STUDIES of DEUTERATED POTASSIUM CHROME ALUM By

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STUDIES OF DEUTERATED POTASSIUM CHROME ALUM

RONALD EUGENE MILLER, B. S., M. S.

A DISSERTATION

PHYSICS

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

DOCTOR OF PHILOSOPHY Aec-1 q^.3

T5 197/

ACKNOWLEDGMENTS

I would like to express my appreciation to Dr. Edward Teller, who suggested that this research be undertaken, for the contributions he made to the interpretation of the results. I would also like to express my sincere appreciation to Dr. B. J. Marshall for his direction of this dissertation and to the Robert A. Welch Foundation of Texas for the financial support given to this project.

11 TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

LIST OF TABLES v

LIST OF FIGURES vi

I. PRELIMINARY CONSIDERATIONS . . . .' 1

Introduction 1

Previous Related Research on Alums 3

II. GROWTH OF SINGLE CRYSTALS 9

Supersaturated Solution 9

Seed Crystal Growth 10

Crystal Growth 11

III. EXPERIMENTAL PROCEDURES AND EQUIPMENT 14

Electronics for Ultrasonic Measurements 14

Crystal Preparation for Ultrasonics Measurements. . . 16

Ultrasonic Cryostat and Dewars 16

Temperature Measurement for Ultrasonics 19

Ultrasonic Transducers 19

Ultrasonic Binders 20

Collection of Attenuation Data 22

Cryostat for Measurement of the Dielectric

Coefficients 23

Electronics for the Dielectric Constants 25

Temperature Measurement for the Dielectric

Constants 25

Dielectric Sample Preparation 25

lii IV

Page

Collection of Dielectric Data 26

Continuous Cooling Method for Measuring the Specific Heat 27

Electronics for Specific Heat 29

IV. RESULTS AND CONCLUSIONS 31

Ultrasonic Attenuation Results 31

Coraplex Dielectric Coefficient Results 33

Specific Heat Results 62

Conclusion 62

Future Work 70

LIST OF REFERENCES 72

APPENDIX I 74 LIST OF TABLES

Table Page

1. Values of a in the [100] Direction at 10 MHz 51

2. Values of a in the [100] Direction at 30 MHz 52

3. Values of a in the [110] Direction at 10 MHz 53

4. Values of a in the [110] Direction at 30 MHz 54

5. Values of a in the [110] Direction at 50 MHz 55

6. Values of a in the [111] Direction at 10 MHz 56

7. Values of a in the [111] Direction at 30 MHz 57

8. Values of K' and K" in the [100] Direction as a Function of Temperature 58

9- Values of K' and K" in the [110] Direction as a Function of Temperature 59 10. Values of K' and K" in the [111] Direction as a Function of Temperature 60

11. Specific Heat Values as a Function of Temperature . . 61 vi

LIST OF FIGURES

Figure Page

1. a Crystal Structure for the Alums 4

2. Directions of the Crystallographic Axes 13

3. Block Diagram of the Apparatus for Ultrasonic Measurements 15

4. Cryostat and Dewars for Ultrasonic Measurements ... 17

5. Cryostat and Dewars for Dielectric Coefficient Measurements 24

6. Block Diagram of the Apparatus for Specific Heat Measurements 30

7. Echo Patterns Showing Increasing Attenuation with Decreasing Temperature 32

8. Ultrasonic Attenuation versus Temperature in the [100] Direction at 10 MHz 34

9- Ultrasonic Attenuation versus Temperature in the [100] Direction at 30 MHz 35

10. Ultrasonic Attenuation versus Temperature in the . [110] Direction at 10 MHz 36

11. Ultrasonic Attenuation versus Temperature in the [110] Direction at 30 MHz 37

12. Ultrasonic Attenuation versus Temperature in the [110] Direction at 50 MHz 38

13. Ultrasonic Attenuation versus Temperature in the [111] Direction at 10 MHz 39

14. Ultrasonic Attenuation versus Temperature in the [111] Direction at 30 MHz 40

15. K' and K" versus Temperature in the [100] Direction at 10 MHz 41

16. K' and K" versus Temperature in the [100] Direction at 30 MHz 42

17. K' and K" versus Temperature in the [100] Direction at 50 MHz 43 vii

Page

18. K' and K" versus Temperature in the [110] Direction at 10 MHz 44

19. K' and K" versus Temperature in the [110] Direction at 30 MHz 45

20. K' and K" versus Temperature in the [110] Direction at 50 MHz 46

21. K' and K" versus Temperature in the [111] Direction at 10 MHz 47

22. K' and K" versus Temperature in the [111] Direction at 30 MHz 48

23. K' and K" versus Temperature in the [111] Direction at 50 MHz 49

24. Specific Heat versus Temperature 50

25. Curve Shapes Expected for K' and K" and C for a Ferroelectric Phase Transition ^ 65 26. Curve Shapes Expected for a Debye-type of Relaxation 66

27. Representation of r 75 CHAPTER I

PRELIMINARY CONSIDERATIONS

Introduction

Potassium chrome alura (KCr(SO ) '12H 0), due to : ts paramagnetic behavior, has long been used for both the production of and thermom- -4 etry in extreraely low temperatures (T - 10 K). As a result, this and other paramagnetic alums have been studied extensiveLy. Howcver, many of their properties are still not well understood. Many alums 12 3 have shown anomalous behavior in some, * ' if not all, of the following properties in the temperature region of 300 to 4.2 K.

(1) Ultrasonic Attenuation

(2) Dielectric Coefficients

(3) Optical Absorption and Transmission

(4) Crystal Structure

(5) Paramagnetic Resonance

(6) Thermal Conductivity

A specific discussion of related research will follow this introduction.

Of particular interest in this dissertation is a study of the ultrasonic attenuation and dielectric coefficients of the alums.

The attenuation of ultrasound in alums generally shows a very large 2 3 peak in the temperature range of 60 to 160 K. * The dielectric coefficients, both real and imaginary, have also shown peaks in this same general temperature range in some of the clear aluras. The two explanations for this behavior have suggested that since raany of the alums becorae ferroelectric, (1) the absorption is due to a ferroelec- 3 4 tric phase transition, * (2) the absorption of ultrasound raight be caused by a hindered rotation of the water molecules due to the inter- action of the perraanent dipole raoments of the water molecules and the dipole moraents produced by the forced motion of the ions in the 2 lattice. This is known as a Debye type of relaxation.

The present work was designed to investigate what effects would be seen in the ultrasonic attenuation and dielectric coefficients if the water molecules were deuterated. If the mass of the water molecules were changed by deuteration then any resonant phenomena involving the rotation or vibration of the water molecules would change its temperature dependence. Furthermore, if the increase in the absorption of ultrasound were due to ferroelectric behavior, an anomaly in the dielectric coefficients would be expected in both the deuterated and undeuterated case. Also this work was designed to check for the possibility of a phase transition that would show up in the measurement of the specific heat.

Measureraents of the ultrasonic attenuation and the dielectric coefficients were conducted in the temperature range of 300 to 4.2 K.

The specific heat was measured over the teraperature range of 80 to

100 K. The results indicate that the effects of deuteration are not of prirae importance. Furthermore, the previously held ideas concern- ing the possible ferroelectric transition and hindered molecular rotation do not seem to be entirely proper for KCr(SO,^^'12D-0 in the light of the absence of anomalies in either the specific heat or the dielectric coefficients. A complete discussion of the results and conclusions can be found in Chapter IV.

Previous Related Research on Aluras

In the following paragraphs a suramary will be raade of the related work conducted on the class of double salts known as alums.

These salts have the general formula AB(SO ) 'l^H^O, where A and B are monovalent and trivalent ions respectively. An excellent summary of the work done on the chrome alums prior to 1951 may be found in the review article by J. Eisenstein.

The crystal structures of aluras are known frora x-ray raeasurements by Lipson and Beevers. * Figure 1 shov^/s a diagram of 1/8 of a unit cell for the alum a structure. The A ions and the B ions form two interpcnctrating face-centered cubic lattices. Each B ion is surrounded by six water molecules which forra an octahedron with axes slightly rotated frora the cubic axis. The octahedron is slightly distorted along a trigonal axis, producing an electric field at the

B ion site. The SO, groups are arranged such that the S atom and one

0 atom lie on a trigonal axis. The other three 0 atoms lie at the other points of a tetrahedron with the S atom in the center. The re- maining six water molecules are spaced at odd points in the lattice and may be considered to be loosely associated with one A ion. Only two of these molecules are shown. The other water molecules may be located by knowing that each one touches two 0 atoms, a water raolecule associated with a B atom, and an A atom.

