THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING OPTICAL OBSERVATION BY TRANSMITTED LIGHT OF THE TETRAGONAL TO CUBIC PHASE TRANSITION IN (Ba,Sr)TiO5 CERAMICS Viola C. Sanvordenker A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Geology and Mineralogy 1963 January, 1964 IP-652

To my parents who, through times of political turmoil, have given me what they value most, an education, ii

ACKNOWLEDGEMENTS The writer wishes to express her thanks to: Professor Reynolds Mo Denning, Chairman of the Doctoral Committee, for his encouragement and counsel throughout the course of this work; Professor Howard Diamond, for many valuable discussions, and for his assistance in the mathematical analysis of the problem; Professors Chester B, Slawson, Paul L. Cloke, and Wilbur C, Bigelow, for their advice; Mr. William Wo Raymond, Mr. Ralph M. Olson, and Miss Constanza Roca, for their technical assistance and helpful discussions; Dr. Keshav So Sanvordenker, her husband, without whose patient support this investigation would not have been completed; Miss Maryon Wells, for her help in typing and editing; the U, So Air Force, Rome Air Development.Center, and Caswell Electronics -Corporation for their financial support during the experimental stages of the work; the Departments of Mineralogy, Chemical and Metallurgical Engineering, and Electrical Engineering for the use of their facilities; the staff of the Industry Program of the College of Engineering, for the final preparation and reproduction of the manuscript, iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTSo.OO.......4.. o o... o o...o.oo.......... io iii LIST OF TABLES... o o o o.............o........................ vii LIST OF FIGURES.............O............. o.......,. viii I INTRODUCTION,....o..................................... 1 II BACKGROUND.........,............................... 5 A. General Structure of Perovskite-type Ferroelectrics.., 5 B. Dielectric Behavior of Single Crystal Barium Titanate and Ceramic Barium Titanate and Barium-Strontium Titanates..,......................................Oo 9 C. Proposed Explanation of the Dielectric Field NonLinearity in Ceramic Barium Titanate and BariumStrontium Titanate.....Q.O.......QOO..OOO............. 12 D. Optical Investigation of (BaSr)TiO3 Ceramic with Variables of Temperature and Electric Field............ 20 III BARIUM TITANATE AND (BaSr) TITANATE MATERIALS,............. 22 A, Structural and Optical Data o.......................... 22 B. Domain Structure and Twinning..o....................... 23 C. Electrical Data,...................6. e............... 26 D. Domain Motion,.......................................... 28 E. Solid Solutions, o...................................... 29 F Theoretical Considerations o.............0...........0.. 31 Go Ceramic Perovskite Ferroelectrics......O...oo.......... 34 Ho Optical Observation of Ceramic Ferroelectrics,..O..,.... 37 IV EXPERIMENTAL PROCEDURE.,,,,.. Q. o.........,........,,.,.. 38 Ao Preparation of SamplesO.....o.ooo.....o...........oo.... 38 B, Measuring Equipmento.o..o.o. o.,..,,...,,o....... 40 1. The Heating Stage.................o............... 40 2, Capacitance Measurements......ooO...O,............ 43 3. Intensity Measurements....4,.,.........,.o.,,.... o 45 V DATA AND OBSERVATIONS... eoeooo oe* o. oo,.,,,,4 o oo. 47 Ao Data, oo..............O. 48 iv

TABLE OF CONTENTS (CONT'D) Page 1o Presentation of Dataa....,o...................... 48 2. Analysis of Curves... o,o,..o................ 62 62 B, Observations......,,...o...,............. 76 1, Short Term Effects,..o..o...o.............. 76 2. Long Term Effects...................~..., 89 VI CALCULATION ON THE LIGHT INTENSITIES TRANSMITTED BY (Ba,Sr) TITANATE CERAMIC THIN SECTIONS..........O........ 92 A. Fraction of Light Transmitted by.a (Ba,Sr)TiO5 Ceramic Thin Section, at Some Temperature T in the Ferroelectric Region.. **..a.,..~..00.,.............. 94 1. Calculation of G,,..................*.....* 96 2. Calculation of the Birefringence of the Equivalent Crystallite,,4,.,.,0.......00.......... 98 3, Calculation of the Relative Intensity for all Wave Lengths from 5200A to 6200A,.................. 100 4, Discussion of the Calculated Relative Transmitted Intensities. o........ o..... o o..............., 101 B. Change with Temperature in the Relative Amount of Light Transmitted by a (Ba,Sr)TiO Thin Section,,............ 105 Co Change with Temperature and Applied DC Field in the Relative Amount of Light Transmitted by a BariumStrontium Titanate Thin Section,...o.....,,.o., o.... o. 113 Do Comparison of the Relative Theoretical and Experimental Light Transmission Curves,,,o,Q,.o e. 115 1, Consideration of Tr, the Peak Incremental Permittivity Temperature of the ceramic at zero Field. 115 2, Consideration of Tr*, the Peak Incremental Permittivity Temperature of a Ceramic Under the Influence of an Electric Field.,.......,,...,,..,o 121 35 Discussion of a,the Variance Parameter of the Ceramic Distribution,,....o......................, 122 4, Comparison of the Experimental and Theoretical Transmitted Light Intensity Curves........o..,.o 126 v

TABLE OF CONTENTS (CONT'D) Page VII CONCLUSIONS...,............ *............,.........*...*..... 139 APPENDIX A....................................... 145 APPENDIX B....... *................................ 155 APPENDIX C....................................................... 163 APPENDIX D.~ ~................................................ 171 BIBLIOGRAPHY...................................................... 172 vi

LIST OF TABLES Table Page 4,1 COMPOSITION AND DIELECTRIC DATA FOR SEVEN BARIUMSTRONTIUM TITANATE CERAMICS,..,,,,.,.......,...,,.oo 39 501 INCREASE IN PERCENT LIGHT TRANSMISSION IN LARGE-GRAINED MIXED TITANATE COMPOSITIONS PRODUCED BY STRONG DC FIELDS.. 70 6,1 RELATIVE LIGHT INTENSITY R (BETWEEN 5200A AND 6200A). TRANSMITTED BY A (BaSr)TiO3 CRYSTALLITE BETWEEN CROSSED POLARS............. *... **o*.........*....*..... 102 6.2 SHIFT IN THE PEAK INCREMENTAL PERMITTIVITY TEMPERATURE WITH FIELD FROM Tr TO Tr*., USED IN THE EVALUATION OF THE THEORETICAL PERCENT LIGHT TRANSMISSION CURVES,..,......,.. 135 6.3 CONSTANTS USED TO EVALUATE THE THEORETICAL PERCENT LIGHT TRANSMISSION CURVES, BASED ON EQUATIONS (6.35) and (6.40). 137 vii

LIST OF FIGURES Figure Page 2.1 Some Typical Ferroelectric Hysteresis.Loops........... 6 2,2 Structure of Cubic Barium Titanate...,...........*.oo 8 203 eA TE Surface for Pure Ceramic Barium Titanate.....,.. 11 2.4 EA TE Surface for a Typical Ba 65 Sr 35 TiO3 Ceramic.. 13 2,5 Dielectric Behavior of Single Crystal Barium Titanate, as a Function of Temperature....o... o....... 4......... 14 3,1 Variation of Birefringence and Axial Ratio of Barium Titanate with Temperature,...44,4...,.......o....,,.. 24 3.2 Variation with Temperature of the Cell Edges and the Cube Root of the Cell Volume of Barium Titanate.,,,.... 24 3.3 Schematic 180~ Domain Pattern in Barium Titanate....,.* 25 3.4 Domain Formation in Tetragonal Barium Titanate........ 27 3.5 90~ and 180~ Walls Near the Surface of a Barium Titanate Crystal.,2.,,,.4.0,.,.4,,,.,4.,4... 0,..44@4 27 536 Nucleation of 90~ Domains in an a-Crystal of Barium Titanate....... * *0* *.* 4....,,,..... 0o..... 30 3.7 Free Energy Curve of BaTiO3 as a Function of Polarization According to Devonshire's Theory,.,4o.,.<Q o..,.. 30 4l1 Sample Configuration of a (Ba,Sr)TiO3 Ceramic,....... 40 4.2 Heating Stage.(a) and Mounted.Sample (b) Used.in the Observation of a Ceramic Thin Section by Transmitted Lightooovoo, oooo ovo.oeo v.^ o oo.o o oo 0 o 40 4~3 Ceramic Thin Section Observed Through a Polarizing Microscope.. 4oooo o.04 o o o 4 4 o. 4..O o. O 4 a o a o 4 42 4,4 Circuit for Capacitance Measurements................ 43 4,5 Circuit for Light Intensity Measurement by a Photocell. 46 viii

LIST OF FIGURES (CONT'D) Figure Page 5o1 Incremental Permittivity vs Temperature of Ceramic Sample 1i with Voltage as a Parameter................ 49 5.2 Percent Light.Transmission vs Temperature of Ceramic Sample 1, with Voltage as a Parameter.................. 50 503 Incremental Permittivity vs Temperature of Ceramic Sample 2, with Voltage as a Parameter,,..,,oo,,,,,,... 51 504 Percent Light Transmission vs Temperature of Ceramic Sample 2, with Voltage as a Parameter.... 04.,,,,,, * 52 505 Incremental Permittivity vs Temperature of Ceramic Sample 3, with Voltage as a Parameter,..o....o........ 53 5~6 Percent Light Transmission vs Temperature of Ceramic Sample 3, with Voltage as a Parameter. o....... o..... 54 5.7 Incremental Permittivity vs Temperature of Ceramic Sample 4, with Voltage as a Parameter O,..o..,.......o 55 5.8 Percent Light Transmission vs Temperature of Ceramic Sample 4, with Voltage as a Parameter.............o.,. 56 5~9 Incremental Permittivity vs Temperature of Ceramic Sample 5, with Voltage as a Parametero,. o o o o o o... 57 5010 Percent Light Transmission vs Temperature of Ceramic Sample 5, with Voltage as a Parameter.o......oo., o. o 58 5o11 Incremental Permittivity vs Temperature of Ceramic Sample 6, with Voltage as a Parameter.,,o......o.....o. 59 5.12 Percent Light Transmission vs Temperature of Ceramic Sample 6, with Voltage as a Parameter o..o..,,...... 60 5o13 Percent Light Transmission vs Temperature of Ceramic Sample 7, with Voltage as a Parameter......,,,/,,,,,,, 61 5.14 Effect of a Strong DC Field on the Optical Anisotropy of a Ferroelectric Ceramic, Sample 35.oo,40..,,,,.,..,. 72 5o015 Effect of a strong DC Field on the Optical Anisotropy of a Fine-Grained Ceramic, Sample 6,,..,,.......,,,,o.. 75 ix

