L ^ ^

MAGNETIC SUSCEPTIBILITIES OF TITANIUM AND

VANADIUM IN CORUNDUM STRUCTURES

by

DONALD JENE ARNOLD, B.S., M.S.

A DISSERTATION

IN

PHYSICS

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

May, 1970

/](L

T3

ACKNO\ LEDGMENTS

I would like to express my sincere appreciation to Dr. Raymond W.

Mires for suggesting the problems and directing the research reported

in this dissertation; to Dr. C. Richard Quade and Alfred R. Smith for

many stimulating discussions; and to F. Allan Wise, William K. Dean

and Morris C. Greenwood, Jr. for assistance in the data analysis.

I would also like to thank Dr. Quade, Dr. C. C. Lin, Dr. L. A.

Boatner and Dr. E. D. Jones for supplying the crystals.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

LIST OF FIGURES v

LIST OF TABLES vi

I. INTRODUCTION 1

II. APPARATUS 4

III. EXPERIMENTAL METHOD 11

Susceptibility Measurements 11

Calibration of Field-Gradient Product 11

Temperature Measurement 13

Experimental Procedure 21

Samples and Suspensions 24

IV. DIAMAGNETISM OF HOST CRYSTALS 27

V. MAGNETIC SUSCEPTIBILITIES OF Al 0 :V " 37

Introduction 37

Experimental Data and Analysis 39

Conclusion 46

3+ VI. MAGNETIC SUSCEPTIBILITIES OF Al 0 :Ti 47

Introduction 47

Theory 49

Experimental Data and Analysis 51

Discussion 60

Conclusion 65

VII. MAGNETIC SUSCEPTIBILITIES OF V 0 67

Introduction 67

• • • 111

IV

Susceptibility Measurements 68

Analysis of Results 69

Conclusion 77

LIST OF REFERENCES 80

APPENDIX I 83

APPENDIX II 86

APPENDIX III 88

LIST OF FIGURES

Figure 1. Schematic Diagram of Apparatus 5

Figure 2. Schematic of Electrical Apparatus 9

Figure 3. F/m Measurements for A1_0 14

Figure 4. Circuit for Ge-Resistance Thermometer 16

Figure 5. Sample Temperature Correction vs. Exchange Gas Pressure 18

Figure 6. Correction for Difference in Temperature Between Sample and Thermocouple 20

2+ Figure 7. ESR Spectrum of MgO:Mn 30

2+ Figure 8. Measured Susceptibility of MgOiMn 32

Figure 9. Measured Susceptibility of "Pure" Al 0„ 33

3+ Figure 10. Energy Levels of V in a Trigonal Field 38

3+ Figure 11. Measured Susceptibilities of Al^O :V 41

3+ Figure 12. Energy Levels of Ti in a Trigonal Field . . . . 50

3+ Figure 13. Measured Susceptibility of Al_0, :Ti ,

H Perpendicular to the c-axis 53

3+ Figure 14. Measured Susceptibility of Al 0 :Ti ,

H Parallel to the c-axis 54

Figure 15. Reciprocal Susceptibility of V 0 , H Perpendicular to the c-axis 71

Figure 16. Reciprocal Susceptibility of V 0 , H Parallel to the c-axis 72

Figure 17. Comparison of Reciprocal Susceptibilities . . . . 74

LIST OF TABLES

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10,

Table 11,

Samples 26

Measured Magnetic Susceptibilities 29

Diamagnetic Susceptibilities 35

3+ Magnetic Susceptibilities of A1^0„:V in Units of (10"^ emu/gm) 40

Parameters Used to Fit Experimental ^ Magnetic Susceptibilities of Al 0 :V 45

3+ Magnetic Susceptibilities of A1„0 :Ti in Units of (10"^ emu/gm) 52

3+ Reported Parameters for Al-0 :Ti 56

3+ Analysis of Experimental Results for A1„0 :Ti . 64

Average Experimental Mass Susceptibilities for V 0 Single Crystal 70

Constants for the Cubic Equation Determined by Least Squares 73

Curie-Weiss Parameters Using x = 2.8 x 10 emu/gm 78

VI

CHAPTER I

INTRODUCTION

A significant part of current research into the interaction of

electrons in solids is concerned with the effect of these interactions

on the optical and magnetic properties of the crystals. Studies of

1 2 the temperature dependence of magnetic susceptibilities, * electron

3 4

spin resonance (esr), and optical spectra provide quantitative data

which can be used to analyze these interactions. The degree of lo­

calization of these electrons, to some extent, determines the method

of analysis. The mutual interactions between electrons, as well as

the interactions with their environment, are also important in in­

terpreting the results. Of course, real systems vary continuously

from the isolated free ion with no interactions, to the systems con­

taining all of these interactions, such as most metals and semicon­

ductors .

One type of system which has received a considerable amount of

attention is the chemical "complex." A chemical complex consists

of a central ion surrounded by a more-or-less symmetrical array of

ions or molecules. The earliest detailed treatment of this system,

called the crystal field theory, was published by Bethe. It was

used by Schlapp and Penney and by Van Vleck to explain magnetic 8 9

properties. Van Vleck and Finklestein * first applied crystal field

theory to electronic absorption spectra in the visible region. Bas­

ically, the theory treats the central ion as being subjected to the

electrostatic field of the surrounding ions or molecules, which are

called ligands. The theory has also been applied to diamagnetic ionic

crystals which contain a small concentration of a magnetic ion impu­

rity. The small concentration is necessary to minimize the inter­

action of the neighboring impurity ions. An outline of the theory

is given in Appendix I. Extensions of the original theory, such as

magnetic pairs, covalency, Jahn-Teller effects and other sophistica­

tions have been incorporated into the more advanced ligand field

theory. The results of these studies provide information on the

structure of the host crystal, including distortion due to the impu­

rity substitution, bonding within the solid, and the effect of the

crystal field on the energy levels and g-factors of the magnetic ion

impurity.

Since the 3d transition metals are the first to be affected by

crystal fields, a considerable amount of work has been published on

them. The theoretical calculations carried out by Orgel and by

12 13

Tanabe, Sugano and Kamimura ' are still useful in the interpreta­

tion of optical spectra. The transition metals, and more recently

the rare earths, as impurities in diamagnetic host crystals are the

subject of much of the current work in atomic and solid state physics.

This study was concerned mainly with trivalent titanium and vanadium

in corundum; however, some additional experiments on other diamag­

netic host crystals and single crystal vanadium sesquioxide (V 0„)

are also included. Some optical spectra and electron spin resonance

experiments were done in addition to the primary measurements of mag­

netic susceptibilities.

Measurements of the temperature dependence of magnetic suscep­

tibilities provide a unique method of combining the results of optical

spectra and electron spin resonance data into a single experiment.

Values for excited state g-factors, small energy spacings and Van

Vleck susceptibilities can be determined for systems for which di­

rect measurements are difficult or impossible. Magnetic susceptibility

measurements fall into two categories, absolute measurements and

anisotropy measurements. The absolute, or principal, susceptibilities

are usually determined for crystalline samples by measuring the force

14

on a magnetic sample in a non-homogeneous magnetic field, and an­

isotropic susceptibilities can be obtained from torque measurements

in a homogeneous magnetic field. The two methods can be coupled

together to obtain a better understanding of the system. The anisot-

3+ 3+

ropy measurements on the Al 0 :Ti and Al 0 :V systems were per­

formed by Smith. As is often the case, the measured principal values

at a reference temperature were used to calibrate the anisotropy

measurements, and the anisotropy measurements were used to corroborate

assumptions about impurities whose contributions must be considered

in these more dilute systems. The apparatus and experimental method

used for the absolute susceptibility measurements are described in

Chapters II and III.

The experimental results and pertinent discussions for each sys­

tem studied are presented in the succeeding chapters, the diamagnetism 34-

of some host crystals in Chapter IV, the Al 0 :V in Chapter V, the 3+

Al 0 :Ti in Chapter VI and the metallic V 0 in Chapter VII.

CHAPTER II

APPARATUS

The magnetic susceptibilities were measured at a given tempera-

14 ture by the Faraday method. This is essentially a measurement of

the force exerted on a magnetic sample by a non-homogeneous magnetic

field. A schematic drawing of the apparatus is shown in Figure 1.

A Cahn RG-2000 electrobalance with a maximum sensitivity of 0.1 yg

was used to measure the magnetic force on the samples. The balance

weighing mechanism (A) was enclosed in a glass envelope (B) which

was fitted on one end with a metal end plate (L) through which elec­

trical connections were made between the balance and its control unit.

A pyrex hangdown tube (C) was fitted to the envelope by a ground-

glass joint, and a fine quartz fiber (D) was used to suspend the

samples from the balance arm. The fibers were in three sections to

facilitate changing samples and were interconnected by small u-shaped

hooks. The hooks between the joints of this quartz fiber were ce­

mented with "secretarial correction ink" to reduce torsion, and the

ink was removed with acetone. Electrical connections into the sample

chamber were made through the tube (E) which was vacuum sealed with

Cenco Softseal Tackiwax. Small leaks were sealed with General Elec­

tric Gevac (TM) vacuum leak sealer. Two hangdown tubes were used

for the low temperature work. These differed only in that one had

installed in it a small tubular copper oven with a carbon resistor

(26 fi, 4 w) as a heating element. The sample was suspended inside

Figure 1. Schematic Diagram of Apparatus.

the tubular oven about 5/8 inch below the upper end. Both tubes were

equipped with a Cu-constantan thermocouple and a Cryocal, Inc. ger­

manium resistance thermometer for temperature measurements between

1-300 K. For measurements above room temperature, a third hangdown

tube containing another copper oven and a chromel-alumel thermocouple

was used. This oven was made of thin-walled copper tubing with 100 ^

of nichrome wire non-inductively wound on the tube for the heating

element. The lower end of this hangdown tube was surrounded by a

silvered vacuum jacket to confine the heat to the sample area and

reduce heating of the magnet poles. In each case, the temperature

sensors were insulated from their surroundings and mounted within

1/4 inch of the sample. In the low temperature hangdov/n tubes, the

thermocouple and resistance thermometer were in thermal contact. The

electrical wires were fed down the hangdown tubes inside 1.7 mm O.D.

pyrex tubing which was pressed against the inside of the hangdown

tubes with spring clips made of 5 mil beryllium-copper sheet.

The low-temperature hangdown tubes fit into a double Dewar sys­

tem used with liquid nitrogen in the outer Dewar (G) and liquid helium

in the inner Dewar (H). The top of the inner Dewar was fitted with

a manifold (F) which was connected to a Model 1397 Duo-Seal vacuum

pump. The line to the vacuum pump had two valves connected in par­

allel. One valve was a small needle valve and one was a 7/8 inch

vacuum valve, and they could be adjusted to control the vapor pres­

sure above the liquid helium. Temperatures to nearly 1 K could be

obtained by pumping on the helium vapor above the liquid when the

level of the liquid was in the tail section of the Dewar where the

surface area was much reduced. A mercury manometer was connected to

this system, and the helium vapor pressure was used as a rough indi­

cation of the temperature.

Tube (J) was connected to a manifold through which the sample

chamber could be evacuated to pressures below 30 y and refilled to

a desired pressure with helium gas for exchanging heat between the

sample and its surroundings, thus allowing thermal equilibrium of

the sample and temperature sensors to be attained. A liquid nitrogen

cold trap was part of the manifold to remove condensable vapors in the

helium gas. An NRC 804 thermocouple vacuum gauge and a mercury manom­

eter were attached to the metal base plate (L) of the glass envelope

to monitor the pressure in the system. The sample and its suspension

were partially counter-balanced by glass tare weights in a quartz

pan (K).

A 2000-G permanent magnet (M) with a 2 1/2 inch gap was used

to supply the non-homogeneous field. The magnetic force is propor­

tional to the field-gradient product and is in the direction of the

gradient. Two types of pole pieces were used with no significant

difference in the results. One set was perfectly cylindrical and

the other set was tapered at a 45** angle. Both were approximately

1/2 inch thick. With these types of pole pieces, two regions of equal,

but opposite, maximum field-gradient product are symmetrically located

on opposite sides of the gap. Measurements made using the difference

of these two maxima are independent of bouyancy forces, pressure

gradients or other effects dependent on the ambient conditions. The

magnet was moved up and down past the sample by means of a motorized

8

hydraulic jack. The ascent speed was not adjustable and was too fast

for the response time of the balance circuit. Data were taken during

the descent of the magnet since this speed could be adjusted to allow

the balance output to follow the magnet. The only exception to this

1 f\

was the high-temperature work on V^O^ which has already been reported.

In this earlier work, a different motor driven magnet was used and

data were taken as the magnet moved in both directions.

The weighing mechanism and cryogenic system were m.ounted on an

elevated platform which was heavily loaded, cross-braced, and attached

to an outer wall of a concrete block building to reduce vibrations.

The glass envelope was mounted on a flat plate, on top of a platform,

which could be raised and lowered by two lab-jacks to facilitate

changing samples. The hangdown tube and cryogenic system were mounted

on slide rails so that they, along with the suspension fiber, could

be decoupled from the envelope and balance arm, respectively, and

slid forward to change samples. A cathetometer was used to place

the cryogenic system and sample back in the same position relative

to the magnet. This visual alignment process was repeated several

times with a reference sample and was reproducible.

A schematic of the electrical apparatus is shown in Figure 2.

The output of the Cahn balance, which is proportional to the magnetic

force, is continuously recorded on a Texas Instruments Servo/riter,

Model PS01W6A, using 10 inch chart paper with 100 divisions. Full

scale deflection was 1 mv with zero center. The recorder dead band

was a function of the scale factor of the Cahn balance, but in all

cases was no more than about one-tenth division. The heater shown

o o o CM

1 CL C4

C rC CO CJ

0) O a CO

rH CO

. Q O }-i +J O (U

.H W

u 0) 4-> CO CU

6 6

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P-. 4J •H

4-> CO X) 0) o

3

(1)

4-J CO CO

U

O

3 00

PIJ g P^ PH

CO

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O

Ch 3 J-4 ^ 4-) >-l O ^

^ c CO

CO 'O <D O

1

(!

i H rH <: 1

<r m m r

1

u (U 4-J

e o •H 4-J

a CU 4-)

o PL

< >

>. +J •H i H •H O CO fn

CO 3 +J CO V< CO CU

<

CO

o •H >-i 4-t O (U

.H W

14-1 O

O • H 4-1 CO

6 CU

x: o

CO

CM

u 3 00

•H

10

in Figure 2 was powered by a Heathkit Model IP-20 regulated power

supply. The emf from all temperature sensors was measured with a

Leeds and Northrup Model 7554-All potentiometer facility. The resist­

ance thermometer circuit is shown in the next chapter.

The force measurements were made by allowing the magnet to pass

by the sample, thus producing a maximum magnetic force in one direc­

tion, then zero force at the center of the gap where the gradient is

zero, then a maximum force in the opposite direction as the gradient

changed sign. This was indicated by a sinusoidal curve on the strip-

chart recorder similar to the point-by-point curve obtained from the

deflection of a quartz helix. The magnetic force is then half the

peak-to-peak amplitude of the curve. Some measurements had to be

made with smaller signal-to-noise ratio than others. In each case,

the noise was of sufficiently high frequency relative to the signal

frequency that a reasonably good average of the superimposed noise

signal could be used as the signal. Under some experimental condi­

tions, a slight drift in apparent sample weight was observed. Data

were taken only if the drift xzas sufficiently linear to allow a quan­

titative correction to be made for it. This drift could not be cor­

related with any particular sample or temperature.

A vacuum Dewar was used for the ice bath reference for the thermo­

couples. The reference junction of the thermocouple and a Weston,

Model 2261 bimetallic, dial type, thermometer were immersed in a

9 mm O.D. pyrex hangdov/n tube containing oil, which was in turn im­

mersed in the ice bath.

CHAPTER III

EXPERIMENTAL METHOD

Susceptibility Measurements

The magnetic susceptibility of a sample is related to the force

measurements described in the previous chapter by

^z = -xH^i7^, (3.1)

where m = sample mass, x - mass susceptibility in the x-direction,

and K dH /9z = field-gradient product in the z-direction. Other terms

in the field-gradient product, i.e., H dH /dz and H dH /9z, as v;ell y y z z

as forces in other directions are made negligible by a judicious choice

of the field geometry at the sample. In principle, such a force meas­

urement is simple, however, in practice there are some problems.