It is known that among the chromiura alums there are three slightly Water molecule associated o with "B" ion

Other water molecule

Figure 1. a Crystal Structure for the Alums. different structures. They are the a, 3, and y structures. These structures are differentiated by the positions of the sulfate ions along the trigonai axis of the unit cell. This positioning is dependent on the size of the raonovalent ion. The positions of the water molecules are also slightly shifted frora a to 3 structure.

Potassium chrome alum is a member of the a-type structure. This structure is known to undergo transitions in the 60 to 160 K range.

The water-containing complexes are thought to be involved in this transition. Indications are such that 3 and y structures are not believed to undergo transition. 7 8 Bleany and Penrose * have investigated the paramagnetic resonance spectra of potassiura chrome alum as well as other alums. The chromium ions sit in an electric field which is predominately cubic in symmetry because of the surrounding octahedron of water molecules.

An electric field contribution from the other neighbors in the lattice is also indicated. The crystalline fields cause a splitting of the electron energy levels of the chromium ion. Measurement of the energy level differences using microwave techniques provide inforraation about the lattice in general and the octahedron of water molecules

Q in particular. Bleany reported that the splitting decreased with temperature in a linear fashion from 290 to 160 K. Below 160 K there was no sudden transition but the splitting remained constant and two side peaks appeared as the temperature was lowered. The side peaks represented two unequal splittings. He interpreted this to mean that some of the ions sit in sites of one field value and the remainder 9 10 occupy sites of another value. Van Vleck ' interpreted the splittings to be caused by a sraall distortion of the water octahedron.

Kraus and Nutting bave raeasured the optical absorption spec- trum of potassium chrome alum. They observed the spectra to be different at 85 K from the spectra at 14 K. At temperatures between

85 and 14 K they observed both types of spectra. A gradual change in the spectra was observed from 85 to 14 K. This suggests a gradual transition in the raaterial as it is cooled below 85 K. 12 Ancenot and Couture studied light transraission by potassium chrorae alum placed between crossed polaroids so that optical phase shifts could be detected. This work was conducted on diluted alums in which chroraiura ions were replaced by aluminum ions. For dilutions in which less than 7% of the chromium ions had been replaced by alum- inura ions the specimen changed frora an optically isotropic medium to a birefringent material at 60 K. When the speciraen was cooled rapidly through 60 K no sudden transition occurred. They interpreted this result to indicate that this alum was structurally in a metastable state when cooled rapidly. The existence of a sharp transition with birefringent properties indicates a lattice structural change at 60 K because birefringence is the result of a lack of a center of symmetry. They also observed structural non-homogeneity of the specimen in the birefringent phase. On this basis they proposed the existence of microdomains. It seeras plausible that if stable and metastable microdomains are present in the birefringent phase then a measureraent of the paramagnetic resonance spectra would produce the two anomalous 8 13 splittings seen by Bleany. The results of Bij1 indicate that the thermal conductivity of potassium chrome alum at helium temperatures depends on the cooling rate at higher temperatures.

Guillien ' * measured the dielectric coefficients for several alums. For most of these alums he found peaks in the neighborhood of 90 K in both the real and imaginary parts, K' and K" respectively. Guillien interpreted these results to be due to Debye type 2 relaxation raechanisms. Marshall, Pederson and Bailey have measured the dielectric coefficients and ultrasonic attenuation of aluminum ammoniura alum and aluminum potassium alum. They found strong peaks in both the ultrasonic attenuation and the coraplex dielectric coefficients at approxiraately the sarae teraperature. They regarded the water raolecules of these aluras as polar raolecules rotating in a mediura of dominating friction similar to the Debye type relaxation mentioned above.

The dielectric coefficients of araraoniura and potassiura chrome 18 alums have been measured by 0. Wohofsky. He found araraoniura chrorae alura to possess a sharp maximum at 83 K. On reheating a thermal hysteresis approximately 10 K wide occurred. Using mixed crystals of amraoniura and potassiura chrorae aluras he found that the temperature dependence quickly disappeared with increasing potassiura content.

For pure potassium chrome alura the dielectric coefficients were found to be alraost temperature independent down to 25 K. The raarked temperature dependence of the dielectric coefficients of amraoniura chrorae alum is considered by Wohofsky to be the result of orientation pol- arization.

The ultrasonic attenuation of ordinary potassium chrome alum has 3 been studied by Norwood. He observed some very large attenuation 8 peaks at low temperatures. Norwood considered the possibility of

.this alum becoming ferroelpctric at low temperatures. However he had no dielectric coefficient measurements to substantiate his con-

clusion.

This work was begun to either confirra or disprove the previously

existing explanations of the large ultrasonic attenuation in ordinary

potassium chrome alum and several other non deuterated alums. The

results of this work indicate that the large ultrasonic attenuation

is not due to either a ferroelectric phase transition or to a Debye

type of relaxation. The results do indicate however that the crystal

is unsuccessfully attempting to make a ferroelectric phase transition

resulting in the ions slightly changing their equilibrium positions.

This would set up a form of space grating within the crystal from

which phonons could be scattered. The large ultrasonic attenuation

could then be due to phonon-phonon scattering due to this space

grating. CHAPTER II

GROWTH OF SINGLE CRYSTALS

Supersaturated Solution

Deuterated potassium chrorae alura single crystals, KCr(SO,)-*120^0, were grown frora a supersaturated solution. In order to prepare a

supersaturated solution of poatssiura chrorae alum in heavy water a

Bench Scale constant temperature bath Model SB-2 was utilized. Sraall

crystals of potassium chrome alum were dissolved in heavy water until

the solution was saturated at room teraperature. An extra araount of

potassiura chrorae alura was added until several hundred graras of the

solute raaterial rested on the bottora of the bottle. This bottle of

solution was then placed in the constant teraperature bath and the

bath regulated at 29 C. In mixing a solution of potassium chrome

alum the teraperature should not be allowed to exceed 30 C. Above

30 C there is a phase change in the solution involving the waters of 19 hydration. This phase change is indicated by a change in color

of the solution frora purple to green. If crystals are precipitated

from such a solution they are partially dehydrated. The bath re-

mained at 29 C for several days. The contents of the bottle were

stirred several times each day in order to thoroughly saturate the

solution at this temperature. To supersaturate the solution the

temperature was lowered to 25 C. The arabient temperature of the

room was less than 25 C so that heat flow was always maintained out

of the bath. This was necessary because the bath has a heating ele- 10 ment but no cooling device, therefore it cannot regulate at a teraperature lower than its environment.

Seed Crystal Growth

A portion of the supersaturated solution was filtered and placed in a watch glass. This watch glass was then covered with another identical watch glass and a small opening was left between the glasses for evaporation. The watch glasses were placed in a reasonably dust free area at roora teraperature. After several days of evaporation this solution precipitated nuraerous seed crystals. The crystals were allowed to grow undisturbed until the eight faces of the sraall seeds could be easily examined with the aid of a magnifying glass.

At this tirae thirteen of the better forraed seeds were chosen for growth. These thirteen seeds were placed in the watch glass and carefully covered with the filtered supersaturated solution and the covering glass was then replaced. In order to preserve the symmetry of the seed crystals they were inspected and turned to rest on a new face every four to six hours. Fresh supersaturated solution was added when needed. Any seeds which precipitated in the watch glass were removed. Any seeds which precipitated onto the growing seeds were carefully reraoved with tissue paper. The seeds were handled gen- tly with tweezers and never touched with the hands. The seeds were cared for in this way for about three weeks until they were coraparable in size to an English pea. At this time the seed crystals were ready for growth into large single crystals. 11

Crystal Growth

To prepare the solution for crystal groxi/th the constant tempcr- ature bath was placed inside an ordinary household refrigerator to furnish a chilled atmosphere that allowed the bath to be operated at 20 C. The constant temperature bath with its jar of reserve supersaturated solution was allowed to stabilize at this new temperature. The solution which had been saturated at 29 C was then without question highly supersaturated at 20 C.

A six-inch battery jar was placed in the constant temperature bath; it contained enough filtered supersatured solution to cover the crystals. Seven of the seed crystals were chosen for crystal growth. These seven were placed on a plexiglass pallet that had been machined to fit inside the battery jar. The pallet was fitted with a handle that extended above the level of the supersaturated solution.

The pallet containing the crystals for growth was lowered into the supersaturated solution and the battery jar was covered with a watch glass.

During the first week of growth the crystals were inspected and turned every four to six hours. As the crystals grew larger they were inspected and turned less frequently. After about two months the crystals required care only twice daily. At the end of three months growth the crystals had a mass of 20 to 25 graras and were practically free of flaws that could be discerned by visual inspec- tion. Crystal growth was then terminated. The crystals were placed in a sealed jar and refrigerated for storage. Potassium chrorae alum will lose its water of hydration if allowed to remain in the 12 atmosphere at room temperature. Care was taken at all tiraes to avoid the loss of the heavy water frora the new deuterated chrome potassium alum single crystals.