LIST OF FIGURES (CONT D) Figure Page 5.16 Effect of Poling on the (101) Twinning Structure of a Ferroelectric Sample, Sample 1.,,,,.oo,~ 79 5.17 Variation of Birefringence with Temperature in Tetragonal Barium Titanate and (Ba,Sr)TiO3 Ceramic.......... 82 5.18a Schematic: Rotation of the Polar Axis in a Ferroelectric Domain Produced by an Applied DC Field.,........., 86 5o18b Field Produced Rotation of the Polar Axes in Antiparallel Domains4 a a a a * a a a a,,., o............ 86 5.19 Extinction of an Anisotropic Area in a Ferroelectric Grain of Sample 1, upon Applying a Strong DC Field...,, 88 5 20 Poling in Sample 1...,,,,.,...o............ 91 6,1 Orientation of the Privileged Directions of a Crystal in the Optical Field of a Polarizing Microscope........ 94 6.2 Relative Intensity R1 of the Equivalent Crystallite with Respect to the Phase Difference Angle..o...o...... 97 6,3 Ellipsoid of Rotation,........................... 98 6.4 Spectral Sensitivity Curve of RCA Photocell 6957,...... 103 605 Birefringence An of a Barium-Strontium Titanate Crystal as a Function of Angle o..... o..... o....00.00 104 606 Assumed Distribution of the Tetragonal to Cubic Phase Transitions of the Ceramic Grains about the Peak Incremental Permittivity Temperature Tr of the Ceramic, 107 6.7 The Error Function erf (x) = e-x,,,.. 107 e 0~00~0~0~P00.000 107 6.8 Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 1,..o.....oo.,...............,. 116 6.9 Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 2.......................,....................... 117 x

LIST OF FIGURES (CONT'D) Figure Page 6010 Comparison.of Theoretical and.Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 3 5................... o o o o., o o,,.... o. 118 6.11 Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 4.,,,.o....oo... o.................. 119 6.12 Polarization as a Function of Temperature, with and without a Biasing Field,o..,,............... o........ 122 6.13 Theoretical Incremental Permittivity vs Temperature Curves for Various Values of the Parameter a,.......... 125 6o14 Comparison of Theoretical and'Normalized Experimental Incremental Permittivity vs Temperature Curves for Sample 1...............,,,,,............................ 127 6. 15 Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs Temperature Curves for Sample 2.............................................. 128 6.16 Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs Temperature Curves for Sample 3............................................. 129 6,17 Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs Temperature Curves for Sample 4.,..o....o oooooo.......... 00oooooooooooo....... 130 6.18 Temperature Dependence of the Optical Transmission of Barium Titanate with Various Applied Fields Near 120~C. 132 xi

I INTRODUCTION Barium titanate has attained importance as a ferroelectric material relatively recently. Toward the end of the Second World War the industrial need for low loss materials of high dielectric constant led to the thorough investigation of titania and titanate compounds in this countryo (1'23) It was found that titania, magnesium titanate, calcium titanate, and strontium titanate form useful capacitor dielectrics. Very high dielectric condensers were also manufactured from barium titanate, but, unlike the other dielectrics, they were surprisingly temperature and field dependent, A series of measurements showed that ferroelectricity was the cause of the dielectric anomaly. Dielectric hysteresis loops were observed on the barium titanate ceramic up to about 120~C, beyond which temperature the material acted like an ordinary dielectric, At about the same time as the study on titania compounds was in (4 5) progress in the UoSA., Wul and Goldman(45) in Russia independently reported on the ferroelectric behavior of barium titanate, The material which in its ceramic form, is piezoelectric when poled, was soon widely used in the manufacture of electro-mechanical transducers and dielectric amplifiers. It proved to be a piezoelectric of high mechanical strength and chemical stability, and a good dielectric, With the growth of single crystals by the Swiss group of Battner, Matthias, Merz, and Scherrer,(6) and later by Remeika(7) it was possible to conduct some basic studies on the new ferroelectric. Within a relatively short time, a wealth of optical, electrical, and mechanical data on barium titanate were gathered. Although barium titanate was not the first -1

-2ferroelectric to be discovered -- the occurence of ferroelectricity was first noticed in Rochelle Salt (sodium potassium tartrate tetrahydrate, NaKC4H406o4H20) in 1921(8) and subsequently in potassium dihydrogen phosphate (KH2P04) in 1935(9) -- its relatively simple structure facilitated the analysis of the dielectric phenomena associated with ferroelectricityo Some of the important aspects are- 1) the large polarization in ferroelectric barium titanate, 2) the high dielectric constants at the phase transitions (barium titanate has 3 ferroelectric phases), 3) the onset of spontaneous polarization at the Curie point, 4) the changes in polarization at the lower transitions, 5) the formation of domains in the ferroelectric phases, and 6) the manner in which domain processes contribute to the changes in capacitance in barium titanate single crystalso Both phenomenological(0 ) and model theories (1 ) have been advanced to account for these and other aspects of ferroelectricity in barium titanateo While much fundamental knowledge was gained from the study of single crystals of barium titanate, research on ceramic barium titanate and barium titanate-type materials proceeded in a relatively empirical manner. The industrial interest in ferroelectric ceramics had increased considerably since the discovery of titania compounds in 1945, Efforts were made to improve the dielectric and piezoelectric properties of barium titanate and barium-strontium titanate ceramics(129) New ferroelectric ceramics were rapidly being discovered and various solid solutions of ferroelectric materials, some with "additives," began to appear on the market. (346) As a result, the manufacturer of dielectric and piezoelectric devices was able to draw from a steadily growing supply of raw materials o

-3An important aim in the manufacture of ceramics has been the production.of materials with reproducible electrical and mechanical properties, It required rigorous standardization of every step in the manufacturing cycle. Other aspects, such as the "poling" and "aging"(47-49) processes in piezoelectric ceramics were thoroughly investigated. ("Poling" refers to the cooling of a piezoelectric material in a strong DC electric field, such that the ceramic has a net piezoelectric vector, Since a poled ceramic is, strictly speaking, not in an equilibrium state, its bulk piezoelectricity will change (lessen) with time. This is referred to as "aging"). The literature on ferroelectric ceramics, which is very extensive, can be roughly classified under: 1. new and improved ferroelectric ceramics,(21-46) 2. the problems encountered in the manufacture and processing of the ceramics,(50-55) and 3. the development of devices using these materials,(5-60) In comparison, with the developmental work, very little has been written on the fundamental processes in feroelectric ceramics, The complexity of the ceramic structure no doubt is in some part responsible for this scarcity of basic information. One of the important properties of ceramic ferroelectrics which has had practical applications (modulating devices and amplifiers), but which has not been thoroughly understood, is the dielectric non-linearity of these materials, i.e., the ability of the capacitance to vary with strong, low frequency fields. It was generally thought that the dielectric non-linearity proceeded by a mechanism of

-4domain motion and domain reorientation, similar to those observed in single crystals of barium titanateo Relatively recently a theory on the "variation of permittivity with electric field in perovskite-like ferroelectrics"(62) was published. It proposed that the non-linear dielectric effect is the result of an induced ferroelectric state in the ceramic rather than of domain motion, This view was substantiated by electrical data obtained on (Ba,Sr)TiO3 ceramic materialso(62) It appeared to the present investigator that a dynamic (i.e,o under conditions of varying temperatures and electric fields), optical study on (Ba,Sr)TiO3 ceramics would enable the direct observation of the processes associated with the dielectric non-linearity in these materials, The present thesis describes the results of such a study, and it attempts to correlate and interpret the numerous optical observations made on thin sections of (Ba,Sr)TiO3o The aim has been to obtain some understanding of the mechanism of the anfield non-linearity in perovskite-type titanates,

II BACKGROUND A. General Structure of Perovskite-type Ferroelectrics All crystalline materials fall into one of the thirty-two crystallographic symmetry classes. Of these, eleven have a center of symmetry, whereas the other twenty-one are non-centrosymmetric. Materials belonging to twenty of the non-centrosymmetric classes have been described as piezoelectric, i.e., they become electrically polarized when subjected to stress or to a temperature gradient. Of the twenty piezoelectric classes, ten are termed the polar or pyroelectric classes. The truly pyroelectric materials are spontaneously polarized in their stressed state. Their inherent polarization can be detected by heating or by cooling the substance uniformly and thereby effecting a change in the existing polarization. Among the pyroelectric crystals there is a smaller group.in which the direction of spontaneous polarization can be reversed by applying an electrical DC field opposite in sign to the polarity of the crystal. Such materials are described as ferroelectric, Their behavior is illustrated in the ferroelectric hysteresis loops in Figures 2,la, 2.lb, and 2,olC Hence a ferroelectric crystal can be broadly defined as having an inherent polarization along some specified crystallographic direction, In addition, the direction of polarization in the crystal can be reversed by a large, external electric field. In the following the perovskite type ceramic ferroelectrics such as polycrystalline barium titanate and solid solutions of barium and strontium titanate are considered in some detail, The adjective "perovskite-type" has been used for many ferroelectric and antiferroelectric materials whose structure can be derived from the ideal perovskite -5

-6P 0 A I ~ECE Figure 2.la. Schematic Ferroelectric Hysteresis Loop OP. w spontaneous polarization OP * remanent polarization cOEc coercive field P P O E Figure 2.1b. Hysteresis Loop of a Barium Figure 2.1c. Hysteresis Loop of a Typical Titanate Single Crystal at Commercial Titanate Ceramic 60 cps (after Merz, Reference at 100 cps 78) Figure 2.1. Some Typical Ferroelectric Hysteresis Loops.

-7structure by small modifications in its atomic positions or even by omissions of atoms not essential to the perovskite framework. (The mineral perovskite, CaTiO3, from which the family name was derived, is orthorhombic, and has the space group Pcmn. It is not an "ideal perovskite," but rather a distorted multiple cell perovskite, and it is not ferroelectric,) As Megaw(63) has described in some detail, ideal perovskitetype structures are cubic and have the formula ABO03 A is a divalent cation, B a tetravalent cation and 0 the oxygen ion. In addition, the A cation must be twelve-coordinated and the cation B six-coordinated, The structure can be considered as a framework of linked B06 octahedra with the A cation occupying the interstices between the octahedra, Barium titanate, whose structure is shown in Figure 2o2, can be classified as an ideal cubic perovskite in its non-ferroelectric form. Its unit cell length is about 4 A and the cell contents is one formula unit. The lower temperature forms of barium titanate, which are ferroelectric, have been classified by Megaw(63) as distorted small-cell perovskiteo These forms are no longer cubic but can be derived from the cubic structure by small distortions in the atomic positions, Their cell 0 edges, if referred to the original cubic unit cell, are still about 4 A long and their cell contents remains one formula unit, The three crystallographic transitions of perovskite-type barium titanate are: 1o the cubic to tetragonal transition at about 120 ~C, 2, the tetragonal to orthorhombic transition near 0~C, and 35 the orthorhombic to rhombohedral transition near -90CoC(63)

-8Ti. ION BI ION / S^_n^^^^ O IONS AT CORNERS?.1-00^~~ /S ImOF OCTAHEDRA Figure 2.2. Structure of Cubic Barium Titanate. (After Megaw, Reference 63)

