TEMPERATURE AND MAGNETIC FIELD DEPENDENCE OF THE SINGLE CRYSTAL ELASTIC CONSTANTS OF NICKEL

Joseph Wayne Brophy

M. S . Thesis Submitted to Iowa State University

Ames Laboratory, ERDA Iowa State University Ames, Iowa 50010

..--.-- NU1 ILC

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TABLE OF CONTENTS

Page

INTRODUCTION

Purpose of the Investigation /

Review of the Problem and Literature Survey

EXPERIMENTAL PROCEDURE

Sample Preparation

Acoustic Measurements

RESULTS

Zero Applied ~ield Measurements

Field Dependent Measurements

DISCUSSION

Magnetic Field Depend.ence

Zero ~ield Temperature Dependence

SUMMARY \

BIBLIOGRAPHY 38

APPENDIX A 40

Pulse-Echo-Overlap System. 40

APPENDIX B 45

Crystal.Elasticity and Wave Propagation 45

ACKNOWLEDGMENTS 47

INTRODUCTION

Purpose of the Investigation

The purpose of this investigation was to apply 'state-of-

the-art' techniques to determine the single crystal elastic

constants of a ferromagnetic material, specifically nickel,

to obtain expertise in working with such materials. Supple-

mental to this was the up-grading of a pulsed.ultrasonic

apparatus to determine sound velocities; the details of the

apparatus can be found in ~ ~ ~ e n d i x A.

The data that were desired were the temperature depend-

ence of the demagnetized elastic constants and to measure, if

possible, the AE effect on single crystal elastic constants

quantitatively. The AE effect is the change in an elastic

modulus from the demagnetized state to the magnetically satur-

ated state. The effect has been thoroughly investigated in

polycrystalline samples.

Nickel was chosen for this investigation because of a

large AE effect, i t has heen reported to be as large as 30%

for polycrystalline samples (1,2), and because the existing

data on nickel are incomplete and contradictory.

Review of the Problem and Literature Survey

The unique feature of bulk ferr~ma~neticbodies is the

presence of domains of differing direction of the magnetiza-

tion vector in the demagnetized state. These domains are

separated by magnetic domain walls of finite width (3) which

can be directly observed by several techniques, such as the

Bitter technique or electron microscopy ( 4 , s ) .

It is precisely the presence of domain walls that creates

difficulties in using ultrasonic techniques to determine the

single crystal elastic constants. The domain walls interact

with the ultrasonic stress waves because the domain walls are

not static (3). The domain walls can move quite readily under

application of stress or magnetic field. In the past the

theory was that the domain walls would attempt to oscillate

sinusoidally with the same frequency as the applied oscilla-

tory stress or magnetic field (1,6,7). As long as the fre-

quency of the applied force is low enough the domain walls can

oscil.S.ate in unison with the signal, which produces a strong

attenuation effect in the signal due to energy losses as the

signal propagates through successive walls. At sufficiently

high frequencies, the domain walls can no longer follow the

period of the applied signal and the attenuation becomes less

severe.

Very recent work by Bostanjoglo (8) with thin film speci-

mens of nickel, using direct observation in an electron micro-

scope, has indicated that, contrary to the existing theories,

the domain walls do not appear to be completely coupled to

ultrasonic excitations. He finds that as the frequency of

.the excitation increases, parts of a domain wall begin to

vibrate with an amplitude about equal to the wall width with

no larger amplitudes being observed. At higher frequencies

large wall portions jumped to neighboring stable positions and

back, but that the motion was not sinusoidal. He also noted

that the positions that the walls jumped to were the same as

the ones reached under an applied static magnetic field. At

sufficiently high frequencies either the domain configurations

were changed or the domain magnetization vectors were changed.

However, the gross effects upon an ultrasonic stress wave

propagating through an unsaturated ferromagnetic sample would

appear to be the same with regards to attenuation and transit

time changes as a function of frequency.

Nickel also exhibits a marked frequency dependence of the

sound velocity in the demagnetized state in'the range 2-20 MHz

(9). The sound velocity increases with stress wave carrier

frequency until a plateau at about 15 MHz is reached. In this

investigation all measurements were made at a carrier fre-

quency of 10 MHz. The difference between values obtained at

15 and 10 MHz is not large, being approximately 0.1% and is

not really significant in terms of the measurements made in

this work. Differences in velocities measured at 15 and 10

MHz for magnetically saturated samples are so small that they

cannot be measured.

Another effect in saturated nickel is the velocity depend-

ence upon the direction of an applied magnetic field. This is

due to magneto-elastic coupling.. The effects are also small,

being on the same order as the frequency dependence. Alers

et al. (10) and Sakurai (11) have both obtained good results --

on this effect for nickel and it was not felt necessary to

try and re-do their work.

There has been a large amount of work done on the single

crystal elastic constants of nickel (1,ll-18); the great bulk

of the work has been done on magnetically saturated crystals

though. Few measurements of demagnetized samples have been

made, due mostly to experimental difficulty in propagating a

signal in an unsaturated state. It was thought appropriate to

try and measure the temperature dependence of a demagnetized

sample to fill the existing gap in the literature and to try

and measure the single crystal AE effect to clear up some.of

the controversy in regards to the magnitude of the effect (1).

Also, the value of the Debye temperature, OD, of the demagnet-

ized state was desired to compare it to the saturated value

calculated by Alers, Neighbors and Sato (14) to see if there

was any notable difference.