The crystals of KCr(SO,) -12^ 0 were in the shape of the octahedron shown in Figure 2, The points of this octahedron are equivalent to a [100] direction. The faces of the crystal are perpendic- ular to a [111] direction. The [110] direction is into an edge of

the octahedron as shown. 13

Figure 2. Directions of the Crystallographic Axes. CHAPTER III

EXPERIMENTAL PROCEDURES AND EQUIPMENT

The experimental procedures can be placed into three categories; ultrasonic attenuation, dielectric coefficients, and specific heat.

Each category will be described in turn.

Electronics for Ultrasonic Measurements

The attenuation of ultrasound was measured by utilizing the 20 single-ended pulse-echo technique. Data were taken along the three major crystallographic axes; the [100], [110] and [111] directions.

The electronic equipment, as shown in Figure 3, consisted of an

Arenberg matched impedance systera, and a Tektronix 547 oscilloscope.

An R. F. electrical pulse of the appropriate frequency of 3-5 microseconds in duration was produced by the pulsed oscillator at a repetition rate variable frora 60Hz up to raore than 2,500Hz. This pulse propagated through a coaxial cable which had a characteristic impedance of 93 ohms, matching that of the Arenberg system, to a coaxial "Tee." A portion of the pulse energy traveled through the

"Tee" to the prearaplifier, wide band amplifier, and through the attenuator to be displayed on the oscilloscope. The oscilloscope was triggered by a separate signal sent directly from the pulsed oscillator to the oscilloscope. The remainder of the energy in the pulse was diverted by the "Tee" through a 93-ohm thin walled stainless steel coaxial transmission line leading to a quartz transducer designed to convert the electrical oscillations into mechanical vi-

14 15

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o •H C O w cO U

w u o m æ

4-1 w •f^ nJ H co (U

cd 60 RJ •H Q

O O iH PQ

u p 60 •H 16 brations. The quartz transducer was mechanically connected to the

crystal using a binder material that forraed a rigid bond between the

crystal and transducer. This bonding coupled the raechanical vibra-

tions of the quartz transducer to the crystal. These raechanical vibrations, or ultrasound, propagated through the crystal. Because

of the large raisraatchi n acoustic impedance the ultrasonic pulse was

reflected at the opposite face of the crystal. This pulse returned

to the transducer in the form of an echo. The transducer served as

a receiver for the returning echo, reconverting a small portion of

the acoustic pulse into an electrical pulse. This pulse, along with

others produced by successive echoes, which continued to reflect back

and forth through the crystal, were araplified and displayed on the

oscilloscope along with the initial pulse. Each successive echo had

less araplitude by an araount that represented the energy loss of a

round trip of the wave through the crystal.

Crystal Preparation for Ultrasonics Measureraents

The crystals were cut to the desired orientation using a Gillings-

Haraco thin sectioning raachine. They were polished by hand using a

fine sand paper at first and then //600 grit aluminum oxide mixed in

ethylene glycol. For good exponential decay of the echoes the end

faces of the crystal must be parallel. The end faces were polished

parallel to within 0.001 inch. Samples were prepared for the [100],

[110], and [111] directions.

Ultrasonic Cryostat and Dewars

Figure 4 shows the ultrasonic cryostat along with inner and 17

Outer Dewar

1" S. S. Tube'

Center Conductor-

Glass Tube

Crystal Holder

Figure 4. Cryostat and Dewars for Ultrasonic Measurements. 18

outer dewars used to raaintain the liquid nitrogen and liquid heliura baths. The cryostat itself was constructed of stainless steel tubing

of 3/8 inch norainal outside diaraeter and wall thickness of 10 mils.

These tubes were held in place by brass plates machined and soldered

to the tubes. Figure 4 also shows a glass tube running frora above

the cryostat to the region of the crystal. This tube was bent at a

right angle and tapered in a raanner sirailar to an eyedropper. This

tube could be raised or lowered and rotated to position the dropper

portion directly over the crystal. Its function was to allow the

two low temperature binders, natural gas and E.P.A., to be applied

to the crystal.

An R.F. transmission line was constructed of 1/4 inch diameter

thin walled stainless steel tubing. It consisted of a length of

brazing rod placed coaxial with the tube and fitted on each end with

BNC connectors. Both ends were held in place by Stycast 2850 GT

epoxy. The cryostat raounting plate was fitted with "0"-ring seals

at both the transmission line and at the glass binder tube. An "0"-

ring was also fitted between the raounting plate and the inner dewar.

The purpose of these seals was not for vacuura but to allow a slight 4 overpressure of dry He gas to be maintained during cooling. The

overpressure prevented moisture-laden air from seeping into the

inner dewar and condensing water vapor on the crystal that would in

turn freeze and prevent the application of good binders at low tem-

peratures.

Both inner and outer dewars were made of double-walled glass

with silvering between the walls to reduce radiation into the cryo- 19 genic liquids and also into the crystal area. Both dewars had narrow unsilvered strips for the purpose of viewinc the crvstal. The inner dewar was fitted by the manufacturer with a stopcock so that its vacuura jacket could be evacuated.

Before each data run this jacket was thoroughly flushed V7ith air at roora teraperature to remove helium that might have diffused through the walls during a previous run and then evacuated with a forepurap.

Teraperature Measureraent for Ultrasonics

The cryostat was equipped with a copper-constantan therraocouple.

The low teraperature junction of the thermocouple was placed in the crystal holder in direct contact with the crystal. The reference end of the thermocouple was raaintained at the tem.perature of a water-ice bath. The voltage between reference and low teraperature junctions was monitored by a Leeds and Northrup six-dial potentiometer facility,

During data runs above 80 K the crystal was cooled at a rate not exceeding 10 degrees per hour to prevent thermal shock to the crystal. Below 80 K where liquid helium was used as the cooling agent the cooling rate was somewhat faster. Cooling the crystal from 80 K to 4.2 K required a little more than an hour. Due to a negligible thermal expansion coefficient at these low teraperatures no fractures of the crystal occurred when cooled at this faster rate.

Ultrasonic Transducers

The quartz wafer transducers used in the raeasureraent of the ultrasonic attenuation were purchased from the Vaipey Corporation. 20

They were 5/8 inch in diameter and polished by Valpey to a thickness suitable for resonance at 10 MHz and its odd harmonics of 30 MHz, and 50 MHz. The transducers were cut to vibrate in a longitudinai mode, often called "x-cut."

/in alternating electric field must be applied across the quartz wafer for it to be used as a transducer; therefore the wafers were specified to have a thin plating of gold completely covering one side and a thin plating of 1/4 inch in diameter on the opposite side con- centric with a thin 1/16 inch rim around the wafer. The rim was in electrical contact with the fully plated side since the edge of the wafer was also plated. Thus the wafer along with the plating formed a parallel plate capacitor. Figure 4 shows the transducer attached with a spring to a female BNC connector. The spring was soldered directly to the 1/4 inch diameter spot and a #34 copper wire furnished a ground path. The transducer, raounted on the feraale BNC connector, was attached to the transraission line which had a male BNC connector.

The transraission line could then be lowered into place on the crystal after a suitable binder had been applied.

Ultrasonic Binders

The pulse-echo technique requires a rigid bonding between the transducer and the crystal. Due to the differential contraction of the crystal, binder, and transducer no single binder was found to hold over the whoie teraperature range of 300 to 4.2 K. Three binders were used. At higher teraperatures Nonaq stopcock grease was used.

Nonaq stopcock grease becomes rigid at approximately 260 K; however, 21 it is suitable for a binder up to 300 K. It will remain rigidly fixed to the transducer and crystal when cooled slowly down to abont

85 K at which time it fractures. Another binder suitable for lower temperatures is an organic raixture of 5 parts ethyl ether, 6 parts isopentane, and 2 parts ethyl alcohol. It is called E.P.A. This raixture was applied at 170 K as a liquid. It becomes very rigid at approximately 140 K. E.P.A. has a tendency to break at approximately

60 to 80 K, but if great care is used in cooling it will soraetimes remain good to liquid helium teraperatures. Natural gas when dried makes a good binder below 80 K. The natural gas was dried by allowing it to flow from a wall outlet through a copper coil immersed in a bath of dry ice and acetone. From this drying bath the gas was allowed to flow slowly down the long glass tube (Figure 4) into the inner dewar. At about 85 K the natural gas liquifies. It was allowed to drop onto the crystal and form a small pool of liquid which covered the surface. The gas flow was then cut off and the glass tube rotated out of the way of the transmission line and transducer. The transducer was then lowered onto the crystal. It was found that the natural gas binder was most easily applied when liquid helium vapor was being slowly transferred into the dewar. When helium vapor was transferred in the usual manner the temperature of the sample rose to about 120 K due to the vapor flowing in a warm transfer line. Natu- ral gas goes into gaseous state about 120 K. However, the temperature of the vapor drops as the transfer line is chilled. If the flow of helium vapor was well regulated, the natural gas binder could be applied as the crystal cooled between the temperatures of 85 and 80 K. 22

A failure to apply natural gas binder in exactly this way resulted in unsatisfactory bonding.

