Pressure dependence of magnetic transitions

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PRESSURE DEPENDENCE OF

MAGNETIC TRANSITIONS





















By
JAMES EDMUND MILTON










A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA

April, 1966














ACKNOWLEDGMENTS


The author is deeply indebted to many people for

aid and encouragement in completing this dissertation. He

wishes to express his gratitude to the many people who have

helped and especially to those listed below.

Dr. Thomas A. Scott, the chairman of the author's

supervisory committee, suggested the problem and was always

ready to give aid when needed.

Mr. Basil McDowell was absolutely indespensible,

he assisted in the design and construction of much of the

equipment used in this study. In addition to this he sup-

plied liquid helium when needed and frequently gave a

friendly helping hand or clever suggestions.

Dr. William S. Goree gave many helpful suggestions

regarding the cryogenic apparatus and designed the pressure

bomb used.

Mr. K. S. Krishnan served as a soundingboard for

many ideas involving experimental techniques and frequently

suggested improvements.

Mr. Guy Ritch and Mr. Frank Ebright were very able

assistants during the final phase of data taking.

Dr. Stanley S. Ballard and Dr. Thomas A. Scott

provided financial support during most of the author's


ii








graduate career. Dr. Knox Millsaps graciously lightened

the author's work load during the writing of this disser-

tation.

Deadlines would not have been met had it not

been for the encouragement and invaluable help given by

Dr. Richard L. Fearn in the preparation of this disser-

tation.

Finally the author wishes to thank Mrs. Jacqueline

Ward who very cheerfully and untiringly typed this manu-

script. Mr. Guy Hardee lent an able hand in drawing the

figures.

This work was financed by Grant No. NSF-GP1866

from the National Science Foundation.



























iii















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii


LIST OF FIGURES v


LIST OF TABLES vii


CHAPTER

I INTRODUCTION 1


II THEORY 22


III APPARATUS AND PROCEDURE 54


IV RESULTS AND CONCLUSIONS 94


REFERENCES 120


BIOGRAPHICAL SKETCH 125













iv











LIST OF FIGURES

Figure Page

1. Rare earth crystal structures. 10

2. Magnetic structures of rare earths. 13

3. Spontaneous magnetization, Weiss theory. 28

4. Hydrogen molecule. 30

5. Wiring diagram for the inductance bridge. 58

6. Inductance bridge. 60

7. Equivalent circuit for the inductance bridge. 63

8. Entrance to high-pressure room. 68

9. Top view of high-pressure room. 69

10. Schematic of high-pressure system. 72

11. Inside view of high-pressure room. 74

12. Schematic of the control panel for the
pressure system. 76

13. Control panel. 78

14. High-pressure gas bomb. 81

15. Electrical seal. 84

16. Bomb plug and seals. 86

17. Cryostat. 88

18. Inductance versus temperature for dysprosium. 96

19. Neel transition for dysprosium. 97

20. Pressure shift of Neel transition for dyspro-
sium. 98

21. Pressure shift of N6el transition for erbium. 100


v








22. Robinson's interaction curve for rare earths. 101

23. Interaction curve for rare earths. 103

24. dIn-T for rare earths. 104
Sln V
25. Dysprosium data. 107

26. Pressure shift of Curie transition for
dysprosium. 108

27. Erbium data taken while cooling. 109

28. Erbium data taken while warming. 110

29. Pressure shift for middle peak on warming
data for erbium. 112

30. Pressure shift for upper peak on warming data
for erbium, 113

31. Pressure shift for upper peak on the cooling
data for erbium. 114

32. &InTC for heavy rare earths. 115
d\nV




























vi













LIST OF TABLES

Table Page

1 Summary of available experimental information

on rare earth magnetic structures. 14


2 Summary of experimental results on the pres-

sure shifts of the magnetic transitions for

the rare earths, 20


3 Sample coils. 64































vii














CHAPTER I

INTRODUCTION

Historical background

Miagnetism has aroused man's curiosity and fired

his imagination for at least several thousand years.

References to the attractive power of the lodestone had

already appeared in Greek writing by 600 B.C.1 The first

kinown reference to the fact that magnetism could also

repel a body is found in the writings of the Roman

Lucretius Carus in the 1st Century B.C.2 It is interesting

that no references to the directive property of the magnet,

as used in the compass, is found in old Greek and Roman

literature, but beginning in the period 1000-1200 A.D. the

history of magnetism is closely associated with the compass

and its use in navigation. The first clear mention of a

magnet used to indicate direction was made by Shen-Kua

(1030-1093), a Chinese mathematician and instrument maker.

By 1100 A.D., the Chinese Chu Yu reports that the compass

was in use by sailors going between Canton and Sumatra. In

1269, Peregrinus de Maricourt reported on experiments made

on a spherical lodestone.3 He explored the surface of the

sphere with small particles of iron and applied the term

pole to the places in which the magnetic power appeared to


1






2

be concentrated. Little progress is reported until 1600

when William Gilbert published his De Magnete which sum-

marized the knowledge of magnetism and reported the

results of many of his ownr experiments. In this book he

propounds his own great contribution, the realization that

the earth itself is a magnet. Also of great importance is

his determination that if magnetized iron is heated to a

bright red it loses its magnetism.

In 1785, Charles A. Coulomb4 established with

some precision the inverse square law of attraction or

repulsion between unlike and like magnetic poles. This

result was taken over by Poisson who became the best inter-

preter of the physical constructs which Coulomb had dis-

covered.5 To magnetism, Poisson brought the concept of the

static potential, which had been so successful in solving

the problems of static electricity. He also assumed mag-

netization to be a molecular phenomenon, but believed that

the molecule became magnetic only when the two fluids it

contained became separated. It was Weber6 who proposed

that each molecule is a permanent magnet, subject to a

frictional force that tends to maintain it in its established

orientation. This theory failed to explain the existance

of residual induction and hystersis.

In 1820, Oersted discovered that an electric

current would affect a magnetic needle. Ampere then inves-

tigated experimentally and mathematically the forces

between currents. He was able to show that a current in a






3
circuit was equivalent to a magnetic shell of calculable

strength. He then put forth the hypothesis that magnetism

arose from currents within the molecules. This theory

stood until it was modified by modern quantum mechanics.

Pierre Curie made the first extensive study of

the thermal properties of magnetic materials.7 As a

result of these studies he was able to establish that the

magnetic susceptibility of a paramagnetic material was

inversely proportional to the absolute temperature.


S= C/T (1.1)

The constant of proportionality was determined for each

material and was always found to be positive. He found a

relatively rapid decrease in the magnetization as each

ferromagnetic material was heated to a critical tempera-

ture, now called the Curie temperature. Above this

temperature, which was different for each material, the

behavior was much like an ordinary paramagnetic substance.

The first important modern development in mag-

netic theory came when Langevin8 used statistical mechanics

to derive the Curie law, equation (1.1). The underlying

assumptions of his theory were that each molecule had a

definite magnetic moment that tended to be aligned by the

applied magnetic field and at the same time disturbed by

thermal agitation. Many years later this derivation was

modified by Brillouin who took into account the quantum

mechanical requirement that the atomic magnetic moments were








restricted to a finite set of orientations relative to the

applied field.9

After the work by Langevin, Weiss10 made the

next big step in developing a modern theory of magnetism.

He assumed that the molecules are exposed both to the

applied field and to a so-called molecular field propor-

tional to the magnetization. It is a consequence of the

Weiss theory that small regions, domains, within a mag-

netic material are magnetized to saturation even though

the net magnetization of the body is zero. This is

possible by having the magnetic moments of the domains

oriented randomly. The Weiss theory is a very successful

one which has been substantiated by many experiments. It

is, however, not a very pleasing one since no explanation

of the origin of the molecular field is given.

In 1925, Uhlenbeck and Goudsmit introduced the

concept of electron spin to explain some discrepancies

between theory and experimental measurements of the spectra

of one-electron atoms and the alkali-metals. This spin has

associated with it both an angular momentum and a magnetic

moment. Following this came the enunciation of the Pauli

exclusion principle. These two developments allowed Dirac

and Heisenbergl1,12 to demonstrate a quantum mechanical

origin for the Weiss molecular field. In their theory one

starts with the Heitler-London model of the hydrogen mole-

cule and considers a Hamiltonian made up wholly of electro-

static terms and kinetic energy terms. The Pauli exclusion








principle enters the discussion only through symmetry

requirements. Further, one must make an assumption

regarding a distribution of energy levels. Heisenberg

assumed this distribution was Gaussian. Using these

conditions, it was possible to reproduce the Weiss theory

and show that the origin of the molecular field was a

quantum mechanical exchange integral. Thus, the most

successful theory propounded had been given a quantum-

mechanical basis and indeed it seemed that magnetism might

at last be understood. But, as is usually the case,

things were not as good as they appeared. Heisenberg's

theory suffered from several serious weaknesses. 1) It

was based upon the hydrogen molecule and contained no

account of lattice periodicity. 2) The results obtained

were very much dependent upon the distribution of energy

levels assumed. 3) The actual calculation of the exchange

parameter was an extremely difficult problem and thus far

has not been resolved.

At about the same time the Heisenberg-Dirac

theory was being developed, Isingl3 proposed a different

method for looking at the problem. The spins were disposed

at regular intervals along the length of a one-dimensional

chain. In accordance with the laws of Uhlenbeck and

Goudsmit each spin was allowed to take on only one of two

possible orientations. It was possible to obtain an exact

solution for this model if it was assumed that each spin

interacted with only a finite number of neighbors.





6


Unfortunately the result indicated that ferromagnetism

should not occur above 0K. Since that time exact two-

dimensional solutions and approximate three-dimensional

solutions have given finite transition temperatures, thus

showing that the failure of the first model was due to

its dimensionality.

Various methods have been employed to try to

improve on the Heisenberg theory. One of the most suc-

cessful of these is the method of spin waves developed by

Bloch and Slater.14'15 This theory starts with the

observation that the eigenvalue of the Heisenberg exchange

coupling can be determined rigorously if the spins of all

but one atom are parallel. Furthermore, approximate

solutions can be found if the number of reversed spins is

small when compared with the number of atoms. Due to the

above assumption this theory is good only for very low

temperatures. It has been quite successful in describing

the variation of magnetization with temperature in this

region.

In 1932, Neel proposed a theory to account for a

type of paramagnetic susceptibility temperature dependence

which did not agree with any of the existing theories.16

He proposed two interpenetrating sublattices undergoing

negative exchange interaction. This theory continues to be

the basis for modern developments in the theory of what is

now called antiferromagnetism.






7

The rare earths, which have become available in

quantity in pure form only since the development of the

ion exchange method of separation, cannot be described

completely by the theories discussed in the preceding

section. They have spurred a renewal in interest in

magnetism on both the experimental and theoretical fronts.

Their physical properties will be discussed in detail in

the following section and the theories developed to

describe them will be examined in a later chapter.



Structure and information on rare earths

The rare earth metals are composed of the fifteen

elements which range from lanthanum to lutetium. The elec-

tronic structure of these elements is normally given by


(4f)n(5s)2(5p)6(5d)l(6s)2


where n increases from 0 to 14 as the atomic number in-

creases from 57 to 71. The outer electronic structure,

which essentially determines their chemical properties, is

the same for all of these elements and they normally appear

in compounds as tripositive ions. Often scandium and

yttrium, atomic numbers 21 and 39 respectively, are grouped

with the rare earths since their external electronic con-

figuration is similar.17,18

The 4f electrons are tightly bound inside the

outer closed shells on the atoms and therefore play only a

small role in chemical bonding. They behave almost as they






8

would in a free ion, giving a resultant angular momentum

due to both spin and orbital motion. Since there is a

magnetic moment associated with this angular momentum, all

rare earth compounds have interesting magnetic properties.

It will be shown in Chapter II that a model

based on ions with 4fn configurations acted on by crystal-

line fields and coupled by exchange interactions is

capable of explaining much of the magnetic phenomena of

the rare earth metals.

Rare earths have been investigated extensively

within the last ten years at the Ames Research Laboratory

and the Oak Ridge National Laboratory. The former of

these has been involved in the separation and purification

of these elements, and with the measurement of specific

heat, thermal expansion, electrical resistivity, magnetic

properties, and other physical properties. The latter

group has performed neutron diffraction studies and

determined the complicated magnetic structures.

The tripositive ion picture, outlined above, is

violated by two of the rare earths, Eu and Yb, which come

immediately before the middle element and the last element

of the series, respectively. These elements should have

4f6 and 4f13 configurations, however, they appear to

prefer to gain the extra correlation energy of a half-

filled or completed shell and take a divalent form with

4f7 and 4f14 configuration. Ytterbium has only a small

paramagnetism as would be expected from a closed shell and






9
europium shows large magnetic moments as it should for

the 4f7 structure. One other element that should be men-

tioned in this respect is Ce, which may be found in a

four-valent state either at low temperature or at high

pressure. This is due to the fact that at the beginning

of the series the 4f and 5d electrons have similar energies.

In this state, as would be expected, Ce is found to

exhibit small paramagnetism. Above 1000K the stable form

is found to be 4f1 and trivalent.

The room temperature crystal structure of the

rare earths tend to fall into two categories, the hexa-

gonal close packed and a double hexagonal structure as

shown in Figure 1. While they have been reported with

various crystal structures, it seems that La, Ce, Pr and

Nd, the light rare earths, usually have the modified

hexagonal structure. Promethium has no stable isotope

and therefore no information is available. Next, Sm has

a very complicated hexagonal structure which repeats after

nine hexagonal layers. The remainder, Gd, Tb, Dy, Ho, Er,

Tm and Lu, have hexagonal close packed structures with c/a

ratios 1.57-1.59. Their magnetic properties, while complex,

show a certain regularity which may be traced to exchange

interactions and crystalline fields.

In the absence of detailed knowledge of the band

structure of any of these elements theoretical work has

been based upon the crude approximation of nearly free

electrons. The effect of lattice symmetry has to some






10















A




A
C B








A B


















Figure 1. Rare earth crystal structures.






11
extent been included by considering the Brillouin zone

structure19 for the heavy rare earth series. The primi-

tive translations tj, t, and t3 are shown in Figure 1.

There are two atoms per unit cell, one at the origin and

one at T= --I(t.2)+ The reciprocal lattice is

also hexagonal and has vectors t and ', with magnitude
4 7
7-T in the basal plane 1200 apart, and "T perpen-
dicular to this plane with magnitude 2i/c

Even when the atoms are triply ionized the

tightly bound 4f electrons are shielded from the crystal-

line field by the 5s2 and 5p6 electrons. Under these

conditions their orbital angular momentum remains

unquenched by the fields of neighboring ions. These

electrons have total orbital angular momentum, L, and

total spin, S, in the ground state as prescribed by

Russel-Saunders coupling and Hund's rules. The energy

difference between the ground state J multiplet and the

first excited J multiplet is usually greater than 0.1 ev,

therefore the excited multiplet plays no role in thermal

properties.

There are basically four types of measurements

that have been made.in order to determine the magnetic

properties of these elements. They are neutron diffraction,

bulk magnetic measurements, specific heat measurements,

and electrical resistivity measurements. A brief dis-

cussion of the information that can be obtained by each of

these methods is given below.






12
Detailed neutron diffraction studies have been

carried out on several of the rare earths.20,21,22,23,24,25,26

The magnetic structures thus determined have been found to

be quite complex. These studies have also given infor-

mation about the magnitude of the ordered moment and its

temperature dependence. The magnetic properties are found

to be highly anisotropic, that is, the moments along the

c-axis are quite different from the moments in the basal

plane. Figure 2 shows some of the types of ordering that

have been found and Table 1 gives the transition tempera-

tures and structures for each element.

Bulk magnetic measurements have shown that the

susceptibility of these elements at high temperatures is

roughly described by the Curie-Weiss law.


3 k (T- )


Here the Weiss constant, 6 indicates the approximate

value of the exchange energy. It is also possible to

obtain the magnitude of the ordered moments from this type

of experiment. If a sufficiently strong magnetic field is

applied to one of the antiferromagnetic structures it is

possible, in some cases, to change it to a ferromagnetic

structure. The field at which this occurs is called the

critical field and can be used to obtain information about

the energy difference between the two states.










I I i I I
I i I I I



< ',-.> <> \^
I I I I
I II

II I I I


I I

- ^^ >< -
'I \>
i I I !I



Ferromagnetic Helix Cone Oblique helix Longitudinal wave
F
Figure 2. Magnetic structures of rare earths.













Table 1

Summary of available experimental information
on rare earth magnetic structures.

Element Highest ordering Order Lower ordering Order
temperature temperature

Ce 12.5 Complex --

Nd 7 Complex 19 Complex

Sm 15

Gd 293 Ferro --

Tb 228 Helix 220 Ferro

Dy 179 Helix 85 Ferro

Ho 132 Helix 20 Cone

Er 85 LW 50 Complex

20 Cone

Tm 56 LW 22




This information was obtained from a review article by Belov52 and Elliott.19


-F:






15
The transitions from one magnetic state to

another are accompanied by sharp peaks in the specific

heat versus temperature curves. These peaks can be located

very accurately and therefore allow accurate determination

of transition temperatures.

The electrical resistance of these elements show

anomalies at the magnetic transitions. These anomalies

have been used to locate the transitions by a number of

investigators.



Experimental work

The way in which magnetic properties of materials

vary with pressure has long been of interest to physicists.

Some of the earliest experiments in this area were done in

an attempt to gain information about the origin of the

earth's magnetic field.27 The more recent ones, however,

have been done in order to try to obtain information on

the volume dependence of the exchange integral. One of the

methods of attack on this problem has been to measure the

shift with pressure of the temperature at which the material

goes from one magnetic state to another. The temperature

of the transition to the ferromagnetic state is called the

Curie temperature, T-, while the temperature of transition

from the paramagnetic to the antiferromagnetic state is

called the Neel temperature, TN. The discussion which

follows will be confined to experiments in which d T/d P

have been determined. There is a vast literature of other






16
types of magnetic experiments including a recent review

by Kouvel.28

One of the earliest attempts in this field was

that of Yeh in 1925.29 He measured the effect of pres-

sure on the magnetic permeability of iron, nickel and

cobalt. This work was followed by that of Steinberger

who, in 1933, did essentially the same experiment with

improved sample annealing techniques0 These experiments

did not specifically set out to measure the shift of the

magnetic transition temperature with pressure. In retro-

spect, however, it can-be recognized that Steinberger

actually induced a phase change from the ferromagnetic

state to the paramagnetic state in a 30Ni 70Fe sample by

the application of pressure at room temperature. From

his data it can be concluded that dTc/d P 0 for this

alloy.

The first actual attempt to measure dTc/dP

was made in 1931 by Adams and Green27 who studied iron,

nickel, magnetite, nickel steel and meteoric iron. They

used the transformer method for detecting the transition.

A primary and a secondary coil were wound on a closed

frame made of the sample material. An alternating voltage

was applied to the primary and the output voltage was

monitored as a function of temperature. The drop in out-

put at Tc is very sharp, and although it does not define

the Curie temperature in the conventional way, the method

is satisfactory for finding a change in Curie point. They






17
used carbon dioxide as the pressure transmitting medium

and therefore achieved truly hydrostatic pressure. The

shift for pressures up to 3.5 kilobars was found to equal

zero for all of their samples. This result has not been

confirmed by other investigators and it is believed that

thermal uncertainties masked the true changes.

Michels et al.31 used the discontinuity in the

(/R) (dR/d T) versus temperature curve to indicate Tc *

The sample material, which was 70Ni 30Cu, exhibited a

broad transition that occurred gradually over 500C.

They concluded from this that it was necessary to deter-

mine a shift of the Curie region. By carefully analyzing

their data they were able to obtain d Tc/dP = + 6.4 x o1"

OK/kilobar. This method requires very accurate resistance

measurements over large temperature intervals and is com-

plicated by the fact that resistance also changes with

pressure. Later a monel alloy was studied by the same

method.32 This transition also was quite broad and

yielded dTc/d P=3xl1o-2 K/kilobar.

Ebert and Kussman33 used large magnetic fields to,

obtain magnetization versus temperature curves so that the

Tc could be determined in the conventional manner.

They then applied pressure and tried to determine dTc/dP

for several pure metals and alloys. The result obtained for

all samples was dTc/d P= O Michels and De Groot34

criticized their result and showed by a thermodynamic

treatment of second order phase transitions that in general







dTc/dP: O They further showed that the experi-

mental method used by Ebert and Kussman was not accurate

enough to show small but significant variations of Tc .

Kornetzki35 took the data obtained by Ebert and Kussman

and re-analyzed them and obtained non-zero values for

d T /d P

In 1954, Patrick36 made a detailed study to

determine d-T /dP of nickel, gadolinium, cobalt, iron,

eight metallic alloys, a ferrite and a perovskite. The

transitions were detected by the transformer method as

developed by Adams and Green. Two pressure systems were

used, one used a gas for the pressure transmitting medium

and the other used a liquid. The pressure was truly

hydrostatic. Patrick's results agreed with those of

Michels et al., and are widely quoted in the literature.

Samara and Giardini37 made measurements on the

shift of Ta in nickel and a nickel iron alloy. A

multianvil pressure system with pyrophyllite as the

transmitting medium was used. Pressures up to 35 kilobars

were generated and the shifts found were in general agree-

ment with those already determined. The transition was

detected by monitoring the self-inductance of a coil which

was wound on the sample.

In addition to the electrical resistivity, self-

inductance and transformer methods of detecting magnetic

transitions, there are two other techniques which have been

used. These transitions can be located by monitoring the






19
mutual inductance between two coils wound on the sample.

Changes in the magnetic moment of the sample show as a

change in this mutual inductance, which can be measured

very accurately by bridge methods. Finally, when the

pressure system permits, a method involving the extraction

of the sample from a magnetic field can be used.

The pressure systems for this type of study fall

into two distinct categories, those whose pressure trans-

mitting medium is a liquid or a gas and those that use a

solid for pressure transmission. The former, of course,

are the only ones which produce truly hydrostatic pres-

sures, however, the latter are able to obtain much higher

pressures. One study has been made by sealing water in

the pressure vessel and then freezing the entrapped water.

The disadvantage of this method is the possibility of

having tremendous pressure gradients inside the pressure

vessel.

The results of all of the investigations of

pressure shifts of the transition temperatures for pure

rare earths as well as the pertinent information about

methods used are summarized in Table 2.

This dissertation deals with the effect of pres-

sure on the Curie transition and the Neel transition in

dyprosium and erbium. These experiments are the second in

a planned series of high pressure studies to be carried

out at the University of Florida. It was necessary to

develop the complete pressure system as well as the methods











Table 2

Summary of experimental results on the pressure
shifts of the magnetic transitions for the rare earths.

Pressure Pressure Detection
Element range medium method K/kilobar oK/kilobar Reference
kilobars

Gadolinium 0-8 Gas Transformer -1.18 36
0-6 Gas Mut. Induct. -1.56 38
0-33 AgCl Transformer -1.60 39
5-52 AgC1 Transformer -1.72 40

Terbium 0-4 Gas Mut. Induct. -.82 38
0-25 AgCl Transformer -1.0 43
0-71 AgC1 Transformer -1.07 40

Dysprosium 0-8.4 Indium Mut. Induct. -.54 41
15-45 Solid Resistance -.60 42
0-4 Gas Mut. Induct. -.60 38
0-25 AgCl Transformer -.40 43
5-77 AgCl Transformer -.66 40
0-7 AgC1 Transformer +1.4 43
7-25 AgCl Transformer -.8 43
0-1.8 Ice Bomb Extraction -3.9 44

Holmium 0-6 Gas Mut. Induct. -.45 38
5-82 AgC1 Transformer -.48 40






21

and equipment necessary for performing the experiments.