EXPERIMENTAL PROCEDURE

Sample P r e p a r a t i o n

Two s i n g l e c r y s t a l bou le s each approximately 2 cm i n

d iamete r and 6 cm i n t o t a l l e n g t h were grown i n a modif ied

Bridgman fu rnace f o r t h i s i n v e s t i g a t i o n . One of t h e bou le s

was grown from J. T . Baker Chemical Company r eagen t grade

n i c k e l s h o t under a p a r t i a l atmosphere of helium i n an A1203

c r u c i b l e . The o t h e r boule was grown from J a r r e l l - A s h n i c k e l

r o d , a l s o under a p a r t i a l atmosphere of helium i n an A1203

c r u c i b l e .

An a n a l y s i s of t h e two c r y s t a l s was performed by t h e

A n a l y t i c a l S e r v i c e Group u s i n g spa rk mass spec t rome t ry , vacuum

f u s i o n and wet chemical t e chn iques . Table 1 l i s t s t h e impur-

i t y c o n t e n t of bo th m a t e r i a l s a f t e r t h e growing p r o c e s s ; Table \

1 a l s o c o n t a i n s an a n a l y s i s of a J a r r e l l - A s h n i c k e l rod i n

t h e a s r ece ived c o n d i t i o n . The a n a l y s e s i n d i c a t e t h a t some A 1

and S i were introduCed i n t o t h e m a t e r i a l s d u r i n g t h e c r y s t a l

growing p roces s . The p u r i t i e s of t h e c r y s t a l s were c a l c u l a t e d

t o be 99.84 a t . % f o r t h e Baker n i c k e l c ' r y s t a l , 99.91 a t . % f o r

t h e J a r r e l l - A s h c r y s t a l and t h e rod was 99.96 a t . % .

Two s i n g l e c r y s t a l s were c u t from each boule . The Baker

n i c k e l bou le was o r i e n t e d and c u t i n such a manner a s t o y i e l d

one c r y s t a l having two s e t s of p a r a l l e l (1101 f a c e s and ano the r

s e t o f p a r a l l e l {1001 f a c e s . The second c r y s t a l had t h r e e s e t s

of p a r a l l e 1 ' { 1 0 0 1 f a c e s . These c r y s t a l s a r e r e f e r r e d t o i n t h e

Table 1. Impurity content in atom ppm of the materials used in this investigation

Impurity Baker nickel Jarrell-Ash nickel Jarrell-Ash nickel

Crystal Crystal Rod

a Wt ppm results.

Table 1. (Continued.)

Impurity Baker nickel Jarrell-Ash nickel Jarrell-Ash nickel

Crystal C.rys t a 1 Rod

text as Ni-B-1 and Ni-B-2, respectively.

The other boule, of Jarrell-Ash nickel, was oriented and

cut to yield a crystal with the same type of orientation as

Ni-B-1; the other crystal cut from the boule had one set of

parallel'~ll1~ faces with the other faces cut to give a

rectangular shape to the sample without regard to their orien-

tation. These two crystals are referred to in the text as

Ni-JA-1 and Ni-JA-2, respectively. Figure 1 gives the sample

shape2 and dimensions.

All orientations were done using standard back-reflection

Laue X-ray techniques with the boule cemented to a three

circle goniometer. Each set of crystal faces used in this

investigation was oriented to within + lo. The faces were cut -

using a Sparkatron spark erosion machine with the crystal

still attached to the goniometer. After cutting, each set of

surfaces was polished flat and parallel to within + 0.0002 cm -

using hand-lapping techniques. After thorough chemical etch-

ings the crystals were individually sealed in evacuated Vycor

capsules and annealed at approximately 62S°C for a minimum of

five hours, then furnace cooled.

Powder X-ray samples were prepared directly from the as-

grown single crystals by filing from the surface of each

boule. In the case of the Jarrell-Ash boule, powder samples 7

were taken from both the outer as-grown surface and from the

interior by filing from a cleaned and polished spark cut sur-

[I To]

Ni- B-2

Ni-JA-I Ni- JA-2

Figure 1. Orientation and shapes of the samples used in this. investigation, . . dimensions are in cm

f a c e . The powders were s e a l e d i n evacuated Vycor c a p s u l e s and

annealed a t 600°C f o r a t l ' e a s t one and a h a l f hours and then

al lowed t o fu rnace coo l .

X-ray d i f f r a c t i o n p a t t e r n s were t aken u s i n g N i - f i l t e r e d

copper K and Ka2 r a d i a t i o n i n a 1 1 4 mm Debye-Scherrer camera. a1

The l a t t i c e parameter f o r each m a t e r i a l was ob t a ined from a

weighted l e a s t - s q u a r e s program u s i n g t h e T a y l o r - S i n c l a i r -

Nelson-Rei ly e r r o r f u n c t i o n ; t h e program was w r i t t e n by D r .

F . X . Kayser of t h e Ames Laboratory . Table. 2 l i s t s t h e l a t -

t i c e parameter a t 2S°C and t h e X-ray d e n s i t y .

Tab le 2 . L a t t i c e parameter and c o r r e c t e d d e n s i t y f o r t h e as-grown Baker n i c k e l and J a r r e l l - A s h n i c k e l c r y s t a l s . .

0

Sample No. a. (A> P (gm/cm3) A t ..% N i

p~ - - - -p - ~~ - -

a Powder sample t aken from t h e o u t e r s u r f a c e of t h e boule.

b ~ o w d e r sample t aken from t h e i n t e r i o r of t h e boule .