Collection of Attenuation Data

Due to the large electrical irapedance raisraatch between the transmission line and the transducer only a small fraction of the electrical energy was actualiy converted into acoustic energy. In order to display the echoes it is necessary to either provide an

R.F. pulse of large amplitude or to have very high gain amplifiers.

A large initial pulse can result in saturating the amplifier which cannot recover before arrival of the first echo. Thus, to avoid distortion of the first echo a high gain and low power level was used. The power output of the oscillator and the gain of the amplifier were adjusted to obtain a suitable set of exponentially decaying echoes on the oscilloscope.

A measurement of the attenuation for data purposes was accom- plished by aligning the top of echo N with a grid mark on the oscilloscope then recording the number of decibels attenuation necessary to successively reduce the amplitude of echoes N-1, N-2, ... 1 to the grid mark. If the measured values of attenuation necessary to reduce the amplitude of successive echoes are labeled a ^ « ^,

a then the attenuation a, for ultrasound in the crystal given in decibels/unit length is

^'N-I ^'N-^ 1 1 a = [-Y- + —2" + • • • +^] 2£(N-1) ^^"^^ where l is the length of the crystal. In the high attenuation regions only one echo was visible. It was then necessary to compare the 23 araplitude of the first echo at one teraperature to its amplitude at a lower temperature. Thus if a (T-) - a.(T^) is the difference in the amplitude of the echo in going from T to T , the value of the attenuation in decibels/unit length at the teraperature T is given by

a (T ) - a (T ) a(Tp + 2í ' ^^"^^

The two equations (3-1) and (3-2) provide a precise operational definition for the data that were taken.

Cryostat for Measurement of the Dielectric Coefficients

Figure 5 shows the apparatus used for raeasureraento f the dielectric coefficients. The cryostat was constructed of plexiglass. It consisted of a mounting plate to which a plexiglass mount was attached at the end of two plexiglass tubes. The crystal holder itself was a machined variable gap parallel plate capacitor. The two sides of the capacitor were spring loaded to hold the crystal in place. Cop- per leads were attached to these plates. The leads were #16 gauge copper. The large size wire was chosen for rigidity. Finally, an arm containing the leads was attached to the mounting plate. The purpose of this arm was to ensure that the leads were rigid and parallel. Thus the stray capacitance of the leads remained constant.

The cryostat could be placed in the dewar in the same way each time the experiment was performed. This preserved calibration of the stray capacitance for which correction was made during data reduc- tion. 24

X Lead Arm Mounting Plate Z_

L Lead Wires "W Inner Dewar 7 Ring ^"0" Ri v////////^ Collar 'Plexiglass Tube Outer Dewar

/ l

Thermocouple—

kWWWVH KWWWVVWN

Figure 5. Cryostat and Dewars for Dielectric Coefficient Measurements, 25

Electronics for the Dielectric Constants

The parallel resistance, R , and the paralle capacitance, C , of the systera shown in Figure 5 was raeasured by using a Boonton Radio

Corporation RX meter type 250-A. The RX meter is a wide frequency range irapedance raeter designed to perrait accurate individual deterrai- nation of the equivalent parallel resistance and parallel reactance of two terrainal networks. In a circuit such as the one used where

the inductance is very small, both R and C can be read directly P P from the RX raeter.

Teraperature Measureraent for the Dielectric Constants

The cryostat was supplied with a copper-constantan thermocouple.

The measuring junction was in direct contact with the crystal. The

reference iunction was placed in a water-ice bath. The voltage be-

tween reference and measuring junctions was measured with a Leeds

and Northrup K-3 potentiometer. The crystal was cooled at a rate

of about 10 K per hour.

Dielectric Saraple Preparation

Crystal saraples were cut using a Gillings-Hamco thin sectioning machine to a thickness of approximately 2 millimeters. They were

further polished by hand using #600 grit aluminum oxide mixed in ethylene glycol until the thickness of the samples was about 1 milli- meter. Care was taken to have the two faces of the sample parallel

to within 0.01 railliraeter. Saraples were prepared for the [100], [110], and [111] directions. These samples were much more delicate than those prepared for ultrasonic attenuation studies and were therefore handled very carefully. 26 Collection of Dielectric Data

To obtain data a prepared saraple was placed between the plates of the capacitor shown in Figure 5. After the lov; temperature junction of the thermocouple was placed in direct contact with the sample the cryostat was then fastened into place. The RX meter was used to measure Cp and Rp as a function of temperatur^ e from 300 to 20 K. Liquid nitrogen was used in the outer dewar as an initial coolant down to about 80 K. Liquid helium vapor was then transferred very slowly into the inner dewar. Data were taken for both cooling and warming.

It was observed that upon warming any frost that collected on the transfer line and portions of the mounting plate would melt. If this water was allowed to run into the area of the leads a variable stray capacitance resulted. This problem was avoided by covering the mounting plate and a portion of the lead arm with a thick covering of spun glass. The spun glass was then enclosed in a plastic sheeting and held in place with masking tape. Thus the leads were well insu- lated and water vapor in the atmosphere could not reach the sensi- tive area. Furthermore, no water could run in through the plastic covering.

The measured values of C and R are associated with the dielec- P P tric coefficients in the following way. The coraplex dielectric coefficient is defined as K = K' - iK" where K' and K" are the real and imaginary parts. For a parallel plate capacitor with capacitance C

K' = -^ C " = -^ -4-, (3-3) e A e A wR o o p where d is the distance bet\ceen plates, A is the area of each plate, 27

R is the parallel resistance, e is the permittivity of space and p o lú is the frequency.

The value of C was obtained by correcting raeasured values of

C for stray capacitance. When C and R are substituted into Equa- P P

tions (3-3) they yield K' and K". Data were taken at 10 MHz, 30MHz,

and 50MHz.

Continuous Cooling Method for Measuring the Specific Heat 21 22 The continuous cooling raethod ' was used for obtaining specific

heat data. These data were taken by establishing the heat leak rate

as a function of temperature and measuring the crystal temperature

as the crystal cooled in an isothermal environment. The specific

heat was determined by writing the usual thermodynaraic expression

for specific heat in terras of Q, the heat leak rate, and T, the time

rate of change of teraperature,

p ra i p

With the crystal in contact with the bottora of the can, shown

in Figure 6, heat was delivered to the crystal at a raeasured rate

until the teraperature was found to be constant in tirae. This was

repeated for several temperatures. The rate of heat leaving the

crystal was thereby established as a function of the temperature.

The crystal was then heated to a stable teraperature and the

heater turned off. As the crystal cooled, data were taken from which

the time rate of change of temperature was determined. The specific

heat was then calculated from Equation (3-4) after standard correc- 28 23 tions were made for a small amount of glyptal that was used to coat the crystal and to thermalTy link the manganin wire heater to the crystal. The correction for the glyptal was less than 1 per cent of the value of the specific heat.

One variation in normal continuous cooling procedures was used.

Normally a vacuum is puraped on the can to help ensure that the sur-

roundings for the crystal are adiabatic. This procedure cannot be

followed for KCr^SO.^^'l^D 0 because the water of hydration will be

reraoved in the pumping process. An attempt was raade to seal one

crystal with a heavy coating of glyptal. When this glyptal-coated

crystal was cooled the coating fractured severely and the water was

reraoved by the pump. An alternative procedure was developed. The

continuous cooling method requires that the heat sink into which the

crystal gives up its energy remain at a constant temperature. To

accoraplish this, helium gas was placed in the can. Helium gas was

passed through copper coils which were submerged in liquid nitrogen

to trap condensable gasses. This gas was introduced into the can

at atmospheric pressure. Then the temperature was elevated and note

taken of teraperature stability. The temperature was found to fluc-

tuate and the fluctuations were taken as an indication of fairly

large thermal disturbances in the helium atraosphere. The pressure

was increased and thermal stability checked at one-half pound per

square inch intervals. The stability was found to be very good at

pressures above two pounds per square inch. During the specific heat

runs the pressure on the helium gas was regulated at three pounds

per square inch. This procedure ensured that the can and the helium 29 atmosphere were in good thermal equilibrium.

Electronics for Specific Heat

Figure 6 shows a block diagram of the specific heat apparatus.