A large high-pressure helium gas system, which is des-

cribed in detail in Chapter III, was constructed. A gas

system was chosen in order to be able to work under truly

hydrostatic pressure at low temperatures. The choice of

samples was based upon the availability of high purity

specimens and also the desire to take advantage of the

ability of the pressure system to work at very low temp-

eratures. When this work was started there were no

published results on pressure shifts in any of the rare

earths. As can be seen in the preceding section there has

recently been a flurry of activity in this field. The

pressure system constructed here is still the only one

capable of studying the lower temperature transitions

under hydrostatic conditions and further studies on

holmium and thulium are underway presently.

The results obtained for erbium for both

d TN /d P and d Tc Id P are new. The results obtained

for dTN /d P for dysprosium are presented as corrob-

orating those which have now been published. The

d T /d P for dysprosium is in marked disagreement with
that presented by Robinson et al.43 which is the only one

published to date. A complete discussion of the results

is given in Chapter IV.














CHAPTER II

THEORY

Introduction

Any discussion on magnetism must be based on

quantum mechanical concepts. In the general discussion

of magnetism which follows the author has relied heavily

on numerous references.45-53 The discussion of rare

earths mainly follows the reviews by Elliott,54 Yosida55

and the books by Van Vleck56 and Chikazumi.57

This discussion can in no way be thought of as

complete, but rather will attempt to describe the methods

that have been most successful in treating the problem

of magnetism. While much progress has been made there

exists, at present, no completely satisfactory theory.



Types of magnetism

This section begins with a brief summary of the

types of magnetism that are observed and some remarks

concerning their origin. The major classifications are

diamagnetism, paramagnetism, ferromagnetism, antiferro-

magnetism and ferrimagnetism.

Diamagnetism is a weak magnetism in which a

magnetization is exhibited opposite to the direction of


22






23
the applied field. It is associated with the tendency

of electric charges to shield the interior of a body from

an applied magnetic field. It can be looked upon as a

manifestation of the well-known Lenz's law, which states

that when the flux through an electrical circuit is changed

an induced current is set up in such a direction as to

oppose this flux change. In a resistanceless circuit

such as the orbit in an atom or in a superconductor the

induced current persists as long as the field is present.

Landau58 has shown that there can also be a diamagnetic

contribution from the conduction electrons in a metal.

Diamagnetism is present in all substances; however, in

all cases except the superconductor it is a small effect

with a susceptibility on the order of -10-5 cm3/mole.

This effect is swamped if any other type of electron

magnetism is present. The superconductors, which exclude

all magnetic fields, exhibit perfect diamagnetism and have

a susceptibility equal to -1/4rr Diamagnetism plays a

small role in the rare earths and will not be mentioned

in the remainder of this discussion.

Paramagnetism arises in materials in which there

are permanent magnetic moments present. Magnetization

results from the orientation of these moments in an

applied field. This orientation is opposed by thermal

agitation and therefore would be expected to be highly

temperature dependent. The permanent moments may arise

from the spin and orbital motion of the electrons or from






24
the nuclei. In the rare earths the nuclear susceptibility

is about 10-6 times the electron susceptibility and will

not be considered in this discussion. The electron para-

magnetic susceptibilities vary from about +10-5 to

+10-2 cm3/mole. The rare earths are accurately described

in the paramagnetic region by the Curie-Weiss law which

will be developed in a following section.

A substance is called ferromagnetic if it pos-

sesses a spontaneous magnetic moment even in the absence

of an applied magnetic field. This moment occurs only

below some critical temperature known as the Curie tempera-

ture. This type of behavior is explained by adding to the

paramagnetic model a strong co-operative effect which tends

to align the permanent moments in a parallel manner.

Since dysprosium and erbium are both ferromagnetic at low

temperatures the theories of this type of magnetism will

be discussed in a following section.

Antiferromagnetism arises from co-operative

effects in a manner similar to ferromagnetism. In this

case, however, the spins tend to align themselves in an

antiparallel manner. The net magnetization is small and

gives susceptibilities on the order of the ones given by

paramagnetism. The temperature dependence of this suscep-

tibility is, however, very different from that of para-

magnetism. More will be said about this phenomenon in

following discussions.






25
The oldest magnetic material known, the lodestone,

is a ferrimagnet. This type of magnetism is characterized

by an antiparallel arrangement of moments but with the

moments of unequal magnitude. This can give a strong

external magnetic field. This type of effect is thought

to arise from the same type of interaction as the anti-

ferromagnetic case. These materials are of great prac-

tical importance since many of them are insulators. None

of the rare earths exhibit this type of magnetism so no

further mention of it will be made.


Quantum mechanical Langevin theory of paramagnetism

Consider a system of N independent atoms in a

magnetic field H There will be 2J + 1 Zeeman levels
for each Y Assume that, as with the rare earths at room

temperature, kT is small compared to the energy gap

between the ground state and the first excited state .

Write the operator equation =9 pJ where

p == eW/(2m) is the Bohr magneton and g is the Lande
factor given by

J(J+I) + S (S+I)- L(LIl)
9 = 2 2J CJ+I)
The energy of interaction between the magnetic moments and

the applied field is given by VV(H)==- = --g p MrH

Using statistical mechanics it can be shown that the

magnetic susceptibility is given by X= k" c)( In1)
H Hystem.
where Z is the partition function for the system.





26
In this system

kT
= { exp 4P

which gives


Sk -t kT exp kT
H Zexp q ps Mr H


After some mathematical manipulation one obtains

HX= B (2.1)

for the magnetic susceptibility, where Bj(x) is
called the Brillouin function and can be written

2j-+ c coth x I co 7h-)
(x)- 221V )X 27 c COth(-f

where

X T= 9 *H (2.2)

If the energy of Zeeman splitting is small compared to
kT then X< probability of occupation for all levels. Under these
conditions By() can be expanded in a power series
and higher order terms neglected to obtain


BT (y) z Jy X (2.3)


By using equations 2.1, 2.2 and 2.3 one can obtain





27



X N T(-+ 1)92- (2.4)
3kT


It can be seen that equation 2.4 is equivalent to
equation 1.1, the Curie law, where C= NJ(L+I)q2'/$k
By using equation 2.1 and the relationship
M= XH a general expression for the magnetization of
a material obeying the above theory can be written as


M= N y, J B (2.5)

In the special case where X << equation 2.1 and 2.4
can be combined to obtain


M= N 9t' ,T'(J I) H (2.6)
3kT





Weiss theory of ferromagnetism
Weiss modified the above theory by adding to
the paramagnetic model an interaction which tended to make
the atomic moments align themselves in a parallel manner.
He defined a molecular field proportional to the magni-
tization of the sample, H m==M, where is the Weiss
constant. Now, using the methods of the previous deri-
vation, one can obtain some useful relationships. If the
magnetic field in equation 2.2 is replaced by an effective






28

field, Heff = H+ 6 M then for ferromagnetic materials


X= 'T (H+M) (2.7)

In order to look for the spontaneous magnetization let
H= 0 and solve equation 2.7 for M to obtain

M= XkT (2.8)

Since M must satisfy both equations 2.5 and 2.8 the
simplest procedure is to investigate its behavior at
various temperatures by graphical methods.
M


N 9 PA J,3. B J(X)
T>TC Tr






Figure 3. Spontaneous magnetization, Weiss theory.

From Figure 3 it can be seen that there is a critical
temperature, Tc below which one gets spontaneous
magnetization due to the M field. As the temperature
increases through Tc this magnetization vanishes.
From equations 2.5 and 2.8 it is possible to obtain the
following expression

M Nq p(_+-I)X x xkT


where it has been assumed that X<< I By taking the






29

derivative with respect to X of both sides of the above

equation and evaluating it at T= Tc and X=O it is

possible to obtain the following relation between the

Curie temperature and the Weiss constant.



Tc= NgA pT() T+(2.9)
3k


Let us now consider a temperature region above

Tc so that there is no spontaneous magnetization.

Then equation 2.5 becomes


M= Ng ZT+1) ( +M) (2.10)
3kT

By using equations 2.9 and 2.10 it is possible to obtain

C- (2.11)
T-FTce

where C= Tc/ Equation 2.11 is known as the Curie-

Weiss law.

This is a very successful phenomenological

theory which describes accurately the results of many

experiments. In deriving it Weiss made no attempt to

explain the origin of the molecular field.


Heisenberg-Dirac theory of ferromagnetism

Heisenberg was the first to show that the Weiss

local field could be given a quantum mechanical origin.
This can be demonstrated by considering the Heitler-London





30
solution for a hydrogen molecule. Consider a simple
system of two atoms, a and b, that have one electron each
and are separated by a distance rab. See Figure 4.

e, ____ ,__ e,



SYrob b

Figure 4. Hydrogen molecule.

Consider the following Hamiltonian.

H=T, -'- +T- e + '. e T e (2.12)
ro Ib2 rafb nit ra 2 rbf

where T denotes the kinetic energy operator and the
subscripts identify the electrons. Consider also the
following relationships.

(T. Cm)l(>= I C>,


(T2- ) = bc 2>


rbi
(T, -e ) (b,> = l bu)>


where, for example, I C(i)) denotes the atomic wave
function for proton a and electron I
With the above wave functions it is possible to
construct symmetric and antisymmetric wave functions for
the system.






31

iL) =VlT ab o)+ lbo^oM

1 A) = I( a, bw I bo,(i)]


The functions IC({ and I b are not orthogonal.
Define the overlap of these functions as L.tE(lb).
With this and the assumption that the atomic wave func-
tions are normalized it is possible to obtain the fol-
lowing.


=I+L~, =0


If the spin is considered it can be seen that there will
be one antisymmetric spin wave function and three sym-
metric spin wave functions.



1->c+> =V(r +



where \% is the spin function. The subscript identifies
the electron and + or denotes spin up or spin down. The
simultaneous wave function must be antisymmetric. There
are then the singlet state, .a)>, and the triplet
state I A A The singlet state has spins paired and
therefore no net magnetic moment. The triplet state is in





32
every way identical to a spin one particle with
Mz== I, -I
By forming <(I HI- it is possible to obtain
the energy shift for the singlet state


E,= 2iK + ,
E2,E4- 1+L

where

e? < ( + e"7 7 le b(?i
b + Ow b &._
re, b rm rb r 1 b r-A a

and


r12b r12 2

K. represents the total electrostatic energy of the two
atoms and J, is the exchange integral. From it follows that the energy shift for the triplet state is

E3 = 2e + P-I
I-- L

Next, consider the energy difference between the singlet
and the triplet state.

2 (J, L K ,)
E, E3= EA E 2

If >0then the triplet state is energetically stable
and the molecule will be magnetic.





33
Let


Ec = 2 + -L.

Using this, it is possible to write


E, Ec + E = E -9


The total spin is a constant of the motion.


S' Ja>= S(S+ )hA')a> =0

S'IAA)= S(S+I)IlAA>= 2t"1I AA>

.S, and S are also constants of the motion with
eigenvalues -- h For the singlet state it can be
shown that

S, Sla3- Iha>
S.S-1a> =

and for the triplet state one obtains



Consider the spin Hamiltonian,

H = Ec-r 2 S2 .-s

It can be seen that it has the same eigenvalues as the
electrostatic Hamiltonian used in the original formulation
of the problem. This gives a spin-spin interaction with






34
a weight factor 9 that arises from electrostatic forces
and symmetry requirements. This part of the Hamiltonian
is called the Dirac-Heisenberg Hamiltonian.


H= -2 i S; (2.13)


One can now give an approximate connection
between the exchange integral and the Weiss constant.45
The assumption is often made that f=0 for all atoms
except the nearest neighbors and that 9=e for all
neighboring pairs. Based on this

wex = -2 eL Sl
-i
where j indicates that the sum is to be taken only over
nearest neighbors. Assume that the instantaneous values
of the neighboring spins can be replaced by their time
averages. Then

Wex = -22 e (5i + Sy
where is the number of nearest neighbors. If the
magnetization is along the z-axis then and


Wex-= 2-e S M (2.14)


This energy should equal the potential energy, V of
the spin L in the Weiss field / M.






35


V= 4 MgSop (2.15)


Thus



Using this and equation 2.9 it is possible to write



2C = -eS(S+i) (2.16)
T3 k





Neel theory of antiferromagnetism

The Heisenberg theory of ferromagnetism is
based upon the assumption that 9>0. When < an

antiparallel arrangement of spins is favored and an anti-
ferromagnetic substance is obtained. This type of system

was investigated by Neel,59 Bitter,60 and Van Vleck,61
and their work forms the foundation for the theory of
antiferromagnetism.

Consider two interpenetrating lattices made up
of sites A, with plus spins, and B, with minus spins.
Assume that there are antiferromagnetic AA, AB, and BB
interactions. Call these interactions Waa, Wab and wbb
respectively. Since A and B are symmetrical waa=wbb= o(
and wab=wba= 3 The effective fields can then be written





36


He = H oMa,- p Mb (2.17)


Hef b H pM 4a b (2.18)


where H is the applied field and o( and P are positive
Weiss constants.
Following the same methods used in the Weiss
theory one can write that in the limit of high temperature
and small X

--= N k (2.19)

where N is the number of A atoms per unit volume. Simi-
larily, if the dipoles on B are identical to the ones on A
then


Mb j He; b1 (2.20)
3kT
Use equations 2.17-2.20 to obtain

M = N g) [2 H -(o
This becomes a scalar equation with the assumption that M
and H are in the same direction.

M 2N J2' J(J+1)/3k C
-H I+ N(ON<+p).9auJ(J.+1)/3k (2.21)






37
This is quite similar to the result obtained for the Weiss
formulation of magnetism.
Next, examine the behavior at T= TN. This
temperature is still far enough away from saturation to
use equations 2.19 and 2.20. With H= O write

Ma=- l(o Ma- fM (2.22)

and

Mt= '-M (Ma+ 3k TN

where H is the magnetic moment per atom, = ( +l).
From these it follows that

TN- J- (3-0( (2.24)

Observe that -T increases as the interaction AB increases
and decreases as AA and BB increases. A relationship can
now be established between TN and 9 by using equations
2.21 and 2.24.

TNL L (2.25)


Experimentally it is found that TN<& which implies that
X> O or that, indeed, there is an antiferromagnetic AA
and BB interaction.






38
Phenomenological discussion of ordering in heavy rare
earths
Equations describing the types of ordering
shown in Figure 2 may be written in the following form.


p= SpjJMcos(*Rn) (2.26)




S= 9pJMsin(- i)R (2.27)




Pn g p M' Sin( Rn+s) (2.28)

where the -axis is taken along the crystallographic c-
axis and p P Pn are the components of the
moments on an atom at Rn. M is, in this case, the
relative saturation along both the X and the V axes
and M' is the relative saturation along the Z -axis.
The vector d is parallel to the o-axis and has a
magnitude, C = 2n/cd where d gives the period of the
magnetic structure.
Equations 2.26 and 2.27, taken together, des-
cribe a helical structure while equation 2.28 alone
describes a longitudinal wave structure. More compli-
cated structures occur and may be described by variations
of the above equations.
Next, examine the results obtained from a
Heisenberg-Dirac form for the Hamiltonian.






39



H=-2 ( (Rn-Rm ).Sn.S (2.29)

Since this exchange energy is, for the rare earths,
usually much smaller than the splitting of the J
multiplets by the spin-orbit coupling, De Gennes2 has
proposed that S for each atom must be projected on the
total momentum J.


S = (-1) -- (2.30)

This comes from a phenomenological approach and has been
examined and shown to be valid by several workers.63,64
Using this expression and the above Hamiltonian one can
obtain the exchange energy for the helical ordering
described by equations 2.26 and 2.27.


Eex =-2 )N (9l-1 J'V M (2.31)

where


-q =Z cos [c*3'(Rn-m)] (2.32)
Rn
For the longitudinal wave the exchange energy is

E ExK =-(f N -I)'J MOI (2.33)

where N is the number of atoms in the crystal.
Note that these structures are energetically
most stable at that % which makes J(e] a maximum. Also,
the spiral state is energetically more stable than the






40
longitudinal wave due to the factor of 2 found in the

exchange energy of the former. However, in considering
stable arrangements it is necessary to look at the free
energy, F= U-TS. A molecular field approximation gives
the same transition temperature for both structures.65,66,67

This transition temperature can be written as

TN= 2 9T)(-i ,T (J+i)/3 k One must look to the
anisotropy energy to determine the relative stability of

the structures.


Anisotropy

The term magnetic anisotropy refers to the
dependence of the internal energy of a crystal on the
direction of the spontaneous magnetization. The energy
associated with this directional dependence is called the
magnetic anisotropy energy. The dominant source of aniso-

tropy in the rare earths is the electrostatic interaction

between the multipole moments of the 4f electrons and the

crystalline electric field. The crystalline potential
for a hexagonal close pack structure takes the form68

V= A1E(3"'- ') + A: (35- 30 ^+3 ')
+A6Z (231'6- 315 '"+ loJr2- 5r') (2.34)

.+ A:Z(x6'- 15 X" V 15 X'*- V" )

where An are the constants determined by the distribution
of charges around the ions and Z is taken along the
c axis. The summation is over the coordinates of all






41
of the electrons. This can be transformed into a more
convenient form by use of the Wigner-Eckert theorem,69


Ha= 2A'
+ A [35J3-30 (J. T(+0+3J (J+1-)25J07-6 J(J+l1

+y A(
4-735 ,'- 525J(J+l)JT +40J'(T+l)'+294 J6'- 60.(jT+)


+- T' r'> ([(Jr+ i-)' + ( )'



where o and p are constants which have been evaluated by
Stevens. The < Y are the mean values of Y" over the 4f
electron distribution and may be computed.70 The An
are very difficult to evaluate and only order of magni-
tude estimates have been obtained.71
If Ha is treated as a perturbation on Way ,
it is found that at high temperatures the first term is
the dominant one, but at low temperatures the higher
order terms also become important. The first term
corresponds to the quadrupole moment and causes the pre-
ferred direction of the ordered moment to be either .
or II to the c-axis depending upon whether o< is positive
or negative. The second and third terms cause the
moments to tend to align parallel to c when they are nega-
tive, but when they are positive the preferred direction
is at an angle from c.





42

For dysprosium and terbium the first term is

dominant and is negative over the whole ordered range.

The ferromagnetic transition in these elements is caused

by an increase in the fourth term with decreasing tempera-

ture. For erbium the third term is positive and fairly

large and makes the conical structure stable at low

temperatures.

This method of combining an exchange inter-

action with crystal anisotropy has given very good quali-

tative results. As yet no quantative calculations have

been made due to the extreme complexity of the problem.


Range of exchange interaction

In order to obtain some idea of the range of the

exchange interaction necessary to stabilize the screw

structure we look at a particular model.72 Assume that

the exchange interaction, between layers of atoms

perpendicular to the c-axis extends as far as second-

neighbor layers. The exchange Hamiltonian now takes the

form


Hy =. Z 2s" Si S i.
L n ot ,i,+-a

where SL is the average spin of an atom in the ith layer.

By summing this exchange Hamiltonian it is possible to

obtain an expression for the exchange energy. By

referring back to equation 2.31 it is possible to see that






43
the assumed spiral configuration will be made most stable
by the values of that maximize 9(). The Q(j) for
this model can be written


Sq) = So + 2 S, cos S + 22 Cosc C.

The value of C which maximizes this expression
is

COS 2 43-
2 4 2
An analysis of the available data for dysprosium has been

made by Enz72 and the values 9,A/k =-24, ,/k =44 and

2/k = -15 obtained. Similar results were obtained from
an analysis of data on erbium. Observe that (g) is
rapidly oscillating and long ranged to produce this spiral
structure. Since the overlap of the 4f electrons on
neighboring atoms must be quite small it would seem that
this long range interaction is due to some other effect.
It is reasonable to consider that the main part of the
exchange interaction is produced by the exchange coupling
between the conduction electrons and the localized spins.


Indirect exchange
Indirect exchange has been extensively investi-
gated;73'74,75 the following discussion closely follows
that of Liu.76 He starts by considering one conduction
electron interacting with the magnetic electrons of one
ion. The interaction Hamiltonian can be written as






44


N
H T -r e .(2.36)
=-i

where t is the position of the conduction electron

and Pi is the position of the ith magnetic electron.

The wave function for the conduction electron is of the

form


^S) = U(J) expr'> ? r (2.37)

where


Hk&I expLLkr-

is a Block function and n is the Pauli spin function.
Since the 4f electrons are well shielded their wave

function can be written in the form


IF( )7 = Ri) Yepe P) (2.38)

The wave function for the entire shell is constructed

from the single particle wave function as prescribed by

Hund's rules and the Pauli exclusion principle. Since

this dissertation deals only with dysprosium and erbium,

for which the 4f shell is over half full, only that case
will be considered.

Liu76 showed that the required wave function of

the shell is


Y'TM =~ C(LSJ; m, M-m AmA t Y,'tsM,Y, t (2.39)
mt






45
where C(LSJ;m,Mv-mn) denotes the vector coupling coef-
ficient and the summation over t refers to a summation
over Young's diagrams.77
The wave function of one conduction electron
and one magnetic shell with no regard to symmetry is

?= YTM(I,...,N) Y(N.) (2.40)

where Nt~")=N'tI e ,SN,). Next, this wave function must be
antisymmetrized with respect to all the N+1 particles.
The resulting wave function is

N
^(hTi l^AO^N)ii- TrA (1, +.1-,- *)'/J (2.41)


The particles are considered to be completely indis-
tinguishable; therefore equation 2.36 must be symmetrized.


H Z e-e (2.42)


where Lt ,..jN+4 and i: 1 Consider the following unsym-
metrized initial and final states.


h e==re(i,...,N) ) (2.43)

"f = "Y;M'.(1,..., N) -N-,,) (2.44)

where

^(N'C) N+ k(I e-piA'PIE ,, 9

"YI,.,) = Uk'(CV.,) exKpo W. rH-r ',,






46
Using the expression for Y~JM equation 2.39, one can

now form the matrix elements of the exchange interaction,

< 't*) H11-Ti V

Liu obtains an expression for HI for the

heavy rare earths which includes direct interaction

between shell electrons, exchange interaction between

shell electrons, direct interaction between conduction

and shell electrons and exchange interaction between

conduction and shell electrons. The last of these is

found to be



H 2 lkk)(9-1) .JT (2.45)


where I(k.k') is the exchange integral, S is the spin

of the conduction electron and J is total angular

momentum of the ion. In order to obtain equation 2.45

Liu made the following approximations. 1) The conduction

electrons are s electrons so their wave functions have

spherical symmetry. 2) The wavelength of the conduction

electron is large compared with the size of the 4f shell

so that expik'-V may be approximated by the leading term

of its power series expansion,

It is very difficult to justify the first approxi-
mation and Liu did not try to show that it held. The

second one can be examined by looking at the radius of the

4f shell as determined by the method of Pauling.78 It is






47

found to be about 0.4 A. Using the free electron approxi-

mation it is found that, for the heavy rare earths,

k .5XIO 8 cm-1 at the Fermi energy. Therefore,

k Y0.6,and the second approximation is seen to be reason-

able.