-9Only the high temperature transition which represents the transition from the cubic, non-ferroelectric, to tetragonal, ferroelectric state is considered in the present investigation. In solid solutions of barium and strontium titanate, the upper transition is linearly shifted to lower temperatures with increasing amounts of strontium titanateo(64,65) Dielectric measurements on solid solutions of barium titanate and strontium titanate taken at the Electromagnetics Materials Laboratory, Electrical Engineering Department, The University of Michigan, show a similar trend for the tetragonal to orthorhombic transition temperature of barium-strontium titanate materials. Similar to barium titanate, these mixed titanate ceramics (the term "mixed" will be used here for solid solutions of barium and strontium titanate) likewise occur in the ideal cubic perovskite and distorted small cell perovskite form. Due to the substitution of the smaller strontium ion in the barium titanate lattice, the average unit cell of the mixed barium strontium titanate ceramic is somewhat smaller than that of pure barium titanate, B. Dielectric Behavior of Single Crystal Barium Titanate and Ceramic Barium Titanate and Barium-Strontium Titanates The ferroelectric hysteresis loops shown in Figures 2ola, 2olb, and 2olc are examples of the polarization versus electric field characteristics of titanate ferroeleetricso Hysteresis loops can be easily observed on a cathode ray oscilloscope by means of a Sawyer and Tower circuit,(66) Figure 2,la shows a schematic hysteresis loop obtained by cycling a ferroelectric sample with an AC field, Ps, the spontaneous polarization, is obtained by extrapolating the slope of the saturation polarization back

-10to the P-axis, Pr, the remanent polarization, is that polarization which remains in the ferroelectric at zero field, Ec, the coercive field, is the field required to remove the remanent polarizationo Figure 2olb shows a square hysteresis loop taken on a "good" single crystal* of' barium titanate at about room temperature. A sudden reversal in the polarization of the crystal occurs when a sufficiently large field of opposite polarity, i.e., the coervice field, is applied to the crystal. Poorer crystals and large grained oriented polycrystalline aggregates will show more rounded loops, indicating that polarization reversal occurs over a narrow range of fields, (Materials with square hysteresis loops are applicable to memory and switching units,) Figure 2,lc shows a hysteresis loop taken on a fine-grained commercial ceramic material, 180~ polarization reversal in the domains of the ferroelectric ceramic occurs over a wide range of fields, with the slope of the curve, dP/dE, being steep near coercive fields and relatively flat at large applied fields. (The narrow loop materials are employed for modulation and as amplifiers,) The dielectric behavior of ceramic barium titanate as a function of temperature and applied DC field is illustrated in Figure 23,, The * A good single crystal is mostly strain-free. It has few imperfections and can easily be poled into an area of uniform polarization by an electric field. On the other hand, "poor" crystals are usually highly twinned, They do not lend themselves to successful poling, or, to the alignment of all the twin vectors in one direction by means of an externally applied DC field.(67,68,69) Twins, or ferroelectric domains, as they are termed in the literature, tend to renucleate at imperfections as soon as the external field is removed,

^ -. - -a -A 1 0 <^ t C 0 0 C: ~. C.o C C' 0~ 0 0, o0 0O 00, 00 0 ^~ ~~ Y eh ^^N^ — ~~~~~~~~~~o1 HlPu ~ ~ ~ A CD) (D 0 OIC,') __~~~~~o 0o ~, i~i ~'

-12incremental permittivity* reaches a maximum of 120~C with zero applied field. The effect of an increasing electric field on the ceramic is to steadily lower the peak incremental permittivity and to raise the temperature at which the peak permittivity is observed. Figure 2.4 shows a surface relating incremental permittivity, applied field, and temperature for a mixed barium-strontium titanate ceramic of composition Bao 65Sr0 35Ti0(62) Here the paraelectric to ferroelectric transition occurs at a lower temperature, and the incremental permittivity peak is somewhat broader than for pure barium titanate, Generally speaking, however, both DC field and temperature have a similar effect on the two ceramicso Co Proposed Explanation of the Dielectric Field Non-Linearity in Ceramic Barium Titanate and Barium-Strontium Titanate It has been observed(69) that whenever a DC field is applied to a multidomain single crystal, the domains tend to align themselves in the direction of the field. The incremental permittivity of such a "poled" crystal, when measured parallel to the field direction, is lower than that of the crystal before poling, This is a result of the large directional variation of the small signal permittivity (or incremental permittivity), EA, in barium titanateo The EA measured parallel to the c-axis is of the order of 200, the EA perpendicular to the axis is about 4000(70) (see Figure 2o5), A crystal with domains oriented in The incremental permittivity EA is the permittivity dP/dE obtained by measuring the dielectric with a very small, high frequency signal. The high frequency measuring signal is superimposed on a large, low frequency AC signal or on a large DC biasing field. The data in Figure 2,4, for example, were obtained by sweeping the ceramic from 0 to 40 kv/cm at a slow, cycling field of 0,5 cps and by measuring it with a high frequency field of 10,000 cps.

*FOT1tJD OT6i 5'S 9'*eg0 Ts T eF xOJ ajznomS'ti* GanTjts ~,'~, o ooot, 0009 i -^T -

-1410 I 8 a 6 40 0 - - 16 -2 0 -180 -140 -100oo -60 -20 o 20 60 o00 140 T,~C Figure 2.5. Dielectric Behavior of Single Crystal Barium Titanate, as a Function of Temperature. (After Merz, Reference 70)

-15different directions will have an incremental permittivity somewhere in between these values. When exposed to a DC field, the domains tend to orient themselves in the direction of the fieldo(68'69) This results in a lowering of the incremental permittivity of the crystal in the field directiono It is well known that a DC biasing field also decreases the incremental permittivity of ceramic barium titanate(71) or bariumstrontium titanateo Before any work had been published which attempted to explain the observed dielectric field non-linearity in the ceramic, McQuarrie(72) proposed three possible explanations for the process. He suggested that the effect might be caused: 1L by a field produced shift of 90~ domain walls in the ceramic, similar to the one observed in single crystals of barium titanate, 2, by a change in the basic polarizability of the material, and 35 by the removal of 180' domain walls, assuming that they did contribute some value to the permittivity of the ceramic in the unbiased state, If the first explanation were correct, McQuarrie reasoned, one might possibly detect a mechanical distortion in the material as a result of the field produced 90~ domain movement, He ruled out the other two hypotheses as not being significant enough to account for the observed, large field-produced changes in the dielectric value of the ceramico

-16As mentioned in the beginning of this section, a field produced alignment of 90~ domains in the ceramic would result in a decrease in the incremental permittivity of the ceramic when measured parallel, and an increase in the.incremental permittivity when measured perpendicular to the field direction. It can be experimentally shown,(75) however, that in the transition region of a ceramic, the incremental permittivity Ec decreases with increasing DC fields, regardless of whether the sample is measured parallel or transverse to the applied field. A theoretical model has been developed(62) which provides an explanation for the field produced dielectric non-linearity in solid solutions of barium-strontium titanate.* The model proposes that the domains in a fine-grained ferroelectric are not free to move with the applied external field, but that they are essentially "frozen in." The observed lowering of the incremental permittivity peak with field in the ferroelectric to paraelectric transition region is considered due to an induced ferroelectric state in those grains in the ceramic which are paraelectric. It is characteristic of the ceramic perovskite ferroelectrics that their transitions to the paraelectric, isotropic state occur over a temperature range rather than at a sharply defined temperature. The reason for this lies in the distribution of Curie temperatures of the crystallites that compose the ceramic. The tetragonal to cubic transition temperature is strongly affected by stresses.(74W75) Stresses occur in the ceramic at imperfections and also result from the approximately 1% linear contraction(7677) which occurs when the ceramic is cooled from the sintering * See section III-G

-17temperature of about 1400OC to about 120~C. Furthermore, the cubic to tetragonal transition is accompanied by a 1% change in the axial ratio of the perovskite-type titanateo(76) Some of the strains are subsequently relieved by complex patterns of twinning within the individual ceramic grainso In a mixed barium-strontium titanate ceramic, small compositional variations in the grains cause a further spread in the Curie temperatures and result in a still broader permittivity peak, Within the region of the peak incremental permittivity, both ferroelectric and paraelectric grains exist side by side, with more ferroelectric grains at the low temperature end of the transition range, and more paraelectric grains at the high temperature end, The lowering of the incremental permittivity with induced ferroelectricity can be simply illustrated on a single crystal of barium titanate. Figure 2,5 represents the dielectric behavior of a single barium titanate crystal with respect to temperature. Tc at 120~C is the Curie temperature, Ti, an arbitrary point on the curve, is 130~C ca and Ce represent the incremental permittivities in the two axial directions of the ferroelectric grain. A crystal of barium titanate at 125~C is paraelectric (see Figure 2~5) and has an CA of about 8000. The formation of a ferroelectric state by applying a field can in a sense be represented by a shift of the entire curve to higher temperatures,* By applying a DC field of 7000 volts/cm to barium titanate at 125~C, the transition temperature is shifted upward by about 10~C to 130~C, (These * Strictly speaking, a field would shift the curve to higher temperatures and also lower the overall permittivity of the biased ceramic,

-18figures are based on calculations by Merz. Merz reports having observed a field produced shift of the Curie temperature on single crystals of barium titanate of 1,4 x 10-3 degrees/volt cm-1)(8) The field exposed crystal is therefore ferroelectric at 125~C, with an e of about 1000 and an ca of about 4500 or less (see footnote p. 17 ) These values are considerably lower than the incremental permittivity of 8000 for the paraelectric crystal at 125~C and zero field, In the following the observations made above on a single crystal of barium titanate will be applied to the case of ceramic barium titanate and mixed barium-strontium titanate compositions. The effect of a DC field on the ceramic at temperatures slightly above its peak incremental permittivity temperature is briefly outlined, Some of the assumptions on which the model of the field-produced dielectric non-linearity in perovskitelike ferroelectrics(62) is based, are thereby illustrated, The ceramic is simply regarded as an aggregate of crystallites. At a temperature slightly above the peak incremental permittivity temperature, most of the grains in the ceramic are paraelectrico However, due to the distribution of the individual Curie temperatures of the grains over a temperature range, some grains have remained ferroelectric. At the temperature in question -- namely, at a temperature somewhat above the peak incremental permittivity temperature -- a distinction can be made between not only two, but three possible states of the ceramic crystallites. 1o There are a number of ferroelectric grains in the ceramic which are considered to be essentially clamped, and on which

-19the field has little effect, They contribute to the total permittivity by a constant fractiono 2. It can be shown from thermodynamic relationships developed for barium titanate(10l78,79) that a narrow temperature range beyond the Curie peak exists in which a ferroelectric state can be induced in a paraelectric single crystal, i,e,, 12~C for barium-titanateo At higher temperatures, the field has a comparatively small effect on the incremental permittivity of the crystal, Hence, in the ceramic, the contribution to the total permittivity from these completely paraelectric grains will be considered another constanto 3. Finally, there remain those paraelectric grains which lie in the "inducible" range and which are sensitive to an applied field, As the field is gradually increased, the incremental permittivity of these crystallites first increases until it reaches a maximum at the induced'Curie" temperature, The phenomenon is illustrated on a single crystal of barium titanate in Figure 2,5 where the permittivity of the crystal at Tc is higher than that at Ti ~ With larger fields, a ferroelectric state is induced in the grains, and the incremental permittivity drops considerably (Figure 2,5), The non-linear behavior of the ceramic can therefore be considered mainly due to this latter group.of grains whose incremental permittivies vary substantially with field, This is very well shown in Figures 235'and 2,4, which represent the dielectric behavior of a barium titanate