The two Hewlett Packard electronic tiraers were operated in the manual gate raode and were triggered by hand at the appropriate teraperatures

to obtain data used to calculate T. A Leeds and Northrup potentioraeter

facility was used to raoniter the voltage between a water-ice reference

junction and the measuring junction of a copper-constantan therrao-

couple. The raeasuring junction was attached directly to the crystal,

The crystal heater power supply contained a 50-ohra standard resistor

in the current circuit across which voltage raeasurementswer e taken

by using a Hewlett Packard 3440A digital voltmeter for the purpose

of determining the values of the current. The digital voltmeter was

also used to measure the voltage drop across the heater coil. The

power delivered to the crystal was calculated from the voltage drop

across the coil and the current values. 30

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pá H O O O Píî O W o o CHAPTER IV

RESULTS AND CONCLUSIONS

Ultrasonic Attenuation Results

Figure 7 shows photographs of the echo pattern for lOMHz waves

propagated in the [110] direction. This series of photographs shows

the decrease in the amplitude for teraperatures approaching the region

of raaxirauraattenuation . In the final photograph for which full ara-

plification was used it is apparent that all echoes have dropped in

amplitude to the extent that nothing but the initial pulse remained

above the base line. In most cases measureraents were made beginning

at 300 K with five to ten echoes. Below 80 K where either natural

gas or E.P.A. was used as a binder four to six echoes were normally

obtained.

Limitations on the electronic equipment allowed a maxiraura of

approxiraately 40 db/echo to be observed. Since the round-trip dis-

tance in the crystals varied from crystal to crystal the maxiraura

obtainable value of attenuation varied frora approxiraately 12 db/cra

in the longer crystals to approximately 30 db/cm in the shorter

crystals.

The nuraber of flaws and dislocations and the other properties

which make up what is sometiraes called the physical purity of a given

single crystal determine the background attenuation for that partic-

ular crystal. This background attenuation will be affected by the

way in which the crystal is grown and the number of times it has

31 32 /

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60 C •H ^ O co co • • c V4 , ' ! 1 ( -•-v^H-r 1 vO CM ' CM ' 1 • i • (U 1 < U ! ' 1 : 60 •H 1X4 33 been cycled to low temperatures. The values of the absolute attenuation will of course be affected by the background attenuation. 3 2

However from the work of Korwood and Marshall it is known that the temperature dependence is not affected by the value of the background attenuation. For these reasons the attenuation results presented in this work should not be considered priraarily as accurate values of the absolute attenuation, but rather as fairly accurate determi- nations of the dependence of the attenuation on teraperature. The attenuation results are shown in Figures 8 through 14. A sraall teraperature hysteresis which occurred in these results can be seen in several of the figures.

The very large peaks in the ultrasonic attenuation are phenomenal.

During two of the data runs after the echoes had become much too small for reliable measurements the attenuation was estiraated from one very small echo to obtain values above 40 db/cra before it corapletely sank into the base line.

Other notable features of the attenuation are the peak near the freezing point of water and a very narrow peak near 60 K visible in several of the figures. A discussion of these features will follow in this chapter.

Complex Dielectric Coefficient Results

The results of the dielectric coefficients K' and K" are found in

Figures 15 through 23. It is apparent that no anomalies are present in the dielectric coefficients. In fact very little teraperature dependence was found and no hysteresis could be seen when the samples were 34

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Table 1

Values of a in the [100] Direction at 10 MHz

T(K) a(db/cra) T(K) a(db/cra)

265.0 8.2 139.0 10.3 261.0 8.4 137.5 11.9 255.1 8.4 135.0 12.7 248.2 8.5 133.0 16.4 238.2 7.0 132.0 19.1 236.0 6.4 130.0 24.2 235.0 6.0 77.0 18.5 234.0 5.5 76.0 15.5 226.0 5.3 74.4 14.3 222.0 5.9 71.0 13.0 212.0 5.1 66.0 12.4 200.0 4.6 58.0 11.4 197.0 4.9 55.0 9.3 191.0 5.9 46.0 7.0 181.0 5.9 33.0 5.2 166.0 6.1 28.0 5.5 160.0 6.1 22.0 5.0 154.0 6.7 21.0 5.8 148.0 7.5 19.0 6.8 142.0 8.2 52

Table 2

Values of a in the [100] Direction at 30 MHz

T(K) a(db/cm) T(K) a(db/cra)

276.0 7.4 190.0 8.2 274.0 6.1 184.0 10.2 250.0 4.4 178.0 12.3 243.0 4.8 174.0 15.2 238.0 4.8 172.0 18.5 232.0 4.8 168.0 21.1 204.0 6.6 166.0 23.3 200.0 6.8 165.0 25.3 196.0 7.2 53

Table 3

Values of a in the [110] Direction at 10 MHz

T(K) a(db/cra) T(K) a(db/cm)

285.0 5.6 143.0 12.5 280.0 6.0 141.0 14.9 278.0 6.8 139.0 16.3 267.0 8.4 136.0 19.5 262.0 7.2 71.3 21.2 259.0 5.5 70.8 20.7 255.0 4.9 70.0 20.1 248.0 5.0 69.6 19-6 244.0 5.0 69.2 19.0 238.0 4.9 68.4 15.6 235.0 4.5 66.0 12.3 222.0 4.9 64.0 10.6 216.0 4.9 62.0 10.6 214.0 4.9 60.0 8.9 200.0 5.0 58.0 8.4 191.0 4.9 54.0 8.4 170.0 5.1 50.0 8.4 165.0 5.5 46.0 8.9 160.0 6.6 42.0 9.5 156.0 7.1 38.0 10.6 150.0 7.9 34.0 10.6 149.0 9.5 32.0 11.17 11

Table 4

Values of a in the [110] Direction at 30 MHz

T(K) a(db/cra) T(K) a(db/cm)

282.0 8.9 172.0 13. ,9 276.0 8.3 166.0 16., 2 269.6 6.7 161.0 20. .6 257.0 7.6 65.0 20. ,0 250.0 6.0 64.2 16.. 2 238.0 6.0 63.8 15, .1 232.0 6.0 63.4 12. .9 224.0 6.7 62.0 10.. 1 213.0 6.9 60.0 7,. 8 204.0 7.9 58.0 7,. 3 186.0 8.9 54.0 7,. 3 177.0 11.1 50.0 8,. 9 175.0 12.2 47.0 9 .2 55

Table 5

Values of a in the [110] Direction at 50 MHz

T(K) a(db/cra) T(K) a(db/cra)

280.0 12.2 182.2 24.0 272.0 12.9 182.0 24.6 268.0 13.4 63.8 19.0 263.0 12.8 63.0 17.9 256.0 12.2 62.0 14.0 248.0 11.7 61.0 14.6 244.0 11.2 60.0 15.4 218.0 11.8 56.0 12.8 210.0 11.4 54.2 11.2 205.0 13.4 50.0 10.6 198.0 15.1 48.0 9.0 197.0 16.2 44.0 8.4 192.0' 17.2 42.0 8.2 191.0 18.4 40.0 7.8 188.4 19.4 38.0 7.2 187.0 20.6 36.0 7.2 186.0 21.3 34.0 7.5 185.0 22.3 32.0 8.3 56

Table 6

Values of a in the [111] Direction at 10 KHz

T(K) a(db/cra) T(K) a(db/c:a)

289. ,2 2.3 128.0 12.4 280. ,0 2.3 126.6 14.0 270. ,0 2.2 89.0 10.8 260. .0 2.2 87.6 9-4 250. ,0 2.3 86.0 8.6 240. .0 2.2 85.0 8.2 230. .0 2.1 84.0 7.3 220. .0 2.2 83.0 6.9 210. .0 2.2 82.0 6.2 200. .0 2.3 74.0 5.7 190. .0 1.9 73.0 5.5 180. .0 2.3 69.0 5.5 175. .0 2.2 66.0 5.2 165. .0 2.6 65.0 4.8 155. .0 3.1 60.0 3.6 150. .0 3.4 54.0 3.0 145. .0 4.2 52.0 2.6 140. .0 4.6 48.0 2.4 135. .0 6.4 46.0 2.6 133. .0 8.0 42.0 2.9 130. .0 10.2 40.0 3.1 57 Table 7

Values of a in the [111] Direction at 30 MHÎ

T(K) a(db/cra) T(K) a(db/cm)

288.8 2.1 199.0 2.3 284.4 2.0 195.0 2.5 279.0 2.0 188.0 3.1 275.0 2.1 195.0 3.4 270.0 2.1 190.0 3.4 264.4 2.0 185.0 3.4 260.0 2.2 180.0 4.6 255.0 2.2 174.0 5.5 249.4 2.4 170.0 5.6 245.0 2.6 163.4 7.4 240.0 2.4 161.6 8.2 235.0 1.8 159.8 8.9 229.0 1.6 158.9 9.2 226.2 1.6 158.0 9.7 225.0 1.6 156.8 10.4 218.8 1.5 155.7 10.9 214.6 1.6 155.0 11.2 210.0 1.9 154,0 11.8 205.0 1.93 153.4 12.4 58