Recently Kaplan and Lyons73 have examined this

second approximation and found that the leading term does

indeed dominate for terbium through erbium and that the

correction by other terms is about 10 per cent.

De Gennes62 has found that since r(k,k') should

be the same for all rare earths Tc or TN should be

proportional to (9-i)J (J-+l). For the heavy rare earths

this reduces to S(-T+I)/J. This is the same result that

Neel obtained in 193879 based on the molecular field

approximation. This relationship is verified experimen-

tally except for ytterbium.


Pressure effects

Using an equation first derived by Neel,

Robinson et al.39 have constructed an interaction curve

for the rare earths in an attempt to predict the effect

of pressure on the transition temperatures. The Neel

equation is

3. 3 (2.46)
k 2 S (JT+)

where 6 is the transition temperature and 7 is the

number of nearest neighbors. Using known values of the






48

right hand side of this equation the quantity 9e/k was

calculated and plotted versus D/2Rwhere D is the inter-

atomic spacing and R is the radius of the 4f shell.

There is some difficulty in using 6, in equation 2.46

for the rare earths. In materials which go directly

from the paramagnetic to the ferromagnetic state Gcf ,

the temperature at which the material actually becomes

ferromagnetic, is a few per cent lower than QOp, the

extrapolated transition temperature, so it makes little

difference which is used. In the heavy rare earths where

antiferromagnetic states are observed these two tempera-

tures are far apart. To account for this, two curves

were plotted, one based on 6ef and one on 6p. The

resulting curve is shown in Figure 22., This analysis

accounts for the magnitude and sign of the dTc/d P found

for gadolinium and terbium and can be used to explain their

result that dTc/ldPfor dysprosium is positive for low

pressures and changes to negative as the pressure is

increased.

Liu has done an analysis of the effect of pres-

sure on Te for ferromagnetic materials.80 He has looked

specifically at gadolinium but was able to draw conclusions
about the behavior of heavy rare earths from his work.

The starting point chosen for this analysis is the indirect

exchange Hamiltonian



+(.47)
2 r i






49
where the first term is the kinetic energy of the conduc-
tion electrons in the scalar effective mass approximation
and the second is the exchange interaction between the
conduction electrons and the ions. The subscript i
refers to the conduction electrons while refers to the
ions. As in the previous discussion of the indirect
exchange the electron is described by a Block function.
The matrix elements of equation 2.47 are written as

I e- L -
-Z S; *S Ikk'exp^(kk-') R,

where


Ik, W= N k 0() If APkcn dr

It has been shown81'74,75 that by second order pertur-
bation theory the exchange interaction can be expressed
by the spin Hamiltonian

Hs= ^ (?,t S, (2.48)

with



.4N (In) R RA 2 k, Rcos(2kFR)- SinL2kR)] (2.49)


where I is the average matrix element for k and k"
approximately equal to kf. Equation 2.49 can alternately
be written as


Jr 7 7 NTlN(E FkR) (2.50)






50
where 2 is the valence of the ion, N(sp) is the density

of states at the Fermi level and F--) is given by

XCo.sx Sin X

De Gennes6 has shown that it is reasonable to assume that

the ferromagnetic state is the ground state for gadolinium.
Following this we can write


Ej = 3' I11i S N(e)C F(2k,,Ri) (2.51)

This equation should now be examined for terms which will

vary with pressure. The summation will be independent

of volume if the electron distribution is isotropic. The

2 and S are independent of volume in the heavy rare

earths. One can now take the logarithm of equation 2.51
and form


D31nE Inlil' Din N(ep) (2.52)
DInV DIn~V in V

The Curie temperature is proportional to the ordering

energy per spin, therefore we may write


01Tc lI12 lI + tn N!-) (2.53)
DInV DInV DinV


The terms on the right side of equation 2.53 are unknown

at the present time. Liu gives some estimates of the

limits that can be expected for them. The thing that
should be noted about them is that they are both functions
only of the electronic properties of the material. Since






51
all of the heavy rare earths have similar electronic

properties,DIn Tc should be the same for each. This is
Din V
a rather strong assumption and should be subjected to

experimental verification. This is discussed further in

Chapter IV.


Thermodynamics of phase transitions

It is generally accepted that the transition

from the paramagnetic to the antiferromagnetic state is

second order while the transition from the antiferromagnetic

state to the ferromagnetic state is first order.52

It is possible to characterize a first order

phase transition by either of the following statements.82

1. There are changes of entropy and volume. 2. The

first order derivatives of the Gibbs function change dis-

continuously. Any phase transition that satisfies these

requirements is known as a phase change of the first order.

The effect of pressure on a first order phase

transition can be determined simply by taking the first

-Tds equation of thermodynamics and integrating it over

the change of phase. The first Tds equation can be

written,





Tds= c.dT + -- (2.54)
c~d +o T






52
Integrating this it is possible that one obtains,


dT uv-tu"
dTP s- S (2.55)
dP +_ s;

In this equation the superscript f refers to the final

phase and L refers to the initial phase.
A second order phase transition is charac-
terized by discontinuous changes in the second order
derivatives of the Gibbs function. There is no change
in entropy associated with this transition. Using the
same superscript notation as before it is possible to
write S'= S at (T, P) and S'+dsi= s' +ds' at
CT+dT, PtdP) These expressions yield


Tds' = Tds' (2.56)

The second TdS equation is now used,

Tds =c,d T dP

By using equation 2.56 and the definition of the volume
expansivity

P ^ ~iDT)P
it is possible to write

cdT-Tu-'dP= c'dT- T r-dP

By re-arranging and using the relation P=3 a and P-
one obtains






53



dT 3T ZA (2.57)
dP P ac

where Ac= 0o and AC = -Cp Equation 2.57 is
known as an Ehrenfest equation. This equation predicts

the pressure shift for the Neel transition and is further
discussed in Chapter IV.














CHAPTER III

APPARATUS AND PROCEDURE


Introduction

A description of the apparatus and of the pro-

cedure involved in the measurements made in this disser-

tation can be roughly divided into four major sections.

The first section is concerned with the detection of the

ferromagnetic-antiferromagnetic and antiferromagnetic-

paramagnetic phase transitions. This task is complicated

by the fact that the sample is contained inside a pressure

bomb which is in turn contained within a temperature

control cryostat. Further complications arise from the

safety requirement that every thing should be operated

remotely. The second section deals with the techniques

involved in the compression, containment and pressure

measurement of helium gas at high pressures and low temp-

eratures. The third section concerns the production and

measurement of temperatures from 50K to 1900K, and the

last section gives a step-by-step breakdown of the procedure

used in performing the experiments.







54






55
Detection of Magnetic Transitions


Several methods have been used to detect magnetic

transitions in rare earths. The most important of these

are neutron diffraction, bulk magnetic measurements,

specific heat measurements and electron .transport property

measurements.

It was decided to look at the bulk magnetic

properties in these experiments since they promised to give

sensitive indications of the transitions, would readily

lend themselves to pressure studies, and did not require

any elaborate instrumentation. Methods of detection of

the transitions by bulk magnetic properties are mentioned

in Chapter I. Several factors had to be considered in

deciding upon the proper method to be used. It was

desired to have as much sensitivity as possible; therefore

a bridge method was selected. A large filling factor was

desirable so the coil was placed inside of the bomb.

Since the working space was limited and since the number

of electrical leads into the high-pressure region should

be minimized, it was decided to use a single coil tech-

nique. After the experiments were well under way Samara

and Giardini37 reported that they had used the same

method. The sample, in the form of a cylinder, was placed

within a solenoid and the self-inductance of the coil was

monitored. There is no simple exact formula for the self-

inductance of a solenoidal coil of practical dimensions.






56

An approximate formula is


S0.8 a0 nr
6a +9b4 10c

where a is the mean radius, b is the length, C is the

radial thickness of the solenoid and n is the number of

turns. The important thing to note is that L is pro-

portional to p the permeability of the core material.

As a ferromagnetic sample is heated through its Curie

temperature, its permeability changes from a large to a

fairly small number. Hence, if the inductance of the coil

is monitored, a large drop is seen as the sample is heated

through its Curie temperature. The transition from the

antiferromagnetic to the paramagnetic state is accompanied

by a peak in the permeability versus temperature curve.

Of course, the inductance of the coil would also be a

function of the thermal expansion of the copper wire and

the sample, and of variations of P due to skin effects.

A blank run was made to insure that the changes in the

coil were not influencing the results. Since the magnetic

changes are quite large it was reasonable to neglect the

other effects.37



Inductance bridge

In order to perform these experiments a very

sensitive self-inductance bridge was needed. The design

of inductance and capacitance bridges has been advanced

considerably in recent years with the development of very






57

accurate ratio transformers.83'84 These instruments

utilize modern high permeability magnetic core materials

and are highly accurate alternating voltage dividers.85

A ratio transformer bridge was built following a design

by Hillhouse and Kline.86 This bridge was capable of

detecting changes of inductance of the sample coil of one

part per million.

The wiring diagram, Figure 5, shows the com-

ponents as connected in the bridge. This design features

the use of commercially available components as listed

below:

1. Audio oscillator, Hewlett-Packard Model 200JR

2. Isolation transformer, Gertsch Model ratio 4-1

3. Ratio transformer, Gertsch Model 1011
4. Decade resistance box, General Radio Type 1432-K

5. Null detector, General Radio 1232-A
6. Standard inductor, General Radio 1482-L

7. Standard inductor, General Radio 1482-H

All of these components, except the null detector and the

audio oscillator, are contained within one cabinet.

Figure 6. All of the external wiring is coaxial cable.

(GR 874-R34) with General Radio shielded connectors. The

switch, S1, allows the isolation transformer to be connected

with a ratio of 4:1 or 1:4. The 600 ohm generator output

impedence can then be transformed to 37 ohms or 9000 ohms.
The purpose for this approximate impedance matching was to






58









\ j -- 81
I





Standard
10ma


Isolatio
Trans.

i n Standard
100ma




----- -2

Ratio
Decade
Transformer Resistor







SUnknown Detector
SInductor Detector






Figure 5. Wiring diagram for the inductance bridge.
























Figure 6. Inductance Bridge.



















MATCEIOPANCE c 0 NDAR TOR -
'3 00 000






0 gS

os I
A 'f 5. ....
S.. ... .. .. ,

~hiDI RES,






61
realize good bridge sensitivity. The ideal ratio between

unknown inductance and standard inductance is 1:1; however,

Hillhouse and Kline found that the accuracy was not
appreciably altered up to a ratio of 10:1. Switch S2
allows for the use of either the 10 mh or the 100 mh
standard inductor. The equivalent circuit for the bridge
is shown in Figure 7. The operating equations for the

bridge are derived below. Standard notation is used with
subscripts 1 and 2 referring to the leads running from

the bridge to the sample coil, S, to the standard, D, to
the decade resistance and X, to the sample coil. The
reading on the ratio transformer, A is that part of the

total voltage that is being applied across the unknown
inductor. Looking at the schematic it is seen that at the
balanced condition, that is, when the current through the
detector equals zero, one can write


e,= E( -A)= I[R,-Rs+ Ro+ wl L,+ Ls+ Lo) (3.1)


e2==EA= I[Rx+ Rz+~wt Lx+L-)l (3.2)


Dividing equation 3.1 by equation 3.2 gives


(I-A[Rx+Ra+w(Lx+L2)]=A[R.+Rs+Ro+jw(L,+Ls+Lo)] (3.3)

Thus


R9= A (Rs+Ro+R,) Ra (3.4)
I- A






62
and


Lx= A (Ls+LD+LI)-La (3.5)


Equations 3.4 and 3.5 constitute the operating equations

for the bridge.
The inductance of the decade resistor is given

by the manufacturer and at the maximum is on the order of

a PH. The inductance of leads 1 and 2 are also on the

order of a pH. During an experimental run the tran-

sitions occur over a small temperature range; therefore

Lp ,1.l and La are small and essentially constant.
The Lx is from 1/2 to 1 Henry and changes in it completely

dominate the picture. The situation with the resistances

is similar. Ri and RI are small and essentially constant

during the determination of the transition temperature.

Using this information, one can truncate the operating

equations and simplify data reduction. The simplified

equations are:



Rx = (Rs+Ro) (3.6)
I-A



L -Ls (3.7)


Hillhouse and Kline have made a detailed error analysis

for this bridge design and found that it is able to inter-

compare inductances at ratios as large as 10:1 to





63

L R.



Ls



Rs


(- A)V




Lo
V





Lx

AV

Rx



L, Ra


Figure 7. Equivalent circuit for the inductance bridge.






64

accuracies an order of magnitude better than the certifi-

cation limits of present standards which is at best t.03

per cent.


Coil and sample description

The two coils used in these experiments were

wound on a teflon core with a Model W coil winder manu-

factured by the Coil Winding Equipment Company. The

dimensions and room temperature characteristics are:


Table 3. Sample coils.

Coil 1 Coil 2

Number of turns 14,000 17,000

Length 13/16 in. 13/16 in.

Inside diameter 1/8 in. 1/8 in.

Outside diameter 7/16 in. 1/2 in.

Resistance 3100 ohms 6000 ohms

Inductance 400 mH 500 mH


The samples were obtained from Leytess Metal and

Chemical Corporation who specified a purity of 99.9 per cent.

When received they were in the form of rods 6 in. long and

.375 in. diameter. These were cut and turned down to a
final sample size of 1/8 in. diameter by 13/16 in. long.

The samples were not annealed after machining.






65
Pressure Generation and Measurement


The purpose of these experiments was to study

the effect of hydrostatic pressure on magnetic transitions

in rare earths. A large high pressure helium gas facility

was constructed to achieve purely hydrostatic pressure

over most of the temperature range covered. In principle,

it is easy to achieve hydrostatic pressure in the fluid or

gaseous phase of helium. In the lower temperature region

approximately hydrostatic pressures may be achieved by

applying the desired pressure to the helium while it is in

the fluid phase and then freezing it at constant pressure.

Further cooling necessitates the calculation of the

pressure from the equation of state of solid helium and

the thermal properties of the high-pressure bomb,7 which

was made of beryllium-copper. This procedure gives very

nearly hydrostatic pressure even though there is some

movement due to the fact that helium has a larger thermal

expansion coefficient than beryllium-copper.

Numerous experimental difficulties arose during

the course of the experiments. By far the largest problem

was leaks in the pressure system. The bomb plug seals

presented the most difficulty since a leak there made

temperature determination and control impossible. Cooling

through the freezing temperature of helium had to be done

very carefully to prevent blocking of the inlet pressure

line before the helium in the bomb was completely solidified.






66

This would have greatly reduced the pressure at the sample

as well as the accuracy with which it was known.



High-pressure room

The safest way in which to conduct high pressure

experiments is not to have any personnel in the vicinity

of the high-pressure equipment. This was done by isolating

all of the high-pressure components in a specially con-

structed, explosion proof room below ground level. This

room was located outside the basement of the low temperature

laboratory. All of the pressure equipment plus cryogenic

apparatus was operated remotely from the adjoining base-

ment. A brief description of this room will now be given.

Figure 8 shows an outside view of the room. The

wall on the left is the outside wall of the Physics building

basement. Figure 9 shows a top view of the room and part

of the laboratory, giving wall details and rough dimensions.

The roof of the room was constructed, from inside out, of

1/2 inch aluminum plate, 4 inches of sand, 9 inches of

reinforced concrete, a 4 inch air gap, 1/2 inch plywood

sheet, and a layer of sand bags resting on this plywood.

Over this was placed another 1/2 inch plywood sheet which

was covered with roll roofing. The outside end of the air

gap was covered with screen and provided ventilation as well

as a path for escaping gas in the event of an accident.

The free volume of the compressed gas was only about one-































Figure 8. Entrance to high-pressure room.































/ .'






; I












/ b.cer collcte e
q7: -AS -' C "

S b ep )arator bh Co;: ..
u: eHelium storage cylinder
100,000 psi intensifier


200,000 psi intensifier

"N --il
-CD High-pressure room


~ .~. .~C: ;. Cryostat

Sc- alInum all around
... .:d: : -aluminu. Sliding door
reservoirs Sliding door :


Lli-tinum g:O -t
Control A; o a - -

Door Fi Door o o oo

Laboratory *0 '

Console for (
instriuaents
i stm ent filled concrete '
10- conduit 0 :0, 0 0
through Pall o\

Figure 9. Top view of high-pressure room.






70
fortieth of the volume of the room and therefore could

not create a significant increase in the pressure of the

room. However, liquid nitrogen and helium dewars as well

as the commercial helium gas tank were left in the room

for remote transfer and if one of these should be rup-

tured by shrapnel it would release enough gas to be

dangerous in a poorly vented room. The door was a 5 1/2

inch thick box made of-1/2 inch aluminum plate and filled

with sand. The box was supported by a steel dolly which

had 6 ball bearing steel wheels that rolled in the channel

of a 6 inch steel I beam. The north and east walls of the

room (Figure 8) consist of 12 inches of steel reinforced

concrete backed by earth. The south wall was constructed,

from inside out, of 1/2 inch aluminum plate, 6 inches of

sand, and a wall of 8 inch solid concrete blocks. The

west wall consists of the outside wall of the Physics

building, 15 inches of reinforced concrete, supplemented

by a 1/2 inch aluminum plate and 6 inches of sand. It

was deemed necessary to add this plate and sand to prevent

spalding of the concrete wall in the event shrapnel struck

the wall.88 The room was designed to contain all shrapnel

and shock waves in the event of a high-pressure gas failure.


High-pressure gas apparatus

The high-pressure system is a three stage system

composed of an Aminco 30,000 psi (H 5968) oil-to-gas






71
separator, a Harwood 100,000 psi intensifier (SA10-8-

1.250-100K), and a Harwood 200,000 psi intensifier

(SA10-6-.875-200K). Figure 10 is a schematic showing

all of the significant components. Figure 11 shows the

relative size and the placement of the components within

the room.

Initial charging was accomplished by a remotely

operated solenoid valve (switch located on the panel).

For safety, a second solenoid valve was used to bleed the

2,000 psi stage of the gas system after charging. The

charging gas flowed through a liquid nitrogen cold trap

and a filter to remove gas and solid impurities. Note

that each stage was separated from its lower pressure

adjoining stage by a one way ball check valve, as shown

in Figure 10. The check valves in the 30,000 psi stage

were Aminco No. 44-6386 while the ones in the other stages

were Harwood ML-603.

The Heise gauge, Model H 26960, located in the

100,000 psi stage, was monitored by a closed circuit tele-

vision system which consists of a Marson television monitor

and a Bell camera. This gauge proved very useful in

locating leaks and controlling bleed down rates in the

system. The pressure during an experiment was always

measured with the Harwood manganin resistance cell. This

pressure was monitored constantly by a Foxboro recorder

but data points were taken with a Carey-Foster bridge.











iRemotely operated eloctrlic
motor driven needle valve


I.ligenin cell Safety pressure-
Srelease valve


Chk valve ._Charging solenoid valve
)-Check valve

/ Filter
To bomb C aheck valve heck valve Blovout
valve /
S-Cold
trap,
SBleed
solenoid
valve






/
2,000 psi helium gas /
storage cylinder
200,000 100,000
psi psi
Separator

From 2,000 psi
oil system JE---- -From 30,000 psi oil system


Figure 10. Schematic of high-pressure system.





























Figure 11. Inside view of high-pressure room.






I Ji. \i U
T^I^I I ^^ J^A^'^''''
^ i- Bt^^^ *"* "'







75
Harwood Manufacturing Company specifies an error of less

than 1 per cent with this cell and bridge. The schematic

also shows the motor driven bleed valve. This was a

Harwood 200,000 psi needle valve driven by an electric

motor through a chain and sprocket arrangement.



Control panel

Figure 12 is a schematic of the control panel

and Figure 13 is a photograph of the actual panel. The

schematic shows that the control panel was divided into

two separate pumping systems; a 30,000 psi oil system for

the gas-to-oil separator, and a 2,000 psi oil system for

the two intensifiers. Sprague Engineering Corporation

air powered pumps were used for both systems. As a

safety feature the air supply was taken through a 115-volt

ac solenoid valve that was normally closed. In the event

of electrical power failure, affecting other components

of the facility, the air supply was automatically stopped

and had to be manually tripped on when power was restored.

The intensifier oil system operated at reasonably low

pressure so it was practical to use one pumping system

with electrically coupled solenoid valves to draw oil from

the proper reservoir and to direct it to the proper inten-

sifier. These solenoid valves were operated by push

button switches on the panel, and pilot lights indicated

which reservoir the oil was taken from and the intensifier

















Sprague

To separator


Oil
reservoir
From air supply solenoid valve



J --------[X------




Sprague Solenoid
pump-7 valve
To intensifier No. 2
Solenoid PTo intensifier No. 1
valve Filter--


Oil reservoirs
From air supply solenoid valve

Figure 12. Schematic of the control panel for the pressure system.

























Figure 13. Control panel.





















lA










































0






m
-- - - - - -
E *






79

to which it was pumped. Each oil reservoir was fitted

with a level indicating sight tube. These were calibrated

to show the actual position of the appropriate piston in

the high-pressure room. Stainless steel tube, type 304,

with 3/8 inch OD was used throughout the intensifier

system, while 30,000 psi Aminco high-pressure tubes and

fittings were used in the separator system.

The photograph of the control panel (Figure 13)

shows the position of the pumping controls. This panel

was designed for maximum safety, efficiency, and conven-

ience of operation. A color coded flow diagram was

painted on the panel to clarify the oil and high-pressure

gas circuits. Also shown on the panel is the Foxboro

Dynalog recorder which-was used to monitor continuously

the Harwood manganin resistance cell located in the 200,000

psi gas stage. A Harwood Carey-Foster bridge (visible on

the shelf in the lower right-hand corner of Figure 13)was

used for accurate readings of this cell. Visual obser-

vations in the room were made with closed circuit television.

The push button and two warning lights in the

upper left-hand corner of the control panel (Figure 13)

remotely controlled the electric motor operated bleed valve

in the 200,000 psi gas stage which bled this high pressure

back into the 30,000 psi gas stage. A safety microswitch

was located at the 30,000 psi stage oil reservoir which

prevented operation of this motor unless the separator was






80

cycled all the way back, thereby providing adequate volume

for the 200,000 psi gas.



High-pressure bomb

In high-pressure, low-temperature experiments

conflicting design considerations occur with the sample

container (bomb). The bomb should be massive and strong

to safely hold the pressure yet it must be small with low

thermal mass so that it will fit a cryostat of reasonable

size and can be cooled to low temperatures without using

an excessive amount of liquid helium. Further, it was

planned to use this bomb for magnetic measurements and

nuclear resonance work, so a non-magnetic material was

required. The desire for a non-magnetic bomb with high

strength led to the choice of BeCu (Berylco 25) for the

bomb material.