The d e n s i t i e s i n Table 2 have been c a l c u l a t e d by c o r r e c t -

ing t h e mass of t h e average me ta l atom i n t h e c r y s t a l b y . u s i n g

t h e a n a l y s e s g iven i n Table 1. The c o n t r i b u t i o n s due t o i n t e r -

s t i t i a l i m p u r i t i e s were n e g l e c t e d i n t h i s c a l c u l a t i o n . The

v a l u e f o r Avagadrots number used i n t h i s c a l c u l a t i o n was

Acoustic Measurements

Velocity of sound measurements were made using a modified

pulse-echo-overlap (PEO) system whose details are given in

Appendix A. Gold plated 10 MHz x and y-cut quartz transducers

were used to introduce longitudinal and shear mechanical waves

into the samples. Nonaq stopcock grease was used as the bond-

ing agent between the transducer and the sample for all tem-

peratures and propagation modes. The sample was held in a

spring loaded copper container for all measurements. Sample

dimensions were measured using a micrometer and parallelism

of the sample faces was checked by several measurements over

the face of the sample.

Samples Ni-B-1 and Ni-B-2 were used for the zero applied

field fixed point temperature data. The temperature was main-

tained by completely immersing the sample and sample holder in

the appropriate bath. The fixed temperature points were taken

at the normal boiling points of helium and nitrogen, at the

normal sublimation point of dry ice and at the ice point of

water. Sample Ni-JA-1 was used for the continuous temperature

dependence measurements without an applied field. The sample

container was non-inductively wound with a manganin resistance

heater. The temperature was controlled by an Artronix 5301

temperature controller using a Minco S209 platinum resistance

thermometer as the sensing element. The sample temperature

was determined by taping a copper-constantan thermocouple

directly to the sample and using a Newport 2400A digital

millivolt meter to read the voltage.

The usual procedure for taking temperature dependent data

was to immerse the sample in liquid nitrogen. This set the

transducer-to-specimen bond and permitted a data point to be

taken at a well determined temperature. The sample was then

placed in a glass double Dewar whose outer jacket was filled

with liquid nitrogen but whose inner chamber was filled with

dry helium gas. The sample was allowed to come to equilibrium

at the inner,chamber temperature, usually near 80 K.

Data points were then taken in 5 K intervals up to what-

ever temperature the signal became too small to measure,

usually near 250 K. The sample was then allowed to cool back

down to 80 K with spot checks taken at selected temperatures

to see if the data were consistent. During the heating and

cooling cycles the rate of temperature change was approximate-

.ly 1 K per minute. After the sample had again reached 80 K

the inner chamber of the Dewar was filled with liquid helium

and a fixed point at the normal boiling point of helium was

taken. The helium was slowly boiled off after equiiibrating

by maintaining a small current in the sample heater. After

the helium was gone the sample temperature was increased at

the rate of about 1 K per minutc with data being taken at 5 K

intervals beyond 20 K. Data were not taken between 4 K and

20 K because the sensitivity of the thermocouple was very low

in this range. Above 20 K points were taken at 5 .K intervals

up to 100 K to insure that the data coincided with previously

obtained values.

The actual sample temperature was known to only about

+ 2.0 K in the low temperature range where the sensitivity - was low and to about - + 1.0 K at higher temperatures. The tem-

perature controller held the temperature stable to approxi-

mately an order of magnitude better than the temperature was

actually known.

A Harvey-Wells Magnion electromagnet with five in,ch

diameter pole pieces with a three inch air-gap was used for

the single crystal elastic constant AE investigation. Data

were taken at room temperature (296 K) and liquid nitrogen

temperature (77 K). The room temperature values were taken

by holding the sample and sample holder in the desired orien-

tation in the center of the magnet gap with styrofoam blocks

so that the sample could not move. The 77 K data were ob-

tained by placing a quartz tube containing the sample and

liquid nitrogen between the pole pieces. The tube was small

enough in diameter so that the sample could not move when

inside it.

The magnetic field strength was measured by a Rawson

rotating coil gaussmeter. The highest field attainable with

this magnet was 14.4 KOe. although in all cases but one any BE

e.ffect observed was completed before the field reached 4.0KOe.

RESULTS

Zero Applied Field ~easurements

Figures 2, 3 and 4 are plots of data for the three inde-

pendent elastic modes for the [I101 propagation direction as a

function of temperature. The data in these figures have not

been corrected for thermal expansion. Figure 2 is a plot of

the longitudinal mode, 1/2(Cll+C12+2C44); Figure 3 is a plot

of the shear'. mode, C44; Figure 4 is a plot of the shear mode,

1/2(Cll-C12). Appendix B contains a brief discussion of

crystal elasticity including the possible- elastic constant

combinations available from specific stress wave propagation

directions.

Table 3 lists the experimentally determined elastic con-

stants with no thermal expansion corrections, also in Table 3

are the values of smoothed elastic constants that have been

corrected for thermal expansion using the data of Nix and

MacNair (19) for pure nickel. In Figure 3 there is an appar-

ent jump in the data near 210 K. The jump in the data is

apparently due to the fact that the sample was held at that

temperature overnight and finishing the data run the next

morning.