-20and a barium-strontium titanate sample with temperature and field. It is seen that the largest changes in the incremental permittivity of the ceramic occur when an electric field is applied to the peak incremental permittivity of the ceramics. The peak marks the temperature at which a maximum number of ceramic grains change from a ferroelectric to a paraelectric state. The farther away from the peak, the more linear the cA vs E characteristics of the material. Do Optical Investigation of (Ba,Sr)TiO3 Ceramic with Variables of Temperature and Electric Field The model of induced ferroelectricity proposes that the observed dielectric field non-linearity in ceramic perovskite-type ferroelectrics is not the result of domain rotation in the ferroelectric grains (the grains are considered to be essentially clamped), but the result of an induced ferroelectric state in those grains which are paraelectric in the region of the ferroelectric to paraelectric transition of the ceramic. It was hoped that an optical study of ceramic barium-strontium titanate by transmitted light would provide a direct means of observing the effect of an electric field in the material. Such an investigation might possibly reveal whether a biased ferroelectric material differed, in some fundamental way, from a biased single crystal. The mineralogical technique of thin sectioning was chosen to prepare several solid solutions of barium and strontium titanate for observations under the polarizing microscope. This technique, rather than the metallographic one of polishing and etching, has the advantage that it permits detection of any immediate changes which varying temperature

-21and/or electric fields may have on the anisotropy of the material. The ferroelectric to paraelectric transition in the perovskite titanates is closely connected with their tetragonal to cubic transition. Hence, when viewed between crossed polars, the changes in the optical anisotropy of the grains can be directly related to changes in the state and structure of the entire ceramic. The results of the optical observation on thin sections of (Ba,Sr)TiO3 are mainly discussed in Section Vo

III BARIUM TITANATE AND (Ba,Sr) TITANATE MATERIALS Various experimental and theoretical observations on barium titanate and barium-strontium titanate materials, which are not essential to the understanding of the present investigation, have been briefly treated below. The reader is referred to this section for more extensive information on several aspects discussed in the thesis. Ao Structural and Optical Data Barium titanate can occur in five modifications, hexagonal C63/mmc, (881) cubic Pm3m,(82) tetragonal P4mm,(82) orthorhombic Bmm2,(83,84) and rhombohedral R3m.(83,84) The transitions occur at 1460~C, 120~C, 0~C, and -90~C respectively. The high temperature transition between hexagonal and cubic is slow, non-ferroelectric, and reconstructive. (The term reconstructive implies here that the transition proceeds by a break-down of the old structure and a subsequent reassembling of the crystalline network to form the second structure.) All other transitions are displacive, i.e., they involve small displacements of the atomic network.(85) The transition at 120~C occurs between a non-ferroelectric and a ferroelectric phase. The remaining two low temperature transitions take place between ferroelectric phases. In the cubic form, barium titanate has the ideal perovskite structure. The titanium ion is situated at the center of an oxygen octahedron, Hence it is six-coordinated, The oxygen octahedra are linked together at their corners.(63) The barium ion, which is 12 coordinated, lies in the large interstices between the oxygen octahedra (Figure 2.2). -22

-23Alternately the barium ion can be visualized as being at the center of a cube, with the titanium ions at the eight corners and the oxygen ions at the midpoint of the twelve edgeso All ferroelectric transitions in barium titanate occur with only small displacements in the ionic positions. A slight stretching of one of the crystal axes of the original perovskite cube gives rise to the tetragonal modification, with the c-axis forming the polar axis (c/a = 1.01). An elongation along the cubic face diagonal yields an orthorhombic form, with the orthorhombic a-axis now representing the ferroelectric axis, Finally, a displacement along the cube body diagonal leads to the rhombohedral, low temperature ferroelectric modification, with [111] being the polar directiono The refractive index of barium titanate is nD = 2.46 at the Curie point. Below 120~C the birefringence of the crystal is strongly dependent on temperature (Figure 3.1)(84) and wavelength.(87) Tetragonal barium titanate is optically negative. At 1100C c -c (nc - na) -0.03 for sodium light, and at room temperature = (n - na) -0.055.(88) The variation of the cell dimension with temperature is shown in Figure.2. (84) 3o2 B, Domain Structure and Twinning Many studies have been made on the very colorful and intricate twinning or domain patterns in ferroelectric, tetragonal barium titanateo(67,88-96) There are essentially two types of twins, the socalled 180~ domains and the 90~ domains, The 180~ domains represent areas of anti-parallel polarization. (Figure 353). The c-axis of the

-24-..... Axial ratio C/R =a * _ - nc 0.07 -[ 0.07 I R1l = nb - ne 0-06-1-010 ne na - - Rx c 004. C o003 -1-005 0:02 - 0.01 - _ Oc Figure 3.1. Variation of Birefringence and Axial Ratio of Barium Titanate with Temperature. (After Kay and Vousden, Reference 84) 4 030 4.020 a'or % o E.01 t C oe....-"-. T I, ta Cubic 4000 o f Barium Ttragona te Orthhorho bic 3-990 - Rhombo3.980 hedral 3'970 -- -100 -50 0 50 100 150 *C Figure 3.2. Variation with Temperature of the Cell Edges and the Cube Root of the Cell Volume of Barium Titanate. (After Kay and Vousden, Reference 84)

-25i I I I I - I!: I II i i I I I I 4 [i t iL liIiIL i Figure 3.3. Schematic 180~ Domain Pattern in Barium Titanate. (After Kaenzig, Reference 97)

-26twin individuals are oriented at 180~ with respect to each other, with (100) forming the composition plane. In 90~ domains the c-axes lie at almost 90~ to each other in a head to tail arrangement, so that no surface charge is built up at the domain walls. The "90~ wall" is a (101) composition plane(91) (Figure 3.4). Combination of the two types of twins occur frequently and lead to complicated domain patterns(92) (Figure 355)~ Twinning in the orthorhombic and rhombohedral structures have been reported by Kay(91) and Forsbergho(88) C. Electrical Data Merz, (78) Cross,(98) and Drougard and Young(99) have reported values for the spontaneous polarization of tetragonal barium titanate, taken on almost perfect single crystals. The room temperature value is 24-26 microcoulomb/cm2. The spontaneous polarization decreases steadily to 16-18 microcoulomb/cm2, as the crystal is heated to the Curie temperature, where it falls off sharply. The coercive force at room temperature is about 500 volts/cm at 60 cycles/sec.(78) The variations of the dielectric constants with temperature, measured perpendicular, a, and parallel, c, to the c-axis are shown a c in Figure 2.o5 Above the 120~C transition, at which the crystal is no longer ferroelectric, the permittivity follows a Curie-Weiss law of the form C = + C (3.1) T-To

o9415' t -o Figure 3.,. Donmain Format io. n in'e-t;raonal.3,r:i.u Titanate. Figure 35.. 90 ~ and 1.O 80 o Walls N ar the Surface oft a B.rilun 1i'tanate Crystal. ( After Hlooton and Merz, ie:eroence 92)

-28E is the dielectric constant, eo the electronic contribution to the dielectric constant, C the Curie constant, T the absolute temperature, and To the extrapolated Curie temperature. The Curie point in barium titanate is easily varied by internal strains, pressure, electric field, and solid solutions. Hydrostatic pressure tends to decrease the temperature of the cubic to tetragonal transition, (74) whereas pressure parallel or perpendicular to the c-axis raises the transition temperature (75) An applied DC electric field also shifts the Curie point to higher temperatures, Merz,(78) Kaenzig and Maikoff(79) and Kawabe(100) have calculated the shift to be 1.43 x 10-3~C/ volt/cm, 1.2 x 10-3~C/volt/cm, and 1,6 x 10-3~C/volt/cm respectively. Do Domain Motion In their study on the dynamic behavior of domains in single crystals of barium titanate, Merz(68) and Little(69) have shown that polarization reversals proceed by the nucleation and growth of spike like domains. When an electric field of opposite polarity to the direction of the polar vector is applied along the c-axis of a single domain crystal, 180~ domains nucleate at the electrodes and grow across the crystal in the form of thin wedges until the polarity of the original crystal is completely reversed. The growth is rapid and stable, the domain wall is thin (around two lattice constants in width(68)), and there is little sideward motion of the domain wall. The widening of the wedges proceeds by a series of dipole flips parallel to the polar axis,(69)

-29When an electric field is applied along the a-axis of a tetragonal crystal, 90~ domains nucleate at one of the electrode surfaces, mostly the cathode. They subsequently propagate diagonally across the crystal, (Figure 536). Unless a spike of this type has traversed the entire crystal, it is usually not stable and is forced out by internal stresses when the field is removed. The stresses arise from the pseudocubic, tetragonal nature of the crystal(6) Little has shown(69) that the direction of the polar axes in neighboring 90~ domains deviate slightly from 90~. Hence the (101) composition plane cannot be strictly considered a 90~ domain wall. The formation of "90~ domains" is therefore accompanied by electrical-mechanical distortions in the crystal. The latter tend to revert the crystal back to its original state, once the field is removed. Existing 90~ domain walls are many lattice spacings thick and sidewise motion of the walls has been observed. When a sufficiently strong DC field is applied to a crystal having a complex pattern of both 90~ and 180~ domains, the domains tend to align themselves in the direction of the field...It is not possible to reverse the overall polarization of a crystal by 90~ domain formation alone. The mechanism by which this switching process takes place consists of a complex sequence of interactions between domain walls, as explained in some detail by Little.(69) More recently studies on domain walls have been continued by several Japanese (101-104) and other workers,(105-107) E. Solid Solutions Ionic substitution in the perovskite structure of barium titanate has a considerable influence on the paraelectric to ferroelectric transition

-30(a) (b) Figure 3.6. Nucleation of 90~ Domains in an a-crystal of Barium Titanate. (After Little, Reference 69) 000oo o0 N so, X BeO 1 -- ni NNNN=..5 X-I__ E_ __ ~__ i.... \!. 1.0 0.5 EP V/cm 50.50 10 10 -I.0 -1.5 2050 100 110 115 I151C Figure 3.7. Free Energy Curve of BaTiO as a Function of Polarization According to Devonshire's Theory. (After Kaenzig and Maikoff, Reference 79)

-31temperature~ The direction of the shift depends on the particular ion substituted. In the solid solution of barium-strontium titanate the transition temperature decreases linearly with increasing concentration of strontium,(64) In solid solutions of barium-lead titanate, on the other hand, the Curie point increases up to 490~ for 100% lead titanateo(6~ Solid solutions of these and other perovskite-type ferroelectric ceramics have been found very useful in industry because they permit some control of the transition temperature as well as the dielectric values of the final product. Fo Theoretical Considerations Several theories of ferroelectricity have been advanced to explain the wealth of information obtained from barium titanate ceramic and single crystals. Among them are the model theories of Slater,(12) Jaynes,(l5) and Mason and Matthias,(ll) and a phenomenological theory by Devonshire. (0) The latter has perhaps been the most successful of all. Since much of the theoretical treatment on ceramic barium-strontium titanate materials is based on Devonshire's thermodynamic description of single crystal barium titanate, some essential points of his theory are outlined belowo The present description is restricted to the tetragonal and cubic phases of barium titanateo In general, a piezoelectric solid can be described by the equation: dA = EdP - Xdx - SdT o (3o2)