Table 8

Values of K' and K" in the [100] Direction as a Function of Teraperature

K' K" T(K) 10 MHz 30 MHz 50 MHz 10 MHz 30 MHz 50 MHz

280 6.50 6.50 6.50 0.25 0.25 1.85 260 6.56 6.50 6.50 0.25 0.25 1.85 240 6.56 6.49 6.45 0.25 0.25 1.80 220 6.57 6.37 6.39 0.25 0.25 1.80 200 6.57 6.35 6.33 0.25 0.25 1.80 180 6.50 6.35 6.33 0.25 0.25 1.80 160 6.45 6.35 6.36 0.25 0.25 1.80 140 6.46 6.26 6.30 0.25 0.25 1.80 120 6.45 6.25 6.30 0.25 0.25 1.80 100 c. /, o 6.20 6.25 0.25 0.25 1.80 80 6.41 6.20 6.25 0.25 0.25 1.80 60 6.49 6.37 6.35 0.25 0.25 1.80 40 6.46 6.41 6.45 0.25 0.25 1.80 20 6.52 6.40 6.45 0.25 0.25 1.80 59

Table 9

Values of K' and K" in the [110] Direction as a Function of Temperature

K' K"

T(K) 10 MHz 30 >fflz 50 MHz 10 MHz 30 MHz 50 i.inz

280 6.49 6.50 6.50 0.25 0.25 1.50 260 6.49 6.50 6.50 0.25 0.25 1.50 240 6.49 6.50 6.50 0.25 0.25 1.50 220 6.50 6.42 6.40 0.25 0.25 1.50 200 6.50 6.37 6.37 0.25 0.25 1.50 180 6.53 6.35 6.32 0.25 0.25 1.50 160 6.57 6.30 6.28 0.25 0.25 1.50 140 6.60 6.25 6.25 0.25 0.25 1.50 120 6.65 6.25 6.25 0.25 0.25 1.50 100 6.62 6.25 6.20 0.25 0.25 1.50 80 6.68 6.22 6.25 0.25 0.25 1.50 60 6.75 6.40 6.30 0.25 0.25 1.50 40 6.83 6.43 6.42 0.25 0.25 1.50 20 6.87 6.46 6.45 0.25 0.25 1.50

m-.# 60

Table 10

Values of K' and K" in the [111] Direction as a Function of Teraperature

K' K" T(K) 10 MHz 30 MHz 50 MHz 10 mz 30 MHz 50 MHz

280 6.50 6.35 6.50 0.25 0.25 2.00 260 6.27 6.10 6.35 0.25 0.25 1.90 240 6.15 6.00 6.25 0.25 0.25 1.90 220 5.96 5.90 6.15 0.25 0.25 1.90 200 5.91 5.75 6.06 0.25 0.25 1.80 180 5.82 5.60 6.00 0.25 0.25 1.80 160 5.80 5.50 5.85 0.25 0.25 1.80 140 5.75 5.45 5.50 0.25 0.25 1.77 120 5.75 5.35 5.25 0.25 0.25 1.72 100 5.75 5.25 5.21 0.25 0.25 1.72 80 5.75 5.15 5.20 0.25 0.25 1.72 60 5.75 5.25 5.36 0.25 0.25 1.72 40 5.80 5.25 5.44 0.25 0.25 1.72 20 5.90 5.35 5.60 0.25 0.25 1.72 61

Table 11

Specific Heat Values as a Function of Teraperature

T(K) C (cal./raole K) P

80.6 135.1 81.0 135.0 82.0 145.0 83.0 150.1 84.0 154.2 85.0 160.1 86.0 160.2 87.0 162.0 88.0 164.8 89.0 168.3 90.0 175.0 91.0 177.8 92.0 181.0 93.0 185.1 94.0 193.6 95.0 200.4 96.0 211.6 97.0 218.8 98.0 229.6 99.0 240.0 100.0 255.0 62 18 warraed. These results are in agreement with the findings of Wohofsky for the [111] direction, using an undeuterated saraple. Note should be taken of the fact that K" v/hich is proportional to the dieiectric

energy loss is a constant independent of the temperature in each case.

Specific Heat Results

The raeasured values of specific heat for the six data runs, all

on the same crystal, are shown in Figure 24. The purpose of this

measurement was to determine if an anomaly would be found in the

teraperature neighborhood of the peak found in the ultrasonic attenu-

ation. No anoraaly was observed.

Conclusion 3

In coraparison to the ultrasonic attenuation results of Nor\>7ood

for undeuterated potassiura chrorae alura the present work shows that

deuteration has no effect on the ultrasonic attenuation. A corapari- 18 son of this work to Wohofsky's results shows that deuteration does not affect the dielectric coefficients.

The narrow absorption peak in the ultrasonic attenuation at

approximately 60 K seeras to be correlated with the optical birefring-

ence at that teraperature. There raust be a structural change involving

at least the water molecules. This change is more than just a rota- 8 9 tion of the water molecules since the microwave results ' indicate

that the water octahedron is distorted and that there are two types

of field environraent for the chroraium ions. This fact along with the 12 optical data of Kraus and Nutting suggests that structural doraains

exist below 60 K. Neither the narrow peak at 60 K nor the rainir..u:a ce of doraains, rouna J.UI. utia.^ alums it is probable that this rainimum is produced by the change in domain scattering as the alura raakes a gradual transition when cooled below 60 K. The suggestion of a gradual transition is further sup- ported by the gradual change in the optical absorption observed by 12 Kraus and Nutting.

The attenuation peaks in the neighborhood of 270 K are due to the water molecules. The water molecules are loosely coupled to the 18 lattice. There is a probability of microscopic pockets of liquid existing in the lattice. These pockets of water would go through a phase transition as the temperature passes through the freezing point of water. If, however, the pockets of water do not exist in the liquid state then the presence of the attenuation peak near the freezing point of water suggests strongly that the water molecules even though they are bound in a solid are so loosely bound that they are attempting to accomplish the familiar liquid-solid phase transition for water.

The large ultrasonic attenuation in the temperature range of 60 to 160 K along with anomalies in the dielectric constants in the same 2 temperature range have been observed in alurainura-araraoniura and alum- 3 inum-potassiura aluras. Norwood observed sirailar ultrasonic attenuation peaks in undeuterated potassiura chrorae alum. Two explanations have previously been put forth to explain this large attenuation.

The first is that the alum goes through a ferroelectric phase transition. The second is that ultrasonic energy is absorbed through a 65

Debye-type relaxation process.

Figure 25 shows the types of cur^^es eicpected in the ultrasonic

attenuation, the dielectric coefficients, and the specific heat for

a ferroelectric phase transition.

"3 " "2 ' "l ' a

JV "

(ã) Ultrasonic attenuation (b) Dielectric coefficients for 0) , (jú^ and a)_. for lú , 0)« and (JO_.

Jigure 25. Curve shapes expected for Ca) a, (h) K' and K", and (c) C for a ferroelectric phase transition. P 66

Figure 26 shows the types of curves expected for a Debye-type of

relaxation of water molecules in hindered rotation.

ÍOj > U)^ > co^ a

(a) Ultrasonic attenuation (b) Dielectric coefficients for b)^ , b)r. and a)„. for (j)^ , u)j and w_.

Figure 26. Curve shapes expected for (a) a, (b) K' and K", and (c) C for a Debye-type of relaxation. P II i

If the results of this work are examined in comparison to the

expected results for a Debye-type relaxatlonj it is seen in^'i^eHi atelv

that they do not meet the requirements for the Debye-type relaxation

process. This theory, first worked out by Debye for polar liquids,

requires a peak in both the real and imaginary parts of the dielec-

tric coefficient. Indeed several alums exhibit this behavior showing

a peak with a tail on the high temperature side characteristic of

the Debye theory. However, neither ordinary nor deuterated potassium

chrome alum have shown anoraalies in the dielectric coefficients.

The question of whether or not potassiura chrorae alum makes a

ferroelectric phase transition certainly seemed to be an open one

until the dielectric coefficient and specific heat results of the 18 present work and the dielectric coefficient results of Wohofsky's

work on ordinary potassium chrome alum became available. Ferroelec-

tric phase transitions fall into two categories. They are the order-

disorder type of transition and a transition of the displacive type.