The reproducibility and internal consistency of the data

were extensively tested at one temperature (77 K) by measuring

all the propagation modes for all the crystals several times

after thermal cycling and bond changes. Table 4 contains the

TEMPERATURE (%) Figure 2. Nickel [I101 longitudinal mode temperature dependence in zero applied

magnetic field

Figure 4. Nickel (1101 shear mode temperature dependence in zero' applied magnetic field

0.56

0.54 CU E 0

0.52- Q) C )r Q

0.50 SY 0 - X

. 0.48- fi 0 I - - 0.46 0 V

N 0.44-

0.42

0.40

I I I 1 I I I 1 1

- - 0 O O O O O O o o o

0 ~ ~ ~ ~ 0 0 0 0 0 0

O O o 0 O O o O O

O 0 o o O o o

- O o o O o o -

- 4

- -

-

- -

I I I I I I I 1 1 0 25 50 75 100 125 150 175 200 225 250

TEMPERATURE ( O K )

Table 3 . Values of t h e raw exper imenta l d a t a t aken i n zero a p p l i e d f i e l d and va lues of the . .smo.o the.d d.at.a c.o.r.re.c.t.ed. . fo r the.rma.1. .e.xp.ans.i.o.n ,. x 1.0 dy.ne.s/.cm2

Temp. Experimental va lues Smoothed v a l u e s ' O K 1 / 2 (Cll+C12 C 4 4 1 / 2 (Cll-cl2) 1 / 2 (Cll+C12 C44 1 / 2 (Cl1-clZ)

+2c44) +2c44.)

Table 4. Data taken ,at 77 K - to determine repeatability and internal c.oa.s.i.s.t.en.cy , .zero applied magnet.ic field

-~~

Sample no. Propagation Elastic mode x1012 dyne/cm 2

d i.r e.c t.i on

Ni-B-1

Ni-B-1

Ni - JA- 1 Ni-B-1

Ni-B-1

Ni-B-1

Ni-JA-1

Ni-JA-1

Ni-B-2

Ni-B-1

Ni-B-1

Ni - JA- 1 Ni-B-1

N ~ - B - I ~ ~ i - ~ - l ~

Ni-B-2

Ni-B-2

Ni-JA-1

N ~ - J A - ~ ~ ~ i - ~ - l ~

~ i - ~ - l ~

~i - JA- la Ni-JA-2

Ni-JA-2

a C11 and C12 were calculated by solving the three simul- taneous equations obtained from the particular [110] direction of propagation.

measured v a l u e s f o r t h e v a r i o u s p ropaga t ion modes, t h e c r y s t a l

and t h e p a r t i c u l a r s e t of f a c e s used . For N i - B - 1 and N i - J A - 1

t h e r e were two p o s s i b l e s e t s of d i f f e r e n t [I101 d i r i c t i o n s and

one [OOl] d i r e c t i o n of p ropaga t ion . The measured v a l u e s of

N i - J A - 2 a r e a l s o inc luded i n t h i s t a b l e . Not a l l of t h e f a c e s

of each c r y s t a l were used because of l i m i t e d s i z e 0.r because

t h e o r i e n t a t i o n was known t o b e skewed by more t han lo .

F i e l d Dependent Measurements

F igu re s 5 t h r u 10 r e p r e s e n t t h e room tempera ture magnet ic

f i e l d dependence of t h e d i r e c t l y measurable e l a s t i c modes.

Graphs o f t h e f i e l d dependence f o r waves p ropaga t ing i n t h e

[ I l l ] d i r e c t i o n a r e n o t inc luded because t h e s i g n a l ampl i tude

a t a l l f i e l d s was t o o sma l l t o a c c u r a t e l y measure. For most

of t h e samples i t was imposs ib le t o p ropaga te any s i g n a l a t

a l l a t room tempera ture w i thou t app ly ing a magnet ic f i e l d ,

approaching 1 k i l o g a u s s , consequent ly many of t h e p o s s i b l e

p ropaga t ion modes f o r v a r i o u s c r y s t a l s a r e n o t r e p r e s e n t e d .

One of t h e i n t e r e s t i n g t h i n g s no ted i n t h i s i n v e s t i g a -

t i o n was t h e behavior of t h e s h e a r v e l o c i t y cor responding t o

1/2(C11-C12). I t 'was found t h a t t h e change i n sound v e l o c i t y

was v i r t u a l l y over when t h e a p p l i e d f i e l d s t r e n g t h reached 4.0

KOe f o r t h e o t h e r modes, n o t s o f o r 1/2(C11-C12). Th is mode

showed a smal l f i e l d dependence o u t t o t h e h i g h e s t f i e l d

a t t a i n a b l e w i t h t h e magnet arrangement.

Tab le 5 c o n t a i n s t h e s a t u r a t e d v a l u e s of t h e room tem-

p e r a t u r e e l a s t i c c o n s t a n t s and t h e v a l u e s r e p o r t e d by o t h e r

Figure 5. Magnetic field dependence of C11 at 300 K, propagation direction is E o o l l , HI 1 [0101

2.60

N

€ 2.56-

\

Q) C )r

u N . - 0 - * 2.52- - -

I I I I I 1 1 I I I

- o FIELD INCREASING -

v FIELD DECREASING -

- -

0 v 0 0 9

- " ' 1 0

N

0 I-'

v

0 0 0

0 0 2 . d v v

V

- H ( .KOe)

Figure 6. Magnetic field dependence of 1/2(Cll+Cl~+ZC44) at 300 K with H I 1 [OOl]

N 3.262,- E \O g

' 0 E! 0 . 3 3.258<- 3 0 N + N - 0 + - 3254 - 0 u

N 12

3.250

I I I I I I ' I I I

v A' 0

- -

o FIELD INCREASING - v FIELD DECREASING

- -

- -

- -

-

0 01 I I I I I I 1 I

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

o FIELD INCREASING v F I E LD DECREASING

Figure 7 . Magnetic f i e l d dependence of C44 a t 300 K , propagation direct ion i s [0011 , H I 1 [0101

' H ( K O e

Figure 8. Magnetic field dependence of C44 at 300 K, propagation direction is P 1 0 1 , H I I [0011

1,

1.27.