Some of these were mentioned in the previous chapter as they pertained

directly to the apparatus. Problems associated with the actual data

are outlined in the following sections.

Calibration of Field-Gradient Product

To obtain the susceptibility from a force measurement, the field-

gradient product must be known. This was accomplished by measuring

the force on a standard sample with known susceptibility. Since high

purity Pt is available, and its susceptibility has been measured accu-

17 18 rately at 20 C, ' it was used in an earlier experiment to deter-

19 mine the Curie-Weiss parameters so that subsequent calibrations

could be done at any temperature. Since this earlier experiment was

11

12

done on the same system as the V 0 measurements, the room temperature

Pt data were used as the calibration for the V 0 . The force data

are shown in Appendix III. Using the Curie-Weiss parameters, the

susceptibility was calculated at the reference temperature to be

X„(296 K) = 0.968 x 10 emu/gm. Then, using the average force, the

field-gradient product was determined from (3.1). The uncertainty

in the field-gradient product was taken as the sum of the average de­

viation due to the force measurement and a 1% uncertainty in the sus-

19 ceptibility. The calibration of the low temperature system was done

using the same method, except that the orientation of the magnet rela­

tive to the balance platform introduced a variation in the field-

gradient product. The force data shown in Appendix III, Run PT-5,

indicate the extremes of the variation and were used to calculate a

field-gradient product for both directions individually. These two

orientations were the same as those used for all of the diamagnetism

studies and were both used in the data reduction for the corresponding

directions. The doped crystals had been measured earlier with two

different unknown orientations. This would effectively introduce an

"instrumentation anisotropy" into the susceptibility for these samples

in addition to any physical anisotropy that may be present. The data

analysis for these crystals was done using an average of the two cali­

brations. A detailed discussion of this anisotropy problem and data

reduction is given in Appendix III.

There is, however,another problem associated with the calibra­

tion which arises when different sized samples are used. This is the

problem of knowing the volume over which the field-gradient product

13

is uniform. To determine this, six different size samples of pure

Al 0- were cut from adjacent spaces in a single crystal boule and

were approximately cubical in shape. The smallest of these had a

volume very near that of the Pt standard sample used for the calibra­

tion. Using this sample, the diamagnetic susceptibility of A1_0

could be determined quite accurately from its temperature dependence.

Also, from (3.1), F/m at room temperature should be a constant for

sample sizes ranging over the volume of uniform field-gradient prod­

uct. Figure 3 shows the results of this experiment. The randomness

of the initial data, after sawing the samples, suggested the possi­

bility that surface contamination, or possibly saw damage, was con­

tributing to the net susceptibility. The samples were lapped and

cleaned with hot methyl alcohol several times and data were taken

after each. This data, also shown in Figure 3, indicate the surface

was contributing to the net susceptibility. This process was repeated

until the data from all samples fell on a horizontal line, indicating

that the field-gradient product was uniform over a volume of approx-

3 imately 200 mm which was larger than any of the actual samples studied

This cleaning process was adopted for all samples in the study which

were cut to size with the diamond saw.

Temperature Measurement

The physical location of the temperature sensors was described

in Chapter II. The CryoCal germanium resistance thermometer was used

to measure tem.peratures up to approximately 50 K. The thermometers

were supplied by CryoCal, Inc., with calibration points at 77.4 K

14

o in

o o r-\

y ^ ^

CO

1 B v_^

OJ

e 3 i H O >

CO

o CM

i H <:

v o I M

CO * J C d) g (U

3 CO CO 0)

s e

pX4

CO

1 ) V4 3 WD

•H P4

O in

(§m/§rl) m/^

15

and 4.2 K, and were further calibrated over the temperature range

1.60-20.0 K by comparison with a calibrated CryoResistor, also manu­

factured by CryoCal, Inc. For temperatures below 1.60 K, a straight

line extrapolation was used; and for temperatures above 20.0 K, an

extrapolation to the liquid nitrogen boiling point was used. From

50 K to room temperature, a Cu-constantan thermocouple was used. The

wires were Leeds and Northrup No. 30, standardized, with limits of

error not exceeding 1%. The agreement between the resistance ther­

mometer and the thermocouple was excellent in the range 50-77 K,

although the sensitivity of the thermometer was much lower in this

range. Because of the lower sensitivity of the thermometer, the

thermocouple was considered the better sensor in this range. Both

were checked at the boiling point of liquid nitrogen. The thermo­

couple reference junction was maintained at 0.0 C by an ice bath.

The stability and accuracy of the ice bath was checked by direct im­

mersion of the measuring junction into liquid nitrogen. The measured

temperature was always well within the 1% error of the boiling point

of nitrogen at the local atmospheric pressure.

A standard four-point emf measurement was used to determine the

thermometer resistance. The circuit is shown in Figure 4. With

SW-1 in the "I" position, SW-2 selects the proper current for each

temperature range, (1) 1-2 K, (2) 2-15 K, and (3) 15-40 K. The cur­

rents are 1 ya, 10 ya and 100 ya, respectively, and were set to the

exact values by adjusting R2, R5, and R8 so that the emf across the

precision resistors, Rl, R4, and R7, was 10 mv. The resistance was

then determined from the emf measured with SW~1 in the "V" position

16

(U o C CO 4-J CO

•H CO 0)

u 0)

B o B V4

I M

• v V v W

c o

• H CO

• H O Q) V4

0) 4-i (U

e o e >^ CU

H

Q) O

CO 4-1 CO

• H CT! d)

C^ I 0)

o u o

•H 3 O >-i

•H CJ

Q) U 3 00

•H

(^ O

17

since the current was known. This is a constant current circuit and

the drain on the battery was sufficiently low so as to be very stable.

The emf was read potentiometrically on the Leeds and Northrup 7554-

All Facility. The resistance was converted to temperature by reading

from a curve of the calibration mentioned previously. An analysis

of errors in this technique, including the calibration of the CryoCal

standard and plotting errors, resulted in temperature errors of

±1.06% in the range 1.5-2.0 K, ±0.8% in the range 2-5 K, and ±1.0%

in the range 5-10 K. Consequently, an error of ±1.0% was used for tem­

peratures obtained from the resistance thermometer. National Bureau

of Standards tables for helium-4 vapor pressures was also used as a

check on the resistance thermometer for temperatures below 4.2 K.

The agreement was always well within the ±1.0% error.

Since the temperature sensors could not be attached directly to

the samples, helium gas was used to allow the sample and temperature

sensors to come to thermal equilibrium. It was necessary to find a

suitable compromise pressure of exchange gas betv een the higher noise

which occurred at higher pressures and the poor thermal contact at

lower pressures. To determine the minimum exchange gas pressure nec­

essary to insure thermal equilibrium between the sample and tempera­

ture sensors, a second resistance thermometer was located at the

position of the sample and the pressure was varied while the two

temperatures were monitored. Figure 5 shows the results. The dif­

ference in temperature between the two thermometers is plotted as a

function of the exchange gas pressure. Data were taken in several

runs and were of two types. The circles represent data taken at the

18

o o rH

o

o

fc^ CM

•

<f

^

CM •

<r V

o o

/ o o o rH

o < o

O n

e o

e^

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o

U 3 CO CO 0) u (^ CO CO

o (U 00

CO

o

CO

>

o •H 4-» o

o u Q) >-i 3 +J CO U CU p . E (1)

H

CU iH

i-CO

c/3

m <u M 3 W)

•H P4

o o

o I o CM

in tH

i n o

( cP) lV

19

boiling point of helium by either adding or removing helium gas and

the triangles are for data taken by filling the system with helium

exchange gas to 10 cm of Hg at the boiling point of the liquid helium

and then pumping on the vapor above the liquid to lower the temperature

in the system. The scatter in the data is mainly attributed to dif­

ferent time intervals between readings which varied between 1 and 10

minutes. The solid line is an approximate average of the circles

and only at the lower pressures is the scatter significant. For

pressures above 1 cm of Hg, the two thermometers agree to within

their limits of error (±1.0%), and no improvement is obtained for

pressures above 10 cm of Hg. The experimental procedure which was

adopted was to allow a minimum of 20 minutes between temperature

changes and to use a minimum pressure of 2 cm of Hg. For the larger

signal levels higher pressures were used. The only exception to the

20 minute time interval was a drift technique developed for data

points at temperatures between the stable points above 4 K, which

is described in the next section.

For the high temperature work on V 0 , it was also necessary to

work at a compromise pressure. In this case, helium exchange gas

at 1 mm of Hg inserted at room temperature was used throughout, and

sufficient time for thermal equilibrium was allowed. This was de­

termined by a constant reading of the chromel-alumel thermocouple.

An experiment similar to the one involving two resistance thermometers

was run using two thermocouples. The results are shown in Figure 6.

In this case, however, a correction to the measured temperature was

made. The excellent agreement to a Curie Law dependence of the mag-

20

o

H

I

300 400 500 600 700

T(K)

Figure 6. Correction for Difference in Temperature Between Sample and Thermocouple.

21

netic susceptibility for Pt and Pd, when this correction was applied,

19 indicated it was accurate and necessary. The solid line is an

average of two runs, and was used as the correction.

Experimental Procedure

Measurements above room temperature were made using the hangdown

tube and heater described in Chapter li. The sample chamber was

first evacuated to less than 30 y at room temperature for several

hours. The chamber was then filled V7ith helium exchange gas to 1 mm

of Hg, and allowed to stabilize as determined from a continuous mon­

itoring of the temperature with a Bristol 10 mv recorder reading the

thermocouple emf. The room temperature data were taken, the heater

was turned on and set to the approximate value necessary to obtain

the desired temperature, and the system was evacuated. This proce­

dure was followed between each set of data taken at each temperature

for the earlier experiments to disturb the system between readings.

Comparison of data taken this way with that taken by turning off the

balance between readings or just doing nothing to disturb the system

indicated that it was an unnecessary precaution.

For measurements below room temperature, the cryogenic system

was used. The inner Dewar and sample chamber were first evacuated

at room temperature. Helium exchange gas was inserted through the

liquid nitrogen trap to a pressure of approximately 2 cm of Hg and

allowed to stabilize. The exception to this was six data points

taken on pure Al 0 in air (See Appendix III). After the room tem­

perature data were taken, the outer Dewar was filled with liquid

nitrogen, and the inner Dewar was filled with He gas to 2 cm of Hg

22

above atmospheric. Data were taken at the boiling point of liquid ni­

trogen after allowing time to stabilize. -Two techniques of obtaining

data above the nitrogen temperature were used. One used the hang­

down tube with the heater, and was done by incrementing the heater

current to obtain the desired temperature. This procedure was te-

dius and required several hours for each temperature. A "drift"

technique was developed which also proved useful for lower tempera­

tures. The liquid nitrogen was allowed to boil away, and the tem­

perature drifted up at a rate of approximately 25 degrees per hour.

Data were taken at about 25 degree intervals, and the temperature

measured before and after the data point. The average of the two

temperatures was used as the sample temperature, and in no case was

the drift as much as 1 degree over the time required for one measure­

ment. Comparison with data taken on the same samples using the heater

indicated no advantage to the heater method. Most of the data were

nearly independent of temperature in this range, and an error of a

few degrees could be easily tolerated.

To obtain data below 77 K, it was necessary to fill the inner

Dewar with liquid helium with the system at 77 K. After allowing

the temperature to stabilize at the boiling point of liquid helium,

data were taken. It was necessary to add helium gas to the sample

chamber, since the pressure dropped when going from 77 K to 4.1 K.

A pressure of 2 cm of Hg was used for the low signal samples, and

4-5 cm of Hg was used for the samples with higher signal levels.

The vacuum pump was then turned on and the vapor above the liquid in

the inner Dewar was removed to a vapor pressure corresponding to

23

approximately 1.5 K. The small needle valve and larger vacuum valve

in parallel in the line between the Dewar manifold and the vacuum

pump could be adjusted to restrict the flow of vapor and thereby ad­

just the vapor pressure to correspond to any temperature betr een 1.5 K

and 4.1 K. Data were taken com.ing up from 1.5 K until the liquid

level dropped into the narrower tail section of the Dewar. It was

found that the temperature could be maintained at about 1.0 K in this

case. As the liquid level dropped to within a few inches of the level

of the sample, extreme turbulence occurred; but, as it dropped well

below the sample, the temperature would rise sharply to about 10 K

and then drift up more slowly. Lower pressures could be used to lower

the temperature at which the drift slowed down, but it was found that

the temperature was not that of the sample. These points did serve

as an indication of the shape of the curve but were not used in the

analysis. In some cases the temperature did not drift slowly enough

to take data until it reached about 15 K. In any event, temperature

readings were made before and after each force measurement, and the

data were accepted if the drift was less than 1 degree. The excessive

turbulence caused by the heater in trying to obtain stable tempera­

tures above 4.1 K, caused noise which completely covered the force

signal.

One major difficulty which was encountered was the presence of

a foreign substance in the sample chamber. It deposited on the sample

and surroundings most readily when the temperature was lovv ered to 1.5 K.

It was assumed to be a gas contaminant which vjas not trapped by the

liquid nitrogen trap, possibly oxygen or argon. The force signals

24

became very large and positive (paramagnetic), and increased with

time as the deposit became visibly thicker. The entire run below

4.1 K was rejected, and the system had to be returned to room tem­

perature and evacuated. Much of the time, the boil-off gas from the

liquid helium storage Dewar was used without incident. Later, a nevj

cylinder of helium gas was obtained and the problem disappeared. It

is possible that this technique could be adapted for measurements of

the susceptibility of solid gases.

Samples and Suspensions

2+ The single crystal samples used in this study were SrCl„, MgO:Mn ,

NaCl, KCl, CaF^, V20^, Al^O^, Al20 :V '*", and AI2O iTi " . The first

two were supplied by Dr. L. A. Boatner ^ ho grew the SrCl„ by the

vertical Bridgman technique. The NaCl and KCl crystals were grown

from the melt at the University of Oklahoma and the CaF„ was natural

flourite from Mexico. Dr. E. D. Jones of Sandia Laboratory supplied

3+

the V_0 single crystal. The A1_0- and Al^O :Ti samples were ob­

tained from the Linde Company through Professor C. R. Quade, who made

20 some preliminary susceptibility measurements on the latter. The

3+ V -doped sample was grown by the flame-fusion method by Dr. D. S.

21 McClure for optical studies on transition metal ions in corundum.

The sample was obtained from Professor C. C. Lin, who had used it

at the University of Oklahoma for susceptibility studies above 4.2 K.

The oxide samples were oriented by x-rays. The crystals were

cleaved or cut into nearly regular rhombohedral-shaped samples with

masses between 150 and 600 mg and were kept in a dry box until meas-

25

ured. A list of the samples is shovm in Table 1. The oxide crystals,

doped and undoped, had to be cut with a diamond saw and were subse­

quently lapped with silicon carbide #240 grit grinding paper and then

2+ cleaned in hot methyl alcohol and distilled water. The Mgo:Mn

sample was obtained in the form used for the measurements and was

only cleaned in alcohol and water. The sawed samples were cut with

the c-axis perpendicular to a face of the crystal. The remaining

samples were cleaved from larger single crystals.

The samples were suspended from the balance arm by a 0.5 mm dia.

quartz fiber, which was in three sections. One set of measurements

was made using a lower section constructed of copper wire (See Appendix

III). Hooks were used to join the sections. These hooks and the wire

suspension loop on the balance arm were all cemented in place with

"secretarial correction ink" to reduce torsion due to the sample ani­

sotropy. This did not prove to be a problem as it was for a quartz

helix apparatus. The lower section of the fibers were formed into a

basket on the bottom to fit the shape of each sample, and were long

enough so that the hook was out of the field. This was necessary to

remove all foreign material possible from the vicinity of the sample.

The quartz fiber was a reproducible system and a correction for the

force due to the fiber was made to all measurements when it was signi­

ficant. A discussion of the correction is given in Appendix III.

Numerous other materials were used for earlier maesurements, but the

low temperature magnetic properties were suspect and in some cases

were obviously dominating the results. This was even true for some

quartz fibers.

