537)
2
Ny :...- 477(l '" '21%1+ W) (52)

2
N=%('—‘s47r) .

I.

where m the ratio of the diameter to the thickness. The
appropriate thickness for N2. is the actual thickness of the
spheroid, because no is oriented along 2. The x and y com—
ponents of the magnetic field are r-f components and therefore
the effective thickness of the spheroids for Nx and Ny is the

skin depth. Therefore for thin spheroids,

ngo
(53)
Ny:== 47f
7r2
Nz:: 'EF

'40

V. RESULTS

Figure 8 shows a typical resonance curve for electro-
plated cobalt. There is only one resonance peak on the
curve. Through the range of temperature possible with the
apparatus, there was no apparent change in this resonance
peak, which came at about 440 gauss.

It was not possible to reach the Curie temperature for
cobalt, which is 1120 degrees centigrade, with the available
apparatus. The highest temperature reached was about 700
degrees centigrade. At this point the apparatus broke down
on each trial. This temperature appeared to be the limit of
the heater coil. Also at temperatures above 700 degrees
centigrade the silver solder melted and gave away to the
pressure of the hydrogen gas in the cavity, and with a hole
in the cavity, the microwave resonance dissappeared.

Figure 9 represents data on 1/T2 as a function of tem-
perature for electroplated cobalt. In the range of temper—
atures in which measurements of l/T2 were made, 1/T2 tended
to reach a nearly constant value of 17 x 109 second“1 as
the temperature increased. All of the samples used gave dif-
ferent values of l/T2 for room temperature, but tended to

flatten at approximately the same Value for l/TZ'

31

._.I_<moo om._.<I_n_Om_.romI_m mom w>m30 woz<zommm I_<Q_d>._.
mmmwdzd. l _
QM Oh 9N O.N . m._ O._

1 a

m6

m wmaoi

 

OBBHOSBV HBMOd

3.2

N
H
._..._<on om._.<.._n_om._.ou.._m mom. I... do uuzmozwamo mmahdmwdzma. m mmaol

oo mm3h<mma2wh

 

 

CON 000 000 00 w 00m OON OO. O
m
0.
2|.— .l
m. m
0 O o 6
I! o s
o o 3
e 0 ON 0—
mm
On
an.

 

33

_APPENDIX

ANI SOTROPY ENERGX IN
ELECTROPLATED COBALT

In this experiment the ferromagnetic material used
were cobalt. The cobalt was used in an electroplated form.
An electroplated material is generally assumed to be poly—
crystalline, that is, there is every possible orientation
of the crystals of the material.

In an uniaxial crystal, such as cobalt, the anisotropy

energy is,

. 2 , 4
= +
Ek Ko +Kl sm 9 Kzsm 9 , (1)

where K0, K1 and K2 are the crystal anisotrOpy constants,
and 6 is the angle between the magnetization vector and
the principal axis of the crystal. To find the anisotropy
energy in a polycrystalline material it is necessary to find

the average of E in all directions,

k
ffEk d“ :: <Ek>ove (2)

After substituting
2fl'7T

I; ngo‘“ K,sIn29+Kzsin48)sIn8'd9'd8 = (Ek>ove.

The coordinate axes can be picked such that 9 = 9'

(3)

3H

Hence

<Ek>ove= O (u)

Therefore the average anisotropy energy in electroplated

cobalt is zero.

B5

‘0 a) ‘0 0‘ Kn c- u: no I4

a,

(3 CI <3 LI :n t: :z p

BIBLIOGRAPHY

. H. E. Griffiths, Nature lig, 670 (1946)

F. Kip and R. D. Arnold, Phys. Rev. 21, 1556 (1949)
Bloembergen, Phys. Rev. IQ, 572 (1950)

W. Healy Jr., Phys. Rev. Qé, 1009 (1952)

D. Spence and J. A. Cowen, 0. O. R. Report (195A)

A. Cowen, Ph.D. Thesis, Michigan State College (1954)
Kittel and E. Abrahams, Rev. of Mod. Phy. 35 233 (1953)
Kittel and E. Abrahams, Phys. Rev. 20, 238 (1953)
Kittel and C. Herring, Phys. Rev.‘zz, 725 (1950)

36

mm.

9n

w

HICHIGQN STQTE UNIV. LIBRQRIES

Ill!
1

312930177

llll

1908