352A is the Helmholtz free energy, E the applied electric field, P the polarization, X the stress, x the strain, S the entropy, and T the temperature of the system, Devonshire assumes that all phases of barium titanate can be considered as strained forms of the cubic structureo He then develops a free energy expression for barium titanate as a function of temperature, polarization and stress. Two cases are considered; first, the case of a free, unstrained crystal, and secondly, that of a strained crystalo For the unstrained crystal, the differential free energy at zero stress becomes A = EdP - SdT o (33) The free energy of the system, which is considered to be zero for the unstressed, unpolarized crystal, can be expanded in terms of increasing powers of polarization. Odd powers in polarization are omitted in order that the free energy function remain the same for any reversal in sign of a polarization component. A(PT) = (Px2 + 2 + pz2) + bll(P + P + 4 y z -12 +P + P 2( +) + b(P z2 + Pzx + x y2) c 6 6 6 + 6(Px + p + 6) 6 x y z! (py~4z4 44 4 (4 4) + -d (P y 4 + P4 4 + P 44) + ~- (3.4) Px, Py, Pz are components of polarization in the x, y, and z directionso The coefficients a, b, c, d, are taken at constant stress (ioeo, the crystal is free to deform). The coefficients are strictly speaking

-33temperature dependent, For the transition from the cubic to tetragonal phase, Equation (3.4) can be simplified to a p2 b p4 + c p6 A(P,T) = a P 2+ + (3.5) e- 4 6 since the polarization vector in the tetragonal phase lies along the z-axis. Hence P = Py = 0, and Pz = P. At the upper transition temperature, a barium titanate crystal goes from an isotropic to a ferroelectric state, that is, it becomes spontaneously polarized. At this transition, both the cubic and tetragonal states are thermodynamically stable forms, as seen from the free energy vs polarization curves (Figure 3.7). Figure 3.7 shows two equal minima in the free energy function at 124o5~C, the transition temperature. For the unstrained, cubic crystal, the free energy, as a function of polarization, is defined as zero, A = a ps2 + b P 4 + p s6 = 0 (3.6) a 2 b 4 c 6 ^ P~ + 7 -- o Ps is the spontaneous polarization. The free energy of the tetragonal form of barium titanate at the Curie transition is also zero. Hence =d aPs + bPs3 + cPs5 = 0 (3.7) dP From (3.6) and (3o7) Devonshire obtained the relations Ps2' - (3~8) and a -- (3.9) 16 c

-34Having determined the experimental values for P and a, (from the dielectric behavior of barium titanate which follows the Curie-Weiss law above the tetragonal to cubic transition temperature), he was able to determine the coefficients b and c o He extended his theoretical analysis to the low temperature forms of barium titanate and was able to determine correctly the transition temperatures of the tetragonal to orthorhombic and orthorhombic to rhombQhedral formso Devonshire has also discussed the case where barium titanate is measured under constant straino He finds that the strains are proportional to the square of the polarization. Furthermore, he predicts that the ferroelectric to paraelectric transition in a clamped single crystal is of the second order rather than the first order, (i.eo, the temperature dependency of the polarization is continuous at the transition temperature, whereas it shows a sharp discontinuity for the unstressed crystal), Subsequently additional coefficients in Devonshire's equations, (e.go, coefficients b, c), were determined by other investigators. Kaenzig and Maikoff(79) published a series of free energy curves of barium titanate as a function of polarization (Figure 357), and Merz(78) showed curves of the variation of polarization with electric fieldo G, Ceramic Perovskite Ferroelectrics Ceramic ferroelectric materials can be crudely visualized as a tightly bonded. aggregate of rather imperfect and randomly oriented single crystalso The interaction of the crystallites in the ceramic is complicated by the occurrence of stresses, internal strains, and in the case of

-35mixed titanate compositions, by small compositional variations among the grains. This results in a ceramic having properties which differ substantially in some respects from that of a single, unclamped crystal. For example, the transition temperature of the ceramic does not occur at a well defined temperature, but rather over a narrow temperature range because the crystallites themselves have slightly differing Curie temperatures. The value of the permittivity peak in the ceramic is not as high as it is in the single crystal. There are other notable differences. A ferroelectric crystal of BaTiO3 responds to an externally applied DC field by orienting its polar axis in the field direction.(69) This occurs within the entire ferroelectric temperature range. (69) A ceramic barium titanate or mixed barium-strontium titanate type ceramic, however, responds to the field mainly in the region of its tetragonal to cubic transition. It is quite insensitive to the field at lower temperatures where the ceramic is actually composed of a larger number of ferroelectric grains (see Figure 2.3, 2~4). Diamond(62) has developed a theory of non-linear processes in mixed titanate ferroelectric ceramics which provides an explanation for the observed behavior in the ceramics. He demonstrates that the change in permittivity with electric field may be explained as an induced ferroelectric state in grains which are paraelectric at zero bias rather than due to domain orientation. The ferroelectric ceramic is represented as a collection of individual grains whose permittivities depend on composition, field, and temperature. The net incremental permittivity of the material

-36<E(TE)> is taken as a statistical average of the permittivities e(T, Tc*, E) of the individual grains composing the ceramic. Hence (Tc-yE-Tr) 2 e(T, Tc, E) e adTc <E(TE)) =- (5.10) Tc -7E-Tr 2 00 _ e a dTc or Tr is the temperature of the peak incremental permittivity, Tc the Curie temperature of the crystal at zero field, Tc is approximately related to Ta-', the Curie temperature of the crystal with field, by the equation Tc = Tc*-yE T is the temperature, and a is the variance of the statistical distribution, a is largely determined by the history of the ceramic, i,e,, how it is prepared, how well it is fired, and how complete the solid solution, etc. Basing his assumptions on Merz s data on the dielectric behavior of barium titanate single crystals(70) (Figure 2.5), Diamond(62) has set up three conditions to describe the permittivity contributions of the ceramic crystallites to the total incremental permittivity of the ceramic in the tetragonal to cubic temperature range. They are: 4hC for T > T T-T - e(Tc TE) = [a for 1 field] for T,<T<Tc* (3511) cav for field c^ for T<T atv oc

-37Equation (3o11) states that (1) for T>Tc the grains follow a CurieWeiss law as for single crystals; (2) for T <T<TC, the grains which were non-ferroelectric in the absence of an electric field now have a ferroelectric state induced in them in accordance with the free energy function. It is assumed that the polar axis of the induced ferroelectric state lies in the field direction; (3) for any grain in which T<Tc e is presumed to be Eav', that is, all ferroelectric domains are considered to be frozen in, Evaluation of the integral leads to results which are in excellent agreement with the experimental dielectric data. The optical examination of ceramic barium-strontium titanate ceramics discussed in the following sections represents an observational check of the non-linear processes with field in the ceramic grains at their ferroelectric to paraelectric transition. Ho Optical Observation of Ceramic Ferroelectrics A number of investigators(l08-115) have gathered much illuminating information on the twinning structure of ferroelectric titanates by employing the metallographic techniques of polishing and etching, and then by examining the surfaces with petrographic as well as electron microscopeso 90~ domains have been found in abundance as well as 180~ domains, By renewed polishing and etching of a sample after exposure to high DC fields, DeVries and Burke(112) were able to transform some antiparallel 180~ domains into a region of uniform polarization. They found however, that the 90~ domains had only been very little affected by the field. This seems to bear out similar observations made on single crystals.

IV EXPERIMENTAL PROCEDURE A, Preparation of Samples Seven compositions of mixed barium-strontium titanate ceramics were chosen for optical observation and capacitance measurements, They are listed by their compositions in Table 4,1. The ferroelectrics all have peak incremental permittivities at or above room temperature, so that they can be observed at the temperature of their tetragonal to cubic transition by simply using a heating stage. In order to prepare a sample for microscopic observation by transmitted light, a thin slab is imbedded in a thermosetting resin and then ground flat on a metallurgical wheel, using a series of graded silicon carbide papers. Subsequently the material is polished with coarse and fine diamond paste, until a minimum of pits and scratches are observed, The sample is then removed from the plastic, is cut into the desired size, and is glued to a microscope slide with the polished side down. Eastman (116) 910 adhesive, a strong, colorless polymerizing cement of index n = 1.45, has proved to be satisfactory for this purpose. The mounted sample is then reimbedded in plastic, with the rough ceramic side exposed, and is ground and polished to 30 - 40 microns thickness, After removing the plastic, the sample is electroded along its edges (Figure 4.1) with evaporated silver or silver paint. Aluminum foil leads are attached to the electrodes, and the sample is then ready to be measured, Unfortunately, thin sections of the dimensions described, namely approximately 1 cm x.1 cm x.004 cm (Figure 4ola), have electrical capacitances of less than lfo Any variation in sample capacitance with -38

-39TABLE 4,1 COMPOSITION AND DIELECTRIC DATA FOR SEVEN BARIUM-STRONTIUM TITANATE CERAMICS Sample Composition, mole % Approximate Peak, Curie Temp, ~C 1 75% BaTiO3 25% SrTiO 3100 49 2 75.0% BaTiO3 24.9% SrTiO0.1% 2 UO(NO3)2 4500 51 3 73.55 BaTiO3 24,5% SrTiO3 2,0% CdO 5500 52 4 85% BaTiO3 15% SrTiO 1110 83 5 Aerovox Corp. Formula 1950 30 6 85% BaTiO3 14% SrTiO3 1% Fe203 1200 48 7 74,25% BaTiO3 24.75% SrTiO3 1o00% Nb205 (6000)* (20)* * Value obtained from a larger ceramic sample of the same compositiono The Sample 7 used for transmitted light intensity measurements had too low a capacitance for accurate determination.

-40Direction of Applied Field -- Electrodes Direction of Trans- ( ) mitted Light (b) Figure 4.1. Sample Configuration of a (Ba,Sr)TiO3 Ceramic. (a) Uniform Thin Section (b) Wedge aomple,Thin End Asbestos //Silver Point / /M Mica / //A Aluminum Foil / BY Slit T h Thermocouple,or I 1Beneath Sample Posts L N ichrT V- -Nichrome Wire (a) (b) Figure 4.2. Heating Stage (a), and Mounted Sample (b),Used in the Observation of a Ceramic Thin Section by Transmitted Light.

-41field or temperature can only be detected with great difficulty because of stray capacitances in the circuit and surrounding media. It was therefore necessary to resort to a new, wedge-shaped geometry. The modified sample is thin enough at one end to allow observation by transmitted light, yet it has a sufficiently high total capacitance to enable accurate electrical measurement (Figure 4,lb), The wedges prepared.were approximately.8 cm long,.1 cm wide and had a thickness of around,1 cm at one end and.004 cm at the other. They were mounted on a glass slide as before, provided with electrodes, and placed on a heating stageO B. Measuring Equipment 1. The Heating Stage A simple stage was devised to observe the ferroelectric to paraelectric transition in the samples. Nichrome wire wound around a strip of mica is attached to an asbestos disk (Figure 4,2a). A slit in the center of the disk allows passage of light to the sample which is placed over the slit (Figure 4.2b), Metal posts on the disk keep the sample in position and act as contacts for the electrical circuitryo An iron-constantan thermocouple is placed in a notch directly beneath the thick end of the sample and the temperature is obtained from the potentiometer readings of the thermocouple voltageo The heating rate of the stage during the course of an experiment is manually regulated by a variaco The entire holder is placed on the stage of a Leitz Panphot microscope (Figure 403).

-42CERAMIC THIN SECTION OBSERVED THROUGH A POLARIZING MICROSCOPE OBSERVER,CAMERA OR PHOTOCELL — OCULAR SYSTEM _ —-ANALYZER - ~OBJECTIVE LENS FERROELECTRIC SYSTEM SECTION LEADS TO 3-^ M~ETER \ aX I HEATING STAGE Q-METER AND VOLTAGE SOURCE CONDENSING LENS SYSTEM LIGHT SOURCE - POLARIZER Figure 4.3. Ceramic Thin Section Observed Through a Polarizing Microscope.