Table 1

Samples

26

Sample

SrCl^

2+

MgO:Mn

NaCl

KCl

CaF^

Rock Salt

AI2O3

3+ Al^O^rTi

Al20^:V 3+

Mass

175,

295,

210,

154,

232,

157,

195.

237,

479,

569,

266,

(mg)

.25

.85

.26

.06

.56

.23

.12

.80

.52

.06

.40

Preparation

Cleaved

Sawed

Cleaved

Cleaved

Cleaved

Cleaved

Sawed

Sawed

Sawed

Sawed

Sawed

CHAPTER IV

DIAMAGNETISM OF HOST CRYSTALS

To extract the information on the magnetic ion in the doped sys­

tems, it was necessary to correct the measured susceptibilities for

the diamagnetism of the host crystal. The diamagnetic susceptibilities

22

of many materials were tabulated by Foex in 1957 from existing pub­

lications, but the samples used were mostly powdered or in solution

and many corrections and assumptions were often necessary to obtain

the desired value. During the analysis of the measurements on the

doped corundum samples, it was found that the tabulated value for

Al 0_ did not fit the results. After eliminating many other possi­

bilities, measurements were made on a single crystal sample of Al 0

to check the tabulated value. A number of other single crystal dia­

magnetic samples were measured to substantiate the results.

Since paramagnetic susceptibilities depend on temperature, meas­

urements were made at two temperatures to be assured that the dia­

magnetic susceptibility was being measured. Two exceptions will be

discussed. Also, since the diamagnetism is isotropic, another check

was made by taking two sets of measurements with the magnet rotated

90° for each. These directions were along and perpendicular to the

c-axis for measurements on the MgO and A1„0- samples. These data

were always in agreement within the experimental error.

The apparatus was equipped with the low-temperature Dewar and

manifold for pumping on the liquid-He vapor. Helium exchange gas at

27

28

approximately 5 cm of Hg pressure was used for thermal contact be­

tween the samples and the temperature sensors for all data except

that taken on the Al 0^ below 77 K. These data were taken with an

exchange gas pressure of 2 cm of Hg. The samples were suspended in

a clean quartz basket, as described in the previous chapter, and a

measured correction for the quartz suspension, amounting to no more

than 5%, was made to the results. The sample chamber was evacuated

to less than 30 microns pressure for 48 hours prior to measurements

on each sample to remove surface contaminants not chemically bound.

The oxide samples were cleaned beforehand, as previously described.

The results of measurements on the single crystal samples taken

at the temperature of boiling nitrogen and at room temperature are

shown in Table 2. The values shown are the mean of 20 determinations,

10 each in the two directions mentioned previously. Since no meas­

urable anisotropy was observed, the two sets were averaged together.

All samples, except the MgO, indicate the absence of any measurable

paramagnetic impurity at these temperatures. The MgO sample was

2+ known from esr experiments to contain a Mn impurity, although its

concentration had not been determined. The characteristic esr spec-

2+ trum for Mn is shown in Figure 7, and was done on this sample by

Dr. L. A. Boatner. Mn has a S , ground state and its suscepti­

bility should follow a Curie Law, i.e..

X = C/T + x,i3 (4.1)

2 2 where C = Ny g S(S + l)/3k, T = absolute temperature, and xJ•.

o die

29

Table 2

Measured Magnetic Susceptibilities

Crystal

SrCl^

KCl

NaCl

CaF (Natural Fluorite)

MgO^

AI2O3

~f\ -x(10 emu/gm) at T = 80 K

0.355

0.522

0.519

0.344

0.287

0.339

~-f\ -x(10 emu/gm) at T - 300 K

0.357

0.527

0.516

0.342

0.364

0.343

a 2+ Known Mn impurity

30

u CU c 4.) CO o

o

CO CU 4J

u 3 O

CM

C)

o e 3 >-i 4J o 0) p .

0:3

Pi CO

w

0)

3 to

•H

31

diamagnetic susceptibility of host crystal. For Mn , g = 2.0 and

S = 5/2. Using these values. Figure 8 shows a least squares fit of

2+ (4.1) to the data with N, the number of Mn ions per gram of sample,

and XJ.^ as parameters. y is the Bohr magneton and k is Boltzmann's Qia o

2+ constant. The value of N corresponds to approximately 0.03% Mn ,

—fi and XJ^„ = -0.393 x 10 emu/gm. This is significantly different

from the measured room temperature value. Plotted points are averages,

but for curve fitting all individual measurements were used.

In addition to the interest in measuring the diamagnetic suscep­

tibility of single crystal Al 0„, it became of some interest to know

the possible effects of a few ppm of common impurities in available

"pure" single crystals of this host. Measurements were made on one

of the six samples used for the field-gradient product studies men­

tioned in Chapter III. Optical spectra were taken on this sample with

a Cary 14 spectrophotometer at 300 K and 77 K and no evidence of an

impurity was found. However, the temperature-dependent susceptibility

shown in Figure 9 clearly shows that some kind of impurity is contri­

buting significantly to the magnetic properties at the lower temper­

atures. These data were found to be isotropic within experimental

error (See Appendix III). This implied that the impurity might be

either Mn or Fe , both of which have S ,„ ground states. This

assumption was not unwarranted, since crystals of this type charac­

teristically contain a few ppm of Mn and Fe. Therefore, a least

squares fit to (4.1) was attempted, using all data given in Appendix

III, and the result is shown in Figure 9 by the solid line. This

curve corresponds to approximately 7 ppm of an S = 5/2 impurity with

32

o o CO

o in CM

o o CM

O i n i H

o o i H

/ - - N

i4 >.—' H

+ CN

ity of MgO:Mn

f H •H

ptib

(U CJ CO

Measured Su

•

Figure 8

o

00 CM

O 1

O CO

o 1

CM CO

O 1

(mS/nms 9-

ro O 1

_OT)X

.36

o !

33

H

CO

CM

<U ^4 3

14-1

o

4-1 •H iH • H

• H 4-) P . (U O CO 3

c/0

' d CU >-i 3 CO CO CU

0)

3 W)

• H 1 4

CM

• O I

CO

o I

o I

(m§/nra3 OT)X

34

-6 Mia X,. = -0.339 X 10 emu/gm. (4.2)

To give some confidence to the assumption of an S = 5/2 impurity,

an esr spectrum was taken by Dr. R. W. Reynolds at 300 K, 77 K, 4.2 K,

and 1 K. At the two lower temperatures, several resonance lines

appeared but they were very weak. While the results were inconclusive,

3+ it did appear that the sample contained Fe since none of the lines

2+

exhibited any hyperfine structure, thus eliminating Mn from con­

sideration. However, the several lines which appeared made it hard

3+

to determine the site symmetry of the Fe ion. The value in (4.2)

was used in analyzing the doped corundum results giving much better

agreement with theory.

Table 3 shows the results for all of the single crystal samples

studied, along with previous results for comparison. It is inter­

esting to note that the natural flourite was pure enough to obtain

the diamagnetic susceptibility at room temperature. A sample of

natural rock salt was also measured and gave the same results as the

grown NaCl crystal. The value obtained for MgO is in good agreement 23

with theoretical values mentioned by Prasad, Dharmatti and Amin.

One other feature which stands out is, except for the SrCl_ for which

no hardness could be found, and the MgO, the differences in the two

columns of Table 3 are greater for the harder materials. Since most

of the earlier work was done on powdered samples, the measurements

had to be corrected for a "packing factor." The softer materials

would pack better and require less correction. The density of the

Table 3

Diamagnetic Susceptibilities

35

Crystal This Experiment

—f\ -x(10 emu/gm)

Previous Results

-x(10 emu/gm)

SrCl,

KCl

NaCl

CaF^

MgO

AI2O3

0.356 ± 0.009

0.524 ± 0.012

0.517 ± 0.013

0.343 ± 0.008

0.393 ± 0.006

0.339 ± 0.005

0.397

0.523

0.518

0.359

0.253

0.363

From tables in Constantes Selectionees Diamagnetisme et Paramagnetisme Relaxation Paramagnetique, Vol. 7 (1957).

36

material was used in the earlier work, and this is much more difficult

to know accurately than the mass, which was used in this work.

The uncertainties in Table 3 are the sum of the average deviation

due to the force measurements and the uncertainty in the calibration,

except for the MgO and the Al 0 . Since a least squares fit to (4.1)

was done, the most probable error from that method of analysis was

used in place of the average deviation.

The diamagnetic susceptibility of single crystal A1_0 has been

measured and the value obtained is considered to be better for the

extraction of paramagnetic susceptibilities from measured suscepti­

bilities when magnetic ions are incorporated into this host crystal.

3+ A much better fit to the A1_0„:V data was obtained using this value.

In addition, the results indicate that a re-evaluation of the tabu­

lated diamagnetic susceptibilities for single crystal samples may be

necessary in the future.

' CHAPTER V

MAGNETIC SUSCEPTIBILITIES OF Al 0 •V '*'

Introduction

Several years ago Brumage, Quade, and Lin reported results of

magnetic susceptibility measurements on vanadium-doped A1„0 in the

temperature range 4-300 K. From these results they were able to de­

termine the zero-field splitting 6 of the trigonal ground state, the

perpendicular g-factor g , the trigonal field splitting A , and the

spin-orbit coupling parameter X. Later, Brumage, Seagraves, and Lin

extended some of these measurements to 1000 K to make a more direct

determination of A . An energy level diagram of the triplet levels

for this system is shown in Figure 10. More recently, Smith and

25

Mires have measured the magnetic anisotropy of this crystal system

using one of Brumage's samples in the temperature range 1-5 K, a

region in which the susceptibility expressions are most sensitive to

6 and g.. This work reports the direct measurements of the parallel

and perpendicular susceptibilities, Xi. and Xi» respectively, in the

same low temperature range. For consistency, however, the measure­

ments have been extended up to 300 K. Not only are these results

compatible with some later, more accurate, anisotropy measurements

performed by Smith, but they indicated the necessity of considering

unwanted impurities in the analysis at low temperatures. Electron

spin resonance measurements on this sample, done by Dr. L. A. Boatner,

exhibited the'same lines observed in the "pure" A1„0„ sample, sub-37

38

Free Ion Cubic Trigonal L S

3+ Figure 10. Energy Levels of V in a Trigonal Field.

39

stantiating this assumption.

The purpose of this experiment was to perform a critical test on

the application of crystal field theory to this system by observing

the individual temperature dependence of the two principal suscepti­

bilities. Also, it was done to provide a reference value for the

anisotropy analysis to make it independent of Ref. 1, and to see if

more accurate crystal field parameters could be obtained.

Experimental Data and Analysis

The experimental results are shov7n in Table 4, and are plotted in

Figure 11 with the filled symbols for x, ^nd the open for x • Above

80 K the filled circles represent both x and x • The uncertainties 1 II

are maximum uncertainties including those due to calibration, noise,

recorder dead-band, basket correction and temperature uncertainties

and representative error bars are shown. The stable temperature data

represent averages of 5 to 10 measurements and the average deviation

was always more like 1-2%. The larger errors represent systematic

errors for the most part and these contribute mostly to the concen­

tration of impurity ions, which is an uninteresting parameter. In

Figure 11, typical errors are shown for the low-temperature points,

which include the uncertainty in the diamagnetic susceptibility of

the host as taken from Chapter IV. For the higher temperature points,

the errors are the size of the points. Both the filled and unfilled

triangles represent very low-pressure drift points and almost cer­

tainly the sample and its surroundings were not in thermal equilibrium,

so these points were not used in the data analysis. Above about 15 K,

40

Table 4

3+ Magnetic Susceptibilities of A1„0„:V

in Units of (10 emu/gm) 2 3

Data without uncertainties are low-pressure drift points (see text).

X . ^ T X,

298.0 K - 0 . 3 1 0 ± 0.017 298.0 K - 0 . 3 0 8 ± 0.017 297.7 - 0 . 3 1 0 ± 0.017 297.7 -0 .302 ± 0.017 238.6 - 0 . 3 0 1 ± 0.016 239 .8 - 0 . 2 9 3 ± 0.016 196.4 - 0 . 2 9 7 ± 0.016 237.0 - 0 . 2 9 3 ± 0.017 193.2 - 0 . 2 9 3 ± 0.016 195.4 - 0 . 2 8 8 ± 0.016 149.6 - 0 . 2 8 2 ± 0.016 194.4 - 0 . 2 8 7 ± 0.016 145.9 - 0 . 2 8 5 ± 0.016 148 .8 -0 .276 ± 0.016 117.0 - 0 . 2 7 7 ± 0.016 146.7 - 0 . 2 8 1 ± 0.016 113.0 - 0 . 2 6 8 ± 0 .015 115.7 -0 .264 ± 0.015

80 .3 - 0 . 2 3 2 ± 0 .013 114.4 -0 .262 + 0.015 57 .8 - 0 . 1 9 9 ± 0.012 8 0 . 1 -0 .226 + 0.013 49.5 - 0 . 1 7 4 ± 0 .011 55 .7 - 0 . 1 8 5 ± 0.012 42.2 - 0 . 1 3 5 ± 0.010 52 .7 -0 .176 + 0 .011 33.6 - 0 . 1 0 4 ± 0.009 39.4 - 0 . 1 3 3 ± 0.010 29.6 - 0 . 0 7 0 ± 0 .008 35.4 - 0 . 1 1 7 ± 0.010 25.6 - 0 . 0 3 2 ± 0.007 28.4 - 0 . 0 7 3 + 0.009 23 .5 - 0 . 0 1 3 ± 0.007 27 .2 - 0 . 0 6 1 + 0.008 17 .8 +0.100 ± 0 .011 2 2 . 1 -0 .026 ± 0.007

• 17.4 +0 .108 ± 0.009 21.2 -0 .007 ± 0.007 16.5 +0.115 20.6 - 0 . 0 1 2 + 0.007 13.2 +0.095 20.2 +0.012 + 0.007 10.0 +0.328 14.7 +0.079 ± 0.012 7.8 +0.447 13.5 +0.103 5.2 +0.774 11.2 +0.109 4.3 +1.167 9.1 +0.143 4.10 +1.217 ± 0.049 7.4 +0.172 3.48 +1.312 ± 0.050 6.3 +0.188 2.91 +1.412 ± 0.056 5.0 +0.173 2.46 +1.472 ± 0.064 4.3 +0.057 2.00 +1.505 ± 0.062 4.10 +0.059 ± 0.014 1.52 +1.557 ± 0.060 3.50 -0.009 ± 0.017 1.48 +1.562 ± 0.058 2.95 -0.076 ± 0.025 1.06 +1.588 ± 0.096 2.44 -0.164 + 0.034 1.02 +1.578 ± 0.064 2.00 -0.181 + 0.028

1.53 -0.195 ± 0.023 1.47 -0.189 + 0.023 1.06 -0.182 + 0.020

41

o o +

CO

CO O

CM

o CO CU

• H 4J •H i H •r-( r P •H 4J a <u o CO 3

CO

<U V4 3 CO CO 0)

<u >-l

3 bO

•H

J L J L J L vO

rH

CM

rH

00 <}•

O O

(m§/nma OI)X

o I

42

the sample chamber drifted up in temperature much more slowly and the

instabilities arising from convection currents were much less violent,

thus allowing for the use of more exchange gas. This is the "drift"

technique described in Chapter III. These points were believed to be

good data and were used in the analysis.

The solid lines in Figure 11 represent a least-squares fit to

all of the data in Appendix III with equations of the form:

X_ = X + C/T + X.. , (5.1) 'in 11,1 dia

—6 where x,. = -0.339 x 10 emu/gm, C/T represents the unwanted iso-

dia

tropic Curie-type impurity and the remaining term is the theoretical

paramagnetic susceptibility for the vanadium impurity; i.e..