A design pressure of 225,000 psi was used so

that the bomb would be reasonably safe with the 200,000

psi gas system. Standard thick-wall cylinder equations89

gave stresses exceeding the yield strength of full hard BeCu

regardless of the wall ratio (outside diameter/inside diam-

eter). These same equations, modified for a double wall

cylinder,89 predict sub yield point maximum stress for the

dimensions shown in Figure 14. The outer cylinder was a

.010-inch interference fit (on diameter) over the inner one.

Assembly was accomplished by cooling the inner cylinder in






81








1" x 14 NF x 1j"







t- -T. =175,000 psi




4.25 .875
.875"
2





S.75"

.65"75




1.3
-- 1..30--*
2.0
-- 3.0" >


Figure 14. High-pressure gas bomb.






82

liquid nitrogen and heating the outer cylinder to 9000F,

then pressing them together. The assembly was then heat

treated at 6000F for 3 hours. Final machining on the

inside bore and sealing surfaces was done after the outer

cylinder was fitted. Figure 14 shows the finished bomb

with dimensions and the tangential stress U at the

critical design points as calculated for an internal

pressure of 225,000 psi. It was very important that all

corners and edges be made round and smooth to reduce

stress concentration.

Before use in the helium gas system, the bomb

was pressurized with a liquid test system to 14 kilobars

and carefully checked and measured for distortion.

The platinum resistance thermometer was installed

on the bomb by means of a band on the outside of the bomb.

This was assembled in the same manner as the b. tself

and was a .03 inch interference fit.



High-pressure seals

In performing these experiments one of the most

difficult problems encountered was the design and fabri-

cation of the electrical and bomb plug seals. The problem

of containing helium gas under pressure at low temperature

is well known to anyone who has worked in this field.

Epoxy seals, which were sufficient for much of the range

covered, had already been developed at this laboratory.90






83

It was desired, however, to develop an electrical seal

which would be easier to work with than this type. The

electrical seal which was finally used is shown in

Figure 15 and is simply a logical next step from the ones

previously developed. The earlier seals depended upon

the bond between the epoxy and the tube to carry the shear

load which prevented the seal from blowing out. The new

seal has a large cross section in the middle so that the

epoxy itself must fail for the seal to blow out.

In this seal, the wires pass through a small

hole filled with an epoxy. (Eccobond 104) The tubing

used in the seal was Harwood 3M and 12H. Standard Harwood

cone and sleeve fittings which were good to pressures

greater than 14 kilobars were used in all cases except

where the 3M tubing mated to the 12H. One non-standard

part, the gland nut, had to be made for this connection.

It was made from type 304 stainless steel and had the

dimensions shown in Figure 15. The 12H tubing was drilled,

tapped and a 600 conical seat was made on each end. The

3M tubing was threaded and coned in the standard manner.

In order to insure that the epoxy bonded to the tubing,

the walls were etched with acid and cleaned with water and

acetone. The wires used were number 36 quadruple formvar

insulated copper. They were cleaned with acetone and

placed inside the tube. Epoxy was mixed and forced into

the tubes with the small stainless steel tube and screw


















10-32 RH- -10-32 LH







10-32 LH-

7/16 x 20 NF-









Figure 15. Electrical seal.






85

shown in Figure 15. The wires were moved slowly back

and forth several times, the excess epoxy was removed and

the seals were baked according to manufacturer's specifi-

cations. The seals were tested on a liquid pressure sys-

tem before being used in the gas facility. All of them

were tested to about 140,000 psi and none blew out. Two

of them were taken to 200,000 psi and while no leaks devel-

oped the wires inside were broken by the epoxy shifting in

the tube. The addition of 5-10 per cent alumina powder to

the epoxy has been reported91 to significantly increase its

strength. This was tried but the one seal made in this way

leaked at a low pressure. This may have been due to incom-

plete filling of the tube during fabrication. More work

should be done on this since it seems that if the epoxy

could be strengthened this should make a simple, inexpensive

seal for gas systems up to 200,000 psi.

The plug seal offered even more difficulty than

the electrical seals. Many variations were tried and dis-

carded. The one finally used is shown in Figure 16 and is

good to at least 100,000 psi. It will probably go higher

but leakage in the intensifier seals have restricted the

experiments to this pressure. It will be noted that it is

a Bridgman unsupported area seal.92 Some workers have

reported that this seal will not work at low temperatures

due to the fact that the indium metal contracts more than

the beryllium copper upon cooling. It was found, however,

that if the bomb was cooled while under pressure no leaks






86



S1 1" hex head'
l_ nut
2

-/


1 14 NF




1^










1/8"
T





3/16"
annealed BeCu 9/
indium 60 9/161
hardened BeCu /
\ /

0.500"


0.625"


Figure 16. Bomb plug and seals.






87

would occur. This seal has the advantage that it seals

with very low torque, is simple to machine, and is

reusable many times. It consists of a lower support

ring made of hardened beryllium copper, a ring of indium,

and an upper extrusion ring of fully annealed beryllium

copper. The plug is made of hardened beryllium copper.





TemDerature Production and Measurement


The basic cryostat system, which was used in all

of the experiments, is shown in Figure 17. Some modifi-

cations, to be described later, were necessary for running

the lowest temperature transition. The basic system

consisted of an outer nitrogen dewar, an inner nitrogen or

helium dewar, and an evacuated can which contained the

bomb. Aluminum foil, which has a low emissivity, was used

to wrap the bomb and line the inner walls of the can. The

inlet pressure line was stainless steel which has a low

thermal conductivity at low temperatures. A non-

inductively would heater coil was connected to the top of

the bomb with woods metal. Three copper-constantan

thermocouples were installed to measure the temperature of

the bomb. The upper and lower ones used a copper lug on

the wall of the can as a reference temperature while the

middle one was brought out of the cryostat to a liquid






88
High-pressure gas

Electrical
seal

Thermocouple and
heater lears _
-..__.. ---- Vacuum
pump










Helium transfer I r
Inner dewar
line

Outer dewar
/
S Heater



Thermocouples




Helium level
resistors





S- **-

i uC s






Figure 17. Cryostat.






89

nitrogen bath for reference. Periodically a platinum

resistance thermometer was installed in the bomb and the

thermocouples were checked. Temperature accuracy was

judged to be .250K and reproductibility was better than

.100K.

In the experiments involving the Neel transi-

tion the inner dewar was filled with liquid nitrogen and

exchange gas was allowed into the can containing the bomb.

When the bomb had cooled to approximately the desired

temperature the can was evacuated and thereafter main-

tained at a pressure of not more than .2 microns. With

the bomb thus isolated the required temperature could be

maintained with the heater.

In the experiment involving the Curie transition

of dysprosium it was necessary to cool the sample down to

about 600K. This was done by pumping on the nitrogen bath

with a large Kinney pump (Model No. KC-46). The tempera-

ture was then controlled by use of the heater.

The ferromagnetic transition in erbium, which

occurs at about 200K, made it necessary to use liquid

helium and also required some modifications in the equip-

ment. Two copper straps (dimensions 3 x .375 x .025 inches)

were soldered to the bottom of the bomb and to the bottom

of the can. In addition to this, a heater coil was wrapped

around the inlet pressure line. With these two modifi-

cations it was possible to maintain a temperature gradient






90

across the bomb sufficient to insure that the helium

froze from the bottom to the top.




Procedure


Procedure for experiments above 600K.

The following procedure is applicable for

experiments above 770K. A modification of the procedure

given at the end of this section allows for operation

down to 600K.


1. Place the sample in the coil and solder the coil leads

onto the electrical leads in the bomb.

2. Clean the seals and install the plug in the bomb.

3. Wrap the bomb with aluminum foil and place the styro-

foam spacer around the bomb.

4. Woods metal the can into place around the bomb. Check

the system for leaks in the joints.

5. Lower the can into the inner dewar and bolt the flange

into place.

6. Carry the dewar system into the pressure room and connect

the pressure lines and all of the electrical leads.

7. Pressurize the system to a few hundred psi.

8. Fill the inner dewar and the thermocouple reference dewar

with nitrogen.

9. After the bomb has been cooled to the desired temperature

range evacuate the can containing the bomb and turn on the

heater to control the temperature.






91
10. Take data points about every 0.10K in the temperature

region near the transition.

11. Top off the thermocouple reference dewar and fill the

cold trap dewar with liquid nitrogen.

12. Connect the nitrogen dewar for remote transfer.

13. Close the blow down valve and plug in power cord.

14. To prepare for first pressure application open the

charging solenoid valve to charge the pressure system

from the 2,000 psi helium storage cylinder. Make sure

that both the intensifier and the separator are cycled

to the bottom of their stroke.

15. Close the charging solenoid valve and open the blow-

down solenoid valve.

16. Pressurize the system with the separator for the first

pressure point.

17. Slowly release the oil pressure in the separator.

18. Control the temperature with the heater and take

readings at this pressure.

19. Activate the pump to the first separator.

20. Go to the desired pressure and take data.

21. After a run is complete bleed the pressure down very

slowly. If the pressure is released rapidly the

intensifier seals may be seriously damaged.



The procedure for work between 600K and 77K is

the same as that outlined above except that the outer dewar

is filled with liquid nitrogen and a vacuum pump is con-

nected to the inner dewar.






92
Procedure for experiment below 600K.

For work below 600K it is necessary to use liquid

helium and the procedure must be altered.


The first three steps are the same as listed before.

4. Fasten the copper thermal shorts to the bomb plug with

woods metal.

5. Woods metal the alternate can, except for bottom

onto the flange around the bomb.

6. Connect the copper straps to the bottom of the can and

seal this onto the can walls. Check for leaks.

7. Insert the helium transfer line through fitting pro-

vided in the flange.

8. Lower the can into the dewar and fasten the top with

bolts.

9. Carry the dewars into the pressure room and connect all

lines and electrical connections.

10. Pressurize the system to a few hundred psi.

11. Fill the inner, outer and thermocouple reference dewars.

Let the system cool to about 770K.

12. Transfer the liquid nitrogen out of the inner dewar and

connect the liquid helium dewar for remote transfer.

13. Start the transfer. A flow meter and a bubbler can be

used in the recovery line from the helium cryostat to

determine the rate at which the transfer is progressing.

14. Set the micrometer needle valve on the bottom of the

control panel so that the pressure gauge on the line






93
supplying helium for the transfer indicates 3 psi or

less. The micrometer valve gives fairly fine control,

and the rate of cool down can be controlled by

adjusting it.

15. After the system has cooled appreciably below 770K

evacuate the can surrounding the bomb.

16. As the temperature of the bomb reaches the freezing

temperature of the helium inside it turn on the heater

on the inlet pressure line and on the top of the bomb.

Transfer slowly and maintain the top of the bomb

several degrees above the bottom. Continue this until

well below the freezing temperature to insure that the

helium freezes from the bottom to the top of the bomb.

17. After the lowest temperature desired is obtained it can

be maintained or allowed to increase slightly by vary-

ing the helium transfer rate.

18. Data can be taken while cooling or warming.

19. If it is desired to increase the pressure it is neces-

sary to warm to a temperature greater than the freezing

temperature of helium at the desired pressure.

20. Pressurize by following 13-17 of the previous section.

21. Steps 13-18 of this section are then repeated.

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PRESSURE DEPENDENCE OF MAGNETIC TRANSITIONS By JAMES EDMUND MILTON A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1966

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ACKNOWLEDGMENTS .-^ The author is deeply indebted to many people for -^ aid and encouragement in completing this dissertation. He wishes to express his gratitude to the many people who have helped and especially to those listed below. Dr. Thomas A. Scott, the chairman of the author's supervisory committee, suggested the problem and was always ready to give aid when needed. I'ir. Basil McDowell was absolutely indespensible, he assisted in the design and construction of much of the ^,i equipment used in this study. In addition to this he supplied liquid helium when needed and frequently gave a friendly helping hand or clever suggestions. Dr. William S. Goree gave many helpful suggestions regarding the cryogenic apparatus and designed the pressure bomb used. Mr. K. S. Krishnan served as a soundingboard for many ideas involving experimental techniques and frequently 7^) suggested improvements. Ilr. Guy Ritch and Mr. Frank Ebright were very able assistants during the final phase of data taking. Dr. Stanley S, Ballard and Dr. Thomas A. Scott provided financial support during most of the author's 11

PAGE 3

t 2) graduate career. Dr. Knox Millsaps graciously lightened the author's work load during the writing of this dissertation. Deadlines would not have been met had it not been for the encouragement and invaluable help given by Dr. Richard L. Fearn in the preparation of this dissertation. Finally the author wishes to thank I
PAGE 4

; ) TABLE OF CONTENTS Page ACKNOWLEDGMENTS H LIST OF FIGURES V LIST OF TABLES vil CHAPTER I INTRODUCTION 1 II THEORY 22 III APPARATUS AND PROCEDURE" $LiIV RESULTS AND CONCLUSIONS 9^4REFERENCES 120 BIOGRAPHICAL SKETCH 125 iv 't^ii ^ail T^ai g f*M 1 .> JOir — C |i p -, s Tr-jr i MTW-

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LIST OP FIGURES Figure Page 1. Rare earth crystal structures, 10 \ 2. Magnetic structures of rare earths, 13 3. Spontaneous magnetization, Weiss theory, 28 ^. Hydrogen molecule. • 30 5. V/iring diagram for the inductance "bridge, 58 6. Inductance "bridge, So 7. Equivalent circuit for the inductance bridge, 63 8. Entrance to high-pressure room. 68 9. Top view of high-pressure room, 69 10, Schematic of high-pressure system, 72 11, Inside view of high-pressure room, 7^ 12. Schematic of the control panel for the pressure system. yS 13. Control panel, 78 1^, High-pressure gas bomb, 81 15' Electrical seal, 8^ 16. Bomb plug and seals, 86 17. Gryostat, 88 ^. 18. Inductance versus temperature for dysprosiijim, 96 19. Neel transition for dysprosium. 97 20. Pressure shift of Neel transition for dysprosium, 98 21. Pressure shift of N4el transition for erbium, 100 a r:^^.a.mii^^tmvi i ^m ^'g : u* mb xt m ^ — *- >*; 11 a ^a J W aKMi t ini irt i Tew M 1 nra MMt^-aM

PAGE 6

) ) 22. Robinson s interaction curve for rare earths. 101 23. Interaction curve for rare earths. 103 2^. '^: '^ '/ for rare earths, 10^^ 25. Dysprosium data. I07 26. Pressure shift of Curie transition for dysprosium. IO8 27. Erbium data taken while cooling. 109 28. ErbiuEi data taken while warming, no 29. Pressure shift for middle peak on warming data for erbium. 112 30. Pressure shift for upper peak on warming data for erbium. 11 3 31. Pressure shift for upper peak on the cooling data for erbium. 114 32. ^I'^Tf f<^3r heavy rare earths. II5 a m V vl --*-— ^— •

PAGE 7

) LIST OP TABLES Table Page 1 Sunmary of available experimental information on rare earth magnetic structures. 14 2 Summary of experimental results on the pressure shifts of the magnetic transitions for the rare earths* 20 3 Sample coils. 64 vii faiWViigw'^^MP^ilMt 'a i iMiiflnii a ^H' gri i ^ i TT i M i l M .. -— t. t*&3i 'jgrfa

PAGE 8

'\ CHAPTER I INTRODUCTION Historical baolcg:round Magnetism has aroused man's curiosity and fired his imagination for at least several thousand years. References to the attractive power of the lodestone had already appeared in Greek X'jriting 'bj 600 B.C.-^ The first knoim reference to the fact that magnetism could also repel a body is found in the writings of the Roman Lucretius Carus in the 1st Century B.C. 2 it is interesting J that no references to the directive property of the magnet, as used in the compass, is found in old Greek and Roman literature, but beginning in the period 1000-1200 A.D. the history of magnetism is closely associated with the compass and its use in navigation. The first clear mention of a magnet used to indicate direction xms made by Shen-Kua (IO3O-IO93), a Chinese mathematician and instrument maker. By 1100 A.D., the Chinese Chu lu reports that the compass was in use by sailors going betvjeen Canton and Siimatra. In 1269, Peregrinus de Maricourt reported on experiments made on a spherical lodestone. 3 He explored the surface of the sphere x^rith small particles of iron and applied the term pole to the places in x\rhiGh the magnetic power appeared to .n"'iiii iufi •i.i"*

PAGE 9

be concentrated. Little progress is reported until I6OO when V/llliam Gilbert published his De Ma?;:nete which summarized the knowledge of magnetism and reported the results of many of his 0^;^.! experiments. In this book he propounds his own great contribution, the realization that ^ the earth itself is a magnet. Also of great importance is his determination that if magnetized iron is heated to a bright red it loses its magnetism. In 1785, Charles A. Coulomb^ established with some precision the inverse square law of attraction or repulsion between unlike and like magnetic poles. This result was taken over by Poisson who became the best interpreter of the physical constructs which Coulomb had disij covered. -5 To magnetism, Poisson brought the concept of the static potential, which had been so successful in solving the problems of static electricity. He also assumed magnetization to be a molecular phenomenon, but believed that the molecule became magnetic only when the two fluids It contained became separated. It was Weber who proposed that each molecule is a permanent magnet, subject to a frictlonal force that tends to maintain it in its established orientation. This theory failed to explain the existanoe ) 01 residual induction and hystersis. In 1820, Oersted discovered that an electric current would affect a magnetic needle. Ampere then investigated experimentally and mathematically the forces between currents. He was able to show that a current in a ia^/t>>fiT r t.W*w^ T e mf mmr rK s r^^,~.

PAGE 10

3 circuit was equivalent to a magnetic shell of calculable strength. He then put forth the hypothesis that magnetism arose from currents within the molecules. This theory stood until it was modified by modern quantum mechanics. Pierre Curie made the first extensive study of ) ^he thermal properties of magnetic materials. 7 As a result of these studies he was able to establish that the magnetic susceptibility of a paramagnetic material was inversely proportional to the absolute temperature. X = C/T (1.1) The constant of proportionality was determined for each material and was always found to be positive. He found a ^relatively rapid decrease in the magnetization as each ferromagnetic material was heated to a critical temperature, now called the Curie temperature. Above this temperature, which was different for each material, the behavior was much like an ordinary paramagnetic substance. The first important modern development in magnetic theory came when Langevin^ used statistical mechanics to derive the Curie law, equation (1.1). The underlying assumptions of his theory were that, each molecule had a ) definite magnetic moment that tended to be aligned by the applied magnetic field and at the same time disturbed by thermal agitation. I-Iany years later this derivation was modified by Brillouin who took into account the quantum mechanical requirement that the atomic magnetic moments Were mAa-^'HliMir^t^i

PAGE 11

4 restricted to a finite set of orientations relative to the applied field. 9 After the work by Langevin, Weiss^^ made the next big step in developing a modern theory of magnetism. He assumed that the molecules are exposed both to the \ applied field and to a so-called molecular field proportional to the magnetization. It is a consequence of the Weiss theory that small regions, domains, within a magnetic material are magnetized to saturation even though the net magnetization of the body is zero. This is possible by having the magnetic moments of the domains oriented randomly. The Weiss theory is a very successful one ^^^hich has been substantiated by many experiments. It v~N ls however, not a very pleasing one since no explanation of the origin of the molecular field is given. In 1925 Uhlenbeck and Goudsmit introduced the concept of electron spin to explain some discrepancies between theory and experimental measurements of the spectra of one-electron atoms and the alkali -metals. This spin has associated with it both an angular momentum and a magnetic moment, Follox^jing this came the enunciation of the Paul! esiclusion principle. These two developments allowed Dirac and Heisenberg^l^2 ^q demonstrate a quantum mechanical origin for the Weiss molecular field. In their theory one starts with the Heitler-London model of the hydrogen molecule and considers a Hamiltonian made up wholly of electrostatic terns and kinetic energy terms. The Pauli exclusion 'V>nii^r J<<'rmf *-'-^rri — ^ n f iMi nriw i T i ViV i r i n~ ii T i T i r i : i i rTir'S \_m *n 1. ni i|.m l i

PAGE 12

principle enters the discussion only through symmetry requirements. Further, one must make an assumption regarding a distribution of energy levels. Heisenberg assumed this distribution was Gaussian. Using these conditions, it was possible to reproduce the Weiss theory ^ and show that the origin of the molecular field was a quantum mechanical ezchange integral. Thus, the most successful theory propounded had been given a quantummechanical basis and indeed it seemed that magnetism might at last be understood. But, as is usually the case, things were not as good as they appeared. Heisenberg's theory suffered from several serious weaknesses. 1) It was based upon the hydrogen molecule and contained no account of lattice periodicity. 2) The results obtained were very much dependent upon the distribution of energy levels assumed. 3) The actual calculation of the exchange parameter was an extremely difficult problem and thus far has not been resolved. At about the same time the Heisenberg-Dirac theory was being developed, Ising^3 proposed a different method for looking at the problem. The spins were disposed at regular intervals along the length of a one-dimensional chain. In accordance with the laws of Uhlenbeck and Goudsmit each spin was allowed to take on only one of two possible orientations. It was possible to obtain an exact solution for this model if it was assumed that each spin Interacted XNrith only a finite number of neighbors. ) ili' r T i i If iTfTri M r fc niaai"^i if r i mri nTr-

PAGE 13

Unfortunately the result indicated that ferromagnetism should not occur above 0K. Since that time exact twodimensional solutions and approximate three-dimensional solutions have given finite transition temperatures, thus showing that the failure of the first model was due to \ its dimensionality. Various methods have been employed to try to improve on the Heisenberg theory. One of the most successful of these is the method of spin waves developed by Bloch and Slater. ^^'^^ This theory starts with the observation that the eigenvalue of the Heisenberg exchange coupling can be determined rigorously if the spins of all but one atom are parallel. Furthermore, approximate solutions can be found if the number of reversed spins is small when compared with the number of atoms. Due to the above assumption this theory is good only for very low temperatures. It has been quite successful in describing the variation of magnetization with temperature in this region. In 1932, Neel proposed a theory to account for a type of paramagnetic susceptibility temperature dependence which did not agree with any of the existing theories.!^ He proposed two interpenetrating sublattices undergoing negative exchange interaction. This theory continues to be the basis for modern developments in the theory of what is noi-r called antiferromagnetism. i ^ ^J^-..' ^--^^.-.-r— .-I ., ^ .-

PAGE 14

I 7 The rare earths, which have become available in quantity in pure form only since the development of the ion exchange method of separation, cannot be described completely by the theories discussed in the preceding section. They have spurred a renewal in interest in J magnetism on both the experimental and theoretical fronts. Their physical properties v;ill be discussed in detail in the following section and the theories developed to describe them will be examined in a later chapter. Structure and information on rare earths The rare earth metals are composed of the fifteen elements which range from lanthanum to lutetium. The elecj.. tronic structure of these elements is normally given by (4f)^(5s)2(5p)6(5i)l(5s)2 where n increases from to 14' as the atomic number increases from 57 to 71. The outer electronic structure, which essentially determines their chemical properties, is the same for all of these elements and they normally appear in compounds as tripositive ions. Often scandium and yttrium, atomic numbers 21 and 39 respectively, are grouped with the rare earths since their external electronic configuration is similar. ''^'' The k-f electrons are tightly bound inside the outer closed shells on the atoms and therefore play only a small role in chemical bonding. They behave almost as they ^ 1 < ^ •iiim i fr MV