-432o Capacitance Measurements Capacitance measurements were taken at 1600 KC on a Bontoon Radio Corporation Q-Meter. A 1300 volt battery box supplied the DC field applied to the ferroelectric wedge. Figure 4~4 shows a schematic of the circuit. A standard resistor, Rstandard 10 me-gohm and a standard capacitor, Cstandard = 110 Elf are introduced as protective measures. The resistor prevents a large current flow through the sample, and the capacitor isolates the Q-meter from the large DC voltage source, The capacitance of the sample holder, Ch, is approximately 24 Elf, the capacitance of the sample, Co, is about 15-6Opi.fo The actual sample capacitance can be easily computed by determining the capacitance Co as measured across the Q-meter terminals and the capacitance of the sample holder (microscope slide with foil leads, heating stage posts, and coax cables leading to the Q-meter, Figure 4.2 and 45)8 Cstandard andard c —— y|!C Co Q-meter B Figure 4,4. Circuit for Capacitance Measurements Let Co be the measured capacitance across the Q-meter terminals C-D, let CST be the standard capacitance, CH the capacitance of the sample holder,

-44and CS the actual capacitance of the sample. Let the impedance across the Q-meter terminals be Z -J - (4.1) ZCD = C= ZAB C (4.) Then the impedance across AB (see Figure 4.4) is ZAB -= j (4.2) o)(CH+Cs) After combining Equations 4.1 and 4.2, -J _ -j -J (4.3) Co HU (C+Cs) CCST Therefore 1 - 1 +1 (4.4)'o CH+CS CST and CST + CH + S (45) co CST (CH + CS) Hlence the actual capacitance of the sample is given by C0 (CST + CH) - CST CH CS = S CST -C (4.6 CST _ Co Prior to taking measurements, the sample is heated and cooled a number of times through its peak permittivity temperature. Capacitance measurements are then taken on the sample at zero field and three different voltages. Simultaneously, the thin end of the material is observed optically and measurements of light transmission are taken with a cadmium sulfide photocell placed directly over the microscope eyepiece (Figure 4.3).

-4535 Intensity Measurements Intensity measurements of the light transmitted by a thin section between crossed polars give an indication of the integrated birefringence of the ferroelectric grains under observation. The measured intensities can be empirically correlated with the proportion of ferroelectric grains in the mixed titanate samples, as shown in Section VI. While observing a barium strontium titanate ceramic, the following assumptions are made: 1o A total extinction of the optical field indicates a completely isotropic, paraelectric material. 2. Maximum brightness of the optical field indicates an extensively ferroelectric material in which all grains are tetragonal. 3. In the region of the peak permittivity the percent of light transmitted by the thin section can be correlated with the percentage of ferroelectric grains in the sample. The light measurements were taken with a cadmium sulfide photocell. Microammeter readings of the light transmitted by a sample were compared with readings obtained from a calibrated Kodak density film strip. For this purpose the strip was placed in the same position as the sample. During the measurements of both film strip and sample, the intensity of the light source was held as nearly constant as possible, The graded densities of the film yielded successive microammeter readings which were then converted to a calibration curve of percent transmission vs microamperes, (See Appendix B).

-46The entire light detecting device consists of the cell, a 75 volt battery, a microammeter, and several resistances (Figure 4,5)~ The light source of the microscope, a tungsten filament lamp, appeared to have a steady light output. As a precaution, a regulating transformer was introduced between the main line and the microscope. At times a slight drift was observed in the voltage output of the mains over a period of several hours. To correct for this change, the photocell was calibrated against the density strip before and after every run, and appropriate adjustments were made in the intensity values of the samples measured. (See Appendix B.) 2 mego 1 meg. 75 v ___ Cell Figure 4,5. Circuit for Light Intensity Measurements by a Photocell.

V DATA AND OBSERVATIONS The material in this section has been classified under two main headings: "Data" and "Observationso" "Data," taken on ceramic wedges of mixed barium-strontium titanate compositions, contain the results of the capacitance versus temperature and the percent light transmission versus temperature measurements, with voltage as a parameter. A distinction is made between large-grained (50 - 150p) and fine-grained (3 - 15p) ceramics because of some important differences in their dielectric as well as optical behavior, (None of the investigated samples consisted of grains of an intermediate size, i.e., approximately 15 - 50 4o) The data are analyzed with respect to two proposed explanations of the field produced non-linear dielectric effect in ceramic titanate ferroelectricso They are: 1o Nonlinearity due to the alignment of ferroelectric domains in the direction of an applied DC field, and 2. Nonlinearity due to a field induced ferroelectric effect in the peak permittivity region of the ceramic. "Observations" include all optical phenomena that were observed in individual grains during the course of this investigation. They are confined to the thin section end of large-grained ceramic wedges where the grains are readily visible by transmitted polarized lighto The fine-grained "thin sections" are almost opaque in transmitted light due to the light scattered at the many grain boundaries in the ceramic, The changes produced in the large-grained samples with varying temperature and field have been classified as (1) short term and (2) long -47

-48term effects, They are defined in the following manner: 1o Short term effects occur instantaneously when a sample is exposed to a change in temperature or to an applied DC field. As soon as temperature and field are returned to their initial condition, the ceramic reverts to its original state. 2o Long term effects are also brought about by an applied voltage or by a temperature change, but they persist for some time after the original values of external condition have been restoredo Both effects are discussed. in some detail and interpretations have been provided wherever possible, A, Data 1, Presentation of Data Figures 5,1 - 5o13 illustrate the data taken on ceramic wedges of barium-strontium titanate compositions, They represent the capacitance versus temperature of the samples, with a DC field as a parametero For the sake of visual clarity, no data points have been shown on the curveso The points lie close together, especially on either side of the peak incremental permittivity temperature, where they tend to crowd the figures, The data for each sample is tabulated in Appendix A, Furthermore, to facilitate comparison of the figures, the heating and cooling curves obtained for each set of measurements have been averaged into a single curve, This is considered permissible for the

D.C. FIELD APPLIED TO SAMPLE 3000 (1) 0 VOLTS/CM. (2) 5,700 VOLTS/CM. (3) 8,700 VOLTS/CM. (4)14,200 VOLTS/CM. _ 12000 w (I) -J (2) H~-~ ~ (3) z W (4) cr_ o 1000 z 20 30 40 50 60 70 80 90 100 11 120 TEMPERATURE, ~C Figure 5.1. Incremental Permittivity vs. Temperature of Ceramic Sample 1, with Voltage as a Parameter.

......... 1-' - l I 1 24 D.C. FIELD APPLIED TO SAMPLE 22 (4) (1) 0 VOLTS/CM. 20- (3) (2) 6,900 VOLTS/CM. (2) - 18 - (I) i\ (3) 9,500 VOLTS/CM. A> ~16\\ (4) 17,000 VOLTS/CM. 00 14 z 12 20 30 40 50 60 70 80 90 100 110 120.I -'8 6 TEMPERATU.RE ~C 2 L0. 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, 0C Figure 5.2. Percent Light Transmission vs. Temperature of Ceramic Sample 1, with Voltage as a Parameter.

5000 I I I I D.C. FIELD APPLIED TO SAMPLE ( I) 0 VOLTS/CM. (2) 6,600 VOLTS/CM. 4000 (3) 9,200 VOLTS/CM. (4) 13,200 VOLTS/CM. 3000 ~-J Hz~ ~ (2) z 2000 (3) 0r (4) z 1000 I i I i 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE, ~C Figure 5.3. Incremental Permittivity vs. Temperature of Ceramic Sample 2, with Voltage as a Parameter.

14 I I I I D.C. FIELD APPLIED TO SAMPLE 19 - (4) 12 - (4) (I) 0 VOLTS/CM, (3) (2) - (2) 5,400 VOLTS/CM. I0 (3) 7,600 VOLTS/CM. z 0 -X (4) 10,900 VOLTS/CM. u,) 2 8 z I Iu l l I - cr r 0 AI i i i i i 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE ~C Figure 5.4. Percent Light Transmission vs. Temperature of Ceramic Sample 2, with Voltage as a Parameter.

5500 D.C. FIELD APPLIED TO SAMPLE (1) 0 VOLTS/CM. (2) 4,500 VOLTS/CM. (3) 7,000 VOLTS/CM. 4500 (4) 10, 500 VOLTS/CM.'U<! I3500 - -J 1500 _ _ ____ i _ i ______ z U 2500 z (3) (4) 1500 II 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, ~C Figure 5.5. Incremental Permittivity vs. Temperature of Ceramic Sample 3, with Voltage as a Parameter.

20 I —-i - i - (4) D.C. FIELD APPLIED TO SAMPLE (I) 0 VOLTS/CM. (3) (2) 4,500 VOLTS/CM. (2) - ^' (2) ^s^~~~ \ \(3) 7,000 VOLTS/CM. 15 (1) (4) 10,700 VOLTS/CM. 0 C, (n z 4 ir -j 10 z w W a. 53 1 20 30 40 50 60 70 80 90 100 10 120 TEMPERATURE, OC Figure 5.6. Percent Light Transmission vs. Temperature of Ceramic Sample 3, with Voltage as a Parameter.

I I I I I I I i I I 1200 D.C. FIELD APPLIED TO SAMPLE (I) 0 VOLTS/CM. 1100 (2) 6,000 VOLTS/CM. 4<: (3) 10,500 VOLTS/CM. 1000 (4) 15,000 VOLTS/CM. 900 w 800 z w uJ 700 0 z (I) 600- (2) (3) (4) 500 I I 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE, ~C Figure 5.7. Incremental Permittivity vs. Temperature of Ceramic Sample 4, with Voltage as a Parameter.

15 I-I i -I - i -! I I D.C. FIELD APPLIED TO SAMPLE (1) 0 VOLTS/CM. (2) 4,200 VOLTS/CM. (3) 7,500 VOLTS/CM. Z 10\ (4) 10,800 VOLTS/CM. 0 U) I z 4) (2) 3: 50. 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE, ~C Figure 5.8. Percent Light Transmission vs. Temperature of Ceramic Sample 4, with Voltage as a Parameter.

D.C. FIELD APPLIED TO SAMPLE (I) 0 VOLTS/CM. 2000 - (2) 12,200 VOLTS/CM. (I)1 V\~~~~~U ^^ —^^ ~~~~~~(3) 17,100 VOLTS/CM. 2^~~ ~~ ^^\^1~~~~~~ 1(4) 24,400 VOLTS/CM. F-~~~~~2 I- ^.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1500 F- ~~~~3) z w 2 W (4) 0 z 1000 I 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE, 0C Figure 5.9. Incremental Permittivity vs. Temperature of Ceramic Sample 5, with Voltage as a Parameter.

(4) 45 - D.C. FIELD APPLIED TO SAMPLE (I) 0 VOLTS/CM. z | (2) 16,600 VOLTS/CM. 0 4.0 Cn (3) (3) 23,300 VOLTS/CM. o) ( \ (4) 33,000 VOLTS/CM. z 3.5 (2) Q 3. 5 J1 2.0 20 30 40 50 60 70 80 90 100 110 120 130 TEMPERATURE, ~C Figure 5.10. Percent Light Transmission vs. Temperature of Ceramic Sample 5, with Voltage as a Parameter.