1.25

1.23

1.21

0 ' -0.4 0.8 1.2 1.6 2.0 2,.4 2.8 3.2 3.6 4.0

I I I I I 1 I I I

o FIELD INCREASING - -

v FIELD DECREASING - -

- -

- -

7 -

- - 0

I v

I I I I I I I I

t- - o FIELD FNCREASING

C v FIELD DECREASING 0.490

Figure 10. Magnetic f i e l d dependence o f 1/2(Cll-C12) a t 300 K showing high f i e l d dependence

Table 5. Values of the saturated elastic constants at 296 K (10 MHz signal frequency)

Worker C44 C12

x1612dyne/cm2 , xl~'~d~ne/crn 2 x l ~ ~ ~ d ~ n e / c r n 2

Present investigation 2. 51S9 1. 2341 1.5368

-\lers et al. (14) -- 2.508

Bozorth et al. (1) -- 2.53

Neighbors et al. (12) -- 2. 5Z8

Epstein and Carlson (15) 2.523

2arma and Reddy (16) 2.503

De Klerk and ~ u s ~ r a v e ~ (13) 2. 465

sakuraia (XI) 2.51

a Used signal frequency of 5 MHz.

workers. At liquid nitrogen temperatures there was essenti-

ally no measurable single crystal AE effect, however, it was

noted that near 1 KOe a marked change in signal amplitude

occurred for all modes.

For the shear modes the magnetic field was applied

parallel to the atomic motion and perpendicular to it. No

measurable change in velocity was observed for C44, but a 0.2%

change was found in 1/2 (Cll-C12). Due to the sample holder

geometry the field could not be applied in any manner except

perpendicular to the particle motion for the longitudinal

velocity measurements. Levy and True11 (18) did attempt to

measure a single crystal AE effect for a field parallel and

a field perpendicular to a longitudinal particle motion, but

their instrument was not sensitive enough to detect the small

velocity changes expected.

DISCUSSION

Magnetic Field Dependence

The attempt to measure single crystal AE effects for var-

ious propagation modes was only partially successful. Since

pulses could not be propagated through the samples at room tem-

perature and zero applied field for most of the possible modes,

it is difficult to say anything quantitative about the effect.

Two approaches were taken to remedy the problem; for those

modes where a signal could be obtained without a field at room

temperature and for those modes where a reasonable estimate of

the demagnetized elastic constant was possible the change was

expressed as

A = [(cS-co)/c0] X 100 (1)

where cS is the particular elastic modulus at saturation and

Co is the same elastic modulus at zero applied field.

Table. 6 lists directly measured and estimated A's for the

crystals. The second approach was to fit the temperature de-

pendent, zero field data to a functional form and extrapolate

the demagnetized data to 300 K to calculate A. The data were

fit to a function of the form

where cYj is the elastic constant value at zero degrees Kelvin,

T is the temperature in degrees Kelvin, Cij is the value of

the elastic constant at any particular T and s and t are

a.dj11st.a.hle parameters. The values of s and t were found by

Table 6.. The change i n t h e e l a s t i c moduli a t 300 K from ze ro . , app.1.i.e.d .f.i.e.l.d .t.o. .s.a.tu.r.a.t.i.on, x 101.2 dyne./cm2

Sample E l a s t i c c o n s t a n t Zero S a t u r a t e d . . mode. . . f i e l d (12.4 KOe) A %

N i - B - 1 1 / 2 (C11+C12+2C44) 3.248 3.262 0.43

11 I t 3.24a 3.258 0.56

--

a Es t ima t ed v a l u e s .

b ~ x t r a p o l a t e d from ze ro f i e l d , t empe ra tu r e dependent d a t a . C ~ f f e c t obse rved , b u t t o o s m a l l t o q u a n t i t a t i v e l y measure.

linearizing Equation 2 and varying t in small steps until the

proper value for the slope was obtained. The linearized form ,

of the equation is;

Thus the correct slope was constrained to be -1.00 and the

intercept of the line yielded In s. The straight lines cal-

culated by varying t were fitted using a linear-least-squares

approach to obtain the slope and intercept for each t. The

data used in the computation were corrected for thermal expan-

sion. The values of the smoothed data given in Table 3 were

computed using Equation 2. The validity of Equation 2 is dis-

cussed by Varshni (20).

The fit to the experimental values obtained from Equation

2 was quite good, the difference between the measured and com-

puted elastic constants was usually on the order of - + 0.0005

x 1012 dyne/cm2. The fitting parameters for the temperature

dependent data are given in Table 7.

Table 7. Fitting parameters for the zero field temperature dependence of the elastic c.on.s.t.ants.

Parameter

I n Table 6 i s a l s o t h e A E e f f e c t change c a l c u l a t e d from

t h e ze ro f i e l d d a t a e x t r a p o l a t e d t o 300 K and t h e d i r e c t l y

measured s a t u r a t i o n va lues a t 300 K . The va lues f o r A

ob ta ined from t h e e x t r a p o l a t i o n tend t o be l a r g e r f o r a p a r -

t i c u l a r mode than t h e e s t i m a t e d o r d i r e c t l y measured A .