PAGE 15

8 would in a free ion, giving a resultant angular moraentum due to both spin and orbital motion. Since there is a magnetic moment associated with this angular momentum, all rare earth compounds have interesting magnetic properties. It will be shown in Chapter II that a model J based on ions with ^f^ configurations acted on by crystalline fields and coupled by exchange interactions is capable of explaining much of the magnetic phenomena of the rare earth metals. Rare earths have been investigated extensively within the last ten years at the Ames Research Laboratory and the Oak Ridge National Laboratory. The former of these has been involved in the separation and purification jof these elements, and with the measurement of specific heat, thermal expansion, electrical resistivity, magnetic properties, and other physical properties. The latter group has performed neutron diffraction studies and determined the complicated magnetic structures. The tripositive ion picture, outlined above, is violated by two of the rare earths, Eu and Yb, which come immediately before the middle element and the last element of the series, respectively. These elements should have '^f and 4f -^ configurations, however, they appear to prefer to gain the extra correlation energy of a halffilled or completed shell and take a divalent form with ^f"^ and ^^f configuration. Ytterbium has only a small paramagnetism as would be expected from a closed shell and

PAGE 16

9 europium shows large magnetic moments as it should for the kt' structure. One other element that should be mentioned in this respect is Ce, which may be found in a four-valent state either at low temperature or at high pressure. This is due to the fact that at the beginning J of the series the ^f and 5d. electrons have similar energies. In this state, as would be expected, Ce is found to exhibit small paramagnetism. Above lOO^K the stable form is found to be ^f^ and trivalent. The room temperature crystal structure of the rare earths tend to fall into two categories, the hexagonal close packed and a double hexagonal structure as shown in Figure 1. While they have been reported with various crystal structures, it seems that La, Ce, Pr and Nd, the light rare earths, usually have the modified hexagonal structure. Promethium has no stable isotope and therefore no information is available. Next, Sm has a very complicated hexagonal structure which repeats after nine hexagonal layers. The remainder, Gd, Tb, Dy, Ho, Er, Tm and Lu, have hexagonal close packed structures with c/a ratios 1.57-1 •59Their magnetic properties, while complex, show a certain regularity which may be traced to exchange interactions and crystalline fields. In the absence of detailed knowledge of the band structure of any of these elements theoretical work has been based upon the crude approximation of nearly free electrons. The effect of lattice symmetry has to some .;

PAGE 17

10 B B Hexagonal close packed structure Double hexagonal structure Figure 1, Rare earth crystal structures.

PAGE 18

11 extent been included by considering the Brillouin zone structure!? for the heavy rare earth series. The primitive translations t, U and ^3 are shown in Figure 1. There are two atoms per unit cell, one at the origin and one at T= -^(t,+ 2t0-f ^ ^^ The reciprocal lattice is also hexagonal and has vectors X. and T with magnitude a/T ^^ the basal plane 120 apart, and ^3 perpendicular to this plane with magnitude 2tt/c EvQn when the atoms are triply ionized the tightly bound ^f electrons are shielded from the crystalline field by the Ss^ and 5p electrons. Under these conditions their orbital angular momentum remains unquenched by the fields of neighboring ions. These electrons have total orbital angular momentum, L, and total spin, S, in the ground state as prescribed by Russel-Saunders coupling and Hund's rules. The energy difference between the ground state J multiplet and the first excited J multiplet is usually greater than 0.1 ev, therefore the excited multiplet plays no role in thermal properties. There are basically four types of measurements that have been made, in order to determine the magnetic properties of these elements. They are neutron diffraction, bulk magnetic measurements, specific heat measurements, and electrical resistivity measurements. A brief discussion of the information that can be obtained by each of these methods is given below. ts*-Av>_ftt Mh fr^:iJwgrt4faaai.t!fa

PAGE 19

12 Detailed neutron diffraction studies have been carried out on several of the rare earths. 2'^2-^ 22,23. 2^,25 26 The magnetic structures thus determined have been found to be quite complex. These studies have also given information about the magnitude of the ordered moment and its temperature dependence. The magnetic properties are found to be highly anisotropic, that is, the moments along the c-axis are quite different from the moments in the basal plane. Figure 2 shows some of the types of ordering that have been found and Table 1 gives the transition temperatures and structures for each element. Bulk magnetic measurements have shown that the susceptibility of these elements at high temperatures is roughly described by the Curie-V/eiss law. ^ 3K(T-<9) Here the Weiss constant, & indicates the approximate value of the exchange energy. It is also possible to obtain the magnitude of the ordered moments from this type of experiment. If a sufficiently strong magnetic field is applied to one of the anti ferromagnetic structures it is possible, in some cases ; to change it to a ferromagnetic structure. The field at which this occurs is called the critical field and can be used to obtain information about the energy difference between the tvro states. ^t— i9i "iiMi i r i tf *iw*< n[ it ri ii i Mt^fci

PAGE 20

13 A A <^ V V 1^ A V A V A A > c8 P 5 o u H ra X, P Jh eU m (^ pi •P o •p to o ^ A ^ A A V V Cfi S o

PAGE 21

ON Ik H ,0 OS B O P i ft o CO 0) ;3 -p o is •p H 03 P w s: o CO &0 c •H (D •n -p O !^ f4 ft is o O +i 1-^ d o &0 (D in U -P O csS -P 1 H 1 1 S-i S-i fl H c 1 1 ft 1 1 u }~i ft 1 e Q) CO s fe f^ ON r-i VP\ CM e o h tl a> I § o CO

PAGE 22

15 The transitions from one magnetic state to another are accompanied by sharp peaks in the specific heat versus temperature curves. These peaks can be located very accurately and therefore allow accurate determination of transition temperatures. The electrical resistance of these elements show anomalies at the magnetic transitions. These anomalies have been used to locate the transitions by a number of investigators. Experimental XTork The way in which magnetic properties of materials vary with pressure has long been of interest to physicists. Some of the earliest experiments in this area were done in an attempt to gain information about the origin of the earth's magnetic field. ^'^ The more recent ones, however, have been done in order to try to obtain information on the volume dependence of the exchange integral. One of the methods of attack on this problem has been to measure the shift X'l'ith pressure of the temperature at which the material goes from one magnetic state to another. The temperature of the transition to the ferromagnetic state is called the Curie temperature, 7c, while the temperature of transition from the paramagnetic to the anti ferromagnetic state is called the Neel temperature, Tn • The discussion which follows will be confined to experiments in which d T/d P have been determined. There is a vast literature of other

PAGE 23

0. X 16 types of magnetic experiments including a recent review by Ko-uvel.28 One of the earliest attempts in this field was that of Yeh in 1925.29 Yi% measured the effect of pressure on the magnetic permeability of iron, nickel and cobalt. This work was followed by that of Steinberger who, in 1933 d-id essentially the same experiment with improved sample annealing techniques ^^ These experiments did not specifically set out to measure the shift of the magnetic transition temperature with pressure. In retrospect, however, it can -be recognized that Steinberger actually induced a phase change from the ferromagnetic state to the paramagnetic state in a 30Ni 70Fe sample by the application of pressure at room temperature. From his data it can be concluded that d 7c /d P < O for this alloy. The first actual attempt to measure dlc/dP was made in I931 by Adams and Green^? who studied iron, nickel, magnetite, nickel steel and meteoric iron. They used the transformer method for detecting the transition. A primary and a secondary coil were wound on a closed frame made of the sample material. An alternating voltage was applied to the primary and the' output voltage was monitored as a function of temperature. The drop in output at Tc is very sharp, and although it does not define the Curie temperature in the conventional way, the method is satisfactory for finding a change in Curie point. They

PAGE 24

V 17 used carbon dioxide as the pressure transmitting medium and therefore achieved truly hydrostatic pressure. The shift for pressures up to 3.5 kilobars was found to equal zero for all of their samples. This result has not been confirmed by other investigators and it is believed that thermal uncertainties masked the true changes. Michels et al.^l used the discontinuity In the 0/RWdR/dT) versus temperature curve to indicate Tc • The sample material, which was 70Ni 30Cu, exhibited a broad transition that occurred gradually over 50C. They concluded from this that it was necessary to determine a shift of the Curie region. By carefully analyzing their data they were able to obtain dTc /dP — +6.4x/o"' K/kilobar. This method requires very accurate resistance measurements over large temperature intervals and is complicated by the fact that resistance also changes with pressure. Later a monel alloy was studied by the same method. ^^ This transition also was quite broad and yielded d Tc/d P = 3x/0'^ K/kilobar. Ebert and Kussman-^-^ used large magnetic fields to, obtain magnetization versus temperature curves so that the. Tc could be determined in the conventional manner. They then applied pressure and tried to determine dTc/dP for several pure metals and alloys. The result obtained for all samples was dTc/dPO Michels and De Groot^^ criticized their result and showed by a thermodynamic treatment of second order phase transitions that in general

PAGE 25

J 18 <^TcMP^ O They further showed that the experimental method used by Ebert and Kussman was not accurate enough to show small but significant variations of Tc Kornetzki35 took the data obtained by Ebert and Kussman and re-analyzed them and obtained non-zero values for d Tc /d P In 195^. Patrick36 j^ade a detailed study to determine dH/dP of nickel, gadolinium, cobalt, iron, eight metallic alloys, a ferrite and a perovskite. The transitions were detected by the transformer method as developed by Adams and Green. Two pressure systems were used, one used a gas for the pressure transmitting medium and the other used a liquid. The pressure was truly hydrostatic. Patrick's results agreed with those of Michels et al. and are widely quoted in the literature. Samara and Glardini37 made measurements on the shift of Tc in nickel and a nickel iron alloy. A multianvil pressure system with pyrophyllite as the transmitting medium was used. Pressures up to 35 kilobars were generated and the shifts found were in general agreement with those already determined. The transition was detected by monitoring the self -inductance of a coil which was wound on the sample. In addition to the electrical resistivity, selfinductance and transformer methods of detecting magnetic transitions, there are two other techniques which have been used. These transitions can be located by monitoring the

PAGE 26

19 mutual inductance betx\reen two coils wound on the sample. Changes in the magnetic moment of the sample show as a change in this mutual inductance, which can be measured very accurately by bridge methods. Finally, when the pressure system permits, a method involving the extraction of the sample from a magnetic field can be used. The pressure systems for this type of study fall into two distinct categories, those whose pressure transmitting medium is a liquid or a gas and those that use a solid for pressure transmission. The former, of course, are the only ones which produce truly hydrostatic pressures, however, the latter are able to obtain much higher pressures. One study has been made by sealing water in the pressure vessel and then freezing the entrapped water. The disadvantage of this method is the possibility of having tremendous pressure gradients inside the pressure vessel. The results of all of the investigations of pressure shifts of the transition temperatures for pure rare earths as well as the pertinent information about methods used are summarized in Table 2. This dissertation deals with the effect of pressure on the Curie transition and the Neel transition in dyprosium and erbium. These experiments are the second in a planned series of high pressure studies to be carried out at the University of Florida. It was necessary to develop the complete pressure system as well as the methods

PAGE 27

20 o 0) VO 00 OnO co o^o H CM CO 0^ O C*-\ C^j:}CO o J o M \ o CM 00 ^ o o o\o U^\OVO^VO I I I I I ^ ^ o H •H O COMD O CM r-i ^\0 O• • • • H rH tH H till O O • • I I ^CO o 00 • • • -^ r-i 1 C^ • + 1 1 CM i-i o +^ o o ^ c6 OS fH JS iH h 7i ^ h EH S EH EM S Eh EH -P ^ 'm ?-i i-\ U +ifs ;so;3assao ^a SIJcJCOOOO-P CO H-PH<(-(o3 ?S0:3hf-iM!HW ^^) aWSBEHEHEHW g:EH <0 in S CO tH CO Ti fH a ^^ '1 to to H H CO H H pi-H tOHHHH O CO H cs5 cS O O 03 O O tA r-\ CflOOOOPQ OS U C5 CiJ M bO O bO 60 -d O O bO bO &0 bO Ci bO ^ CM CO\D r^V^ ^ Cvl CNCO -* ^ CM £V OCM rH MD CO i I I III I 1 t I I I I I II o o o lA ooo o u^o o xpvo iN-o ovr\ -p ;3 H O $ e Eh a H 1 O ^H ft CO !>> Q a H a H O

PAGE 28

21 and equipment necessary for performing the experiments. A large high-pressure helium gas system, which is described in detail in Chapter III, was constructed. A gas system was chosen in order to be able to work under truly hydrostatic pressure at low temperatures. The choice of samples was based upon the availability of high purity specimens and also the desire to take advantage of the ability of the pressure system to work at very low temperatures. l^^len this work was started there were no published results on pressure shifts in any of the rare earths. As can be seen in the preceding section there has recently been a flurry of activity in this field. The pressure system constructed here is still the only one capable of studying the lower temperature transitions imder hydrostatic conditions and further studies on holmium and thulium are underway presently. The results obtained for erbium for both 6Tn/dP and dTc/dP are new. The results obtained for d Tn /d P for dysprosium are presented as corroborating those which have now been published. The d Tc /d P for dysprosium is in marked disagreement with that presented by Robinson et al.^^ which is the only one published to date. A complete discussion of the results is given in Chapter IV,

PAGE 29

CHAPTER II THEORY ^ Introduction Any discussion on magnetism must be based on quantum mechanical concepts. In the general discussion of magnetism which follows the author has relied heavily on numerous references. ^5-53 The discussion of rare earths mainly follows the reviews by Elliott, ^^ Yosida-5-5 and the books by Van Vleck-^o and Chikazumi.-5'I' This discussion can in no way be thought of as complete, but rather will attempt to describe the methods that have been most successful in treating the problem of magnetism. While much progress has been made there exists, at present, no completely satisfactory theory. Types of map:netism This section begins with a brief summary of the types of magnetism that are observed and some remarks concerning their originThe major classifications are diamagnetism, paramagnetism, ferromagnetism, antiferromagnetism and ferrimagnetism. Diamagnetism is a weak magnetism in which a magnetization is exhibited opposite to the direction of 22

PAGE 30

23 the applied field. It is associated with the tendency of electric charges to" shield the interior of a body from an applied magnetic field. It can be looked upon as a manifestation of the well-known Lenz's law, which states that when the flux through an electrical circuit is changed an induced current is set up in such a direction as to oppose this flux change. In a resistanceless circuit such as the orbit in an atom or in a superconductor the induced current persists as long as the field is present. Landau58 has shown that there can also be a diamagnetic contribution from the conduction electrons in a metal. Diamagnetism is present in all substances; however, in all cases except the superconductor it is a small effect with a susceptibility on the order of -10-5 cm3/mole. This effect is swamped if any other type of electron magnetism is present. The superconductors, which exclude all magnetic fields, exhibit perfect diamagnetism and have a susceptibility equal to -l/kn Diamagnetism plays a small role in the rare earths and will not be mentioned in the remainder of this discussion. Paramagnetism arises in materials in which there are permanent magnetic moments present. Magnetization results from the orientation of these moments in an applied field. This orientation is opposed by thermal agitation and therefore would be expected to be highly temperature dependent. The permanent moments may arise from the spin and orbital motion of the electrons or from jWifli*^jMp I

PAGE 31

21^ the nuclei. In the rare earths the nuclear susceptibility is about 10-^ times the electron susceptibility and will not be considered in this discussion. The electron paramagnetic susceptibilities vary from about +10-5 to +10-2 cm3/mole. The rare earths are accurately described ) in the paramagnetic region by the Curie-Weiss law which will be developed in a following section. A substance is called ferromagnetic if it possesses a spontaneous magnetic moment even in the absence of an applied magnetic field. This moment occurs only below some critical temperature known as the Curie temperature. This type of behavior is explained by adding to the paramagnetic model a strong co-operative effect which tends j to align the permanent moments in a parallel manner. Si^e dysprosium and erbium are both ferromagnetic at low temperatures the theories of this type of magnetism will be discussed in a following section. Antiferromagnetism arises from co-operative effects in a manner similar to ferromagnetism. In this case, however, the spins tend to align themselves in an antiparallel manner. The net magnetization is small and gives susceptibilities on the order of the ones given by paramagnetism. The temperature dependence of this susceptibility is, however, very different from that of paramagnetism. More will be said about this phenomenon in following discussions. Mae>4CMili^Me

PAGE 32

25 The oldest magnetic material known, the lodestone, is a ferrimagnet. This type of magnetism is characterized by an antiparallel arrangement of moments "but with the moments of unequal magnitude. This can give a strong external magnetic field. This type of effect is thought to arise from the same type of interaction as the antiferromagnetic case. These materials are of great practical Importance since many of them are insulators. None of the rare earths exhibit this type of magnetism so no further mention of it will be made. Quantum mechanical Langevin theory of paramagnetism Consider a system of M independent atoms in a magnetic field H • There will be 2 J + I Zeeman levels for each J Assume that, as with the rare earths at room temperature, kT is small compared to the energy gap between the ground state and the first excited state J"' V/rite the operator equation "p ^ Pt>J where jUfj = efi/(2mc)is the Bohr magneton and a is the Lande q factor given by The energy of interaction between the magnetic moments and the applied field is given by W(h) = -pH = "O l;(j MtH Using statistical mechanics it can be shown that the magnetic susceptibility is given by X= 4t^ ^^'" ^^ Where 2 =fS~"' is the partition function for the system.

PAGE 33

26 In this system 2 = "lexp ^^i^f Mt = -J which gives x = NkT H H' After some mathematical manipulation one obtains for the magnetic susceptibility, where Bj-(x) is called the Brillouin function and can be written B.,,= ^ coihiWU -27 cothiif) (2.1) where (2.2) If the energy of Zeeman splitting is small compared to kT then Xl and one obtains a nearly equal probability of occupation for all levels. Under these conditions Bjo^) can be expanded In a power series and higher order terms neglected to obtain B,7 {y) 3J X (2.3) By using equations 2.1, 2.2 and 2.3 one cdn obtain

PAGE 34

27 X= Nr(J^^Oq>' (3.,, It can be seen that equation 2,k is equivalent to equation 1.1 the Curie law, where C = N J( J+l)g'^p/3 k By using equation 2.1 and the relationship |V1=XH a general expression for the magnetization of a material obeying the above theory can be written as In the special case where X 1 equation 2.1 and 2A can be combined to obtain J K I I Weiss theory of ferromap:netism Weiss modified the above theory by adding to the paramagnetic model an interaction which tended to make I the atomic moments align themselves in a parallel manner. ^ He defined a molecular field proportional to the magnltization of the sample, Hm = ^M. where V is the Weiss constant. Now, using the methods of the previous deriI vation, one can obtain some useful relationships. If the magnetic field in equation 2.2 is replaced by an effective

PAGE 35

J y 28 field, Heff = H + 2f M then for ferromagnetic materials X= ^^fH^-^M) (2.7) In order to look for the spontaneous magnetization let 1-1 =• O and solve equation 2.? for M to obtain Since M must satisfy both equations 2.5 and 2.8 the simplest procedure is to investigate its behavior at various temperatures by graphical methods. M (2.8) Figure 3 Spontaneous magnetization, Weiss theory. From Figure 3 it can be seen that there is a critical temperature, Tc below which one gets spontaneous magnetization due to the -^ M field. As the temperature increases through Tc this magnetization vanishes. From equations 2.5 and 2.8 it is possible to obtain the follomng expression 3 __ XkT where it has been assumed that X I By talcing the

PAGE 36

29 derivative with respect to X of both sides of the above equation and evaluating it at T= Tc and X — O it is possible to obtain the following relation between the Curie temperature and the Weiss constant. -r. N9Vj(J-H)?r (2.9) Let us now consider a temperature region above Tc so that there is no spontaneous magnetization. Then equation 2.5 becomes M= Ng>,%m^)) (H4-j^M) (2.10) By using equations 2.9 and 2.10 it is possible to obtain (2.11) T-Tc where C = Tc// • Equation 2.11 is known as the CurieWeiss law. This is a very successful phenomenological theory which describes accurately the results of many experiments. In deriving it Weiss made no attempt to explain the origin of the molecular field. Heisenberg-Dirac theory of ferromagnetism Heisenberg was the first to show that the Weiss local field could be given a quantum mechanical origin. This can be demonstrated by considering the Heitler-London

PAGE 37

30 solution for a hydrogen molecule. Consider a simple system of two atoms, a and b that have one electron each and are separated by a distance /ab. See Figure 4. Figure 4-, Hydrogen molecule. Consider the following Hamiltonian, H= T, --f+Ta— ^ ^^-T-^--T^ (2.12) roi fba fab Til laa Tbi where "T denotes the kinetic energy operator and the subscripts identify the electrons. Consider also the following relationships. (t, --^,)la(o> = £laio> (T--^)lbc,,>=elU,> (T,--g^Jlaw)= ela.> where, for example, l^(i>^ denotes the atomic wave fTinction for proton CL and electron | With the above wave functions it is possible to construct symmetric and antisymmetric wave functions for the system. t itt i ia#' r aifc r

PAGE 38

31 The functions \C(y and lb) are not orthogonal. Define the overlap of these functions as Ls = l+L\ = ,-<-. If the spin is considered it can be seen that there will be one antisymmetric spin wave function and three symmetric spin wave functions. \ ia> = vT ^ >!.(+) Ha '-^~n.^>n.uO where s'] is the spin function. The subscript identifies the electron and + or denotes spin up or spin down. The simultaneous wave function must be antisymmetric. There are then the singlet state, Ua]> and the triplet state lA^)^. The singlet state has spins paired and therefore no net magnetic moment. The triplet state is in

PAGE 39

32 every way identical to a spin one particle with Mz= 1, O, -I By forming <^lHl^"> it is possible to obtain the energy shift for the singlet state where and T / \ \ Q" L. ^^ ^^ ^"^ 1 L_ /\ K. represents the total electrostatic energy of the two atoms and J*, is the exchange integral. From <.A\H\A'> it follows that the energy shift for the triplet state is Next, consider the energy difference between the singlet and the triplet state. E, E. = A E = ^'fl^K^'^ = 2 ^ If 0>Cthen the triplet state is energetically stable and the molecule will be magnetic.

PAGE 40

33 Lei t-c — ^€.+ )— L** Using this, it is possible to write The total spin is a constant of the motion. 5, and Si are also constants of the motion with eigenvalues -|h For the singlet state it can be shown that and for the triplet state one obtains Consider the spin Hamiltonian, It can be seen that it has the same eigenvalues as the electrostatic Hamiltonian used in the original formulation of the problem. This gives a spin-spin interaction with ^W^fltrM^anil' I' l l B Mtifc Tji i a n M w^ i nn M l> MiW** ln' M L' ''t:' >> 3 'wi.' *^ i<>^ ^i>i

PAGE 41

34 a weight factor Q that arises from electrostatic forces and symmetry requirements. This part of the Hamlltonian is called the Dirac-Heisenberg Hamiltonian, H= -2fl,,S."§;(2.13) One can now give an approximate connection between the exchange integral and the Weiss constant. ^-5 The assumption is often made that Ci-O for all atoms except the nearest neighbors and that Q J'e ^o^ all neighboring pairs. Based on this Wex ^-2geY^SrS, where -^ indicates that the sum is to be taken only over nearest neighbors. Assume that the instantaneous values of the neighboring spins can be replaced by their time averages. Then where Z is the number of nearest neighbors. If the magnetization is along the z-axis then < Sy,-/^ =• (^7 and This energy should equal the potential energy, V of the spin l in the Weiss field ^ M —^-^^...^.l: ---^.^ ...-,.-. ^,,^—11 ^.>1^^_-^^ -^. _--..