Xji = 2NBy2(g2/kT) exp(-5/kT)

+ 2NBy2{[2a'^/A + 2 (a + 2)2a2x2/A_,3 ] (5.2) o c i

+ 2[2a'^/A^ + (a + 2)2a2x2/A^3] exp(-6/kT)},

X^ = 2NBy2(g2/5)[i-exp(-5/kT)]

+ 2NBy2[a'^/A + a2(l-10a2A2/A^2)/^ ] ^o c T T

+ 4NBy2 exp(-6/kT){a' /A + o c

+ a2[l-(3a2 + 2a - 4)X2/A2]/A^},

(5.3)

43

where

B = [1 + 2 exp(-6/kT)] \ (5.4)

The symbols are identical to those in Ref. 1. Theoretical expressions

for gI and 6 are given by

and

g^ = g - 2a2A/A^ - a'2A/A^ + a2(a - 1)A2/A2

- aa' X2/A A^ + 2a3(a + l)X^/tl (5.5) c i i

6 = X[a^e - 2a3e2 _ ct e3] - (3/2)a'^A2/A . (5.6)

Substitution of these into (5.2) and (5.3) yields expressions in which

> gii» ct» ct , A , A , and N are unknown parameters. ESR measurements

26 ' give g = 1.915; crystal field theory gives values for a and a as

1.29 and 1.52, respectively; optical data gives A = 17,400 cm ;

-1 24 and from high temperature susceptibility measurements A = 1,100 cm

Substitution of these values into (5.2) and (5.3), and then in turn

putting these results into (5.1), the theoretical equations for the

total measured susceptibilities in the parallel and perpendicular

directions are obtained. The only remaining unknowns are N, X, and C.

The value of C is determined by considering (5.2). At low tempera-

3+ tures, the paramagnetic susceptibility of the V -ion approaches a

44

constant given by the second line of (5.2). An estimate of the size

of that term can be obtained by substituting the values of II and A

from Ref. 1, and the above listed parameters into the expression.

-9 This value is approximately 2 x 10 , and can be neglected, leaving

C + Xi. to account for the total measured susceptibility at 1 K. dia ^ ^

—A Subtracting xj• gives C = 0.16 x 10 emu/gm. This corresponds to

approximately 10 ppm of an impurity with g = 2, S = 5/2 and is con­

sistent with the undoped sample and with guaranteed purity specifi­

cations supplied with other crystals by the manufacturer.

. The method of least squares was then applied to the two sets of

data independently, for the two directions of magnetic field with N

and X as parameters. The limits of error were established by taking

the sum of the squares of the experimental uncertainties as the max­

imum least squares residue. The results are shown in Table 5. The

two sets of data were then considered simultaneously, and the limits

of error were established by either residue exceeding the sum of the

squares of the experimental uncertainties individually. This essen­

tially excludes values of the parameters for one data set which are

outside the limits of the other. The values of g and X, and their

limits of error are found from (5.5) and (5.6). These results are

also shown in Table 5, along with the results from Ref. 1 and Ref. 15

All of the least squares analyses were performed on an IBM-360 com­

puter.

Final Results

The quantitative results for the parameters shown in Table 5

45

m (U

iH

cO H

+ CO

CO

o C M

IW

o CO d)

• H 4-i • H i H • H . Q • H 4-) P . <U O CO 3

C O

o • H 4J CU CJ W3 CO

iH CO 4->

C! (U

•H U (U P.

W

+J •H

0) CO

tD

CO »^ <u 4-1 <u 6 CO ^) CO

P-i

H

K? I 6 o

e £>0 3 6 (U

CM O I P. . H

J3 I B o

CU a j-i 3 o

CA)

U O [5

CO • H

.00

7

o +1

.71

9

TH iH O

o +1

.73

0

00

6

o +1

.72

4

00

5

o +1

.72

0

00

2

o +1

.72

1

CM

rH

CM CM CM

i H

in CO o +1

in CN 00

CM

o +1

r

CO

O

+1

o 00

CM

O

+1

CO

00

r-{

O

+ 1

CM iH 00

vj

« CM

+1

CO •

CM CTi

VD •

CO

+1

<]-•

00 00

o •

CM

+1

in • o a\

CO

in

+1

in cr>

cr> • o +1

<r •

r-\

cr>

VD 1 1

o iH y^

O tH •

O

+1

CO <f •

1 1

o iH /'-\ O r-{ • o +1

00 CO •

1 1

o r-{ /--\ o r-{ • o +1

o -;!-•

vD I O iH

X

CM r-

vD 1 1 O tH / ^ O iH •

O

+1

00 CO •

CM

X ^z

X

X 4J O cq

tH

• M-l CU P^

m tH

• IH (U (^

C M

<U ( ^

CO

46

which utilizes both sets of data simultaneously are the final results

2 of this experiment. The value of Ny is smaller than Brumage reported

for the same crystal. Thi-s could be due to the temperature range

over which it was determined, the heat treatment it received, or to

the fact that only a piece of the original crystal was used and the

3+ V concentration may not have been distributed uniformly throughout

the sample. The solid curves of Figure 11 are plotted using these

new parameters, and it can be seen that they fit the data points

exceptionally well over the entire temperature range.

Conclusion

No significant improvement in the values of previously reported

parameters was obtained; however, the measurements have been extended

down to 1 K and the simple crystal field theory is adequate to explain

the results if an S-state impurity is included. The agreement with

parameters determined from anisotropy measurements which do not in­

clude contributions from the S-state impurity, and the presence of

lines in the esr spectrum similar to those in the "pure" A1„0 sample

corroborates the assumption of an S-state impurity. In addition, the

necessity of using an improved value for the diamagnetic susceptibility

of Al 0„ was evident in the analysis.

These measurements do provide a set of parameters determined

completely from absolute susceptibility measurements which agree V7ith

those from previous experiments. The three independent experiments

using quite different methods of analysis provide a high degree of

reliability to the parameters.

CHAPTER VI

MAGNETIC SUSCEPTIBILITIES OF Al 0 •Ti' '

Introduction

3+ The transition-metal ion Ti has a single 3d-electron in the

octahedral environment of the corundum lattice, and for this reason

one might expect that it would have been the first such impurity to

receive any attention. However, only very recently has much effort

3+ been made to understand the Al„0^:Ti system. This was probably

due to tv70 reasons: (1) good single crystals for experimental work

have been extremely hard to obtain, and (2) the suggestion that a dy-

27-29 namic Jahn-Teller effect may be important in this system has

confounded the theoretical aspects of the problem. The first sig-

21 nificant experimental results were reported in 1962 by McClure on

the optical spectra. The paramagnetic resonance spectrum had been

30 reported earlier by Kornienko and Prokhorov. These were followed

by a theoretical treatment of the paramagnetic spectrum by Gladney

31 and Swalen, and magnetic susceptibility measurements by Cottrell,

20 Andreadakis, and Quade. Both of these used crystal field theory with

no dynamic Jahn-Teller effect for the analysis of the results. The

limited experimental data severely hampered the analyses. The far-

32 infrared spectra reported by Nelson, Wong, and Schawlow were the

first observation of experimental energy spacings within the two

trigonal manifolds when the free ion is incorporated into the corun-

33 dum structure. A very short time later, Macfarlane, Wong, and Sturge

47

48

used a second-order dynamic Jahn-Teller calculation to explain the

energy spacings as well as to calculate the g-factors for the spin-

orbit split trigonal ground state and excited state. Macfarlane's

g-factors for the ground spin-orbit state were in good agreement \7ith

30 spin-resonance results. From the Zeeman structure of the far-infrared

34 spectra, Joyce and Richards obtained reasonable agreement with

Macfarlane's calculated g-factors for the excited spin-orbit state.

These results are taken to substantiate the presence of the dynamic

Jahn-Teller effect. However, reasonably good agreement with the ex­

perimentally determined quantities can also be obtained from the

crystal field theory using different values for the trigonal and spin-

orbit parameters from those used by Macfarlane to discredit that theory.

The end result is that there is still some disagreement as to the

validity of the proof for a dynamic Jahn-Teller effect.

In an attempt to clarify this situation, a complete study of the

3+ magnetic susceptibilities of Al^O :Ti over the temperature range

1-300 K has been performed. The sample was from the same crystal

20

used for the earlier magnetic measurements, and the results agree

within the experimental error. It was hoped that the parameters which

could be obtained from an analysis of this study, such as the excited

state g-factors and Van Vleck susceptibilities, could be used to

determine the extent to which the simple crystal field theory could

predict the temperature dependence of the magnetic susceptibility.

Also, the previously reported parameters could be verified by in­

corporating them into the analysis.

49

Theory

The energy level scheme for a single 3d-electron in an octahedral

field with a small trigonal distortion is shown in Figure 12. The

three lowest-lying levels are all Kramer's doublets and all split in

a magnetic field. The dynamic Jahn-Teller effect would not alter

this splitting scheme, but would reduce the level separations. As

mentioned earlier, Macfarlane has calculated the energy-level split­

tings and g-factors and, in principle, his results could be used to

obtain expressions for the susceptibilities. Expressions for the

susceptibilities \-jere given in the earlier measurements of magnetic

susceptibilities, but the excited trigonal level was neglected and

its contribution is significant above 50 K. Such expressions contain

a large number of parameters which have not been firmly established

and the mathematical methods of data analysis do not work well in this

situation. This makes an analysis similar to that used for the

3+ A1„0^:V system impossible. Instead, a more phenomenological approach

is taken, in which only the measured parameters are used.

The general analysis of a collection of non-interacting magnetic

moments is given in Appendix II. The application of (II.5) to a level

scheme composed of three Kramer's doublets gives the paramagnetic

susceptibility as

Ny2g2 Ny2g2 6

Xp = ^ ^ - ^ ^ ^vv + ^-^k^ + ^vv ? ^"P^- kY^ ^ o 1

Ny2g2 6

'1/2

50

'E g

/T 'E X

y

y

'3/2

'D

/

I I

\ A

\

\

\

\

\

\

\

\ 2g

/

/ /

/

/

/ /

\ \ \ \ \

2^1/2 W,

y y

y

\

1^1/2 ...

t

1 3/2

Free Ion

Cubic Field

Trigonal Field

Spin-Orbit

3+ Figure 12. Energy Levels of Ti in a Trigonal Field.

51

where

-1 B = [1 + exp(-6i/kT) + exp(-62/kT)] \ (6.2)

and

6i = W? - W°, 62 = W^ - W°. (6.3)

The usual substitutions, W! = ±l/2g.y and x = -2NW", have been 1 1 0 vv. 1

1

made. In (6.1), g^ is the spectroscopic splitting factor, y is the

Bohr magneton, k is Boltzmann's constant, T is the absolute tempera­

ture, x^^ is the Van Vleck susceptibility, and N is the concentration 3+ ^

of Ti ions per gram of sample. Both x and x are given by (6.1) I* ^

by substituting the proper values for the g-factors and the Van Vleck

susceptibilities.

Experimental Data and Analysis

The average measured susceptibilities for the titanium-doped

corundum sample are shown in Table 6. As in the vanadium data, the

uncertainties are estimated uncertainties, and the stable temperature

data indicate the measurements are more accurate than shown. The average

data are plotted in Figures 13 and 14 along with the theoretical fit to

the data. Representative error bars are shown which include the un­

certainty in the diamagnetic susceptibility of the host. The single

crystal sample was cut from the center of a boule and exhibited light-

and dark-colored striations which occurred as narrow parallel layers

52

Table 6

Magnetic Susceptibilities of Al 0 :Ti in Units of (10 emu/gm)

T

298.6 K 296.6 294.4 250.2 200 .1 149.9

98.9 81 .1 79 .8 77.2 70.3 69.9 68.6 60.6 58.9 4 7 . 1 45 .8 43.9 29 .1 24.4 23.2 21.4 17 .1 11.9 11.2 10.9

4 .14 4.12 4.09 4 .08 3.49 2.62 2.02 2.00 1.55 1.54 1.40 1.06

a X.

X

- 0 . 3 3 0 ± 0 .013 - 0 . 3 3 5 ± 0 .013 - 0 . 3 3 5 ± 0.017 - 0 . 3 3 5 ± 0 .015 - 0 . 3 3 2 ± 0.014 - 0 . 3 3 0 ± 0.014 - 0 . 3 2 9 ± 0.014 - 0 . 3 2 1 ± 0.014 - 0 . 3 1 7 ± 0 .013 - 0 . 3 1 5 ± 0 .013 - 0 . 3 1 2 ± 0 .013 - 0 . 3 1 1 ± 0 .013 - 0 . 3 1 3 ± 0 .013 - 0 . 3 1 2 ± 0 .013 - 0 . 3 0 8 ± 0.013 - 0 . 3 0 4 ± 0 .013 - 0 . 3 0 4 ± 0 .013 - 0 . 3 1 2 ± 0.014 - 0 . 2 9 3 ± 0 .013 - 0 . 2 8 9 ± 0 .013 - 0 . 2 7 3 ± 0.012 - 0 . 2 7 2 ± 0.012 - 0 . 2 7 1 ± 0.012 - 0 . 2 5 0 ± 0 .013 - 0 . 2 4 5 ± 0.012 - 0 . 2 4 5 ± 0.010 - 0 . 1 6 0 ± 0.009 - 0 . 1 5 9 ± 0.010 - 0 . 1 4 1 ± 0 .013 - 0 . 1 5 9 ± 0.010 - 0 . 1 1 5 ± 0.010 - 0 . 0 6 1 ± 0 .028 - 0 . 0 3 7 +0 .038 +0.082 ± 0 .021 +0 .071 ± 0.015 +0.229 +0.130 ± 0.016

T

298.5 K 296.6 294 .3 250.2 200 .3 150.0

98.7 81.0 79 .8 77.0 75.2 73 .8 70 .1 61 .3 57 .5 48 .8 39.4 30.8 29.2 25 .8 22.9 20.7 18.0 11 .8 11.4 10.4

4.14 4.12 4.09 4 .08 3.46 2.60 2.02 1.97 1.55 1.53 1.43

a X.,

II

- 0 .324 ± 0.013 -0 .327 ± 0 .013 - 0 . 3 2 5 ± 0.015 -0 .327 ± 0.015 - 0 . 3 2 2 ± 0.014 -0 .324 ± 0.014 - 0 . 3 2 2 ± 0.014 -0 .316 ± 0.014 - 0 . 3 1 0 ± 0.014 - 0 . 3 0 4 ± 0 .013 - 0 . 3 1 3 ± 0.014 -0 .310 ± 0.014 -0 .304 ± 0 .013 -0 .304 ± 0.014 - 0 . 2 9 8 ± 0.013 -0 .296 ± 0.013 -0 .296 ± 0 .013 - 0 . 2 8 8 ± 0.012 - 0 . 2 8 7 ± 0.013 - 0 . 2 7 8 ± 0.013 - 0 . 2 7 8 ± 0.012 -0 .276 ± 0.012 -0 .280 ± 0 .021 -0 .244 ± 0.012 -0 .240 ± 0 .011 - 0 . 2 3 3 ± 0.012 - 0 . 1 1 8 ± 0 .008 -0 .116 ± 0.008 -0 .104 ± 0.010 -0 .122 ± 0.012 - 0 . 0 7 4 ± 0.015 +0.006 ± 0 .013 +0.065 +0.048 ± 0.015 +0.140 ± 0 .018 +0.129 ± 0.014 +0.308

Data without uncertainties are low pressure points (see text).

o o vD

CO ^ O

53

CO • H X CO I

O

CU

o o

o 4-)

u CO

tH 3 O

•H

<u Cu 5-1 (U

P-i

+ CO

1

J CM • O

»-CHo

t H

* O

.. 1 1 o

•

o

I 1 r-{

o 1

J 1 CM

O 1

1 1 CO

o 1

1 <

c 1

t H

1-

>

Fig

ure

•H H

CO D CM

<4-l

o

> 4-1 •H iH •H .P •H 4-) P4 0) o CO 3

CA)

0) >-< 3 CO CO CU

s CO

(m§/nmB 01 ) ^

54

CO • H

CO I

o (U

o

Q)

CO V4 CO

+ CO • H

H

CO

CM

O

>^ +J • H rH •H .P

0) O CO 3

CO

<u u 3 CO CO 0)

tH

<U V4 3 to

•H P4

(mS/nms C l ) ^

55

perpendicular to the c-axis of the sample. It was suspected that

these striations resulted from a non-uniform distribution of Ti

ions with the darker layers being more concentrated that the lighter

ones. This assumption was supported by the initial data of the ani­

sotropic susceptibility which showed a strong field dependence over

the entire temperature range. The sample was heat treated at 1200 C

3+ for 24 hours in an attempt to diffuse the Ti ions throughout the

sample. Although this baking process did not remove the striations,

it did remove the field dependence except at temperatures of 1.5 K

or lo N er where presumably weak coupling energy is greater than the

thermal energy.