The magnitude of t h e s i n g l e c r y s t a l A E e f f e c t f o r t h e

e x t r a p o l a t e d c a s e s ag rees f a i r l y w e l l w i t h t h e d a t a of Bozorth

e t a l . ( I ) , excep t t h a t t hey f i n d t h a t t h e changes i n t h e -- s h e a r c o n s t a n t s a r e s i g n i f i c a n t l y l a r g e r t han t h a t of t h e

l o n g i t u d i n a l c o n s t a n t . They f i n d t h a t t h e changes i n t h e

s h e a r v a l u e s a r e between t h r e e and f o u r t imes a s l a r g e a s

t h e change i n t h e l o n g i t u d i n a l mode they measured. However,

one must t a k e i n t o account t h a t t h e domain c o n f i g u r a t i o n f o r

a g iven sample shape may be q u i t e d i f f e r e n t from t h a t of

ano the r ( 3 ) , s o t h a t on ly approximat ions may be made f o r t h e

r e l a t i v e change between p ropaga t ion modes of d i f f e r e n t samples.

No a t t emp t was made t o c a l c u l a t e demagnetizing f a c t o r s

f o r t h e sample geometr ies used i n t h i s i n v e s t i g a t i o n a s t h e

problem i s ve ry complex. The re fo re , t h e a c t u a l f i e l d s een by

an atom i n t h e c r y s t a l i s n o t known. However, t h e agreement

between o t h e r work, which used d i s k shaped samples ( I ) , and

t h e p r e s e n t work u s i n g roughly cub ic samples r e g a r d i n g t h e

f i e l d a t which t h e A E e f f e c t was completed i s q u i t e good.

Genera l ly t h e f i e l d a t which t h e e l a s t i c c o n s t a n t appeared t o

s t o p changing was n e a r 1 . 5 KOe. For 1/2(Cll-CI2) t h e e l a s t i c

constant showed a rapid change, as did all the other measured

modes, up to near 1.5 KOe but then instead of remaining a

virtual constant there was, a small field dependence out to

the highest fieid attainable. The significance of this is

not known.

An attempt was also made to measure the AE effect at

liquid nitrogen temperature. The magnitude of the change was

very small for a given sample. As mentioned in the previous

section a marked amplitude change occurred at this temperature

near 1.0 KOe for all propagation modes but there were virtu-

ally no velocity changes. The values of the saturated elastic

constants at 77 K are listed in Table 8.

Table 8. Elastic constants of nickel at 77 K in a saturating magnetic field, x 1012 dyne/cm2

Investigator

Present work

Alers et al. (14) -- 3 . 3 6 2

Sarma and Keddy (17)

Present work 3 . 3634 1.2975 0.5255 (.demagnetized)

The values of the present work in Table 8 represent aver-

ages of all the measurements from all of the samples for both

the saturated and demagnetized data. As can be seen from the

average values an apparent measurable AE effect does exist at

77 K but the effect for any particular measurement was small.

The uncertainty for the average values of the elastic con-

stant combinations is roughly - + 0 . 0 0 4 ~ 1 0 ~ ~ dyne/cm2, with the

largest uncertainty in 1/2(C11-C12) and the smallest uncer-

tainty in C44.

Zero Field Temperature Dependence

.The most striking difference between the present investi-

gation and the temperature dependence of the.magnetically

saturated elastic constants done by Alers, Neighbors and Sato

(14) is the difference in C12,. Even making allowances for the

AE effect the difference at all temperatures is large, being

on the order of 1.5 to 2%. The overall accuracy of the elas-

tic constants measured in this investigation is estimated to

be - + 0.5% and probably better than that at the fixed tempera-

ture points. The temperature dependent values of CI1, C44 and

C12 found in this investigation were extrapolated to the Curie

2 temperature (631 K) with the results being 2.332x1012 dyne/cm , 1 . 069x1012 dyns/cm2 and 1 . 5 2 5 ~ 1 0 ~ ~ dyne/cm2, respectively.

The differences between the extrapolated and measured .values

of Alers -- et al. (14) are 1.30%, -3.26% and 3.11%, respectively,

which is adequate considering the length of the extrapolation.

As was mentioned in the previous section there was an

apparent "annealing" phenomenon observed in the absence of a

magnetic field. The effect was most prominent in C44, see

Figure 3 near 210 K. 'I'he effect was first noticed when the

sample was held at the same temperature for a few hours or,

more and then completing the data run. If the sample was then

cooled back down from the break in the data, the line of new

data was displaced roughly parallel to that observed previous-

ly.. The effect was looked for and found in the other shear

mode and the longitudinal mode. The magnitude of the effect

,was much smaller in the latter cases. The effect did show the

same characteristics upon cooling and heating as did C44. In

Figure 2 near 200 K and in Figure 4 near 165. K .small jumps in

the data can be seen due to holding the sample at a constant

temperature for a period of time. The data used in the fitting

function, especially for C44, was chosen to minimize this

effect.

One possible explanation is a slow rearrangement of the

domain configuration. This is proposed because the PEO con-

tinued to input stress waves into the sample throughout the

time the sample was being held at temperature. However, the

changes in velocity are quite small, being almost within the

estimated experimental error.

The bulk modulus and Debye temperature were calculated

for the demagnetized case. The bulk modulus at 300 K was cal-

culated for the demagnetized state using the extrapolated

elastic constants while the saturated state bulk modulus was

calculated from the averaged values of the saturated measure-

12 ments. The demagnetized bulk modulus at 300 K is 1.8479x10 .

dyne/cm2, while that for the saturated case is 1.8633x1012

dyne/cm2.

The Debye temperature was calculated from the demagnet-

ized constants using the 4.2 K values as being good 'approxi-

mations of the zero Kelvin values. The Debye temperature was

calculated using the method of Overton and Schuch (21). The

calculated temperature was found to be 471.6 K for the demag-

netized state'as.compared to 476 calculated by Alers' et -- al.

(14) from their saturated measurements.