PAGE 42

35 V= ^Mt^SzijJii (2.15) Thus Using this and equation 2.9 it is possible to write T, = 2^g|S^^-'^ (2.16) Neel theoTj of antif erromapinetisni The Heisenberg theory of ferromagnetism is based upon the assumption that 0>O. When 0<^ an antiparallel arrangement of spins is favored and an antiferromagnetic substance is obtained. This type of system was' investigated by Neel,59 Bitter, ^ and Van Vleck, -^ and their work forms the foundation for the theory of antiferromagnetism. Consider two interpenetrating lattices made up of sites A, with plus spins, and B, with minus spins. Assume that there are antiferromagnetlc AA, AB, and BB interactions. Call these interactions w^g^, w^i^ and wi-,-]-, respectively. Since A and B are symmetrical w^a=^b= o( and Wa^T3=wt)a= & • The effective fields can then be written •t^tiffrnfaromm^ietmi^tM^* -it r

PAGE 43

36 Hea = H 0( Ma-(3 Mb (2.17) He-ff b -H-(3Mo--aMb (2.18) where H is the applied field and (X and e> are positive Weiss constants. Following the same methods used in the Weiss theory one can write that In the limit of high temperature and small X M^^MidJinA H^„^ (2.19) where N is the number of A atoms per unit volume. Similarily, if the dipoles on B are identical to the ones on A then 3 k r Use equations 2.17-2.20 to obtain M = ^3>;iy^" [2 H -(o. .(3) ka] This becomes a scalar equation with the assumption that M and H are in the same direction. Y^ M __ 2NQyaJ{j^l)/3k C ^g 21) -'^''W-1 — rT|iiii'-"lr't"^?ir-T i ii T .rr rT M i f <' i r n ~ i n ii i r i |. -) iir 'i j i ri ji n' i i-M uiiii iw i i iniia m wii n i -^n^ iw ii la^ r ii r i w iiif rr-n';Trtnn un mi

PAGE 44

37 This is quite similar to the result obtained for the Weiss formulation of magnetism. Next, examine the behavior at T~ Th' This temperature is still far enough away from saturation to use equations 2.19 and 2.20. With U~ write y" Ma = -J^UWio+^W\b>) (2.22) and JVib = -^^^ {(3Ma+(xM,J (2.23) where u is the magnetic moment per atom, U = qVa J"CT+1). From these it follows that T^J = -f^ i^-Oi) (2.24) } Observe that Tw Increases as the interaction AB Increases and decreases as AA and BB Increases. A relationship can now be established between Tn and Q by using equations I 2.21 and 2.24. Jj (^-^ (2.25) Experimentally it is found that Tiv/ < <9 which implies that j 0(> O or that, indeed, there is an antif erromagnetic AA ,) and BB interaction. •

PAGE 45

/-'> ^ .. 38 Phenomenolop:lcal discussion of orderlnp: in heavy rare earths Equations describing the types of ordering shown in Figure 2 may be written in the following form, |J* = gju^ JMCOS (5' Rn) (2.26) jUa^ = gjJ^ JMsin(^. RJ (2.27) |J* = gp,3 J M' Sin(^-Rn + g) (2.28) where the 2 -axis is taken along the crystallographic cX H ^ axis and jJ^ jU„ jj^ are the components of the moments on an atom at f?n. M is, in this case, the relative saturation along both the X and the w axes and M is the relative saturation along the 2 -axis. The vector Ci is parallel to the o-axis and has a magnitude, q^—Zn/cd where d gives the period of the magnetic structure. Equations 2.26 and 2.27, taken together, describe a helical structure while equation 2.28 alone describes a longitudinal wave structure. More complicated structures occur and may be described by variations of the above equations. Next, examine the results obtained from a Heisenberg-Dirac form for the Hamiltonian. •t.'V tfJii i~i-rc" -1 1

PAGE 46

39 / ^ H = -2^(R.-R^) Sn-S^ (2.29) since this exchange energy is, for the rare earths, usually much smaller than the splitting of the J" \ „y multlplets by the spln-orblt coupling, De Gennes"^ has proposed that S for each atom must be projected on the total momentum J S=(3-')J (2.30) This comes from a phenomenologlcal approach and has been examined and shown to be valid by several workers. 3o^ Using this expression and the above Hamlltonian one can -^ obtain the exchange energy for the helical ordering described by equations 2.26 and 2.2?. Esx=-2^(|)N(9-I)VM' (2.31) where ^(?)=^P,.-„)Cos[^-(R.-R-o^] (2-32) For the longitudinal wave the exchange energy is 2) Ze.^-^(i) NLg-\?J'U (2.33) where N is the number of atoms in the crystal. Note that these structures are energetically most stable at that "a which makes vJ(|) a maximum. Also, the spiral state is energetically more sta'ble than the n4ga-<^-~'1MBJlT^it~ii.-*T7fcMiT jn^ n.ti!if.,

PAGE 47

^0 longitudinal wave due to the factor of 2 found in the exchange energy of the former. However, in considering stable arrangements it is necessary to look at the free energy, F= U-JS. A molecular field approximation gives the same transition temperature for both structures. -5 7 This transition temperature can be written as Th= 2 Q[q)i^-if J iJ+\)/3k One must look to the anisotropy energy to determine the relative stability of the structures, Anisotropy The term magnetic anisotropy refers to the dependence of the internal energy of a crystal on the direction of the spontaneous magnetization. The energy associated with this directional dependence is called the magnetic anisotropy energy. The dominant source of anisotropy in the rare earths is the electrostatic interaction between the multipole moments of the ^f electrons and the crystalline electric field. The crystalline potential for a hexagonal close pack structure takes the form where A^ are the constants determined by the distribution of charges around the ions and 2 is taken along the c axis. The summation is over the coordinates of all •Ma#3BS^^

PAGE 48

J 41 of the electrons. This can be transformed into a more convenient form by use of the Wigner-Eckert theorem, ^ + pA:[3Sj;-30j/ J(j+0+3J^(J+l)V25J.-6J(J+l)] +jA;b3lj/-3l5j(jtl)j2+lo5jYjfifj;-5J'(J^0^ (2.35) where o( and s are constants which have been evaluated by Stevens, The < K"j^ are the mean values of \'" over the ^-f electron distribution and may be computed."^*-* The An are very difficult to evaluate and only order of magnitude estimates have been obtained.'-'If Ha is treated as a perturbation on Nay it is found that at high temperatures the first term is the dominant one, but at low temperatures the higher order terms also become important. The first term corresponds to the quadrupole moment and causes the preferred direction of the ordered moment to be either _!. or I! to the c-axls depending upon whether c<. is positive or negative. The second and third terms cause the moments to tend to align parallel to c when they are negative, but when they are positive the preferred direction is at an angle from c

PAGE 49

^n-^ -^^ i i^2 For dysprosium and terbium the first term is dominant and is negative over the whole ordered range. The ferromagnetic transition in these elements is caused by an increase in the fourth term with decreasing temperature. For erbium the third term is positive and fairly large and makes the conical structure stable at low temperatures. This method of combining an exchange interaction with crystal anisotropy has given very good qualitative results. As yet no quantative calculations have been made due to the extreme complexity of the problem. Ranp:e of exchange interaction In order to obtain some idea of the range of the exchange interaction necessary to stabilize the screw structure we look at a particular model. "^^ Assume that the exchange interaction, between layers of atoms perpendicular to the c-axis extends as far as secondneighbor layers. The exchange Hamiltonian now takes the form He. = -II 2f S,-S.-., L n = Q,i.+-^ where Si, is the average spin of an atom in the ith layer. By summing this exchange Hamiltonian it is possible to obtain an expression for the exchange energy. Ey referring back to equation 2.3I it is possible to see that

PAGE 50

r'^*^ ^3 the assumed spiral configuration will be made most stable by the values of ^ that maximize H^^j The 0(3) for this model can be written ^(|) = ^0 + 2 g.cos^ + 2g^cosc^c The value of O which maximizes this expression is S^ Ji. COS 2 4J, An analysis of the available data for dysprosium has been made by Enz"^^ and the values Ot/k —-Z^, 0, /k = 44 and Qi/k^ ~IS obtained. Similar results were obtained from an analysis of data on erbium. Observe that 0((i) is rapidly oscillating and long ranged to produce this spiral structure. Since the overlap of the ^f electrons on neighboring atoms must be quite small it would seem that this long range interaction is due to some other effect. It is reasonable to consider that the main part of the exchange interaction is produced by the exchange coupling between the conduction electrons and the localized spins. i Indirect exchange j_j Indirect exchange has been extensively investigated;'^^* 7^. 75 the following discussion closely follows that of Liu.'''" He starts by considering one conduction electron interacting with the magnetic electrons of one ion. The interaction Hamiltonian can be written as hrs~r^' Si rS aii S£^n -* i jT i i^ ^ *" • i IAnimti ^ n tii r i wy ^m <* hi ••mn i. .iMt^^M^i^ w i crfi i T ^ i ^. ^ ^^ A^ g

PAGE 51

.) ) Itk L =1 where Tn+i is the position of the conduction electron and r.is the position of the ith magnetic electron. The wave function for the conduction electron is of the form T(r,s) = jUkC?) expl^k-r] rj (2.37) where is a Block function and T^ is the Pauli spin function. Since the 4f electrons are well shielded their wave function can be written in the form ^ipCn^ = ^mY,^(o,^) ri (2.38) The wave function for the entire shell is constructed from the single particle wave function as prescribed by Hund's rules and the Pauli exclusion principle. Since this dissertation deals only with dysprosium and erbium, for i^hich the ^f shell is over half full, only that case will be considered. Liu^ shoxTOd that the required wave function of the shell is ^irA~Y^ C(LSJ; m, M-m^ At'YL^,tT5,NN-m,t (2.39) — *im(fr^l.i>qiar% J

PAGE 52

i^5 x-rhere C(LSJ;hfijVl-nr> ) denotes the vector coupling coefficient and the sunmation over t refers to a surrmiation over Young's diagrams.'' The vxave function of one conduction electron and one magnetic shell with no regard to symmetry is ^= Yxm(i,...,n)^(n^.O (2.40) where "^nih) ~ "^^t^Kvi ,sn+i) Next, this wave function must be antisymmetrized with respect to all the N+l particles. The resulting wave function is N (N+-0^ L'Yj-/v\Cl,-,N')'^(lvH-0— ^ Tj/v\(l,...,t-i,Nt-i, ii.i^...N")'^i.y (2.4l) The particles are considered to be completely Indistinguishable; therefore equation 2.36 must be symmetrized. u ~7 _^^\^ (2.i^2) where i->i^^ 1 wii — |

PAGE 53

if6 Using the expression for Tjtm equation 2.39. one can now form the matrix elements of the exchange interaction, Liu obtains an expression for Hx for the heavy rare earths which includes direct interaction ^ between shell electrons, exchange interaction between shell electrons, direct Interaction between conduction and shell electrons and exchange interaction between conduction and shell electrons. The last of these is found to be i H = 2 I(M')(g-l) S'J (2.^5) Jjjwhere ICk.k'") is the exchange integral, S is the spin of the conduction electron and J is total angular momentum of the ion. In order to obtain equation 2. '4-5 Liu made the following approximations. 1) The conduction electrons are s electrons so their wave functions have spherical symmetry. 2) The wavelength of the conduction electron is large compared with the size of the i^f shell so that eKp^ik'l}) may be approximated by the leading term of its power series expansion. It is very difficult to justify the first approximation and Liu did not try to show that it held. The second one can be examined by looking at the radius of the ^f shell as determined by the method of Pauling. 78 it is 5"

PAGE 54

I ^7 fo\ind to be about 0.4 A. Using the free electron approximation it is found that, for the heavy rare earths, k~ \.S X 10^ cm"-'at the Fermi energy. Therefore, k '"F;0.6,and the second approximation is seen to be reasonable. Recently Kaplan and Lyons'''3 have examined this second approximation and found that the leading term does indeed dominate for terbium through erbium and that the correction by other terms is about 10 per cent. De Gennes2 has found that since I(k,k') should be the same for all rare earths Tc or Tw should be proportional to (^-i)^J ( vT+l). For the heavy rare earths this reduces to S'CJ'+O/J. This is the same result that Neel obtained in 1938"''9 based on the molecular field approximation. This relationship is verified experimentally except for ytterbium. Pressure effects Using an equation first derived by Neel, Robinson et al.39 have constructed an interaction curve for the rare earths in an attempt to predict the effect of pressure on the transition temperatures. The Neel equation is J ^ ^^g^j (2.46) k 2iiS'(J'+0 where (^ is the transition temperature and 2 is the number of nearest neighbors. Using known values of the -' T -——'—>—— ——T~

PAGE 55

k8 right hand side of this equation the quantity Oe/k was calculated and plotted versus D/2R where D is the interatomic spacing and R is the radius of the ^f shell. There is some difficulty in using &^ in equation 2.^6 for the rare earths. In materials which go directly from' the paramagnetic to the ferromagnetic state Oct the temperature at which the material actually becomes ferromagnetic, is a few per cent lower than Ocp the extrapolated transition temperature, so it makes little difference which is used. In the heavy rare earths where antiferrom.agnetic states are observed these two temperatures are far apart. To account for this, two curves were plotted, one based on Od and one on Ocp The resulting curve is shown in Figure 22. ^ This analysis accounts for the magnitude and sign of the dTc/d P found for gadolinium and terbium and can be used to explain their result that dTc/^Pfor dysprosivim is positive for low pressures and changes to negative as the pressure is increased. Liu has done an analysis of the effect of pressure on Tc for ferromagnetic materials. ^0 He has looked specifically at gadolinium but was able to draw conclusions about the behavior of heavy rare earths from his work. The starting point chosen for this analysis is the Indirect exchange Hamiltonian

PAGE 56

^9 where the first term is the kinetic energy of the conduction electrons in the scalar effective mass approximation and the second is the exchange interaction between the conduction electrons and the ions. The subscript i refers to the conduction electrons while /j refers to the ions. As in the previous discussion of the indirect exchange the electron is described by a Block function. The matrix elements of equation 2,4? are written as It has been shown^^ "^^ "^-^ that by second order perturbation theory the exchange interaction can be expressed by the spin Hamiltonian With ^^^^"' InC2^p-n-R[2k.RC05{2krR)-Sin(2i<,R)] (2.^9) Where 1 is the average matrix element for k and k approximately equal to k^ Equation 2.^9 can alternately be xwitten as JCr^l^ttrN(e,)F(2,,R) (2.50) men J ^m i—i m^ — i ,tmmm rtr^nKT iTTWfe -: a c iej -;

PAGE 57

50 where H is the valence of the ion, N(£f) is the density of states at the Fermi level and F^) is given by De Gennes"'^ has shown that it is reasonable to assume that the ferromagnetic state is the ground state for gadolinium. Following this we can write Ey= fill's* N(£.)^ F(2k.Ri) (2.51) This equation should now be examined for terms which will vary with pressure. The summation will be independent of volume if the electron distribution is isotropic. The 2 and S are independent of volume in the heavy rare earths. One can now take the logarithm of equation 2,51 ^ and form ainE __ Dlnlll" Din N(£f) /g 52^ ^\r^y ~ 7)ln\/ "^ 2>|n V ~ The Curie temperature is proportional to the ordering energy per spin, therefore we may write 0\r\Tc. DIolTI' Din N/ce/r) / ^^^ DInV ~ ainV DInV ^^oj; The terms on the right side of equation 2.53 are unknown at the present time* Liu gives some estimates of the limits that can be expected for them. The thing that should be noted about them is that they are both functions only of the electronic properties of the material. Since 1 iiM •^-r~

PAGE 58

J 51 all of the heavy rare earths have similar electronic properties, gj|^ ^^ should be the same for each. This is a rather strong assumption and should be sub;5ected to experimental verification. This is discussed further in Chapter IV. Thermodynamics of phase transitions It is generally accepted that the transition from the paramagnetic to the antiferromagnetic state is second order while the transition from the antiferromagnetic state to the ferromagnetic state is first order. 52 It is possible to characterize a first order phase transition by either of the following statements. ^^ i; There are changes of entropy and volume. 2. The first order derivatives of the Gibbs function change discontinuously. Any phase transition that satisfies these requirements is known as a phase change of the first order. The effect of pressure on a first order phase transition can be determined simply by taking the first TdS equation of thermodynamics and integrating it over the change of phase. The first Tds equation can be written, Tds = c^dT -t-Y^j du(2.5^)

PAGE 59

52 Integrating this it is possible that one obtains, dT --^ ^-^ (2.55) In this equation the superscript f refers to the final phase and l refers to the initial phase. A second order phase transition is characterized by discontinuous changes in the second order derivatives of the Gibbs function. There is no change in entropy associated with this transition. Using the same superscript notation as before it is possible to write S'= S^ at (T, P) and S'-^ds = s'+-as^ at Cr+dT, P + dP) These expressions yield Tds' = Tds' (2.56) The second TdS equation is now used, Tds=c,dT-(^j_dP By using equation 2.56 and the definition of the volume expansivity it is possible to write CpdT-Tu-p'dP oldT' Tu-(3^dP 3y re-arranging and using the relation B = 3oi and P= — one obtains

PAGE 60

) 53 _dT yr _A^ (2.57) where A(X-o(^-a:' and AC^ct-C^, Equation 2.5? is knoim as an Ehrenfest equation. This equation predicts the pressure shift for the Neel transition and is further discussed in Chapter IV. \ J

PAGE 61

) CHAPTER III APPARATUS A1^"D PROCEDURE Introduotlon A description of the apparatus and of the procedure involved in the measurements made in this dissertation can be roughly divided into four major sections. The first section is concerned with the detection of the ferromagnetic-antiferromagnetic and antif erromagneticparamagnetic phase transitions. This task is complicated by the fact that the sample is contained inside a pressure bomb which is in turn contained within a temperature control cryostat. Further complications arise from the safety requirement that every thing should be operated remotely. The second section deals with the techniques involved in the compressions containment and pressure measurement of helium gas at high pressures and low temperatures. The third section concerns the production and measurement of temperatures from 5K to 190K, and the last section gives a step-by-step breakdown of the procedure used in performing the experiments. 5^

PAGE 62

55 Detection of Ma,g;r.etlc Transitions Several methods have been used to detect magnetic transitions In rare earths. The most Important of these are neutron diffraction, bulk magnetic measurements, specific heat measurements and electron .transport property ^ measurements. It was decided to look at the bulk magnetic properties in these experiments since they promised to give sensitive indications of the transitions, would readily lend themselves to pressure studies, and did not require any elaborate instrumentation. Methods of detection of the transitions by bulk magnetic properties are mentioned in Chapter I. Several factors had to be considered in ^''"' deciding upon the proper method to be used. It was desired to have as much sensitivity as possible; therefore a bridge method was selected. A large filling factor was desirable so the coil was placed inside of the bomb. Since the working space was limited and since the number 1 1 of electrical leads into the highpressure region should 1 be minimized, it was decided to use a single coil technique. After the experiments were well under way Samara and Giardinl37 reported that they had used the same method. The sample, in the form of a cylinder, was placed within a solenoid and the self -inductance of the coll was monitored. There is no simple exact formula for the selfinductance of a solenoldal coil of practical dimensions.

PAGE 63

5^ An approximate formula is 60. +^b4 iOc where CI is the mean radius b is the length, C is the radial thickness of the solenoid and n is the number of turns. The important thing to note is that L is proportional to /J the permeability of the core material. As a ferromagnetic sample is heated through its Curie temperature, its permeability changes from a large to a fairly small number. Hence, if the inductance of the coil is monitored, a large drop is seen as the sample is heated through its Curie temperature. The transition from the antiferromagnetic to the paramagnetic state is accompanied by a peak in the permeability versus temperature curve. Of course, the inductance of the coil would also be a function of the thermal expansion of the copper wire and the sample, and of variations of jJ due to skin effects. A blank run was made to insure that the changes in the coil were not influencing the results. Since the magnetic changes are quite large it was reasonable to neglect the other effects. 37 Inductance bridge In order to perform these experiments a very sensitive self-inductance bridge was needed. The design of inductance and capacitance bridges has been advanced considerably in recent years with the development of very

PAGE 64

57 accurate ratio transformers. "^toitThese instruments utilize modern high permeability magnetic core materials and are highly accurate alternating voltage dividers. ^5 A ratio transformer bridge was built following a design by Hillhouse and Kline. This bridge was capable of detecting changes of Inductance of the sample coll of one part per million. The wiring diagram, Figure 5, shows the components as connected in the bridge. This design features the use of commercially available components as listed below: 1. Audio oscillator, Hewlett-Packard Model 200JR 2. Isolation transformer, Gertsch Model ratio 4-1 3. Ratio transformer, Gertsch Model 1011 if-. Decade resistance box. General Radio Type 1432-K 5. Null detector, General Radio 1232-A 6. Standard inductor. General Radio 1482-L 7. Standard inductor, General Radio 1482-H All of these components, except the null detector and the audio oscillator, are contained within one cabinet. Figure 6. All of the external wiring is coaxial cable. (GR 874-R34) with General Radio shielded connectors. The switch, S-j^, allows the isolation transformer to be connected with a ratio of 4:1 or 1:4. The 60O ohm generator output impedence ceui then be transformed to 37 ohms or 9OOO ohms. The purpose for this approximate impedance matching was to

PAGE 65

58 &fv OQOQ I ^-S* ^OOO0 ^OQ -TV^ 0 6 Ratio Transformer i^— — ^^^— ^^ Unknown Inductor Standard lOOma Decade Resistor ) & -—6 M' Detector Figure 5. Wiring diagram for the inductance bridge.

PAGE 66

c m o OS § H 0) I •H

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-^: I 61 realize good bridge sensitivity. The ideal ratio between unknown inductance and standard inductance is 1:1; however, Hillhouse and Kline found that the accuracy was not appreciably altered up to a ratio of 10:1. Switch 33 allows for the use of either the 10 mh or the 100 mh standard inductor. The equivalent circuit for the bridge is shown in Figure ?• The operating equations for the bridge are derived below. Standard notation is used with subscripts 1 and 2 referring to the leads running from the bridge to the sample coil, S, to the standard, D, to the decade resistance and X, to the sample coil. The reading on the ratio transformer 1 A is that part of the total voltage that is being applied across the unknown inductor. Looking at the schematic it is seen that at the balanced condition, that is, when the current through the detector equals zero, one can write e,= E(l-A')=l[R.-fRs + Ro+^u;(L, + Ls+Lp)] (3.1) ez=EA= I[Rx + R2+^u;iLx+L^)] (3.2) Dividing equation 3.I by equation 3.2 gives (I-A[Rx+Ri+c=j^ fRs + Ro + RO-Ra. (3.^)

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62 and Lx= j:r^{Ls + LD + L.)-Lz (3.5) Equations 3.4and 3.5 constitute the operating equations for the "bridge. J ^ The inductance of the decade resistor is given by the manufacturer and at the maximum is on the order of a |jH. The inductance of leads 1 and 2 are also on the order of a juH. During an experimental run the transitions occur over a small temperature range; therefore Lp ,1.. and La are small and essentially constant. The Lx is from 1/2 to 1 Henry and changes in it completely dominate the picture. The situation with the resistances f^ is similar. Ri and Ri are small and essentially constant during the determination of the transition temperature. Using this information, one can truncate the operating equations and simplify data reduction. The simplified equations are: Rx = -j3a(Rs + Rd) (3.6) J L^=-pf^ Ls (3.7) Hillhouse and Kline have made a detailed error analysis for this bridge design and found that it is able to intercompare inductances at ratios as large as 10:1 to r irt'iriir-rmr"

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63 L R. J ) L; R. Figure ?. Equivalent circuit for the inductance bridge.