To fit the data, the expression for the total measured suscep­

tibility must include contributions from the diamagnetic host and

from an unwanted S-state impurity. The measured susceptibility then

has the form

X., = X^. + X,. + C/T, (6.4) ll»l Plhl ^1^

—ft where x.. = -(0.339 ± 0.005) x 10 emu/gm from Chapter IV, and C

0.13.

is the Curie constant associated with the S-state impurity and must

be treated as an adjustable parameter. The presence of the S-state

impurity was indicated by the low-temperature behavior of the perpen-

dicular magnetic susceptibility. Electron spin resonance and far-34

infrared spectra indicate the perpendicular g-factors for the first

two states are small or zero. Since the perpendicular susceptibility

did not approach a small constant at low-temperatures, there must be

56

Table 7

3+ Reported Parameters for Al^O :Ti

Parameter

^o„

«ox

111

^11

211

^21

*1

S

Value

1.067

<0.1

2.0

0.0

1.9

2.0

37.8 cm"

107.5 cm"

-1

-1

Type Exp't.

ESR

ESR

IR

IR

(Calc.)

(Calc.)

IR

IR

Ref.

26

26

34

34

33

33

32,34

32,34

57

another source of magnetic moment. This was confirmed by an esr

spectrum of the sample in which similar lines to those which apppeared

in the spectrum of the pure Al 0„ were observed, in addition to the

Ti " line at 1.07.

The parameters in (6.1) which have been reported from other meas­

urements, or calculations, are given in Table 7. If these parameters

are used in (6.4), with (6.1), the only unknown parameters left are

C, N, and the Van Vleck susceptibilities. Although g_ and g- have

not .been reported experimentally, the presence of the unwanted im-

3+ purity and the small temperature dependence of the Ti contribution

above 50 K makes the fit to the data insensitive to these two param­

eters. The analysis was done in three stages using all of the data in

Appendix III in the temperature ranges (1) T ^ 12 K, (2) T ^ 50 K, and

(3) T < 300 K. In all analyses, the data below 2.5 K were neglected.

The small signals obtained at these low temperatures caused considerably

more scatter in the data, and these data, as well as those from the ani­

sotropy measurements, indicated the possibility of pair formation. The

analysis is described below.

Region (1): For T ^ 12 K, only the ground state contributes to

X and the susceptibilities can be written as P

X = X + X^. + C/T, (6.5) ^1 '^w ' dia

oi

2 2 Ny g

X = ,,V" + X + X^. + C/T. (6.6) 11 4kT ^ w ., ^dia

oil

58

(6.5) was used in a least squares program to determine x and C. vv oi

Then (6.6) was used to obtain x and N. The results were vv

oil

X = (0.031 ± 0.019) X 1 0 ~ em_u/gm, ^^ol

c

X = (0.024 ± 0.018) X 10 emu/gm, ^ oii

-.6 C = (0.641 ± 0.037) X 10 K-emu/gm

N = (1.06 ± 0.21) X 10^^ Ti " ions/gm sample. (6.7)

Substituting these parameters into (6.5) and (6.6) give curve a. in

Figures 13 and 14.

Region (2): For T < 50 K, the ground and first excited states

contribute to x and the susceptibilities become

X = [x + X exp(-5 /kT)][l + exp(-6 /kT)]' 1 vv vvijL 1 ^

+ XJ. + C/T, (6.8) ' dia

and

Ny2g2 Ny2g2

X„ = t-Atr" - X,, + ( ^ i ^ - X, ) exp(-a,/kT)] " Oil HI

X [1 + exp(-6 /kT)]"^ + x^^3 + C/T. (6.9)

59

Using the parameters obtained from the fit in Region (1), a least

squares program with x as parameters gave ^ li,ll

X^^ = -(0.015 ± 0.015) X 10 ^ emu/gm, li

X^^ = (0.013 ± 0.013) X 10 ^ emu/gm. (6.10) ^111

These theoretical results are shown as curve b in Figures 13 and 14.

Region (3): For T ^ 300 K, (6.1) must be used in (6.4) with

the only undetermined parameters being x and x . Of course vv„ w

2i '''211

the g-factors for the second excited state should be included as ad­

justable parameters, but it was found that the least squares fit was

essentially independent of these, indicating that

I X I > Ny2g2/kT. " 2 o^2

This rather insensitive region gave

X = -(0.025 ± 0.020) X 10 ^ emu/gm, ^^2i

X = -(0.025 ± 0.020) X 10 emu/gm. (6.11) ^ 211

These final results are shown as curve _c in Figures 13 and 14. The

three curves are overlays below their intersections. The exceptional

fit to the data belies the rather large uncertainties for the deter­

mined parameters in (6.7), (6.10) and (6.11). The largest single

60

contribution to these uncertainties comes from the uncertainty in the

field-gradient product. The remainder comes from the uncertainty in

XJ. and the random scatter of the data, dia

For. the fit below 12 K, (6.5) and (6.6) are linear equations of

the form x = b/T + a. Direct calculations of a and b for the two di­

rections were made and the most probable error due to the scatter of

the data was determined by standard formulas. To obtain the Van Vleck

susceptibilities, the diamagnetic susceptibility must be subtracted

from the value of the intercept. The uncertainty in x ,. was added to '^ •' dia

the random error. Since an error analysis for the excited states was

difficult to perform and the Boltzmann factor makes contributions

rather small, the uncertainty was assumed to be the same percentage

as that obtained for the ground state. Finally, this process was

repeated twice, once each for the extremes of the uncertainty in the

field gradient product, and the change in the parameters for the best

fit was added as the contribution from this source of error.

Discussion

There are tv70 important features to be noted from the fit to

the data as shown in Figures 13 and 14. First, the two regional curves

fit the data for their respective regions very well, but deviate from

the data above their regions. In the perpendicular case, the addition

of the first excited state fits fairly ivzell, but there is only a small

correction for the parallel case. The addition of the second excited

state produces excellent agreement with the data. Of course, the

fact that the inclusion of this upper state causes only a small cor-

61

rection is manifest in the large uncertainties for the parameters

determined for this contribution.

Second, the curve does not fit the data below about 2.5 K. The

deviation might be explained by pair formation at these temperatures,

and the field dependence of the magnetic anisotropy at about 1.5 K

seems to substantiate this. The deviation from a 1/T dependence at

3S low temperatures for the anisotropy data was fit by a Neel-Van Vleck

model, but the large scatter in this data would severely limit any

conclusions which could be made from such an analysis.

Once the parameters in (6.1) are determined, theoretical expres­

sions can be used to determine the crystal field parameters. As stated

earlier, Macfarlane has calculated some of the measurable parameters

using a second-order dynamic Jahn-Teller calculation. Because of the

limited amount of data used to analyze the expressions obtained from

20 the standard crystal field theory by Cottrell, Andreadakis, and Quade,

it was felt that this should be done again with the results of this

experiment. The theoretical equations from the above reference which

were used in this analysis are

? 2

X„ = N P ^ { ~ f [1 + exp(-6,/kT)] + ^ ^ } , (6.12)

X i " Np2{B[^exp(-6j/kT) + f^^

1 T c T (6.13)

62

where

4/(2a^, + X)^ + IX'^

^o„ = A^ + a^ ' ^01 0' (6.14)

1„ = tl + r V ^ - (A +aJ(2A -A)l' (6.15) c i c i 1

4A ^°'T "*" ' 11= - 2A-^T(^ - rn^)> (6.16)

i c i

4a + X

c i 1

i c T

B = [1 + exp(-6j/kT)] \ (6.19)

X is the spin-orbit constant, A and a are the trigonal field param­

eters, and A is the cubic field parameter. (6.12) and (6.13) can

c

be rearranged so the Van Vleck susceptibilities for the first two

states can be recognized as

4Ny2 X -T—T—y (6.20) vv A + a^

oil c T

63

4Ny' o

X" ^ = A + a ' (6-21) vv - . _ 111 c T

X = Ny2(-^ + -^ + —-4 ), (6.22) ^w o 6 A^ A + a^ ^ ^

Oi 1 T c T

X = Ny2(- |£- + •f- + — 4 ). (6.23) vvj_j_ o 6 A^ A^ + a^

Using the measured Van Vleck susceptibilities and the parameters from

Table 7, there are a number of approaches which can be used to obtain

values for the theoretical parameters. The value A + a„ = 19,400 cm

^ c T

21

from the analysis of the optical spectra by McClure was used through­

out.

The first method employed was to use the measured values for the

Van Vleck susceptibilities to determine p directly. Subtracting

(6.23) from (6.22) and equating this to the measured difference gives 4Ny^p^/6, = (4.6 ± 3.4) x 10~^, which gives p = 0.97 ± 0.36. Then

o 1

from (6.14), with 80 < A < 154 cm , the trigonal field parameter

a is found to be-(2590 ± 75) cm , where the negative sign is taken

to be consistent with (6.16) giving a small value for g . Combining

these results into (6.18) yields X/A - 0.04 ± 1.5, which is to say

that this approach is undoubtedly too sensitive to the value of p.

The results of this approach are shown in Table 8 as column (1), where

the rest of the equations have been evaluated, and the uncertainties

omitted.

To indicate that a rather reasonable set of parameters can be

64

Table 8

3+ Analysis of Experimental Results for Al 0 :Ti

P a r a m e t e r

a^ (cm

A^ (cm

X (cm

^oll

%l l

^ 1 1

6 (cm

P

b ^vv

oil

Hu

Oi

' " '11

•h

• ' )

• ' )

• ' )

See text.

Units:

( 1 ) ^

- 2 5 9 0

2190

84

1 .067

2 . 9 2

- 0 . 1 0

3 7 . 8

0 . 9 7 2

9 . 5 X lO"

9 . 5 X l O '

2 . 3 X l O '

- 2 . 3 X lO"

: ( emu/gm) .

-11

-11

-8

-8

(2)

- 2 6 5 0

415

120

1 .067

2 . 8 8

- 0 . 8 6

3 7 . 8

0 . 9 4 5

9 . 5 X

9 . 5 X

2 . 3 X

- 2 . 1 X

a

10 '

l O '

lO"

l O '

-11

-11

-8

-8

E x p e r i m e n t a l R e s u l t s

1 .067

2 . 0

0 . 0

3 7 . 8

0 . 9 7 ± 0 . 3 6

( 2 . 4 ± 1 .8) X lO"'^

( 1 . 3 ± 1 .3) X 10~^

( 3 . 1 ± 1 .9) X 10~^

- ( 1 . 5 ± 1 .5) X l O " ^

65

used to fit the .same measured parameters, it was assumed that

X = 120 cm , which is 80% of the free ion value and a reasonable

34 reduction for this host. Then a and A can be determined from 6

and g . These results are shown in column (2) of Table 8 and, except

for a slightly larger negative value for g , fit the parameters equally

well. The experimental values are shown for comparison.

The parallel Van Vleck susceptibilities give very poor agreement

and are independent of the adjustable parameters. It would require

3+ two orders of magnitude higher concentration of Ti ions to account

for the discrepancy. One other point which was considered was to

require agreement with g,,,. This forced A to be about 36 cm and

X to be about 70 cm . g is then found to be extremely large, and

p is at the lower extreme of its experimental limit, at about 0.53.

This is an excessive reduction in the spin-orbit parameter, and the

small A is inconsistent with the optical spectra. The inclusion of

orbital reduction in the Zeeman term is the only thing which appreciably

affects the value of a required to fit the parameters.

Conclusion

3+ The magnetic susceptibilities of Al^O :Ti have been measured

and analyzed on the basis of the crystal field theory of Cottrell,

20 Andreadakis, and Quade with limited success. The presence of a

rather large contribution to the total susceptibility due to an un­

wanted impurity has seriously limited the analysis. The concentra-

3+ . . tion of Ti ions in this crystal is near the solid solubility limit,

according to the manufacturer, and very little improvement can be

66

gained in attempting to increase the relative contribution from the

titanium. It may be possible to reduce the impurity concentration;

however, considerable work will be necessary to determine if this is

possible. This crystal did contain the largest concentration of

impurity ion, assuming it is the same species in all of the Al^O^

host crystals used for these experiments. Recent studies on heat

treatments of titanium-doped Al^O may provide a technique for im-

44 proving the crystals for these studies.

Comparison of the parameters calculated from the crystal field

33

theory, shown in Table 8, with those of Macfarlane, Wong, and Sturge

using a dynamic Jahn-Teller effect seems to indicate that either

method can be used to fit about the same number of parameters. Of

course, the Van Vleck susceptibilities were not given in Ref. 33, and

these should be compared. However, the larger number of adjustable

parameters in the method using a dynamic Jahn-Teller effect would

give it a distinct advantage in fitting more calculated parameters.

It does not seem that there is sufficient justification to favor one

theory over the other. In both cases an orbital reduction of 20% or

more can be used and this is not an unreasonable amount of covalency 34

to expect in this system.

Although the desired result of clarifying the theoretical situ­

ation could not be attained, the level scheme of Figure 12 has been

substantiated and estimates of the Van Vleck susceptibilities for all

three trigonal states have been obtained.

CHAPTER VII

MAGNETIC SUSCEPTIBILITIES OF V 0

Introduction

The magnetic susceptibility of V„0„ has been measured previously

by three research groups. Foex and Wucher obtained data for poly-

crystalline V„0„ which showed a temperature-independent susceptibility

over the temperature range from about 400 K to 500 K. On either side

of this region, the susceptibility exhibited a non-Curie-Weiss be-

37 havior. Carr and Foner measured the susceptibility for a single

crystal of V 0„ from liquid helium temperature to about 350 K. There­

fore, the temperature-independent region is not included in their

38 measurements. More recently, Jones has measured the susceptibility

on single crystals by the pendulum magnetometer method from about

180 K to about 300 K, and using Knight-shift measurements the data was

extrapolated to higher temperatures. The result was a very Curie-

Weiss behavior over the entire temperature range and does not have a

temperature-independent region. It is apparent that these three meas­

urements do not provide compatible information over the extended tem­

perature range. Furthermore, at those temperatures common to all three

measurements, significant quantitative discrepancies exist. It be­

comes worthwhile, therefore, to resolve these differences in the ex­

isting experimental data and to extend the measured single crystal

data beyond the high-temperature anomaly.

Single crystal magnetic data through the high-temperature tran-

67

68

sition is also important in vievz of the recent theory of the semi­

conductor-to-metal transition at 160 K in V 0 . Feinleib and Paul

use the results of transport measurements through this transition to

predict an extremely narrow-band approximation for the metallic region.

They also show that by extrapolating their resistivity curve through

the high-temperature anomaly, the results are consistent with the in-

41 terpretation by other V7orkers that this is an antiferromagnetic-

42 to-paramagnetic transition. In fact, it has been shown that the

qualitative features of the high-temperature transition can be pro­

duced in the theoretical expression for magnetic susceptibility by

coupling the c-axis pairs together antiferromagnetically, but only

if the orbital contribution is included.

The magnetic susceptibilities along and perpendicular to the

trigonal axis of a single crystal of V„0 have been measured between

300 K and 700 K. The samples were supplied by Dr. E. D. Jones.

Susceptibility Measurements

The sample was cut from a single-crystal chip, had a mass of

237.8 mg, and was determined by mass analysis to be within 0.1% of

the stoichiometric composition. Laue back-reflection techniques were

used to locate the trigonal, or rhombohedral, axis. The orientation

was probably accurate to within 1° of this axis. As described in

Chapter III, the measurements were made in a low-pressure (1 mm of

Hg) helium atmosphere up and down the temperature range several times.

At each temperature, 4 to 6 independent measurements were made by

mechanically disturbing the system between each reading. Excellent

69

reproducibility was observed in all the measurements which indicated,

in addition to the stability of the equipment, that no free oxygen was

present to change the oxidation state of the sample in spite of the

several hours that it remained at the high temperatures. A second

sample, cut from the same chip, was also measured at various tempera­

tures with no appreciable change in the results. This indicated that

the thermal history of the first sample had no effect on its magnetic

properties.