SUMMARY

A series of nickel single crystals of different starting

material and orientation were prepared to measure the magnetic

field dependence of the single crystal elastic constants and

to measure the temperature dependence of the elastic constants

in the demagnitized state. The measurements were performed

using a pulsed ultrasonic method of determining sound wave

velocities in solids.

Three things of interest found in this investigation

were; (1) the apparent magnetic field dependence of 1/2(Cll-

C12) at high magnetic fields; (2) the large difference between

the previously considered "best" value of C12 and the value

obtained in this investigation; (3) the apparent "annealing"

effect of the elastic constants in the demagnetized state.

Measurements of quantitative values for the single crystal AE

effect were only partially successful due to signal attenua-

tion at zero applied magnetic field. Extrapolation of the

demagnetized data using Varshni's functional form (20) was

used to estimate the single crystal AE effect and to calculate

the room temperature zero applied field elastic constants and

bulk modulus.

The Debye temperature was calculated from the low temper-

ature demagnetized data and compared to the value calculated

by Alers -- et al. (14) from saturated data and no significant

difference was found.

BIBLIOGRAPHY

1. R. M. Bozorth, W. P. Mason and H. J. McSkimin, 'Be'll Sys't'em Te'ch.' J. ,' '30, - 970 (1951) .

2. R. M. Bozorth, "Ferromagnetksm," D. Van Nostrand Co., New York, N.Y., 1951, pp. 268-269.

3. C. Kittel,' Rev.' Mo'd'.' Phys., '21, 54.1 (1949). - 4. H. J. Williams and J. G. Walker, Phys. Rev., - 83, 634

(1951).

5. R. Carey and E. D. Issac, "Magnetic Domains and Tech- niques for Their Observation," Academic Press, New York, N.Y., 1966.

6. R. Becker and W; Doring, "Ferromagnetismum," Springer, Berlin, Germany, 1933.

7: W. P. Mason, Rev. Mod. Phys., 25, 136 (1953). - 8. 0. Bostanjoglo, Phys. Stat. Sol. (a), 25, K9 (1974). 9. J. De Klerk, Proc. Phys. Soc. London, - 73, 337 (1959).

10. G. A. Alers, J. R. Neighbors and H. Sato, J. phys. Chem. Solids,, - 9, 21 (1958) .

11. J. Sakurai, J. Phys. Soc. Japan, - 19, 311 (1964).

12. J. R. Neighbors, F. W. Bratten and C. S. Smith, J. Appl. Phys., 23, 389 (1952).

13. .J. De Klerk and,M. J. P. Musgrave, Proc. Phys. Soc. London, - B68, 81 (1955).

14. G. A. Alers, J. R. Neighbors and H. Sato, J. Phys. Chem. Solids, - 13, 40 (1960).

15. S. G. Epstein and 0. N. Carlson, Act'a Met., 13, 487 (1965).

-

16. V. P. N. Sarma and P. J. Reddy, Phil. Mag 27, 769 (1973).

- 9 -

17 . V. P. N. Sarma and P. J. Reddy ,' 'Phy's .' 'S'ta't . Sol'. '('a) ,' '16 , - 413 (1973).

18. S. Levy and R. True11 ,' 'Rev.' Mo'd.' 'Ph'ys . ,' '25, - 140 (1953) . 19. F. C. Nix and D. MacNair, Phys'.' 'Rev. ,' '60, - 597 (1941).

20. Y. P. Varshni ,' 'Phy's..' 'Rev.' B ,' - 2, 3952 (1970).

21. W. C. Overton and A . F. Schuch, USAEC Report LA-3615-MS, Los Alamos, New Mexico, Dec. 1966.

22. G. A. Alers, "Physical Acoustics," Vol. IV-A, W. P. Mason, Ed., Academic Press, New York, N.Y., 1965, pp. 277-297.

23. J. E. May, Jr., IRE Na'tl. Conv. Rev., - 6, Pt. 2, ,134 (1958).

24. E. P. Papadakis, 'J.' Acou's't. Soc. Amer., - 24, 1045 (1967).

25. D. H. Chung, D. J. Silversmith and B. B. Chick,"Rev'. 'Sci. In'st. ,' - 40, 718 (1969).

26. F. R. Eshelman, USAEC Report IS-2594, Ames, Iowa, Jan. 1972.

27. P. D. Waterman, Phys. Rev., - 113, 1240 (1959).

28. A. E. H. Love, "A Treatise on the Mathematical Theory of Elasticity," Dover Publications, New York, N.Y., 1944.

29. J, F , Nye, "Physical Properties of Crystals," Oxford University Press, London, England, 1957, pp. 131-138.

30. C. Kittel, " Introduction to Solid State Physics," 4th ed. , John Wiley and Sons, New York, N.Y., 1971, pp. 144- 155.

31. Y. Shirakawa, Y. Tanji, H. Moriya and I. Oguma, J. Jap. Inst. Metals Sendai, - 33, 116 (1969).

32. J. Sakurai, M. Fujii, Y. Nakamura and H. Takaki, 'J.' Phys. Soc. Japan, - 19, 308 (1964).

40

APPENDIX A

Pulse-Echo-Overlap System

It is generally accepted that the easiest and most accu-

rate way to determine the adiabatic single crystal elastic

constants is by measuring the velocity of sound in the sample.

Various electronic schemes have been devised to introduce

so.und waves in samples and to measure their velocity (22).

The ease in using the pulse-echo-overlap system (PEO)

outlined by May (23) and developed by others (24,25,26) and

the elimination of an empirical determination of the transit

time error produced by the transducer-to-specimen bond thick-

ness made the PEO the most promising for this work.