D.C. FIELD APPLIED TO SAMPLE 1200p (I) 0 VOLTS/CM. (2) 10,000 VOLTS/CM. (3) 14,400 VOLTS/CM. 1 I100 (4) 19,000 VOLTS/CM. 1- 1000 s () — -- A== (3) H 800 I I- (4) Z 700 600 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, 0C Figure 5.I1- Incremental Permittivity vs. Temperature of Ceramic Sample 6, with Voltage as a Parameter. with Volt~age as a, Parameter.

6 - D.C. FIELD APPLIED TO SAMPLE (I) 0 VOLTS/CM. 5 - (2) 10,900 VOLTS/CM. (3) 15,600 VOLTS/CM. (4) 4 (4)20,600 VOLTS/CM. 00 4 (3) Tn (2) 2 () 0| Q< 3 0. a. 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, ~C Figure 5.12. Polarization as a Function of Temperature, with and without a Biasing Field.

5 D.C. FIELD APPLIED TO SAMPLE (I) O VOLTS/CM. (2) 7,000 VOLTS/CM. Z (3)16,800 VOLTS/CM. 0 C) l) 4 2E (3) o') z 0 1- 3 z 0 UJ a. 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, oc Figure 5.13. Percent Light Transmission vs. Temperature of Ceramic Sample 7, with Voltage as a Parameter.

-62following reasons: 1, Although a temperature hysteresis is observed in the samples, the effect is relatively small. Depending on the composition, it ranges from 0 - 3~ between heating and cooling cycles. 2o The peak incremental permittivity in all samples is somewhat larger during the cooling cycle than the heating cycle, regardless of whether a field is applied to the sample or noto Condensing the heating and cooling curves into a single curve therefore does not change the relative dielectric behavior of the samples with respect to each other, 3o The light transmission data do not show any consistent hysteresis effect, The differences in the heating and cooling curves lie within the range of the estimated experimental error, The percent light transmission data is therefore presented as the best curve drawn between the points of measurement, Appendix A shows tables of the point by point capacitance and percent light transmission measurements taken on the ceramic sampleso Appendix C contains a sample calculation which illustrates the method by which the data were reduced, 2, Analysis of Curves From the permittivity versus temperature curves with voltage as a parameter it can be seen that an applied DC field has the effect of: 1) reducing the value of the peak incremental permittivity, and

-63 2) shifting the peak incremental permittivity to higher temperatures. The effect of the field on the percent light transmission of the ceramics, the large grained samples in particular, is 1) a considerable increase in anisotropy, especially in the region of the tetragonal to cubic transition, 2) a shift of the curve of percent light transmission toward higher temperatureso The samples measured can be divided into two categories: one containing ceramics with large grains, the other containing fine-grained material. Thin sections of the first category in which the grain size varies from 50 - 150l are composed of a monolayer of grains. Thin sections of the second category made from the fine grained material with grain sizes ranging from about 3 to 15., consist of several layers of grains. Both groups show a similar trend in the dielectric and optical behavior with temperature and fieldo However, the light transmitted by the fine-grained samples is substantially reduced due to the large amount of scattering that takes place in the fine-grained complex, Sample 1 (see Figures 5ol and 5.2), Sample 2 (see Figures 5.3 and 5~4), Sample 3 (see Figures 5.5 and 5.6), and Sample 4 (see Figures 507 and 5.8) belong to the first large-grained category. Sample 5, a commercial barium-strontium titanate ceramic (see Figures 5.9 and 5.10), Sample 6 (see figures 5o11 and 5,12), and Sample 7 (see Figure 5.13) represent the fine-grained category. For a listing of the chemical composition of the samples see Table 4,1,

-64ao Large Grained Ceramics Figures 5.1 - 5.8 represent the capacitance and percent light transmission data taken on four large-grained ceramic sampleso It is seen in Figures 5,1, 5~3, 5.5 and 5.7, that large fields applied to the samples markedly reduce their peak permittivitieso The permittivity and percent light transmission curves also experience a shift to higher temperatures with increasing fields. Two mechanisms have been proposed in the literature to account for the field produced, non-linear dielectric behavior in perovskite-type ferroelectrics, such as solid solutions of barium-strontium titanate (see Section II-C and Section III-G, They will be briefly discussed here as 1) the mechanism of domain alignment, and 2) the mechanism of induced ferroelectricityo According to the mechanism of domain alignment, the polar vectors in domains of ferroelectric ceramic grains tend to orient themselves in the direction of an applied field, In this respect, the ceramic is thought to behave similar to a single crystal of barium titanateo A twinned barium. titanate crystal, for example, can be ideally transformed into one of uniform polarization by applying sufficiently large fields (68,69) As a result of the proposed "alignment" of the polar vectors in the field direction, a pronounced decrease in the permittivity of.the ceramic can be expected throughout the tetragonal, ferroelectric regiono This follows from the dielectric anisotropy of the material. The permittivity along the c-axis of a single crystal of barium titanate permittivity ec; 200, and and the permittivity along the a-axis ca \ 4000,

-65The second model attributes the lowering of the ceramic peak permittivity to a field induced ferroelectric state in the paraelectric grains in the ceramic. The domains in the ferroelectric grains are considered relatively static, 90~ domain alignment does not occur according to this theory, although the existence of 180~ domain alignment, i.e., a 180~ reversal of the polar vector in the domains is recognized. 180~ domain switching has been actually observed in ceramics by DeVries and Burke (112) However, a 180~ reversal of the polar vectors in the ceramic does not change the permittivity of the ferroelectric, 180~ domain alignment alone therefore does not explain the observed field non-linearity in the ceramic. According to the second mechanism, the Curie temperatures of the grains composing the ceramic are distributed about the peak permittivity temperature in a Gaussian distribution, the exact shape of which is determined by the history of the material, The grains do not all have the identical transition temperature; their transitions occur over a temperature range. Within this range, both ferroelectric and paraelectric grains can be found side by side, and a large external field produces a. ferroelectric state in the paraelectric grains. The polar vector formed in the grains is considered to lie in the direction of the applied field. It can be shown that the total imcremental permittivity of the ceramic drops as a result of the applied field in this temperature region, The model also predicts the observed shift in the peak permittivity temperature with field. This shift has a parallel in the increase in the tetragonal to cubic transition temperature of a single crystal of barium titanate as a result of a large external field,(78)

-66In the following the dielectric and optical results obtained on the large-grained samples are examined in detail with respect to any evidence of either domain alignment or an induced ferroelectric state in the ceramic grains. In order to compare the four different ceramic compositions, a reference point, namely the peak incremental permittivity temperature of each composition, is chosen.Criteria are set up at temperature intervals both below and above the peak incremental permittivity temperature. These aim at distinguishing between the possible effects of domain alignment and/or induced ferroelectricity on the behavior of the ceramicso They are compared with the actually observed changes in the incremental permittivity and percent light transmission data that has been brought about by an applied field. The temperature points at which the ceramics are examined in detail are: 1) 50~C below the peak incremental permittivity temperature 2) 25~C below the peak incremental permittivity temperature 5) 6~C above the peak incremental permittivity temperature 4) 25~C above the peak incremental permittivity temperature 5) 50~C above the peak incremental permittivity temperature 1. 50~C Below the Peak Incremental Permittivity Temperature At 50~C below the peak incremental permittivity temperature, it may be safely assumed that almost all grains are ferroelectrico It has been estimated that at least half of the crystallites composing the largegrained samples have transitions that occur within + 15~C of the peak incremental permittivity temperature of the ceramic.

-67If the domains tend to reorient themselves in the direction of the field as a result of 90~ domain motion, a substantial increase in the percent light transmission of the thin section can be expected. The increase in the percent light transmission is the result of a rotation of the crystallites into a position of maximum anisotropy with respect to the crossed polars.* On the other hand, if the domains are sufficiently clamped so that 90~ domain alignment with the accompanying mechanical strains cannot occur, then the effect of the field would be very small, 2, 25~C Below the Peak Incremental Permittivity Temperature The ceramic compositions here are largely ferroelectric, However, as a result of the distribution of transition temperatures of the individual crystallites, there may be a smaller number of grains that are paraelectric, For the case of domain orientation, the same arguments as above holdo For the case of induced ferroelectricity, a small overall decrease in the incremental permittivity and increase in the percent light transmission of the ceramic can be expected, due to the effect of the field on the paraelectric grains, 35 6~C Above the Peak Incremental Permittivity Temperature Slightly beyond the peak incremental permittivity temperature, the greater part of grains composing the ceramic are in the cubic state, If 90~ domain orientation occurs with field, it is restricted to a small, ferroelectric fraction of the grains. As a result, the capacitance of the sample is expected to decrease and the percent light transmission is expected to increase by a small amount. On the other hand, if * The direction of the applied DC field is at about 45~ with respect to the privileged directions of the polars,

-68ferroelectricity is induced in the paraelectric grains, it will be seen as optical anisotropy in the formerly cubic grains. This will result in a large increase in the percent of light transmitted by the thin section as compared to the zero field caseo The overall incremental permittivity of the sample is generally expected to decrease with strong applied fields. The above anticipated results from the mechanisms of both domain alignment and induced ferroelectricity differ from each other mainly in a matter of degree, It may therefore be difficult to interpret the data in terms of either one or the other postulateo In the latter part of this section some microscopic observations on individual grains are described, which provide a useful aid to the analysis of the incremental permittivity and light transmission curves. 4. 25~C Above the Peak Incremental Permittivity Temperature The ceramic is composed mostly of cubic grains at this temperature The process of domain alignment can hardly take place. In terms of induced ferroelectricity, the effect of the field is also moderate, since ferroelectricity can no longer be induced in a large number of grainso (The range within which a ferroelectric state can be induced in a single crystal of barium titanate is derived from the thermodynamic relationships of the system (see Figure 6o12)o 5o 50~C Above the Peak Incremental Permittivity Temperature At this temperature, almost all grains are isotropic. A strong field is expected to produce little change in the optical and dielectric properties of the dielectric,

-69In the following paragraphs the dielectric and optical data obtained from the large-grained samples are analyzed, particularly with respect to the criteria established for distinguishing between the processes of domain alignment and an induced ferroelectric state in the ceramic grains. The curves of the incremental permittivity versus temperature and DC field are shown in Figures 5.1, 553, 5.5 and 5.7. There is a relatively small decrease in the incremental permittivity with field at 50~C (Figure 5,7), and at 25~C (Figures 5.1, 553, 5~5 and 5,7) below the peak incremental permittivity temperature. The change at 6~C (Figures 5o1, 503, 505 and 507) above the peak incremental permittivity temperature is very large. Changes at 25~C and 50~C above the peak incremental permittivity temperature are respectively small and very small (Figure 5,1, 5.3, 5~5 and 5~7). From these observations it can be concluded that the incremental permittivity data taken on the large-grained ceramic compositions approximates the requirements proposed for the mechanism of induced ferroelectricityo The data show little evidence of any large field produced changes in the strictly ferroelectric region of the incremental permittivity curves, Such a result would be expected from the mechanism of domain alignment, At no point are the changes in the incremental permittivity very large except at the permittivity peak. In Table 5.1, the percent increase in light transmission is listed for thin sections of the large-grained ceramic at several temperatures, In general, the greatest change in light transmission with field occurs slightly above the peak permittivity temperature, in which case the samples are