The magnetic susceptibilities along and perpendicular to the

trigonal c-axis are shown in Table 9, and the reciprocal mass sus­

ceptibilities are plotted in Figures 15 and 16. In each case, the

data points represent the average of at least four independent meas­

urements. Also, the points in the table and figures represent meas­

ured mass susceptibility with no corrections for diamagnetism or or­

bital paramagnetism. The individual data in Appendix III were used

in all curve fitting. The maximum uncertainty in the absolute value

of the susceptibility was estimated to be 1%, but the uncertainty in

the relative values is better than 0.5%.

Analysis of Results

In Figures 15 and 16, the solid lines were drawn by fitting all

the data.by a least squares method to a cubic equation of the form

(X X 10 )~^ = a + bT + cT^ + dT^ (7.1)

The constants for (7.1) are given in Table 10. These results were

then used to construct the solid curves in Figure 17 which shows a

comparison of these results with previously published data for single

70

Table 9

Average Experimental Mass Susceptibilities

for V^O Single Crystal

T

(K)

298.2 317.6 341.6 372.5 397.1 418.6 440.3 465.9 490.0 515.5 538.3 564.0 573.3 586.4 612.6 636.6 659.9 668.7

X X lo6 II

(emu/gm)

12.42 12 .23 12.00 11 .83 11.67 11.60 11.56 11 .55 11.62 11 .73 11.78 11.74 11.70 11.66 11 .53 11.40 11.24 11.19

T

(K)

297 .8 340 .1 373.7 394.0 420.0 442.9 466 .1 490 .8 517.0 540 .8 564.5 588 .8 611.2 660.7

X^ X lo6

(emu/gm)

12.58 12.14 11.86 11.72 11.62 11.50 11.49 11.47 11.48 11.49 11.47 11 .31 11 .21 10.88

71

H

CD • H X CO I

a

o 4-1

u CO

iH 3 a

•H

C 0) O.

(U (l4

o CN

>

O

>% 4-J

4J

<u o CO 3

CO

CO a o >-i a.

•H a

Pi

m

3 00 •H

^ (mS/nma) X/ QT

o o

72

• - N

• OJ > CO

CO 4-J CO Q

CO <U u CO 3 cr CO

4-J CO CO Q)

t H

0\

o •H nO 3

C_)

^ CO

CO CO

•H 0)

1 CU

•H J-I 3 u

-0^

o o

o o i n

^x!

J I I

O

o

o o CO

CO • H X CO I o tu

4-»

OJ

CO S-i CO

P-I

f O

o CN

> o > 4-1 • H rH •H

^ • H 4-t

CU O CO 3

C/0

CO o o 5-4

a •H

0)

vO

(U >-4

3 oo

•H fe

J I J L

o CT> CO

^_(m§/nma) X/ QT

o 0 0

73

Table 10

Constants for the Cubic Equation

Determined by Least Squares

—fi —fi

Xj (10 emu/gm) ^\\^^^ emu/gm)

-1.869 X 10~^ -2.969 x 10~^

+6.266 X lO"" +7.201 x lO""

-1.252 X 10"6 -1.478 x 10'"6

+8.426 X 10"- ° +9.984 x lO"''"

74

i n o C3^

in

00 o 00

CO CU

•H 4-1 •H tH •H rCi •H 4-) Qu CU O CO 3

CO

CO o o V4

•H o CU

( ^

o c o CO

• H 5-1 CO Pu

o o

r^

(U

3 00

•H fe

^_(m§/nm3) X/ QT

75

crystal V-O . The differences are obvious, but it is interesting to

compare the data v;ith Jones' results below 400 K. The agreement is

quite good, as would be expected since the same crystal was used; but

beyond this temperature, the results begin to deviate considerably

from Jones' curve. In fact, the single crystal is seen to exhibit

a behavior similar to the polycrystalline sample of Foex and Wucher.

The individual cubic curves are shown as the solid lines in Figures

15 and 16.

38 Following Jones, the total susceptibility x(T) can be written

as

X(T) = X^(T) + X^, (7.2)

where XJ(T) is the d-band susceptibility, x is the total temperature-

independent susceptibility, which includes the diamagnetic contribu­

tion and the orbital susceptibility x u- Since V„0- is metallic in

^ •' orb 2 3

the temperature range of interest and in view of Ref. 39, XJ(T) should

be calculated from the principles of band theory. This would require

a knowledge of the density of states for V 0 which is not yet known.

Therefore, in order to discuss these experimental results and to

compare them with other data, a more naive approach is adopted, that

of fitting sections of the data by a Curie-Weiss law of the form

X ( T ) = ^ + X „ . (7-3)

where 0 is the Weiss temperature and C is the Curie constant which,

for localized moments, is given by

76

In (7.4), N is the number of vanadium atoms/gm sample, y .. the effec-ef f

tive magnetic moment per vanadium atom, and k is Boltzmann's constant.

(7.3) quite properly can be regarded as a first-order approximation

since the narrow-band implies an approach to localized moments. Thus,

y rr is not completely without meaning for this case.

In fitting the data, (7.3) was used in the m.ethod of least squares,

and one set of parameters was found for the region below 400 K and

another set for the region above 550 K. In any case, though, it was

not possible to determine C, 6, and x iri this way, so a choice for

X was made. Aside from the diamagnetism (which was about 2% of x(T)

and V7as neglected), x is an orbital paramagnetism which originates,

in metals, from the motion of the collective electrons in the field.

3+ In the magnetically dilute A.1 0 :V system, x is the Van Vleck sus­ceptibility X which originates from the orbital motion of localized

vv

electrons and is anisotropic. The narrow-band approximation and the

observed anisotropy shown in Figure 17 indicate that x has charac­

teristics of both collective and localized electrons. However, for

the purpose of calculating C and 0, the isotropic value of x = 2.8 x 10

38

cgs/emu obtained by Jones was used. The anisotropy was then re­

flected in different values for C (and, therefore, y ^.j.) and 0 for err

the two directions. At least, that allo\\7S a comparison between the

two temperature regions if not between the two directions. This value

of X is very close to the value for x obtained for the parallel o vv

-6

77

direction by extrapolating from the dilute crystal to the pure crys-

38 tal, thus making the parallel results appear more meaningful. The

results, using the above x > are shown in Table 11 and also in Fig­

ures 15 and 16 as the dotted lines. In Table 10 they are compared

with Jones' results and are in general agreement with them as would

be expected. They may also be compared with other previously pub-

lished results. For example, Kosuge, Tanada and Kachi, obtained

3fi 0 = -72 K and 1 ££ = 2.63, and Foex and Wucher give 0 = -1100 K.

Conclusion

The magnetic susceptibilities of single crystal V 0 have been

measured beyond the high-temperature transition. The experimental

results support the following concluding remarks:

(1) Single crystal V^0„ does indeed have a high-temperature

38 transition similar to the polycrystalline sample. x is practically

independent of temperature from about 450 K to 550 K, however, x has

a slight "Van der Waals-type" loop in this temperature interval. On

this basis, the NMR Knight-shift could be expected to have a similar

behavior.

(2) A detectable anisotropy exists above the transition just

37 as it does above the 160 K transition except x and x have crossed.

1 II

(3) A Curie-Weiss law with x included can be fitted to the data

approximately above and below the high-temperature transition but

with a different set of values for C and 0.

(4) Any theory which is to explain the entire reciprocal sus-

Table 11

Curie-Weiss Parameters Using —ft

X = 2.8 X 10 emu/gm o

78

(emu/gm) 0 (K)

^eff (Bohr Magnetons)

(a) Below 400 K

11

Jones

0.00926

0.00922

0.00877

-649

•660

-600

2.352

2.347

2.37

(b) Above 550 K

0.01134

0.01183

-743

•740

2.603

2.659

79

ceptibility curve must be at least cubic in temperature. In the band

approximation this would imply a rather complicated temperature de­

pendence for the density of states function.

LIST OF REFERENCES

1. W. H. Brumage, C. R. Quade, and C. C. Lin, Phys. Rev. 131, 949 (1963).

2. D. J. Arnold and R. W. Mires, Bull. Amer. Phys. Soc, Ser. II, _12, 200 (1967).

3. W. Low, Paramagnetic Resonance in Solids, (Solid State Physics, Suppl. 2, Seitz and Turnbull, eds.; New York: Academic Press, 1960).

4. D. S. McClure, Electronic Spectra of Molecules and Ions in Crys­tals , (Solid State Reprints; New York: Academic Press, 1959).

5. H. A. Bethe, Ann. Physik(5) _3, 133 (1929).

6. R. Schlapp and W. G. Penney, Phys. Rev. 4^, 194 (1932); Phys. Rev. _4^, 666 (1932).

7. J. H. Van Vleck, Phys. Rev. 41., 208 (1932).

8. J. H. Van Vleck, J. Chem. Phys. , 787 (1940).

9. R. Finklestein and J. H. Van Vleck, J. Chem. Phys. 8, 790 (1940)

10. C. J. Ballhausen, Introduction to Ligand Field Theory, (New York: McGraw-Hill Book Co., Inc., 1962).

11. L. E. Orgel, J. Chem. Phys. , 1004 (1955); J. Chem. Phys. 2_3, 1819 (1955); J. Chem. Phys. _22, 1824 (1955).

12. Y. Tanabe and S. Sugano, J. Phys. Soc. Japan 9, 753 (1954); J. Phys. Soc. Japan _9, 766 (1954).

13. Y. Tanabe and H. Kamimura, J. Phys. Soc. Japan 13 , 394 (1958).

14. L. N. Mulay, Magnetic Susceptibility, (New York: Interscience

Publ., 1963).

15. A. R. Smith, Ph. D. Dissertation, Texas Tech University, 1970.

16. D. J. Arnold and R. W. Mires, J. Chem. Phys. 48 , 2231 (1968).

17. F. E. Senftle, M. D. Lee, A. A. Monkewicz, J. W. Mayo, and T. Pankey, Rev. Sci. Instr. 29 , /!29 (1958).

80

81

18. F. E. Hoare and J. C. Walling, Proc. Phys. Soc. (London) 64B, 337 (1951).

19. D. J. Arnold, M. S. Thesis, Texas Tech University, 1967.

20. T. H. E. Cottrell, N. C. Andreadakis, and C. R. Quade, Phys. Letters 2i> 7 (1966).

21. D. S. McClure, J. Chem. Phys. , 2757 (1962).

22. G. Foex, Constantes Selectionees Diamagnetisme et Paramagnetisme Relaxation Paramagnetique, vol. 7, (1957).

23. M. Prasad, S. S. Dharmatti, and H. V. Amin, Proc Indian Acad. Sci. A2^, 312 (1948).

24. W. H. Brumage, E. C. Seagraves, and C. C. Lin, J. Chem. Phys. 42, 3326 (1965).

25. A. R. Smith and R. W. Mires, Phys. Rev. 1^, 265 (1968).

26. K. M. Zverev and A. M. Prokhorov, J. Exptl. Theoret. Phys. (U.S.S.R.) 3^, 1023 (1958) [translation: Soviet Phys.—JETP 3^, 707 (1959)].

27. H. A. Jahn and E. Teller, Proc. Roy. Soc. (London) A161, 220 (1937).

28. H. A. Jahn, Proc. Roy. Soc. (London) A164, 117 (1938).

29. F. S. Ham, Phys. Rev. jl^, A1727 (1965).

30. L. S. Kornienko and A. M. Prokhorov, J. Exptl. Theoret. Phys. (U.S.S.R.) _38, 1651 (1960) [translation: Soviet Phys.—JETP U, 1189 (I960)].

31. H. M. Gladney and J. D. Swalen, J. Chem. Phys. _42, 1999 (1965).

32. E. D. Nelson, J. Y. Wong, and A. L. Schawlow, Phys. Rev. 1^,

298 (1967).

33. R. M. Macfarlane, J. Y. Wong, and M. D. Sturge, Phys. Rev. 1^,

250 (1968).

34. R. R. Joyce and P. L. Richards, Phys. Rev. 179, 375 (1969).

35. J. H. Van Vleck, J. Chem. Phys. % 85 (1941).

36. M. Foex and J. Wucher, Compt. Rend. M., 184 (1955).

82

37. P. H. Carr and S. Foner, J. Appl. Phys. Suppl. _31, 344 (1960).

38. E. D. Jones, Phys. Rev. j ^ , A978 (1965).

39. D. Adler and H. Brooks, Phys. Rev. 1^, 826 (1967).

40. J. Feinleib and W. Paul, Phys. Rev. 155 , 841 (1967).

41. J. Wucher, Compt. Rend. 241, 288 (1955); S. Teranishi and K. Tamara, J. Chem. Phys. , 1217 (1957).

42. R. W. Mires, Ph. D. Dissertation, University of Oklahoma, 1964.

43. K. Kosuge, T. Takada, and S. Kachi, J. Phys. Soc. Japan 18, 318 (1963).

44. G. A. Keig, J. Crystal Growth 2 , 356 (1968).

45. E. U. Condon and G. H. Shortley, Theory of Atomic Spectra, (Cambridge: University Press, 1951).

46. J. S. Griffith, Theory of Transition-Metal Ions, (Cambridge: University Press, 1964).

47. J. H. Van Vleck, Theory of Electric and Magnetic Susceptibilities, (London: Oxford University Press, 1932).

83

APPENDIX I

CRYSTAL FIELD THEORY

For the purposes of optical spectra investigations, the Kamil-

tonian for a free ion may be approximated by

H = I [-1/2 V2 + V(r )] + I -i-, (I.l) ° i=l "- ^ i<j=l ""ij

where n is the number of electrons outside closed subshells, V(r.)

is the electrostatic potential for the ith electron in the field of

the nucleus and the core electrons, and 1/r,. is the electrostatic

interaction of the outer shell electrons. Since the integrals over

the radial part of the wave functions will be left as parameters, no

assumptions need to be made regarding the electrostatic field, except

that it be a "central field." The electrostatic interaction between

outer shell electrons is treated by perturbation theory. It is nor­

mal to add an additional perturbation for spin-orbit interactions.

H' = I C(r,)l, • t (1.2) i=l

but since the first transition series normally fits the intermediate

field case, it will not be considered here. It is introduced as a

perturbation after the effect of the ligand field. Other perturba­

tions such as spin-spin, spin-other orbit and orbit-orbit are sr.all

84

enough to be neglected. The eigenvalues of the Hamiltonian operator

in (I.l) produce the well-known multiplet structure of free ions.

This case is known as Russel-Saunders coupling, and the eigenfunctions

are antisymmetric combinations of one-electron v/ave functions, such

as the Slater determinantal wave functions.

For d-electrons in an octahedral cubic field along the trigonal

axis, the potential may be written

V = I A° r'![Y° + c . ^ i4 1 4 ^

1=1

~ (Y3 - Y-3)]. (1.3)

The small trigonal component is written

n V = y [B° r?Y^ + B° r'^Y°]. (1.4) t >^ 12 1 2 l4 1 4

1=1

The Y™ are spherical harmonics and are functions of the angular co­

ordinates of each electron in the sum. For magnetic studies, the

spin-orbit coupling and the interaction with an external magnetic

field are

XL • S + y (L + g S) . H, (1.5) o s

where X is the spin-orbit coupling constant, y^ is the Bohr magneton,

and R = 2.0023. The Hamiltonian usually used is then written ^s

u ^ H + V + V + X L - S + y(L + g ?) • ti. (1.6) o C t O S

85

The effect of covalency which occurs as a result of overlap of the

magnetic electrons with the bonding orbitals of the ligands can be

introduced by an orbital reduction factor, k, multiplied by the an-

3 gular momentum operator. This effectively replaces the process of

using wave functions which are linear combinations of the magnetic

ion orbitals with the orbitals of the ligands.