Figure A1 is a block diagram of the PEO ultrasonic system

that was constructed at this laboratory. The components of

the system are: a General Radio 1330-A bridge oscillator; a

Computer Measurements Company 7266 Counter; a General Radio

1217-B pulse generator to serve as the pulse shaper; a Hewlett-

Packard 222A pulse-delay generator; a Hewlett-Packard 18QD

oscilloscope with 1820C and 1801A plug-in modules; an Arenberg

attenuator; a frequency divider and pulsed oscillator con-

structed by the Ames Laboratory Instrumentation Group. Fig-

ures A2 and A3 show the circuit diagrams of the frequency

divider and pulsed oscillator, respectively.

The system functions by using the bridge oscillator fre-

quency as a relative time base to synchronize all the com-

COUNTER P

TRIGGER

FREQUENCY

PULSED

7 OSCILLOSCOPE * j INPUT I TRANSDUCER b

Figure A l . PEO block diagram

4vDC MVlDE BY 4VDC

INWr HI' 213904

- - IOMlj/20 IoMfd/20V

II + 1 1 -

2N3804

I KC) lo&

1 - OUTPUT - w-

(NOT USED) RESET ( W T USED)

- - STANCOR P-6134

HEAT SINK . .

Figure AZ'.' Frequency divider schematic diagram

LRCR ctcTERMlWES CUM. FREQUENCY

ALL RESISTOS I /2 WATT UQPT WHERE wOmD

Figure A3. Pulsed oscillator schematic diagram

ponents to the same repetition rate. The frequency of the

sine-wave generated by the bridge oscillator is divided by

10, 100 or 1000 by the frequency divider; the pulsed oscilla-

tor and pulse-delay generator are then synchronized together

by a -4 V spike output from the divider.

When measuring the transit time the oscilloscope is

triggered externally by thebridge oscillator and the appro-

priate portion of the oscilloscope^ trace is intensified by

the pulse-delay generator. To get an overlap, the horizontal

time base of the oscilloscope is adjusted and the final over-

lap is accomplished by adjusting the bridge oscillator fre-

quency until there is a one-to-one cycle correspondence in

each echo packet.

Generally, there is not a precise one-to-one cycle cor-

respondence even in adjacent echoes due to dispersion, of the

signal by orientation errors (27), dislocations and other

imperfections in the sample. This difficulty is taken care

of by overlapping a large number of echoes in one group and

using an average transit time which gives the group the best

cycle correspondence rather than using the transit time ob-

tained from overlapping only two adjacent echoes.

The estimated accuracy of the system is one part in ten

thousand with respect to the transit time. However, due to

uncertainties in the sample dimensions, orientation errors,

etc.,the total accuracy of the measured sound velocities is

estimated to be on the order of 0.25%.

APPENDIX B

Crys ta l E l a s t i c i t y And l a v e Propagation

For s u f f i c i e n t l y small s t r a i n s t h e s t r e s s is d i r e c t l y .

propor t ional t o t h e s t r a i n . This behavior is expressed by

Hooke's law (28) ;

- ' i j - 'ijkkekk (PA)

Where the u i j t s a r e the t ensor s t r e s s components, t h e r k k t s a r e

a r e t h e t ensor s t r a i n components and t h e C i j k Q ' s a r e t h e

e l a s t i c cons tants . Due t o c o n s t r a i n t s placed upon the c rys -

t a l we have t h e condi t ion

- - - ' i jkk 'i j ek - ' j ike (2A)

This reduces t h e number of poss ib le independent e l a s t i c

cons tants from 81 t o 36. The no ta t ion can be condensed t o a

mat r ix form,

'i = C . . E

11 j ( 3A)

I t can be f u r t h e r shown t h a t C i j - - C j i . This leaves

only 2 1 poss ib le independent cons tants f o r t h e lowest sym-

metry c r y s t a l .

Nye (29) and K i t t e l (30) have shown t h a t f o r a cubic

c r y s t a l t h e r e a r e only t h r e e independent e l a s t i c cons tan t s ,

labe led C l l , C l 2 and C 4 4 . The equat ions of motion of a s t r e s s

wave i n a cubic c r y s t a l f o r var ious o r i e n t a t i o n s have been

t r e a t e d by K i t t e l . He shows t h a t t h e fol lowing combinations

of t h e e l a s t i c cons tants can be obtained from v e l o c i t y of

sound measurements if the density, of the sample is known.

For a wave propagating in the. [I101 direction there 'are

three possible solutions, for a longitudinal wave

2 pvL = 1/2(Cll+C12+2C44) = CL ( 4A)

for a transverse wave polarized in the [OOl] direction

for a transverse wave polarized in the [ll~] direction

Propagation in the [loo] or [Ill] direction only allows

measurements of two combinations of elastic constants becausk

only one transverse mode is possible. For waves propagating

down [loo] the equations are

2 PVL = CI1 (7A)

where vL and vT are the longitudinal and transverse velocities, i

respectively. For waves propagating down [Ill] the equations

are

where vL and vT are, again, the longitudinal and transverse

velocities, respectively and p is the crystal density in all

the equations.

1 would first like to thank Dr. F. X. Kayser for the

time and guidance he provided in this investigation; and to

thank Mr. G. L. Stowe for growing the single crystals used

in the project.

A special thanks is extended to Mr. R. Prior and Mr. G.

Holland of the Ames Lab. Instrument Group for the time and

effort to build and keep running portions of the pulse-echo-

overlap system.

I also would like to thank Dr. C. W. Chen for the gener-

ous use of.the magnet required in this investigation.