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64 accuracies an order of magnitude better than the certification limits of present standards which is at best t.OJ per cent. J Coil and sample description The two coils used in these experiments were wound on a teflon core with a Model W coil winder manufactured by the Coil Winding Equipment Company. The dimensions and room temperature characteristics are: Table 3. Sample coils. / Number of turns Length Inside diameter Outside diameter Resistance Inductance Coil 1 14,000 13/16 in. 1/8 in. 7/16 in. 3100 ohms 400 mH Coll 2 17,000 13/16 in. 1/8 in. 1/2 in. 6000 ohms 500 mH The samples were obtained from Leytess Metal and Chemical Corporation who specified a purity of 99.9 per cent. When received they were in the form of rods 6 in. long and .375 in. diameter. These were cut and turned down to a final sample size of 1/8 in. diameter by 13/I6 in. long. The samples were not annealed after machining. '~~i7-T~~~rViM*'i'r rt 'ntr~^ — — .—

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^5 Pressure Generation and Measurement The purpose of these experiments was to study the effect of hydrostatic pressure on magnetic transitions in rare earths. A large high pressure helium gas facility V7as constructed to achieve purely hydrostatic pressure over most of the temperature range covered. In principle, it is easy to achieve hydrostatic pressure in the fluid or gaseous phase of helium. In the lower temperature region approximately hydrostatic pressures may be achieved by applying the desired pressure to the helium while it is in the fluid phase and then freezing it at constant pressure. Further cooling necessitates the calculation of the pressure from the equation of state of solid helium and J the thermal properties of the high-pressure bomb,^'' which was made of beryllium-copper. This procedure gives very nearly hydrostatic pressure even though there is some movement due to the fact that helium has a larger thermal expansion coefficient than beryllium-copper. Numerous experimental difficulties arose during the course of the experiments. By far the largest problem was leaks in the pressure system. The bomb plug seals J. presented the most difficulty since a leak there made temperature determination and control impossible. Cooling through the freezing temperature of helium had to be done very carefully to prevent blocking of the inlet pressure line before the helium in the bomb was completely solidified.

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66 This would have greatly reduced the pressure at the sample as well as the accuracy with which it was known. Hip;h-pressure room The safest way in which to conduct high pressure J experiments is not to have any personnel in the vicinity of the high-pressure equipment. This was done by isolating all of the high-pressure components in a specially constructed, explosion proof room below ground level. This room was located outside the basement of the low temperature laboratory. All of the pressure equipment plus cryogenic apparatus was operated remotely from the adjoining basement. A brief description of this room V7ill now be given. Figure 8 shows an outside view of the room. The wall on the left is the outside wall of the Physics building basement. Figure 9 shovjs a top view of the room and part of the laboratory, giving wall details and rough dimensions. The roof of the room was constructed, from inside out, of 1/2 inch aluminum plate, k inches of sand, 9 inches of reinforced concrete, a ^ inch air gap, 1/2 inch plywood sheet, and a layer of sand bags resting on this plywood. Over this was placed another 1/2 inch plywood sheet which was covered with roll roofing. The outside end of the air gap was covered with screen and provided ventilation as well as a path for escaping gas in the event of an accident. The free volume of the compressed gas was only about oner -f"

PAGE 74

4 i O m 0) h i -ri o o i -P a ,# v CO (D to

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69 .; A_,/ gfea^;^^s;:^:^:-:g^:e?a^^-g^?gQ-§>l^ ~CS ;/ :^" ,-y-,-^^:^.^c?;-c?-.
PAGE 77

70 fortieth of the volume of the room and therefore could not create a significant increase in the pressure of the room. However, liquid nitrogen and helium dewars as well as the commercial helium gas tank were left in the room for remote transfer and if one of these should be rup] tured by shrapnel it would release enough gas to be dangerous in a poorly vented room. The door was a 5 1/2 inch thick box made of -1/2 inch alximinum plate and filled with sand. The box was supported by a steel dolly which had 6 ball bearing steel wheels that rolled in the channel of a 6 inch steel I beam. The north and east walls of the room (Figure 8) consist of 12 inches of steel reinforced concrete backed by earth. The south wall was constructed, from Inside out, of 1/2 inch aluminum plate, 6 inches of sand, and a wall of 8 inch solid concrete blocks. The west wall consists of the outside wall of the Physics building, 15 inches of reinforced concrete, supplemented by a 1/2 inch aluminum plate and 6 inches of sand. It was deemed necessary to add this plate and sand to prevent Spalding of the concrete wall in the event shrapnel struck the wall.^S The room was designed to contain all shrapnel and shock waves in the event of a high-pressure gas failure. Kip;h"pressure gas apparatus The high-pressure system is a three stage system composed of an Aminco 30,000 psi (H 5968) oil-to-gas J

PAGE 78

/ "^ 71 separator, a Harwood 100,000 psi Intenslfier (SAlO-81.250-lOOK), and a Harwood 200,000 psi intenslfier (SAIO-6-.875-2OOK). Figure 10 is a schematic showing all of the significant coEponents. Figure 11 shows the relative size and the placement of the components within 'j the room. Initial charging was accomplished by a remotelyoperated solenoid valve (switch located on the panel). For safety, a second solenoid valve was used to bleed the 2,000 psi stage of the gas system after charging. The charging gas flowed through a liquid nitrogen cold trap and a filter to remove gas and solid impurities. Mote that each stage was separated from its lower pressure adjoining stage by a one way ball check valve, as shown in Figure 10. The check valves in the 30,000 psi stage were Amine o No. ^^-6386 while the ones in the other stages were Harwood I4L-603. The Helse gauge. Model H 2696O, located in the 100,000 psi stage, was monitored by a closed circuit television system which consists of a Marson television monitor and a Bell camera. This gauge proved very useful in locating leaks and controlling bleed down rates in the system. The pressure during an experiment was always measured with the Harwood manganin resistance cell. This pressure was monitored constantly by a Poxboro recorder but data points were taken with a Carey-Poster bridge. ii — — <^'— v-T-T-^ -Ti ^^n mn m imii

PAGE 79

72 ) CO (>> CQ CO CQ
PAGE 80

a o o u (0 CO to u p< s cO o s ... ^.--^ ^...^-^.i-^^ iU.. ,5v

PAGE 81

) )

PAGE 82

^ 75 Harwood Hajiufacturing Company specifies an error of less than 1 per cent with this cell and bridge. The schematic also shows the motor driven bleed valve. This was a Harwood 200,000 psi needle valve driven by an electric motor through a chain and sprocket arrangement. Control panel Figure 12 is a schematic of the control panel and Figure 13 is a photograph of the actual panel. The schematic shows that the control panel was divided into two separate pumping systems; a 30 000 psi oil system for the gas-to-oil separator, and a 2,000 psi oil system for the two intensif iers. Sprague Engineering Corporation air powered pumps were used for both systems. As a safety feature the air supply was taken through a 115-volt ac solenoid valve that was normally closed. In the event of electrical power failure, affecting other components of the facility, the air supply was automatically stopped and had to be manually tripped on when power was restored. The intensifier oil system operated at reasonably low pressure so it was practical to use one pumping system with electrically coupled solenoid valves to draw oil from the proper reservoir and to direct it to the proper intensifier. These solenoid valves were operated by push button svjitches on the panel, and pilot lights indicated which reservoir the oil was taken from and the intensifier

PAGE 83

OJ H 1^ J U o E o Eh u\K /? en ft 0) •r-1 d a H o H &< ft -P4 .u ^1 o •H OJ o u S 0) -p CQ 0) CO Kl G) ?^ ft (D -p (h O
PAGE 84

H ft Q h 6 O ^ •H

PAGE 85

)

PAGE 86

79 to which it was pumped. Each oil reservoir was fitted with a level indicating sight tube. These were calibrated to shov; the actual position of the appropriate piston in the high-pressure room. Stainless steel tube, type 30^, with 3/8 inch OD was used throughout the intensifier systems while 30*000 psi Aminco high-pressure tubes and fittings were used in the separator system. The photograph of the control panel (Figure 13) shows the position of the pumping controls. This panel was designed for maximum safety, efficiency, and convenience of operation. A color coded flow diagram was painted on the panel to clarify the oil and high-pressure gas circuits. Also shown on the panel is the Foxboro Dynalog recorder whichwas used to monitor continuously the Harwood manganin resistance cell located in the 200,000 psi gas stage. A Harwood Carey-Poster bridge (visible on the shelf in the lower right-hand corner of Figure 13)was used for accurate readings of this cell. Visual observations in the room were made with closed circuit television. The push button and two warning lights in the upper left-hand corner of the control panel (Figure 13) remotely controlled the electric motor operated bleed valve in the 200,000 psi gas stage which bled this high pressure back into the 30,000 psi gas stage. A safety microswitch was located at the 30,000 psi stage oil reservoir which prevented operation of this motor unless the separator was ••|aiU-:**rwl f r=wg*.aMi>. '>^ tt^ — i m gr B-gtt-JM

PAGE 87

80 I cycled all the way back, thereby providing adequate volume for the 200,000 psi gas. HiRh-pressure bomb In highpressure, low-temperature experiments / "" J conflicting design considerations occur with the sample container (bomb). The bomb should be massive and strong to safely hold the pressure yet it must be small with low thermal mass so that it will fit a cryostat of reasonable size and can be cooled to low temperatures without using an excessive amount of liquid helium. Further, it was planned to use this bomb for magnetic measurements and nuclear resonance work, so a non-magnetic material was required. The desire for a non-magnetic bomb with high strength led to the choice of BeCu (Berylco 25) for the bomb material. A design pressure of 225,000 psi was used so that the bomb would be reasonably safe with the 200,000 I psi gas system. Standard thick-v/all cylinder equations9 gave stresses exceeding the yield strength of full hard BeCu regardless of the x^rall ratio (outside diameter/inside diam-. eter). These same equations, modified for a double wall cylinder, ^ predict sub yield point maxim-urn stress for the dimensions shown in Figure 1^. The outer cylinder was a .010-inch interference fit (on diameter) over the inner one. Assembly was accomplished by cooling the inner cylinder in

PAGE 88

81 1" X 1^ KF X li" ^.25" .65'' .Q75" ClTft =175.000 psi Fip^ure 1^. Righ-pressure gas bomb. fW^U>^t^*wtjb. ^u<>4juS^;'

PAGE 89

/'^A 82 liquid nitrogen and heating the outer cylinder to 900P, then pressing them together. The assembly was then heat treated at 600F for 3 hours. Final machining on the inside bore and sealing surfaces was done after the outer cylinder was fitted. Figure 1^ shows the finished bomb with dimensions and the tangential stress (7^ at the critical design points as calculated for an Internal pressure of 225.000 psi. It was very important that all corners and edges be made round and smooth to reduce stress concentration. Before use in the helium gas system, the bomb was pressurized with a liquid test system to 1^ kllobars and carefully checked and measured for distortion. The platinum resistance thermometer was installed on the bomb by means of a band on the outside of the bomb. This was assembled in the same manner as the bo., itself and was a .03 inch interference fit. Hl.g:h -pressure seals In performing these experiments one of the most difficult problems encountered was the design and fabrication of the electrical and bomb plug seals. The problem • of containing helium gas under pressure at low temperature is well known to anyone who has worked in this field. Epoxy seals, which were sufficient for much of the range covered, had already been developed at this laboratory. 90 m 'f *, tttt ^^ mmm c t**'**

PAGE 90

83 It was desired, however, to develop an electrical seal which would be easier to work with than this type. The electrical seal which was finally used is shown in Figure 15 and is simply a logical next step from the ones previously developed. The earlier seals depended upon y the bond betx-j-een the epoxy and the tube to carry the shear load vihich prevented the seal from blowing out. The new seal has a large cross section in the middle so that the epoxy itself must fail for the seal to blow out. In this seal, the wires pass through a small hole filled with an epoxy. (Sccobond 10^) The tubing used in the seal was Karwood 3^ a^d 12H. Standard Harwood cone and sleeve fittings which were good to pressures greater than 14kilobars were used in all cases except where the 3M tubing mated to the 12H. One non-standard part, the gland nut, had to be made for this connection. It was made from type 30^ stainless steel and had the dimensions shown in Figure 15The 12H tubing was drilled, tapped and a 60 conical seat was made on each end. The 3M tubing was threaded and coned in the standard manner. In order to insure that the epoxy bonded to the tubing, the walls were etched with acid and cleaned with water and t I acetone. The wires used were number 36 quadruple formvar j insulated copper. They were cleaned with acetone and placed inside the tube. Epoxy was mixed and forced into the tubes with the small stainless steel tube and screw

PAGE 91

8^ m H 05 0) CO H OS o •P O 0) H W H O rH

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85 shown In Figure 15 The wires were moved slowly back and forth several times, the excess epoxy was removed and the seals were baked according to manufacturer's specifications. The seals were tested on a liquid pressure system before being used In the gas facility. All of them were tested to about 1^1-0,000 psl and none blew out. Two of them were taken to 200,000 psl and while no leaks developed the wires Inside were broken by the epoxy shifting in the tube. The addition of 5-10 per cent alvunlna powder to the epoxy has been reported^l to significantly Increase its strength. This was tried but the one seal made in this way leaked at a low pressure. This may have been due to incomplete filling of the tube during fabrication. More work should be done on this since it seems that if the epoxy could be strengthened this should make a simple, inexpensive seal for gas systems up to 200,000 psl. The plug seal offered even more difficulty than the electrical seals. Many variations were tried and discarded. The one finally used is shown in Figure 16 and is good to at least 100,000 psl. It will probably go higher but leakage in the intenslfier seals have restricted the experiments to this pressure. It will be noted that it is a Bridgman unsupported area seal. Some workers have reported that this seal will not work at low temperatures due to the fact that the indium metal contracts more than the beryllium copper upon cooling. It was found, however, that if the bomb was cooled while under pressure no leaks

PAGE 93

86 1' 2 Y 1^ V -V -Y 1" hex head nut 1-1^ NP 1/8" annealed BeCu indi\n hardened BeCu 0,500^^6o --^YA \ / \ / V 4 i 3/16" 9/16" 0.625" Figure 16. Bomb plug and seals.

PAGE 94

87 would occur. This seal has the advantage that it seals with very lox^ torque, is simple to machine, and is reusable many times. It consists of a lower support ring made of hardened beryllium copper, a ring of indium, and an upper extrusion ring of fully annealed beryllitm ) copper. The plug is made of hardened beryllium copper. Temnerature Production and Measurement The basic cryostat system, which was used in all of the experiments, is show-n in Figure 1?. Some modifications, to be described later, were necessary for running the lowest temperature transition. The basic system consisted of an outer nitrogen dewar, an inner nitrogen or helixm dex^rar, and an evacuated can x-rhich contained the bomb. Aluminum foil, which has a low emissivity, was used to X"jrap the bomb and line the inner x-ralls of the can. The inlet pressure line was stainless steel which has a low thermal conductivity at low temperatures, A noninductively would heater coil was connected to the top of the bomb with woods metal. Three copper-constantan thermocouples were installed to measure the temperature of the bomb. The upper and lower ones used a copper lug on the X'rall of the can as a reference temperature while the middle one was brought out of the cryostat to a liquid

PAGE 95

88 Hip;h-pressure gas I Thermocouple and heater lear'*^ Helium transfer line Electrical seal >G<^ > Vacuum pujnp Inner dewar Outer dewar Heater Thermocouples Helium level resistors Figure 1?. Cryostat.

PAGE 96

y 89 nitrogen bath for reference. Periodically a platinum resistance thermometer was installed in the bomb and the thermocouples were checked. Temperature accuracy was judged to be -.25K and reproductibility was better than .10K. In the experiments involving the Neel transition the inner dewar was filled with liquid nitrogen and exchange gas was allowed into the can containing the bomb. V/hen the bomb had cooled to approximately the desired temperature the can was evacuated and thereafter maintained at a pressure of not more than .2 microns. With the bomb thus isolated the required temperature could be maintained with the heater. In the experiment involving the Curie transition of dysprosium it was necessary to cool the sample down to about 60K. This was done by pumping on the nitrogen bath with a large Kinney pump (Model No. KC-i^6) The temperature was then controlled by use of the heater. The ferromagnetic transition in erbium, vrhich occurs at about 20K, made it necessary to use liquid helium and also required some modifications in the equipment. Two copper straps (dimensions 3 x .375 x .025 inches) were soldered to the bottom of the bomb and to the bottom of the can. In addition to this, a heater coil was wrapped around the inlet pressure line. With these two modifications it was possible to maintain a temperature gradient

PAGE 97

90 across the bomb sufficient to insure that the helium froze from the bottom to the top. frocedure Proced'are for experinents above 60K. The following procedure is applicable for experiments above 77'^K A modification of the procedure given at the end of this section allows for operation down to SO^li. 1. Place the sample in the coil and solder the coil leads onto the electrical leads in the bomb. 2. Clean the seals and install the plug in the bomb. 3. Wrap the bomb with aluminum foil and place the styrofoam spacer around the bomb. ^. Woods metal the can into place around the bomb. Check the system for leaks in the joints. 5. Lower the can into the inner dewar and bolt the flange into place. 6. Carry the dewar system into the pressure room and connect the pressure lines and all of the electrical leads. 7. Pressurize the system to a few h\mdred psi. 8. Fill the inner dewar and the thermocouple reference dewar with nitrogen. 9. After the bomb has been cooled to the desired temperature range evacuate the can containing the bomb and turn on the heater to control the temperature.

PAGE 98

91 10. Take data points about every 0.1K in the temperature region near the transition. 11. Top off the thermocouple reference dewar and fill the cold trap dewar with liquid nitrogen, 12. Connect the nitrogen dewar for remote transfer. 13. Close the blow down valve and plug in power cord. ik. To prepare for first pressure application open the charging solenoid valve to charge the pressure system from the 2,000 psi helium storage cylinder. Make s^xce that both the intensifier and the separator are cycled to the bottom of their stroke. 15. Close the charging solenoid valve and open the blowdown solenoid valve. 16. Pressurize the system with the separator for the first pressure point. 17. Slowly release the oil pressure in the separator, 18. Control the temperature with the heater and take readings at this pressure. 19. Activate the pump to the first separator, 20. Go to the desired pressure and take data. 21. After a run is complete bleed the pressure down very slowly. If the pressure is released rapidly the intensifier seals may be seriously damaged. The procedure for work between SO'^K and 77K is the same as that outlined above except that the outer dewar is filled with liquid nitrogen and a vacuum pump is connected to the inner dewar. I miTWiW I "T^iirn I 1 -|i n Ttla I

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92 Procedure for exTPerlment below 60K. For work below 60K it is necessary to use liquid helium and the procedure must be altered. The first three steps are the same as listed before. ^. Fasten the copper thermal shorts to the bomb plug with woods metal. 5. VJoods metal the alternate can, except for bottom onto the flange around the bomb. 6. Connect the copper straps to the bottom of the can and seal this onto the can walls. Check for leaks. 7. Insert the helium transfer line through fitting provided in the flange. 8. Lower the can into the dewar and fasten the top with bolts. 9. Carry the dewars into the pressure room and connect all lines and electrical connections. 10. Pressurize the system to a few hundred psi. 11. Fill the inner, outer and thermocouple reference dewars. Let the system cool to about 77K, 12. Transfer the liquid nitrogen out of the inner dewar and connect the liquid helium dewar for remote transfer. 13. Start the transfer. A flow meter and a bubbler can be used in the recovery line from the helium cryostat to determine the rate at which the transfer is progressing. l^J-. Set the micrometer needle valve on the bottom of the control panel so that the pressure gauge on the line a^inr J usiSmS

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93 supplying helium for the transfer indicates 3 psi or less. The micrometer valve gives fairly fine control, and the rate of cool down can be controlled by adjusting it. 15. After the system has cooled appreciably below 77K ^ evacuate the can surrounding the bomb. 16, As the temperature of the bomb reaches the freezing temperature of the helium inside it turn on the heater on the inlet pressure line and on the top of the bomb. Transfer slowly and maintain the top of the bomb several degrees above the bottom. Continue this until well below the freezing temperature to insure that the helium freezes from the bottom to the top of the bomb. y-. 17. After the lowest temperature desired is obtained it can be maintained or allowed to increase slightly by varying the helium transfer rate. 18. Data can be taken while cooling or warming. 19. If it is desired to increase the pressure it is necessary to warm to a temperature greater than the freezing temperature of heliiim at the desired pressure. 20. Pressurize by following 13-1? of the previous section. 21. Steps 13-18 of this section are then repeated.

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CHAPTER IV RESULTS AND CONCLUSIONS Introduction The data taken in this study fall quite naturally into two categories. The higher temperature transitions (Neel transitions) and the lower temperature transitions (Curie transitions). The Neel transitions are, as discussed in Chapter II, mainly caused by the indirect exchange mechanism and are second order phase transitions. The Curie transitions are due to the temperature variations of the anisotropy energy and are first order phase transitions. The first section of this chapter presents the data obtained on the Neel transitions followed by a comparison with the available theories. The second section does the same with the Curie transition data. The final section is a discussion of the conclusions drawn from the results. Neel Transitions Results The Neel transition occurs at about 179K for dysprosivim and was the first one studied. This transition 94 IIJU'P I >Mili 1^1— '""

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95 was found to be sharp and reversible. Figure 18 shows a run for dysprosium at atmospheric pressure over a temperature range great enough to show both transitions. In determining the Neel transition temperature it was necessary to traverse only about one degree in temperature. The height of the peak was depressed and the location shifted by the application of pressure. Figure 19 shows a typical run at atmospheric pressure and at 25.000 psi. The peak was located in the following way. In the region near the maximiim, pairs of temperatures corresponding to points of the same inductance on either side of the maximum were read from the graph. The locus of the mean of these pairs was extrapolated to intersect the experimental curve. This procedure is indicated in Figure 19* It was necessary to rerun the atmospheric pressure peak each time the apparatus was reassembled as the location of the transition varied somewhat from one assembly to the next. This was probably due to small variations in the way the thermal gradients arranged themselves each time. The shifts were obtained for each pressure at least twice and in some cases numerous times. The value of the shift was found to be reproducible to within .Oi^K at a given pressure. The pressure shift for dysprosium is presented in Figure 20. Each of the determinations fell within the datum point shown. The average shift was found to be dT^^/dP = -0.440.Dl ^K/kilobar.