The eigenfunctions of the unperturbed Hamiltonian, H , are

45 Clebsch-Gordan type combinations of one-electron orbitals. Various

perturbation techniques are then used to solve the complete Hamil­

tonian in (1.6) which are suitable to the particular system under

investigation. The theoretical analysis for trivalent titanium has

31 been done by Gladney and Swalen, and by Cottrell, Andreadakis, and

20 Quade; and the analysis for trivalent vanadium has been done by

Brumage, Quade, and Lin. Most of the earlier work (pre-1960) on

3 4 transition metal ions can be found in reference works by Low, McClure )

and Griffith."^6

86

APPENDIX II

MAGNETIC SUSCEPTIBILITY

The magnetic susceptibility of a medium is given by

M X = •|7> (II.1)

where M is the magnetization and H is the magnetic field. Now the

magnetization of N atoms or molecules per gram of sample is

M = Nm, (II.2)

where m is the mean magnetic moment of a single atom. If the atoms

are assumed to be sufficiently non-interacting to allow the use of

Boltzmann statistics, the mean magnetic moment is

y m. h 1

e -Wj /kT

•^ = r -W, /kT ' ( -3)

V7here W. is the energy of a stationary state of the system with mag­

netic moment m,; k is Boltzmann's constant. If the energy is expanded

in a power series in terms of the magnetic field, vje can write the

magnetic moment as

87

8W. 1 ,,' «.j

m. = i = •• afT =-"i - 2w. (II.4)

where W ^ W . +W.H + W,H is the expansion to second order. Sub­

stituting (II.4) and the series for W. into (II.3), the exponential

can be expanded for temperatures high enough for the thermal energy

to be much larger than the magnetic energy; i.e., kT >> W.f/. Now

if only the term which is linear in the field is retained, this is

substituted into (II.2), and the result substituted into (II.1), the

familiar form for magnetic susceptibility from Van Vleck is obtained.

wl^ _o

— (II.5) V -W /kT

1

where the sum is now over the levels and g. is the degeneracy of each

level. Retaining only the linear term in the expansion of (II.3) is

equivalent to assuming a paramagnetic, or field independent, suscep­

tibility; i.e., no permanent magnetic moment and no contributions from

second order effects. These are both valid for the systems being

studied.

88

APPENDIX III

TABLES OF DATA

The following tables contain all of the raw force data for the 3+ 3+

Pt calibration, pure Al 0 , Al 0 :V , Al 0 :Ti and V 0 . These are

measured forces and must be corrected for the sample suspension before

they can be used to calculate sample susceptibilities with exceptions

as indicated in the following two paragraphs. The correction is a meas­

ured force for the entire suspension system with the sample removed,

and it must be subtracted from the tabulated forces.

The earliest experiment was done on the V 0_ sauiple and its cali­

bration. For the calibration of the small magnet, a 1 gram Pt cylinder

was suspended by a quartz fiber and basket. The measured force correc­

tion was -5 yg. The V 0 sample was suspended by a quartz fiber and a

small quartz hook. Because the sample forces were much larger and the

suspension hook contained much less mass than the basket, the correction

to the V„0„ forces was negligible.

.3+

The next set of measurements were Runs T-2 and T-3 on the Al20^:Ti

sample. These were done on the low temperature system using a modified

three section fiber similar to that described in Chapter III, p. 25.

The modification consisted of using No. 34 AWG copper wire for the bottom

section of the suspension fiber and basket. The copper vrire was used

because the susceptibility of copper is much smaller than that for quartz,

and the mass of that portion o^ the suspension in the field could be

89

made small. To be assured that the suspension correction could be

neglected for these measurements, attempts were made to measure the

force on the empty container. However, without the weight of the

sample, the wire would not hang straight and the force could not be

measured directly. Even so, the correction was assumed negligible

for these two runs.

To check this assumption, an all quartz suspension like that

described on p. 25 was used for Runs T-4 and T-5. The measured force

correction for this suspension was -(2 ± 0.4) yg. Comparison of the

average force at room temperature from Run T-2 (F = -115.4 yg) with

the average corrected forces from T-4 (F = -114.9 yg) and T-5

(F = -116.2 yg) at about the same temperature indicates the neglect

of the correction for the copper wire was justified.

Similar suspension fibers and baskets to those used in Runs T-4

and T-5 were used for the rest of the measurements reported in this

dissertation. The only difference was the shape of the basket which

depended upon sample shape. The magnitude of the force correction

was assumed to be the same, i.e., -(2.0 ± 0.4) yg.

The net magnetic susceptibilities of the sample can then be

obtained from the corrected forces by a calibration constant, K, by

writing

X = KF, (III.l)

where

90

* = mF (T )' (I"-l) r r

and m = mass of the reference sample, x (T ) = magnetic susceptibility

of the reference sample at the reference temperature T , m = mass of

the sample being studied, and F (T ) = average measured force on the

reference sample at the reference temperature T . This is equivalent

to finding the field-gradient product from (3.1) for the reference

sample, and substituting that value back into (3.1) when it is applied

to the measurements on the unknown sample. As long as the units for

the forces and masses in (III.l) and (III.2) are the same, respectively,

the susceptibilities x ^^'^ X will have the same units. The two step

process of calculating the field-gradient product first, and then cal­

culating K for each sample, was actually used in practice; however, for

simplicity, only K is given in each of the following data tables. The

method of determining the proper calibration constants for all data

reduction is described in the following two paragraphs.

The calculation of sample susceptibility from the V 0 force meas­

urements was straightforward and was done as described in the preceding

paragraph for each individual force measurement shown in the following

tables as V 0 , Run #6. The calibration data for the V^O measurements

are listed under "Run #2 on small magnet."

The calibrations for all of the remaining experiments described in

this dissertation were taken from data listed under "Run PT-5 on 2000-G

3+ magnet." The Runs T-2, T-3, T-4 and T-5 on the Al^O^:^ sample. Run

3+ V-8 on the Al 0 :V sample, and Runs A-3 and A-4 on the pure AI2O2

91

sample were all performed in that order before the calibration Run

PT-5. The rotational variation in the field-gradient product referred

to on p. 12 was evident from the difference in the measured forces for

the parallel and perpendicular directions of the magnet in Run PT-5.

These two orientations were the same as the corresponding directions

for Runs A-3 and A-4 and were found to be the extremes of the varia­

tion. Therefore, to remove this "instrumentation anisotropy," seperate

calibration constants, K, and K.., were used to calculate sample suscep­

tibilities from corrected force measurements for the corresponding data

from Runs A-3 and A-4. This was also done for all of the diamagnetism

studies reported in Chapter IV, which were performed after the cali­

bration Run PT-5 using the same corresponding orientations. Since the

precise orientations of the magnet corresponding to the parallel and

.3+ perpendicular force measurements for all of the A1^0^:Ti and the

3+ Al^O :V data were unknown, an exact calibration was not done. It was

felt that the average of the two calibration constants calculated from

the parallel and perpendicular force data in Run PT-5 was more nearly

3+ 3+ correct for the Al 0 :Ti and kl^O^iV data reduction, and this

average was used.

All individual force measurements listed in the raw data tables,

after being reduced to magnetic susceptibility values according to the

preceding description, were used in the curve fitting processes de­

scribed in their respective chapters with the following exceptions.

3+ The exceptions are: (1) all data below 2.5 K for all Al20^:Ti , and

3+ (2) all low pressure "drift" points in Run V-8 on the Al20^:V which

are no ted in the raw data tables. To obtain the susceptibility tables

92

in the respective chapters, measurements within a degree of each other

3+ were averaged, with the exception of Al 0 :Ti data from different

runs near 4 K. None of this averaging involved different runs except

3+ for points below 2.5 K for the A1_0 :Ti table, and these were not

used in any analyses. These average values of susceptibility were

used to plot the points on the graphs in the respective chapters.

The uncertainties on the tabulated force measurements are the

estimated maximum random uncertainties due to the sum of the recorder

dead-band, chart paper reading error, and noise. The contribution

from the noise was determined as half of the peak-to-peak noise ampli­

tude. The chart paper reading error includes a contribution from an

uncertainty due to the drift correction for those readings made with

a base-line drift, as described in Chapter II. The uncertainties in

the V_0_ force measurements were omitted since they were the same in

all cases, i.e., ±10 yg.

There are some fluctuations in data at the same temperature which

could not be explained. An example is the room temperature measure­

ments for pure A1„0 , Runs A-3 and A-4. As noted, the first three

were done in air and the rest in He exchange gas. This should not have

produced this much change. As an example of the fluctuation observed

by rotating the magnet away after a set of measurements and returning

to the same position for another set of measurements, the parallel

force measurements made at approximately 302.5 K for the pure Al20^

can be used. The first five were done, then the magnet was rotated

to make the five perpendicular force measurements at about the same

temperature, and then the magnet was rotated back to the parallel

93

direction and the next three taken. The first five average F.. =

-(106.2 ± 0.7) yg and the next three average F = -(105.3 ± 0.3) yg,

where the average deviation is used as an uncertainty. The difference

of these two average forces is within the estimated random uncertainty

for the force measurements and their average deviations actually make

them overlap. One other check made on this property of the magnet

system, i.e., to return to the same force after the magnet has been

moved, gave a deviation of only 0.2%.

Platinum Calibration

94

Run #2 on small magnet mass = 1.0751 g

(K) (yg) (K) (yg)

296 .1 296 .1 295.2 296 .3 296 .3 296.5 296.5 295.7 295.7 296.2 296,7 296 .8 296.8

+798.0 +799.0 +812.5 +813.0 +809.0 +806.5 +811.5 +811.0 +811.0 +821.0 +811.0 +809.0 +806.0

+ +

+

+

+

+

+

+

+

+

+

+

+

2.0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0 2 .0

296.7 296.9 297 .3 297 .3 297.4 297.4 296.6 296.6 296.5 296.5 297.0 297.0

+811.0 +807.0 +826.0 +820.0 +827.0 +820.0 +826.0 +826.0 +828.0 +830.0 +812.0 +814.0

± 2,0 ± 2.0 ± 2.0 ± 2 .0 ± 2.0 ± 2 .0 ± 2.0 ± 2.0 ± 2 .0 ± 2 .0 ± 2 .0 ± 2 .0

Run PT-5 on 2000-G magnet mass = 552.26 ir.s

Parallel Perpendicular

(K) (yg) (K) (yg)

296.8 296 .8 296 .8 296 .8 296 .8 296.9 296.9 296.9 296.9 296.9

+339.0 ± 1.0 +339.0 ± 1.0 +338.5 ± 1.0 +337.0 ± 1.0 +336.5 ± 1.0 +337.5 ± 1.0 +337.0 ± 1.0 +336.5 ± 1.0 +338.0 ± 1.0 +337.5 ± 1 . 0

296.9 296.9 297.0 297.0 297.0 297.0 297.0 297.0 297.0 297.0

+329.5 ± +326.5 ± +328.5 ± +329.0 ± +328.0 ± +327.0 ± +327.0 ± +330.0 ± +328.0 ± +326.5 ±

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Runs A—3 and A-4

Parallel

Pure Al 0

K,j = 3.29 K = 3.39

95

mass = 479.52 mg

Perpendicular

(K) (yg) (K) (yg)

299.9 299.9 299,9 302,5 302.5 302.5 302,5 302,6 302,6 302.6 302,6

79,7 79,7 79.7 79.7 79.7

4 .08 4 . 0 8 4 ,08 4 ,08 4 ,08 4 .10 4 .10 4 .10 4 .10 4 .10 1.49

. 1.49 1,49 1,49 1.50 1.06

- 1 0 8 , 6 - 1 0 8 , 0 - 1 0 8 , 6 - 1 0 6 , 0 - 1 0 5 , 8 - 1 0 5 , 0 - 1 0 7 , 6 - 1 0 6 , 6 - 1 0 4 . 8 - 1 0 5 . 6 - 1 0 5 . 6 - 1 0 4 . 0 - 1 0 5 . 0 - 1 0 5 . 2 - 1 0 4 . 0 - 1 0 4 . 4 - 95 .4 - 95 .4 - 95 .0 - 95 .2 - 9 4 . 8 - 96 .6 - 97 .6 - 98 ,2 - 97 ,4 - 96 .8 - 85,2 - 86,6 - 85 ,2 - 83,2 - 88,8 - 79.4

+ +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+ +

+

+

+

+

+

+

+

+

+

+

+

0 , 8 * 0,8:^c 0 , 8 * 0 ,8 1,2 1,2 0 ,8 0 , 8 0 ,8 0 ,8 0 .8 0 .4 0 .4 0 .4 0.4 0 .4 0 .8 0 . 8 0 .8 0 .8 0 .8 0 .8 0 . 8 0 .8 0 .8 0 .8 2 .0 2 .0 2 .0 2 .0 3,0 3.0

299 .8 299 .8 299.9 302.6 302.6 302.6 302.6 302.7

79.7 79.7 79.7 79.7 79.7

4 .08 4 .08 4 .08 4 .08 4 .08 4.10 4 .10 4.10 4 .10 4.10 1.50 1.49 1.49 1.49 1.48 1.06

- 1 0 6 . 0 - 1 0 4 . 4 - 1 0 5 . 0 -101 .4 - 1 0 1 . 0 -100 .6 -100 .6 - 1 0 0 . 2 - 1 0 2 . 8 -101 .4 -102 .2 - 1 0 2 . 2 - 1 0 2 . 2 - 92 .4 - 9 3 . 8 - 9 0 . 8 - 92.6 - 93 .4 - 94 .0 - 93 .4 - 94 .0 - 93 .6 - 9 4 . 8 - 82.6 - 83 .8 - 81.0 - 81.4 - 81.4 - 77.6

± 0 . 8 * ± 0 . 8 * ± 0 . 8 * ± 1.2 ± 1.6 ± 1.2 ± 1.2 ± 1.2 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 3.0 ± 3.0 ± 3.0 ± 3.0 ± 3.0 ± 1.2

* In air, all others in He exchange gas.

Al203:V 3+

96

Run V-8, sample //I, after bake K = 6.01 mass = 266.40 mg

Parallel Perpendicular

(K)

298.0 298.0 298.0 298.0 298.0 298,0 298,0 298,0 298,0 298.0 297.7 297.7 239 .8 237.0 195.4 194.4 148.8 146.7 115.7 114.4

8 0 . 1 80 .1 8 0 . 1 8 0 . 1 8 0 . 1 8 0 . 1

. 8 0 . 1 8 0 , 1 55 ,7 52 ,7 39 ,4 35 ,4 28 ,4 27 ,2 2 2 , 1 2 1 . 1 20 ,6 20 .2 14 .7 13 .5 11 .2

9 . 1

(yg)

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

—

-

—

—

+ + + +

53 .6 53 .5 53 .6 52 .9 53 .0 5 3 . 1 53 .0 52 .9 53 .0 5 3 . 1 5 2 . 1 52 .2 50 .7 50 .7 49 .9 49 .7 47.9 4 8 . 8 45 .9 45 .5 39.4 39.6 39.4 39.7 39 .5 3 9 . 3 39.9 39 .8 32 .8 31.2 24 .2 21 .4 14 .2 12 .2

6 .4 3.2 4 . 0 0 .0

11 .2 1 5 . 1 1 6 . 1 21 .8

+ +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

1.0 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .6 0 .6 0 .4 0 .6 0 .6 0 .4 0 .4 0 .4 0.4 0 .4 0 . 3 0 . 3 0 . 3 0 .3 0 . 3 0 . 3 0 . 3 0 . 3 0.2 0 .2 0 .2 0 .2 0 .3 0 . 3 0 .2 0 . 3 0 .4 0 . 3 0 .8

(K)

298.0 298.0 298.0 298.0 298.0 298.0 298.0 298.0 298.0 298.0 297.7 297.7 297.7 297.7 239.0 238.2 196.4 193.2 149.6 145.9 117.0 113.0

80 .4 80 .3 80 .3 80 .3 80 .3 80.2 80 .2 80.2 80 .2 80.2 5 7 . 8 4 9 , 5 42 ,2 33 ,6 29 ,6 25 .6 23 .5 23 .5 17 .8 17 .4

(yg)

- 5 3 . - 5 3 . - 5 3 . - 5 3 . - 5 3 . - 5 3 . - 54 . - 5 3 . - 5 3 . - 5 3 . - 54 . - 52 . - 5 3 . - 5 3 . - 52 , - 52 , - 5 1 , - 50, - 48 , - 49, - 48 . - 46. - 40. - 40. - 40. - 40. - 40. - 40. - 40. - 40 - 41 - 40 - 35 - 31 - 24 - 19 - 13 - 7 - 3 - 5 + 14 + 15

9 ± 6 ± 2 ± 5 ± 6 ± 3 ± 0 ± 8 ± 5 ± 5 ± 1 ± 9 ± 7 ± 3 ± 0 ± 0 ± 4 ± 8 ± 9 ± 4 ± 1 ± 6 ± 2 ± 8 ± 6 ± 2 ± 5 ± 8 ± 2 ±

.8 ±

.1 ±

.4 ±

.1 ±

.0 ±

.4 ±

.3 ±

.6 ±

.3 ±

.0 ±

.4 ±

.7 ±

.9 ±

0.4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0.4 0 .4 0.4 0 .4 0 .4 0 .4 0.4 0 .4 0 .4 0 .4 0 . 3 0 . 3 0 . 3 0 . 3 0 . 3 0 . 3 0 . 3 0 . 3 0 . 3 0 .3 0 .2 0 .2 0.2 0.2 0 . 3 0 . 3 0 . 3 0 .2 0 .6 0 .2

Al203:V 3+ (cont.) 97

Para l l e l Perpendicular

(K) (yg) (K) (yg)

7.4 6.3 5.0 4.3 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4,10 3,50 3.49 2.96 2.94 2.44 2.44 2.00 2.00 1.54 1.53 1.53 1,53 1.53 1,47 1,47 1.06 1,05 1.05 1,06

+ + + + + + + + + + + + + -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

—

26 29 26 7 6 6 5 7 6 8 8 10 11 3 4

14. 15. 28. 30. 32. 31. 34. 32. 36. 34. 34. 33. 33. 32. 29, 33, 33.