The dielectric constant and the specific heat are particularly sensi-

tive to these types of phase transitions. The present results show

no evidence of a complete ferroelectric phase transition. Further- 4 more Jona and Shirane have reported that no ferroelectric hysteresis

loops have been found in chrome potassium alum. It is concluded

therefore that this alum does not become ferroelectric. 7 8 Bleany and co-workers ' found that the Stark splitting of the

electron energy levels of the chromiura ion is linear with tempera-

ture down to 160 K. As the teraperature was lowered further the split-

tiag remained the saine but two side peaks gradually appeareJ. This « • i i

result suggests that sorae chromiura ions sit in one electric field

while the reraainder sit in anoLher field. Van Vleck^'"''^ intcrpretcd

that this was accoraplished by a distortion of the water octahedron

shown in Figure 1. This sort of distortion would add to the anharm-

onicity of the energy of the ions, especially the chromium ions and

the ions in the water octahedron. It is easy to visualize the chrom-

ium ion sitting in a predominantly cubic field surrounded by an octa-

hedron of six water molecules. Such an arrangeraent is highly sym-

metric and should produce a very small Stark splitting; this is indeed 8 — 1

the case. Bleany has reported the splitting to be 0.03 cm at

160 K. The restoring force on this ion should be alraost linear with

displacement from its equilibriura position and its energy proportional

to the square of the displaceraent. That is to say the energy should

be very nearly harraonic.

If, however, as Van Vleck has suggested, the water octahedron

undergoes distortion as the teraperature is lowered below 160 K then

the high degree of symmetry would be lowered creating a new potential

well for the chromium ion. This new potential well would be unsym-

metrical and would most likely create a new equilibrium position for

the chromiura ion. Bleany's results suggest that not only one equil-

ibriura position is possible but the new. potential well is of such a

shape as to create two closely lying equilibrium positions each in

a slightly different electric field. The evidence for this is of

course the two side peaks observed by Bleany. The physical dimensions

of these displaced equilibriura positions for both the chroraium ions

and the water molecules is small and cannot be classified as an order- 69 disorder or a displacive type transition required for the crystal to become ferroelectric. On the other hand, the anharmonic contribution to the energy of the water chromium ion system as well as the other ions through nearest neighbor interactions is not necessarily small. It is concluded therefore that while this alum approaches a condition in which it would undergo a phase transition and become ferroelectric below 160 K it does not succeed in making the ferroelectric transition as exhibited by several alums; however, in doing so the anharmonic contribution to the potential energy of raolecular oscillators is enhanced. An expression of the energy of an ion rel- 2 3 ative to its equilibriura position should take the form ax + bx + 4 cx + . . . where terms in the displacement x are called anharmonic when they are cubic or greater.

The existence of anharmonic terms in the energy leads one directly into phonon-phonon interactions. Appendix I shows how such terras produce phonon-phonon interactions.

It is concluded therefore that the ions in the crystal in the process of unsuccessfully attempting to make a ferroelectric phase transition slightly change their equilibrium positions. This would set up a form of space grating within the crystal frora which phonons could be scattered. The large ultrasonic attenuation could then be due to phonon-phonon scattering due to this space grating.

As a sunraiation of the conclusions of this work the following stateraents can be raade. The effects of deuteration seem to have no effect on the ultrasonic attenuation. The peaks near 270 K are associated with the waters of hydralioii making a phase change. The 70

small peak near 60 K is a structual change involving the water raole-

cules and is associated with the optical birefringence at this temperature. The very large ultrasonic attenuation in the temperature

range from 60 to 160 K cannot be explained by a Debye type relaxa-

tion process or a complete ferroelectric phase transition. Arguments have been made for the existence of an anharmonic contribution to

the energy of the oscillators in the crystal. This leads to phonon-

phonon interactions. It has been concluded that a space grating is

set up in the crystal from which phonons could be scattered by pho-

non-phonon processes.

Future Work

The following is a list of work needed in future investigations

of aluras:

1. An experiraent designed to detect the presence of the pro-

posed scattered phonons. This would be accoraplished by propagating

a phonon of given frequency with one transducer and detecting the

scattered phonons using transducers cut for different frequencies.

2. Accurate x-ray structural analysis studies at temperatures

in the 60 to 160 K range to deterraine if this crystal was approach-

ing an order-disorder or a displacive transition.

3. The specific heat in the above teraperature range is planned

for several of the clear alums. Alums which do undergo a phase

transition would show an anoraaly in the specific heat. This could

also be done for raixed crystals. A mixture ranging frora pure ammo-

nium chrome alum to pure potassium chrome alum wouid indicate what 71

role the substitution of a larger ion plays in the ferroelectric

transition.

4. The velocity of ultrasound and the elastic constants is needed in the high attenuation region. The results of such measure-

ments would of course be raost interesting when an alura which does

undergo a ferroelectric transition is diluted with one which does

not.

5. Extension of this work to higher and lower frequencies would

give inforraation about the frequency dependence of the ultrasonic ab-

sorption. The frequency dependence could raost accurately be deter-

rained at lower frequencies where the raaxiraura values of the peaks

could be obtained. Such a frequency dependence is iraportant in de-

termining the contribution to the scattering from domains. This

inforraation would be especially useful below the 60 K transition.

6. The dielectric coefficients of several of the alums are also

planned to be investigated. LIST OF REFERENCES

1. J. Eisenstein, Rev. of Mod. Phys. 2^, 74(1952).

2. B. J. Marshall, D. 0. Pederson, and W. E. Bailey, J. App. Phvs., ^, 2116(1967).

3. M. H. Norwood, Ph. D. thesis, Rice University, Houston, Texas (1960).

4. F. Jona and G. Shirane, Ferroelectric Crystals (Macmillan, NPV: York, 1962), p. 326.

5. H. Lipson and C. Á. Beevers, Proc. Roy. Soc. (London) A148, 664(1935).

6. H. Lipson, Proc. Roy. Soc. (London) A151, 347(1935).

7. B. Bleany and R. P. Penrose, Proc. Phys. Soc. (London) 60, 395(1948).

8. B. Bleany, Proc. Roy. Soc. (London) A204, 203(1950).

9. J. H. Van Vleck, J. Chem. Phys. T^, 61(1939).

10. J. H. Van Vleck, Phys. Rev., 52» ^26(1940).

11. D. L. Kraus and G. C. Nutting, J. Chem. Phys. , 9^, 133(1941).

12. C. Ancenat and Couture, Le Journal de Physique et Le Radium, ^, 47(1960).

13. D. Bijl, Physica, 1_4, 684(1949).

14. R. Guillien, Comptes Rendes, 209» 21(1939).

15. , Comptes Rendes, m., 991(1941).

16. , Comptes Rendes, 2^7» 443(1943).

17. P. Debye, Polar Molecules, Dover Publications, New York.

18. W. 0. Wohofsky, Ber. Bunsengesell. Phys. Chem. (Gerûiany) 7^» 631(1966).

19. M. M. T. Anous, R. S. Bradley and J. Colvin, Jour. Chera. Soc. (London) 3348(1951).

20. C. F. Squire and C. V. Briscoe, Phys. Rev. 106^, 1175(1958).

72 73

21. F. H. Crawford, Heat Thermodynamics and Statistical Physics, (Hartcourt, New York, 1963), p. 169.

22. J. F. Cochran, C. A. Shiffman, and J. E. Neighbors, Review of Scientific Instrmnents, ^1» 499(1966).

23. N. Pearlm.an and P. H. Keesom, Phys. Rev. , ^, 398(1952).

24. R. E. Peierls, Quantum Theory of Solids, (Oxford Press, London, 1965), Ch. II. APPENDIX I

A theory of the physical properties of solids would be practically irapossible if the most stable structure for most solids were not a regular crystalline lattice. The N-body problem, associated with N atoras in the crystal is reduced to raanageableproportion s by the existence of translational syraraetry. This raeans that there exists basic translational vectors, a , a and a such that the atoraic structure remains invariant under translation through any vector which is the sum of integral multiples of these vectors. Furtherraore, we can define the physical arrangement of the whole crystal if we specify the contents of a "unit cell." The unit cell is raade up of the sraal- lest triad of the vectors a^, a- and a such that the crystal appears to be made up of stacks of these unit cells. Thus, by knowing the unit cell we also know the crystalline structure.

Consider a crystal composed of a number N unit cells, each of which is a parallelpiped bounded by the three non-coplanar vectors

ã^, a^ and a„, which are the basic translational vectors of the lattice. If we now choose an origin on some arbitrary atom then we can denote the equilibriura position vector of the nth unit cell by

a = n,a^ + n„a_ + n„a_ n 11 II 3 3 where n-, n and n are integers. If there are "r" atoras within the unit cell then the location of the r atoras in the unit cell are given by the vectors d. where j distinguishes the different atoms in the unit cell and take on the values 0, 1, 2, ..., r - 1. For convenience

74 75 we choose the origin of coordinates in such a way that d (j=0) = 0.