PAGE 103

ss ) ?s •r\ CQ O ^i ft CO l>J •d fs O M 9 t-t ^-i^ CD ^ Ui o CO cO
PAGE 104

97 i^ CO CD H H I O H CO > H Q) bO Q S 13 •H O r^ Ph CO O o -H H p H 0) 0 ON -H Q) to CO Cs] j3u^ C<1 CO CV OD .h.^ GOUBaoaDaT

PAGE 105

98 J u o rH CQ Ph H I t I S ;3 H CQ o W O Cm O •H +i CO H CD 21 o -p ;c CQ W W 0) ^1 "O u }Iq 8J:nc).^j:8d:iaaq. lafM ut sSusqo

PAGE 106

99 The data for the Neel transition for erbium, whioh occurs at about 85K, were also sharp and reversible. The shift for erbium was found to be d Tn/cjP^O. 26 O.oi '^K/kilobar and is presented in Figure 21. Comparison of results with theory It is possible, as shown in Chapter II, to calculate the value of dT^^/dP by using the theory of second order phase transitions. By using the expression, dP P ACp xfl-here Ao(93 is the height of the thermal expansion anomaly, ACp9^ is the height of the specific heat anomaly, and p 95 is the density, it was possible to calculate dTN/d P = -C?. 45 KAilobar for dysprosium. The excellent agreement between the calculated value and the experimental value is perhaps better than should be expected since the data for Ac< are not that accurate. The Ao( is unfortunately unknown for erbium so it is not possible to calculate the shift in this case. The interaction curve of Robinson et al. as shovm in Figure 22 would seem to predict a positive value for the shift in dysprosium and erbium. As pointed out in Chapter II there is some uncertainty as to just what value for the transition temperature should be used in this theory. In any case our data do not agree with this positive value.

PAGE 107

100 ; O r-l •H (!) w CO H Pm o I H I H I U o xa § H O o CO a) w h H CO 60 ^ t ;^Q 9j:nq.Ba:9duioq. X89]\i ut aSu^sqo

PAGE 108

101 CO C\J )U CM o • 0^ r ft CO C3 !^ o (!) > O o O 03 U 0) CO o to o CM CM u

PAGE 109

102 McWhan^O has suggested that it would be desirable to construct an interaction curve for the highest transition from which the dependence of the ordering temperature upon J, Land S has been eliminated. This can be done by dividing T hy ig-\)'^JiJH) the De Gennes function. ^2 Such an interaction curve is presented in Figure 23, Without detailed Imowledge of the compressibility of dysprosium and erbium in this temperature range it is not possible to accurately calculate what the shifts would be according to this theory. It can be seen, however, that the proper sign is predicted. 3y using Bridgman's room temperature compressibilities, A=2.74and 2. 63Xio'''cm2Ag for dysprosium and erbium respectively, it was possible to calculate dTsz/dP^-d^^ *^K/kilobar for dysprosium and dTN^P=-0.:iq OK/kilobar for erbium. In view of the uncertainties involved in using these compressibility data at lox-x temperatures the agreement seems to be quite good. The important point is that not only is the proper sign predicted but also the shift for erbium is predicted to be smaller than that for dysprosium. McVman has calculated ^~~ for all of the experi
PAGE 110

103 J vo ; 3J CO o CM 0^ o en cc cv o o+xK,(/-^; +i h 03 a) Pm CS h u o <1h o cd,lc > -p^a fH Ki ^ H;^ o ^r (-> t .>M or^ o •HIO -H "-'i -p Oi^i ^ -p(o 5 c3-P fn ^ o o cuia p 4f d ^ SH i—l l-HQ • m
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lOif ^1 !>CO ON o o ra •H [>n
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V 105 All of the points except gadolinium are calculated for Neel temperatures. Curie Transitions Results The ferromagnetic transition for dysprosium was much more difficult to work with than the Neel transition. The data taken X'fhile cooling were found to differ by about lO^K from data taken while warming. Even more disturbing was the fact that if the sample was taken to a temperature near the transition and held there the inductance values would continue to change for a long time. A test was run at 77K and it was found that the inductance was still changing slowly at the end of three and a half hours. This result made it undesirable to use the method of extrapolating the straight line portion of the curve to where it intercepted the temperature axis for determining the transition temperature. The extrapolation method has been used quite successfully to determine pressure shifts of transition with some materials. It was found that if dysprosium was cooled to about 60Kt far below its transition temperature, and then the data were taken while warming the sample the results were reproducible and the inflection point of the inductance versus temperature curve occurred at 85K. This value is in

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' 106 good agreement with other magnetic measurements. -^9 Several heating rates were tried and were fotmd not to change the location of the transition. All of the data used in determining dTc/dP xsrere taken at a warming rate of about 6 degrees per hour. The experimental curves are y) presented in Figure 25 anddTc/^P is given in Figure 26. Over the pressure range studied the shift was found to bedTc/d P= 1.2 K/kilobar. The data on erbium for its low temperature transition is presented in Figures 27 and 28. As can be seen the shape of the curve for cooling data is completely different from that of the warming data. The reasons for this difference is not understood at the present time. A similar situation was found by Adams and Green^'^when they studied meteoric iron. The upper peak on the warming curve corresponds to an anomaly found at 52*^K by both neutron diffraction and electrical resistivity experiments. ^9 From the neutron diffraction work it has been concluded that between 80K and 52K only the c component of the magnetic moment is modulated sinusoidally along the c-axis. Sinusoidal modulation of the perpendicular component begins • at temperatures lower than 52K. I I The loxTOr peak in the warming curve occurs at 20K and would therefore seem to correspond to the ferromagnetic transition. The fact that the middle peak on the warming curve has not been reported in high field magnetic measurements is perhaps not too surprising since these experiments

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107 J s P O T-rm 3ouuq.oupuT

PAGE 115

108 J o H •H > xi o 03 C3 U o o p -H CO CM to >Iq 9a:nq.Ba:dm9q. tj:uo

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109 CO ft CO ft ft ft o o o o o o o o o o o r—i' O O O v^ vn VOi o o o en CJ H ^ ^•^ v^ Htn 9 0uBq.onpui

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110 ft CQ ft to ft CO ft o o o • o o o o o o o o ft M •H o (M 1 o o o o o o o o J O 60 C •H B i H Si a o & -p a p a: CO fcO •H o o \o v^ v> VOi o O: O' o ^ cn w H \r\ y> ^A vr^ Epu 9ou^q.onpui

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y 111 do not detect the 52K transition either. It is curious that it has not been seen by the neutron diffraction studies since it is about the same magnitude as the other two peaks. The upper cooling peak probably corresponds to the 52 degree transition with a shift due to rate effects. The tremendous peak found at 2^.2K in the cooling curve is quite surprising. The pressure shifts of these peaks are presented in Figures 29, 30 and 31. The large lower peak in the cooling curve showed no shift with pressure while all of the others did, A blank run was made in order to insure that none of the anomalies that were observed were due to the apparatus Comparison of results with theory The theory, as developed, by Liu, for the shift in the paramagnetic Curie temperature for rare earths can only be examined qualitatively here. Unfortunately the data were not taken in a way that allows an accurate determination of this temperature and even if it had been, the compressibilities are not known in this region of temperature. Swenson has calculated ^! ^ by using the temperature at which the material actually goes ferromagnetic instead of the paramagnetic Curie temperature. The results of his calculations with the one for dysprosium and erbium from this work added are given in Figure 32. The results do not show the constant value predicted by Liu's theory but this could be due to not having data on the paramagnetic Curie temperature.

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112 ) 3^5 Pressure kilobar Figure 29. Pressure shift for middle peak on warming data for erbium.

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113 b4 O OS ft S EH 5 ^ 5 Pressure kilobars Figure 30* Pressure shift for upper peak on warming data for erbium.

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11^ O pi P 03 U (D ft s o Eh 3 ^5 Pressure kilobar Figure 31 Pressure shift for upper peak on the cooling data for erbium.

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115 ra -p u f. > o s: u o o Q P'> O o Eh pi Cvi H O H A^IP ir\ IL'JIP

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116 Conclusions The agreement between the experimental and thermodynamically calculated shift for the Neel transition in dysprosium is quite encouraging. It would be desirable to have a more accurate determination of the height of the thermal expansion anomaly for dysprosium in order to determine if this result is valid. Of course the same information is needed for erbium. The qualitative predictions based upon the interaction curve, Figure 22, formulated by Robinson et al.^3 are found to be violated by the data for both dysprosium and erbium. The interaction curve given by Figure 23 which was constructed following a suggestion by McWhan^^ predicts shifts of the magnetic transition temperatures that are in fair agreement with the experimental ones. This curve is based, however, on a phenomenological approach and is therefore not as pleasing as one based upon first principles. In it no account is taken of the strong anisotropic forces which exist in the antiferromagnetic and ferromagnetic states of the rare earths. There has been, as yet, no explanation given for the grouping of the values of -^ J; as given in Figure 2^. It is hoped that in the near future some theoretical basis can be given for this result.

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117 While Liu's theory for the pressure dependence of the magnetic transitions in the heavy rare earths is open to objection on several points, it is based upon first principles and is a start in the explanation of these phenomena. The present theories indicate that ani^ sotropy has a powerful influence on the magnetic properties of the rare earths. This would indicate that some further modifications must be made to make Liu's theory more realistic. He has based his theory on the premise that the indirect exchange mechanism causes the transitions, and that the ground state of the system due to the indirect exchange is ferromagnetic. This is not in keeping with the mainstream of the theoretical development which indicates the ferromagnetic transitions are probably caused by the anisotropic influences. Of course, as more information becomes available on the conduction electron properties there will probably be a need to modify the form chosen to represent their wave functions. This may quite likely bring in more terms with pressure dependence, for, instance 4-'F(2kFRi) in equation 2.51 may have to be considered. In fairness it must be said that the above is speculative since no accurately determined data are available to test Liu's prediction. Here it is perhaps appropriate to quote from a paper by Monfort and Swenson?^ in regard to these predictions. "There is need for precise data in order to verify this prediction, and except for gadolinium, the existing data for the rare earths merely serve to give an

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].. 118 order of magnitude for what is a very small effect." In addition to precise data ondT/dP It will be necessary to have compressibility data in the region of the transition. If single crystals could be obtained it vxould be very useful to do pressure studies on them since according to Landau's thermodynamic theory-52 ^p-^""" gives the ratio of the magnetic anisotropy energy to the magnetic exchange interaction energy. Where ^p is the paramagnetic Curie temperature and II refers to measurements made parallel to the c-axis and _L perpendicular to it. This would seem to give a powerful tool to check the qualitative predictions of present theories. The results for dTN/dP for dysprosium are in general agreement with those reported previously and shown in Table 1. The dTN/dPfor erbium is new but seems to be quite reasonable according to phenomenological theories. The results on dTc/dP for dysprosium are in agreement with some unpublished data by Swenson but in marked disagreement with those of Hobinson et al. ,^3 i^ho find a positive shift for low pressures and a negative shift at high pressure. The appearance of the new peaks in the erbium data at low temperature are not understood and it is recommended that further study be given to this point. In viev; of the fact that anisotropy is important and internal strains in the material could set up disturbances in the internal potentials of the material it would

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. 119 seem very desirable to rerim the low temperature transition in erbium with a fully annealed sample to see if the previously unreported peaks are still present. The accomplishments of this work are listed below, 1. A high-pressure gas system capable of generating hydrostatic pressiures at low temperatures was designed, constructed and made operational. It should be a basic tool in many experiments in future years. 2. A new method for studying pressure shifts of magnetic transitions was developed. The necessary equipment for doing these studies was constructed and brought to the point where they work dependably. 3. Accurate measurements were made for dT /c/P for dysprosium and erbium anddl^/dP for dysprosium. Preliminary results were obtained for 6Tc /d F for erbitim. ^. The way was opened and experiments have been suggested which should be very fruitful in further checking theories of magnetic transitions. 5 Publications related to this dissertation: A note describing fabrication of a simple, inexpensive high pressure gas electrical seal is being prepared; also a paper is in progress describing the experimental results of this dissertation.

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.--'s. REFERENCES 1. G. Sarton, Introduction to t he History of Scienc e (Baltimore, 192?), Vol. I. 2. Lucretius Carus, De. rerum natura (circa 70 B.C.), 3. Peregrinus de Maricourt, "Epistola Petri Peregrin! de Poucancourt Militem de Magnete (1269), 4-, C, A, Coulomb, Collection des Memoires Relatlfs a la Physique 1 (188^), 5. S. D. Poisson, Mem, de I'Acad, 5, 2^-7. 6. W. VJeber, Abhandlungen der Kg. Sachs Gesellschaft der Wissens, i,p. 572 (1852). 7. P. Curie, Ann. Chim. Phys, (7) ^, 289 (I895). } / 8 P. Langevin, Ann, Chim, Phys. (8) ^, 70 (I905). 9. L. Brlllouin, J. Phys. Radium 8, 7-^ (I927), 10, P. Weiss, J, de Phys. (^) 6, 66I (I907), 11, P, A, M. Dirac, Proc, Roy. Soc. 117A 6IO (1928). 12, W. Heisenberg, Z. Physik, ;}Q, ^11 (I926). 13, E. Ising, Z, Physik 21' 253 (1925). 1^. P, Bloch, Z, Physik 6I, 206 (I93O), 15. J. C, Slater, Phys. Rev. 21 509 (1930), 16, L. Neel, Ann, Phys. (Paris) l^.' ^^ (1932). 17. K. A. Gschneidner, Jr., "Crystallography of the Rare Earth Metals," The Rare Earths, ed. P. H. Spedding and A. H. Doane (John Wiley & Sons, Inc., New York, I96I), Chapt. Ik, p. 192, 18, K. A. Gschneidner, Jr, Rare Earth Alloys (D. van Nostrand Company, Inc., New York, I96I)', Chapt, 1, p, 6. 120

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' 121 19. R. J. Elliott, "Theory of Maf^netism in the Rare Earth Metals," Magnetism, Vol. IIA, ed. G. T. Rado and H, Suhl (Academic Press, I965), p. 389. 20. M. K, Wilkinson et al., J. Appl. Phys. 31, k^S (I96I). 21. W. Koehler et al. Rare Earth Research (Macmillan, New York, I96I), p. 1^9. 22. W. Koehler, J. Appl. Phys. ^[1, 20S (I96I). 23. W. Koehler et al. J. Phys. Soc. Japan l^, 32s (I962). 2i|-. W. Koehler et al. J. Appl. Phys. 21 ^8S (I96I). 25. W. Koehler et al. Phys. Rev. 126, 1672 (I962). 26. G. Will et al., J. Appl. Phys. 25.> 10^5 (196-^). 27. L. H. Adaras and J. W. Green, Phil. Kag. 12, 36I (I93I). 28. J. S. Kouvel, "Magnetic Properties of Solids Under Pressure," Solids Under Pressure ed. W. Paul and D. M. V/arschauer (McGraw-Hill, New York, I963), pp. 277-295. 29. C. Yeh, Proc. Am. Acad. Arts Sci. 60, 503 (I925). 30. R. L. Steinberger, J. Appl, Phys. k, I53 (I933). 31. A. Michels et al. Physica ^1:, IOO7 (1937). 32. A. Michels and J. Strijland, Physica 8, 53 (19^1-1). 33. P. Ebert and A. Kussman Physik. Z. 2£ 598 (I938). 3^. A. Michels and S. R. de Groot, Physica 16, 2^9 (I950). 35. M. Kornetzki, Z. Physik £8, 289 (1935). 36. L Patrick, Phys. Rev. 2^, 38^ (195^). 37. G. A. Samara and A. A. Giardini, Rev. Sci. Instr. 36, 108 (1965). 38. D. Bloch and R. Pauthenet, Compt. Rend. 2^, 1222 (I962) 39. L. B. Robinson et al. Phys. Rev. r^, AI87 (196^^). ^0. D. B. McWhan and A. L. Stevens, "The Effect of Pressure on the Magnetic Properties and Crystal Structure of Gd, Tb, Dy and Ho." (Unpublished paper presented at A.P.S. meeting. New York, I965.)

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122 ^1. P. Landry and R. Stevenson, Can. J. Phys. ^1, 1273 (1963). — k2, P. G. Severs and G. Jura, Science 1^5 575 (1964'). ^3. L. Bo Robinson et al. Phys. Rev. 2M_, 5^8 (I966). ^k, K. P. Belov et al. Soviet Phys. JETP 4^, 26 (I963). ^5. A. J. Dekker, Solid State Physios (Prentlce-Kall, Inc., Englewood Cliffs, N. J., 1957). Chapt. I9, pp. J^64-488. 46. F. Seitz, The Modern Theory of Solids (McGraw-Hill Book Company, inc.. New York, 1940), Chapt. I6, pp. 576-627. 47. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., New York, 195^), 2nd ed. Chapt. 15, pp. 402-450. 48. J. B. Goodenough, Magnetism and the Chemical Bond (Interscience Publishers, New York, I963), Chapt. 2, pp. 75-156' 49. R. M. Bozorth, Ferromagnetism (D. van Nostrand Company, Inc., New York, 1951). 50. P. H. Spedding-et al. "Some Physical Properties of the •Rare Earth Metals," Progress in Low Temperature Physics Vol. 2, ed. C, J. Gorter (Interscience Publishers Inc., New York, 1957), PP. 368-394. 51. D, C. Mattls, The Theory of Magnetism (Harper & Row, New York, I965), Chapt. 2, pp. 31-55. 52. K. P. Belov et al. Soviet Phys.-Usp. 2 179 (1964). 53o L. B. Robinson et al. "Magnetic Behavior of Rare Earth Metals at High Pressures," Physics of Solids at High Pressures ed. C. T. Tomizuka and R. M. Emrick (Academic Press, New York, I965), pp. 272-294. 54, R. J. Elliott, "Theory of Magnetism in the Rare Earth :l,Metals," Magnetism Vol. IIA, ed. G. T. Rado and H. Suhl (Academic Press, I965), p. 389. 55* K. Yosida, "Magnetic Structures of Heavy Rare-Earth Metals," Progress in Low Temperature Physics Vol, 4, ed. C. J. Gorter (Interscience Publishers, Inc., New York, 1964), pp. 265-295. 56, J, H. van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford University Press, London, 1932), Chapt. 2, pp. 27-41.

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123 57. S. Chikazumi, Physics of Ma,g:netism (John VJiley & Sons, I Inc., New Iork7l9wT Chapt. 20, pp. ^i^O-^58. 58L. D. Landau, Z. Physlk 6^, 629 (1930). 59. L. N^el, Ann. Fhys. (Paris) ^, 232 (I936). 60. F. Bitter, Phys. Rev. ^, 79 (I937). 61. J. H, van Vleck, J. Ghem. Phys. £, 85 (19'+1). 62. P. G. de Gennes, Compt. Rend. 2^, I836 (I958). j 63. R. Brout and E, Suhl, Phys. Rev. Letters 2, 387 (1959). 6^)-. G. S. Anderson and S. Legvold, Phys. Rev. Letters 1. 322 (1958). 65. T. A. Kaplan, Phys, Rev. 12^, 329 (I96I). 66. R. J. Elliott, Phys. Rev. 12^^, 32^6 (I96I), 67. K. Yosida and H. Mlwa, Progr. Theoret. Phys. (Kyoto) 26, 693 (1961). ^ V J. y 68. R. J. Elliott, Phys. Rev. 12if, 3^6 (I96I). 69. K, W. H. Stevens, Proc. Phys. Soc. (London) A 65. 209 (1952). ^ -^' ^y 70 A. J. Freeman and R. E. V/atson, Phys. Rev. 122., 2058 ( j-yOc. J 71. H._Miwa and K. losida, J. Phys. Soc. Japan 17, Suppl. B-i, 5 (1962). 72. U, Enz, Physica 26, 698 (I96O). 73. T. A. Kaplan and D. H. Lyons, Phys. Rev. 129, 2072 (1963). — 7^. T. Kasuya, Progr. Theoret. Phys. (Kyoto) I6, 1^5 (I956). ) 75' K. Yosida, Phys. Rev. IO6 893 (1957). 76. S. H. Liu, Phys. Rev. 121, ^51 (I96I). 77. L. D. Landau and E. M. Lifshitz, Quantum Mechanics trans. J. B. Sykes and J. S. Bell (Addison-Wesley Publishing Company, Inc., 1958), Chapt. 9, p. 211. 78. L. Pauling, Proc. Roy. Soc. (London) A ll4, 181 (I927). E> ni'.'pgrr:, dfc!^fcMrie~n I >wriiitWl— lr>itJ

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12^ 79. L. Neel, Compt. Bend. 206, ^\-9 (I938). 80. S. H. Liu, Phys. Rev. 122, I889 (I962). 81. M. A. Huderman and C. Kittel, Phys. Rev. £6. 99 (195^). 82. M. VJ. Zemansky. Heat and Thermodynamics (McGraw-Hill Book Company, Inc., New York, 1957). Chapt. 15, pp. 317-338. 83. A, M. Thompson, IRE Trans, on Instrumentation 1, 24-5 (1958). 8^. H. C. McGregor et al. IRE Trans, on Instrumentation 1, 253 (1958). 85. R. S. Travis and J. A. Zugel "Ratio Transformers, Theory, Design and an Application," Precision K easurements Course (The Pioeing Company, Aerospace Division, Seattle, Washington, I962), Session 26. 86. D. L. Hillhouse and H. v;. IQine, IRE Trans, on Instrumentation £, 251 (i960). 87. J. S. Dugdale and F. E. Simon, Proc. Roy. Soc. (London) A 218, 291 (1953). 88. 0, L. Anderson, Some Safety Problems Associated with High Pressure Equipment (Unpublished technical memorandum. Case 381^1-3. Bell Telephone Laboratories, 195^). 89. S. Timoshenlco, Strength of Materials (D. van Nostrand Company, Inc., New Jersey, 1956), part 2, p. 205. 90. W. S. Goree, B. McDowell and T. A. Scott, Rev. Sci. instr. 2k* 99 (I965). 91. L. A. Davis et al., Rev. Sci. Instr. 15, 368 (196^). 92. P. W. Bridgeman, The Physics of High Pressure (G. Bell & Sons, Ltd., London, 1958). 93. P. Barson et al. Phys. Rev. 10^, -^18 (1957). 9^. M. Griffel et al., J. Chem. Phys. 21, 75 (1956). 95. P. H. Spedding et al. "Some Physical Properties of the Rare Earth Metals," Progress in Low Temperature Physics Vol. 2, ed. C. J. Gorter (Interscienoe Publishers Inc., New York, 1957). PP. 370-371. 96. C. E. Honfort III and C. A. Swenson, J. Phys. Chem, Solids 26, 623 (1965).

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BIOGRAPHICAL SKETCH James E. Milton was born May 12, 193^. at Floralaj Alabama. In June 1951 he was graduated from Columbia High School in Lake City, Florida. He attended the University of Florida intermittently from 1951 195^From 1955 until 1958 he served in the United States Army. Following his discharge from the Army, he returned to the University of Florida. In June 19^0, he received the degree of Bachelor of Aeronautical Engineering. From September I960, until the present time he has pursued his work toward the degree of Doctor of Philosophy. James E. Milton is married to the former Mary Eleanor Jernigan and is the father of two children.

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. April, 1966 u f^TLe^. Dean, Collerfe//ofArts and Sciences Dean, Graduate School Supervisory Committee Chairman 1 I //