.6

.2

.7

.4

.5

.2

.2

.8

.6

.1

.0 8 .0 0 0 4 0 7 0 4 8 9 2 3 3 5 8 1 5 5 0 9

+

+

± +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

1,0 1,0 1,0 1,0 1,0 2,0 2.0 1.0 1.0 2.0 2.0 3.0 3.0 4.0 4.0 3.0 3.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 0.3 2.0 1.6 1.0

Note: Data points without uncertainties are low pressure drift points and were not used in curve fitting.

16.5 13.2 10.0 7.8 5.2 4.3 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 3.48 3.47 2.91 2.91 2.46 2.46 2.00 2.00 1.52 1.52 1.52 1.52 1.52 1.53 1.53 1.48 1.47 1.06 1.06 1.05 1.03 1,02 1,02 1.02 1.02

+ 17.1 + 13.8 + 52.5 + 76.3 +126.8 +192.0 +199.0 +199.5 +199.0 +200.0 +200.0 +201.0 +201.0 +199.5 +199.5 +202.5 +203.5 +200.0 +200.0 +217.0 +215.5 +230.5 +235.0 +242.0 +243.5 +246.5 +250.0 +256.5 +256.5 +256.0 +256.0 +257.5 +258.5 +257.7 +258.5 +257.0 +265. +261.5 +260.0 +260.0 +260.0 +260.0 +262.0 +260.0

+

+

+

+

+

+

+

+

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 2.5 2.5 2.0 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 15. 2.0 5.0 2.0 2.0 2.0 2.0 2.0

Al203:Ti 3+

Run T - 2 , sample # 1 , a f t e r bake

98

K = 2.82 mass = 569.06 mg

P a r a l l e l P e r p e n d i c u l a r

(K) (yg) (K) (yg)

294 .3 294 .3 294 .3 294 .3 294.4 250.2 250.2 250.2 250 .1 200.4 200.4 200 .3 200.2 150.0 150 .1 150.0 149.9

98 .6 98 .6 98 .7 9 8 . 8 81 .0 81 .0 81 .0 80.9 80.9

4 .14 • 4 .14

4 .14 4 .14 4 .14 4 .14 4 .14 4 .14 2 .02 2 .02 1.55 1.55 1.55 1.55 1.55 1.43

- 1 1 5 . 8 ± - 1 1 5 . 6 ± - 1 1 5 . 2 ± - 1 1 4 . 6 ± - 1 1 5 . 8 ± - 1 1 5 . 2 ± - 1 1 5 . 6 ± - 1 1 7 . 6 ± - 1 1 6 . 6 ± - 1 1 4 . 0 ± - 1 1 4 . 0 ± - 1 1 4 . 6 ± - 1 1 5 . 0 ± - 1 1 3 . 6 ± - 1 1 6 . 8 ± - 1 1 6 . 0 ± - 1 1 3 . 6 ± - 1 1 4 . 2 ± - 1 1 3 . 8 ± - 1 1 5 . 8 ± - 1 1 4 . 0 ± - 1 1 1 . 6 ± - 1 1 1 . 6 ± - 1 1 2 . 6 ± - 1 1 2 . 2 ± - 1 1 2 . 6 ± - 4 4 . 1 ± - 42 .0 ± - 40 .8 ± - 4 0 . 8 ± - 42 .4 ± - 4 0 . 8 ± - 42 .2 ± - 41 .8 ± + 25 .2 + 27 .2 + 45 .2 ± + 44 .2 ± + 44 .0 ± + 4 4 . 8 ± + 45 .0 ± +109.6

1.0 1.0 1.0 1.0 1.0 0 . 8 0 .8 0 .8 0 .8 0 .6 0 .6 0 .6 0.6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .6 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4 0 .4

0 .4 0 .4 0 .4 0 .4 0 .4

294.4 294.4 294.4 294.4 294.4 250.2 250 .1 250 .1 250.2 200 .1 200 .1 200 .1 200 .1 150.0 149 .8 149 .8 149.9

98.9 98.9 98.9 98.9 81.2 81.2 81 .1 81 .1 81 .1

4 .14 4 .14 4.14 4.14 4.14 4.14 4.14 4.14 2 .01 1.99 1.55 1.55 1.55 1.55 1.40

-118 .6 ± 1.6 - 1 2 0 . 0 ± 1.6 - 1 1 9 . 8 ± 1.6 -118 .2 ± 1.6 -118 .2 ± 1.6 -120 .0 ± 0 .8 - 1 1 7 . 8 ± 0 .8 -119 .4 ± 0 .8 - 1 1 8 . 8 ± 0 .8 -118 .4 ± 0 .6 - 1 1 8 . 0 ± 0 .6 -118 .0 ± 0.6 -118 .0 ± 0 .6 - 1 1 7 . 8 ± 0.6 -116 .6 ± 0 . 6 -116 .4 ± 0 . 6 - 1 1 7 . 8 ± 0.6 -116 .4 ± 0.6 -116 .6 ± 0.6 - 1 1 6 . 8 ± 0 .6 - 1 1 7 . 0 ± 0.6 - 1 1 4 . 4 ± 0 . 6 - 1 1 3 . 2 ± 0.6 -114 .6 ± 0.6 - 1 1 3 . 2 ± 0 .6 - 1 1 4 . 0 ± 0 .6 - 55.4 ± 0 .4 - 57 .2 ± 0 .4 - 56 .0 ± 0 .4 - 56 .2 ± 0 .4 - 5 7 . 8 ± 0.4 - 56 .8 ± 0.4 - 57.6 ± 0 .4 - 56 .6 ± 0.4 + 14 .0 + 13 .0 + 22.6 ± 2.0 + 23 .0 ± 2 .0 + 22.6 ± 2 .0 + 22 .0 ± 2 .0 + 81 .2

99

Al^O^tTi^"^ (cont.)

Run T-3, sample #1, after bake K = 2.82 mass = 569.06 mg

Parallel Perpendicular

(K) (yg) (K) (yg)

77.0 7 0 . 1 5 7 . 5 4 8 . 8 39 .4 3 0 . 8 22 .9 20 .7 1 1 . 8 11 .4 10 .4

4 .12 4 .12 4 .12 4 .12 4 .12 2 .02 2 .02 1.54 1.55 1.55

- 1 0 7 . 8 - 1 0 8 . 0 - 1 0 6 . 0 - 1 0 5 . 0 - 1 0 5 . 0 - 1 0 2 . 2 - 9 8 . 8 - 98 .0 - 86 .8 - 85.4 - 82 .8 - 40 .4 - 41 .4 - 42 .2 - 41 .6 - 40.6 + 20. + 20. + 56. + 60 . + 59 .

± 0 .4 ± 0 .4 ± 0 .4 ± 0 .4 ± 0 .6 ± 0 .4 ± 0 .4 ± 0 .4 ± 0 .8 ± 0 .4 ± 0 .8 ± 0 . 8 ± 0 . 8 ± 0 .8 ± 0 .8 ± 0 .8 ± 4 . ± 3 . ± 4 . ± 4 . ± 4 .

77.2 69 .9 60 .6 4 7 . 1 43 .9 2 9 . 1 24.4 1 7 . 1 11.9 11 .2 10.9

4.12 4 .12 4 .12 4 .12 4 .12 2.02 2.02 1.54 1.54 1.54

-112 .0 -110 .6 -111 .0 -108 .0 -111 .0 -104 .2 -102 .6 - 96 .2 - 89 .0 - 87.0 - 87 .0 - 57.2 - 56.6 - 56 .8 - 56.4 - 54.6 - 12 - 14 + 38 + 37 + 38

± 0 .4 ± 0 .4 ± 0 .4 ± 0 .4 ± 0.6 ± 0 .4 ± 0 .4 ± 0 .4 ± 0 .8 ± 0 .6 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8

100

Al203:Ti 3+

( c o n t . )

Run T-4 , sample / / I , a f t e r bake K = 2.82 mass = 569.06 mg

P a r a l l e l P e r p e n d i c u l a r

(K) (yg) (K) (yg)

298.5 298.5 298.5 298.5 298.5

6 1 . 3 29 .2 2 5 . 8 18 .0

4 .09 4 .09 4.09 4 .09 4 .09 3.45 3.46 3.47 2 .60 2 .60 1.52 1.52 1.52 1.52 1.53

- 1 1 7 . 0 - 1 1 6 . 4 - 1 1 6 . 0 - 1 1 7 . 8 - 1 1 7 . 4 - 1 1 0 . 0 - 1 0 3 . 8 - 1 0 0 . 8 - 1 0 1 . 6 - 38.2 - 40 .6 - 39.2 - 38 .4 - 37 .8 - 30 .0 - 23.4 - 32.0

0 .0 0 .0

+ 48 .2 + 47 .8 + 47 .0 + 4 7 . 8 + 45 .2

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

0 .8 0 . 8 0 . 8 0 .8 0 .8 1.2 1.0 1.2 4 .0 2 .0 4 .0 2 .0 2 .0 0 .8 4 .0 4 .0 4 .0 4 .0 4 .0 2 .0 2 .0 2 .0 2 .0 2.0

298.6 298.6 298.6 298.6 298.6

58.9 45 .8 23.2 21.4

4.09 4.09 4.09 4.09 4.09 3.48 3.49 3.50 2.62 2 .61 1.53 1.53 1.53 1.53 1.53 1.02

-120 .0 -118 .4 -119 .4 -119 .2 - 1 1 8 . 8 -111 .4 - 1 0 9 . 8 - 99 .0 - 98 .8 - 56 .0 - 54 .8 - 52 .4 - 49.6 - 47 .8 - 42.0 - 42.2 - 44.2 - 26. - 2 1 . + 27.6 + 26.4 + 28.6 + 27.0 + 25.4 + 44 .0

± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 0 .8 ± 1.0 ± 1.0 ± 0 .8 ± 4 .0 ± 1.2 ± 2.4 ± 2 .0 ± 4 .0 ± 2.0 ± 2.0 ± 4 .0 ± 10. ± 8. ± 4 .0 ± 4 .0 ± 4 .0 ± 2 .0 ± 2 .0 ± 4 .0

101

Al203:Ti 3+ ( c o n t . )

Run T - 5 , sample # 1 , a f t e r bake

P a r a l l e l

K = 2.82 mass = 569.06 mg

P e r p e n d i c u l a r

(K)

296.6 296.6 296.6 296.6 296.6

79 .8 79 .8 79 .8 79 .8 79 .8 75.2 73 .8

4 .08 4 .08 1.97 1.97 1.54 1.54

(yg)

- 1 1 9 . 0 ± 0 .8 - 1 1 7 . 4 ± 0 .8 - 1 1 9 . 0 ± 0 .8 - 1 1 8 . 4 ± 0 .8 - 1 1 7 . 0 ± 0 .8 - 1 1 2 . 4 ± 0 .8 - 1 1 2 . 0 ± 0 .8 - 1 1 2 . 4 ± 0 .8 - 1 1 2 . 0 ± 0 .8 - 1 1 1 . 4 ± 0 .8 - 1 1 3 . 2 ± 0 .8 - 1 1 2 . 0 ± 0 .8 - 46 .0 ± 2 .0 - 44 .6 ± 2 .0 + 15 .0 ± 4 .0 + 15 .0 ± 4 .0 + 34.0 ± 2 .0 + 34.6 ± 2 .0

(K)

296.6 296.6 296.6 296.6 296.6

79 .8 79 .8 79 .8 79 .8 79 .8 68 .6 70.5

4 .08 4 .08 1.97 1.54 1.54

(yg)

-121 .2 ± 0 .8 -121 .4 ± 0 .8 -120 .4 ± 0 .8 -121 .4 - 1 2 0 . 8 -113 .4 -115 .2

0.8 0.8 0.8 0.8

-115 .2 ± 0 .8 -115 .2 ± 0 .8 -114 .4 -113 .2

0.8 0.8

-113 .0 ± 0 .8 - 58.2 ± 1.6 - 58 .4 ± 1.6 - 1 1 . 8 + 13.4 ± 2.0 + 13 .4 ± 2 .0

^2°3

102

Run #6, sample y/4 K = 5.38 mass = 237.80 mg

Parallel Perpendicular

(K)

298.2 298 .3 298.4 298 .3 298 .3 298 .1 298 .1 320 .3 319.7 319.2 318.7 318.4 318.2 344.5 344.5 344.7 344.8 345 .1 345 .1 377.7 377.4 376.9 376 .8 378 .1 377.4 403.9 403.6 403 .3 402.7 402.4 402.6 427 .3 426.4 425.6 425.2 424.7 424.6 447.7 448.6 448.2 447 .7 447 .5

(yg)

2310 2320 2310 2310 2310 2315 2315 2280 2280 2275 2275 2280 2280 2230 2235 2240 2240 2235 2235 2200 2195 2210 2200 2205 2210 2170 2170 2165 2180 2180 2180 2160 2150 2150 2160 2170 2170 2155 2150 2160 2160 2145

(K)

297.7 297 .8 298.0 297.9 342.9 343.2 343.2 343.4 379.0 378.7 378.5 378.6 399.8 400 .1 399.9 399.8 426.7 427 .1 427.2 426.9 450 .8 450.6 450.7 450.5 474.6 475.0 474.6 474.6 499.7 500 .3 500.2 500 .3 526.9 526.9 527 .1 526 .8 550 .8 551.4 551.5 551 .3 575.5 576.0

(yg)

2355 2340 2345 2335 2275 2255 2265 2255 2210 2205 2215 2205 2190 2180 2190 2175 2170 2160 2165 2160 2155 2135 2145 2135 2155 2135 2140 2135 2140 2130 2140 2135 2150 2130 2145 2130 2150 2140 2135 2135 2140 2140

Run //6 (cont.)

V2O3 (cont.)

103

Parallel Perpendicular

(K) (yg) (K) (yg)

448.1 474.0 474.5

474.8 474.4 474.7 499.7 499.4 499.2 499.2 498.7 499.3 525.5 524.3 526.0 525.9 525.4 525.2 548.8 548.8 548.7 548.7 548.7 548.6 575.3 575.2 575.1 574.8 574.7 574.4 584.6 584.5 584.3 584.3 599.7 598.1 597.0 596.1 624.5 624.1 624.3

2150 2150 2145 2155 2150 2155 2155 2165 2160 2160 2165 2175 2160 2185 2180 2195 2175 2190 2185 2190 2195 2200 2195 2195 2195 2180 2185 2190 2185 2195 2185 2180 2180 2175 2180 2180 2160 2175 2170 2150 2145 2155

575.2 574.9 600.0 600.0 599.8 600.5 622.9 622.9 623.0 622.4 674.4 673.7 672.0 671.5 648.3 648.8 648.8 649.2

2140 2130 2105 2110 2110 2105 2100 2085 2095 2070 2035 2020 2030 2020 2075 2060 2070 2045

104

Run #6 (cont.)

Parallel

(K) (yg)

624.0 2145 649.2 2135 649.0 2110 647.9 2125 647.9 2120 671.9 2100 671.6 2090 672.7 2100 671.9 2085

V2O3 (cont.)