Thus, in general the equilibrium position vector of the jth atom of the nth unit cell is given by

-o -o , -:o r . = a + d,. nj n j

If this atom is slightly displaced from. equilibrium then

r . = a + d. . nj n 3

J^^l^

Jigure 27. Representation of r . nj 76

We can represent the displacement, of the jth atom in the nth

unit cell, from equilibrium by

- - -o u . = r . - r . . nj nj nj

The total potential energy U of the crystal is assuraed to be

sorae function of the instantaneous positions of all atoms. If U is

expanded in a Taylor's series in powers of the atomic displacements

u . we get nj ^

u-u=i/2y yA.,., ,, ° nn' jj' -'^J^ J ^3 n'j'

+ ^^^ l l B .,^...M , , „ „ + 1/36 l nn'n" jj'j" ~^J^ ^ ^ ^ ^J " J ^3 nn'n"n"'

jjijnj.M ^njn'j'n"j"n"'j"' %j \'j' V j" V'j"' '^ " ' '

We have discarded the linear term since all first derivatives

of U must vanish in the equilibrium condition. Here A, B and C are

tensors of second, third and fourth rank respectively, linking the

components of the vectors u which follow.

In the harmonic approxiraation we consider only the terras quad-

ratic in u. Thus, the tensors B and C are discarded. For the

harraonic approxiraations we can write

V - V = 1/2 l l l A ^ u . u_ ,., o ^i . . , ^o ot6 anj Bn'j ' nn jj a,3 ^.^,.. where we are using the a and 3 coraponents of u. Also 2 . _ 931 njn'j^ Í•J , .' , d U a. u u ^,.B , nj n'j' 77 and A „ u^ is just the force exerted in the a-direction on the a3 6 njn'j' n'j' atom at r . when the atom at r ,., is displaced a unit distance in nj n'j' the 3 direction.

It can be shown by setting up the normal modes of vibration that for the harmonic approximation there is no energy transferred from one norraal raode to another. Without energy transfer frora one norraal mode to another there can be no phonon-phonon interactions.

In order to get coupling between the norraal raodes and allow for energy to be scattered by phonon-phonon interactions it is necessary to include the anharmonic terras in the Taylor's expansion of the potential energy. Normally the cubic term is sufficient to account for raeasurable phonon-phonon interactions. The quartic terras give an extremely sraall contribution.

In dealing with phonon-phonon interactions it is custoraary to 24 work with annihilation and creation operators Q and Q* respectively.

When Q operates on a phonon systera Y it reduces the phonons in that systera by one, i.e. QH' = /tl/2Ma) /n H' _^. Also for the creation process Q*H' = /Ti/2Ma) /n + 1 "^^^y

Here M = mass of the crystal consisting of N unit cells, u) is the frequency of the phonons in the particular normal mode of the system and n represents the number of phonons in that particular normal mode.

Now considering phonon-phonon interactions we find that the

anharmonic contributions to the Harailtonian are generally sraall in

comparison to the whole Hamiltonian. We can generally treat thera 78 as perturbations of the harmonic, independent states, giving rise to transitions between them. In other words the anharmonic perturbations couples energy between the so-called normal modss.

We shall have to employ the standard formula for time dependent perturbation theory

-^ = — || ô(e^ - c.)

^^if where —7— is the transition probability for a transition to occur from the initial state |i> of energy e. to a final state |f> of energy e^. The H' represents the part of the Hamiltonian i^hich produces the transition. Thus the H' would be the cubic and the quartic terras in the Taylor's expansion. Since the cubic term will be the dominant one past the quadratic this is the only one that we shall consider.

H' = U = 1/6 l l B , , „ „ u u , , u ., „. c nn'n" ii'i" ^J^^^J ^J ^ ^ ^ ^

If we express u . in terms of normal coordinates q and the propagation nj vector f of the wave we get

u . = ) q-p ^ (t) e V (f,s). nj -^^ ^f,s J

Here s denotes the particular mode of vibration, i.e. longitudinal or transverse. Also v. is the displaceraent of the jth atora in the cell n = 0 at t = 0.

We then write

U = 1/6 l b(f,s;f',s';f",s") q^ q^,^, ^7-....

ss's" 79 where

b(f,a;f'a';f"a") = l l B . ,., „.„ e^"^^ V^ ^n^- '^) nn ,n !i j. J. J, ... ^njn j "n^ j

X V (f,s) V ,(f',s') V „(f",s"). J J J

The annihilation and creation operators are expressed in terras of the

normal coordinates, q, by

and

Q*(^'^^ °21.(f.s) tqf3H-la.(f,s)qg^]. '

24 - - Furthermore it can be shown that q-p = Q(f,s) + Q(f,-s) and Q*(f,s) = Q(-f,s) and a)(f,s) = -a)(f,-s). Thus

U = 1/6 5; l b(f,s;f',s';f",s")[Q(f,s) + Q(f,-s)] ^ ff'f" ss's" :

i [Q(f',s') + Q(f',-s')J[Q(f",s") + Q(f",-s")].

If we denote o = ±1,±2,±3 ...

U = 1/6 5; l b(f,a;f'a';f",a")[Q(f,a)Q(f'a')Q(f"a"). ^ ff'f" oa'o''

The significance of Q(f,a), Q(f',a') and Q(f",a") is obvious since

they represent either annihilation or creation operators depending

upon the signs of the f's and a's. This combination acting upon any

given phonon state has the effect of either increasing or reducing

by unity the numbor of quanta in the m.odes representcd by (f,a). 80

(f',a') and (f",a").

In phonon-phonon collisions it is necessary to conserve both pseudo-momentum and energy. For a three phonou process these two conservation laws take the respective forms for one annihilation and two creations:

(1) f + f' - f" = K and

(2) a)(f,s) + a)(f',s') - a)(f",s") = 0 where K = translation vector in the reciprocal lattice.

We denote our initial state by |i> = [N ,N ,N-...> and the final state by If> = |N',N',N'...> where the N's represent the nuraber of phonons in each raode. Using our general operator we get

1 - ° o - I I s . / Cî O -z «tN'(l^f,|a|)+ (-J-^a,/2,„, Q(T^ f,o) N;,N',N'...> = [ ' ' 2mCt,a) 1 K'

N^...(N(-r|T- f,|a|) - -A-) ...> .

Here we see that if o is negatlve then we have a creation process

Q(-||j f,a) N|,N^...> = Q*(-f,|a|) N|,N^...>

= [ IfN|zL£Í + i]l/2| . Nl. ..[N(-f,o) + !]...>

If the sign of o is positive we get an annihilation process 81

Q(-y~|- f,a)|N|,N^...> = Q(f,a)|N|,N^...>

= [^í^^^^^l^/'lN'N' ..[N(f,a)-1]...>. 2Ma)(f,a)

Only the raodes that are affected by the operators will be changed, i.e. N = N;; N = % but N(-r^ f,|a|) => [N(j^ f,|a|) - -A-]. 1 iz z \o \ \o \ '' \o \ For a one annihilation and two creation process we get

K^7 ..f'o'.f" ,..N|2pN'(f,s)[N'(f's') + l][N'(f"s") + 1] ^lil-'-ir ^o|Dv,i,s,is,r,s.;|L _ "^^ 4M-^ a3(f,s)a)(f's')a)(f"s")

ô(-a)(f,s) + a)(f's') + a)(f"s"))

^ (f's') (f,s) f (f"s")

We see therefore that the transition probability depends on

|b(f,s;f',s';f",s") l^. We recall that these coefficients are related

to the B . ,., „.„ coefficients by njn'j n j

b(f,s;f',s';f",s") = l l B , , „ „ e ^ n nn'n" jj'j" ~^^^ ^ "" ^

X v.(f,s) v.,(f',s') V „(f",s"). J j -'

Also recalling the equation for the cubic (an harraonic) terms

U = 1/6 y y B .,.,„.„ u . u , , u I, „ c ^Ju.A.u njn'j'n j nj n j nj nn n jj j

we can see that the coupling between normal vibrations, the b coef-

ficients, which enter the transition probability in squared form 82 are dependent on the ^nin'i'n"!" ^^^f^icients. These B's are in turn dependent on the lattice bonds. Thus it can be seen that if the lattice changes in such a way as to increase U thcn naturaliv c •' the B coefficients increase in the above expression. This in turn increases the b's which are the coupling between normal raodes.

Therefore it follows that an increase in the coupling between normal modes increases the transition probability,

from a state where (f,s) no longer exists but (f',s') and (f",s") have appeared.

Thus motion of the ions which produce an increase in the anharmonic contribution to the energy set up a spatial grating effect from which phonons can be scattered in phonon-phonon interactions.