TABLE 5,1 INCREASE IN PERCENT LIGHT TRANSMISSIO N IN LARGEGRAINED MIXED TITANATE COMPOSITIONS PRODUCED BY STRONG DC FIELDS Increase in the Absolute % Light Transmission |Maximum |Applied Produced by the Field, Given at Temperatures on Permittivity Max, DC Sample Either Side of the Peak EA Temperature* Temperature, Field, Thickness, Sample ~C Volts/cm Microns -50~ -25~ + 6~ + 25c + 50~ 1 49 17,000 4o --- 2.9 6.6 35 1o2 2 51 10o900 35 --- 2.0 3.6.8 o1 3 52 10,700 35 -- 3.8 8.6 2,2 o5 4 83 _10,800 30 0.4 535.4 * The temperature of the peak incremental permittivity of each sample is taken as its reference temperature (0~)o The negative temperatures indicate the ferroelectric side of the % light transmission curves. The positive temperatures indicate the paraelectric side. See figures 5.2, 5,o4 5.6, 5,80

-71largely optically isotropic. This is seen very.well in the light transmission curves (see Figures 5o2, 5,4, 5,6 and 5.8). The increase in light intensity with field in the ferroelectric region of the compositions is distinctly noticeable, except for Sample 4, in which the effect is smallo In this connection it was observed that strong fields have little effect on the birefringence of a pure barium titanate ceramic at room temperature (barium titanate has a tetragonal to cubic transition at 120~C)o Changes were noted only when barium titanate was heated to about 90~Co Sample 4, which has a peak incremental permittivity temperature at 83~C, is similar to barium titanate in that a field produces few optical changes at temperatures well below the peak incremental permittivity temperature (Figure 5.8). In the region of the tetragonal to cubic transition the effect of the field on the percent light transmission is large. The optical data indicate the existence of an essentially isotropic ceramic at 6~C above the peak incremental permittivity temperatureo After applying to a strong field, however, the percent light transmission approaches that of the ceramic in the ferroelectric region. Photographs taken on Sample 3 (Figure 5014) show the field effect on the ceramic at room temperature and at 55~Co At the lower temperature (Figures 5.14a, 5.15b) the ceramic is ferroelectric, at the higher (Figures 5,14c, 5o14d) it is at its peak incremental permitivity temperature. The field induced optical anistropy makes the sample appear as bright at 55~C as it had been at room temperature. At temperatures considerably above that of the peak permittivity, the field effect is very small,

-'TKA - -N M ~ ~ ~ ~..........~~~~~. I *~ a) 25 oC, zoro app.ioci t'iolx b) 2500C, 127000 voit~i/om I. M OM E-,kR,: 0*i c) >VOe, zexro alpQH~dcle vij) 550(1 1A2000 rx 51.lu~ ~lk.lc )'~(Tect oX' at 8tf~ong ))0 iFioJ~doa -t~h Qptiel Anis~rotropy~ oX' a }!'~orrooltrio Goxsafa~tc, $u ~1, ~I5 )?oEalk inrcx~enoa1 J), pemitt iv$ity at 53jQ~f, 0( al~l~ -Mp botwooolln Mrtredxnrs Mg~3T~

-73At 25~C below the peak incremental permittivity temperature, a small, distinct field produced increase in intensity transmitted by the ceramic leads one to believe that the anisotropy of the ferroelectric grains is affected by strong fields, The influence of a field on the optical behavior of individual grains is discussed in the latter part of this section0 bo Fine-Grained Samples The fine-grained Samples 5 and 6 (Figures 5.9 and 5o11) are characterized by very broad incremental permittivity peaks and a finegrained structure. Incremental permittivity data for Sample 7 are lacking, Accurate capacitance measurements could not be made on Sample 7, since the only sample available was a thin section, not a wedge. Sample 7 is nevertheless included in the data because it represents one of the few fine-grained ferroelectric ceramics prepared at the Electro-Materials Laboratory, Electrical Engineering Department, The University of Michigan. In many ways it resembles the commercial ceramic Sample 5. The dielectric data of the fine-grained ceramics show the typical lowering of the incremental permittivity with field and a discernible shifting of the peak permittivity to higher temperaturesO However, the percent-transmission data at best only give a small indication of the actual optical behavior of the ceramic, since the light traversing several layers of the ferroelectric is absorbed and scattered in the material. This explains the very low percent transmission values measured. No evidence of domain structure can be observed in the individual grains in the ferroelectric region. In Sample 6 which "peaks" around

-74480~C entire anisotropic patches covering large areas of the sample can be seen (Figure 5o15a). These patches are greatly enlarged when a field is applied (Figure 5.15b)o As the thin section is heated, the number of patches gradually decreases, but the sample never extinguishes completely. (It appears that a strain-induced anisotropy exists in the fine-grained ceramics which is independent of temperature.) When a strong DC field is applied to Sample 6 at 78~C, 300C above its peak incremental permittivity temperature, the microscope field becomes brighter, although not to the same degree as at the peak incremental permittivity temperature, At about 1100C the field induced anisotropy can no longer be detected. The strain anisotropy, however, still persists. Samples 5 and 7 behave in a similar manner In an x-ray and electron diffraction study on barium titanate, composed of colloidal particles, Anliker, Brugger and Kaenzig(ll7) have found that the ferroelectric to cubic phase is increasingly spread out over a wider temperature range as the particle size decreaseso They report that the spontaneous strain, which is normally observed in the tetragonal phase, does not vanish completely even at several hundred degrees above the-Curie point of the microscopic crystal0 They attribute 0 the anomalous effect to a highly strained surface layer of about 100 A. The spontaneous tetragonal strain in this layer is independent of temperature and larger than the spontaneous strain in the interior of the small particle. Fatuzzo and Merz(ll8) have recently investigated the surfaces of single crystals, and have found a thin, electrically biased and mechanically strained layer which has a much smaller permittivity than the bulk

-75a) Peak incremental pexrmitti.v.lty b) Peak incremental permitti-. vity temperature, zero app.:l.ied fi.eld temperat-ure, 15000 vol.ts/cm c) 30~ above the peak incremental d) 50~ above the peak i. ncrementa. pe''rmi~.t:ti.v~i.ty -temperatcure, vzo permkit-ivi'ty temperature, app:l.ied fiend 15000 vol.ts/cm' F.:gutre 5'.'5.,Effect of a StirongDC Fi..e..d on the O)ti.cal Aniso....o-tropy of a F.ine-.s.Gra!.ned SCeramnfc, S:ample 6. SaEple between crossed pol.ars (Mag. no' M55X)

-76of the crystal. Io I. Ivanchik(ll9) mentions that ferroelectricity disappears completely if the thickness of a particle grain is less than 200 Ao This seems to contradict the findings of Anliker et alo(17) who report the persistence of the Curie point in fine-grained particles upto very high temperatureso The high spontaneous strain at the surface of the small particles gives rise to.their anomalous ferroelectric behavioro It is possible that the discrepancy arises from a difference in terminology. Ivanchik may not have chosen to describe the observed anomalous dielectric behavior as "ferroelectricityo" The fine-grained, mixed titanate materials investigated here are composed of grains 3 - 15.in diameter. They consist of considerably larger particles than would be considered colloidal. However, the broad peak in the dielectric data and the observed strain anisotropy seem to suggest that surface effects are responsible for their anomalous ferroelectric behavior. Bo Observations In. the search for some indication of domain alignment and/or induced ferroelectricity in the compositions investigated, many field produced phenomena were observed in individual grains which could not be isolated by the integrated light measurements of the photocell. They are discussed in some detail belowo The effects are classified as short term and long term. 1.Short Term Effects Optical changes in ferroelectric thin sections which are produced by a short exposure to an external DC fiel nd disappear when the field is removed are considered short term effectso

-77As mentioned previously, the effect of a strong DC field on the optical behavior of a ferroelectric titanate thin section at temperatures substantially below that of the peak incremental permittivity appears to be negligible, The room temperature observations on ceramic barium titanate and Sample 4 which "peak" at 120~C and 83~C respectively show no distinct changes in birefringence or in crystallographic orientation with field. However, in the temperature range approximately 25~ below the peak incremental permittivity, a number of field induced effects are observed in the large grained compositions: 1) The field noticesbly increases the interference colors in some grains. In Sample 1, 404 in thickness, the field changes a first order yellow into a first order red-blue. (This corresponds to a change of birefringence of o010 to o015 with fieldo) Similarly, in Sample 3, 355 thick, a first order yellow changes to a second order blue-green with fieldo (The corresponding increase in birefringence is almost o010) 2) Twinning striations tend to disappear when the sample is heated close to its peak incremental permittivity temperatureo These striations are due to 90~ domains or twinning across the (101) planeo On heating, the parallel twin lines gradually become less well defined, and then disappear completely, Often the area which the striations have occupied at low temperatures appears to be of uniform color when heated. The color is usually a first order white or yellow.

-78Then, when a high DC field is applied to the sample, the twinning striations may reappear in their original direction and number, When heating the biased ceramic further, close to its transition temperatures, the field produced striations disappear. They are similarly replaced by areas of uniform interference coloro The reverse effect, an actual disappearing of twin lines with field, is very rarely observed, In Figures 5o16a and 5.16e grain 1, the twinning striations are faint lines which do not all extend across the grain. As seen in Figure 5,16b, they disappear completely with the applied field, In the neighboring grain, Figures 5,16a and 5,16e, grain 2, where the striations are at 90~ to the former, the field seems to lengthen the twinning lamellaeo 5) Although the field tends to increase the overall light transmitted by a sample, some isolated grains become dark when a field is applied, 4) In some instances the "brightening" of a grain is not instantaneous, but seems to saturate over a time period that is noticeable by eye, The period is about 53 to 1 second, Also, a movement of dark, irregular lines within a grain has been frequently observed, Field induced effects of this latter type are very difficult to analyze since they are not repro-. ducible, A detailed discussion of the above effects follows,

-79I'::: ~ ~ ~ ~ ~ ~ ~ ~ ~ *v 4~j~~ *~ ~i~~~~~~~~~~~~~~~~~~~~~~~~~:i ~.I:~~~~~.'....n~.n tu'u o C. 14 mple bet,.,teen para].*.e]. po~ars. (Maf~.c:~:;~'j20X) fri.. M N'~~~~1~h "~ ** ii.- ~ * j'. N'~~ +' ~~~~~~~~~~~~~~~~~~~~~:' ~~~~~~~::'' i P: i~~~i ib~~. ~'~*i~ i* 1* ~ ~~~~~~~~~~~ *'\ i~~~'4::::.:X*'**..~~~~~~~~~~~~~~~~9~~~ -:i ~ ~ ~ ~ ~ *z**'i*NN:* V:i ~ ~.*V*~~*.s\ N..' N ~~~ir~~~~~t~":~~~~~ ~:'N:N c) Afr~tor pol~ing. Pooin tolinporaLtuo dl) Aft~er peA ing. Pioomn'L 1pera lure, no f:Leid app~l~i.edt fie~ld ap'pliedcc F'igure 5. 16. Vflfeet; of' Polijng, on'Lhe (:i:oi)'m inni~ng C~rue~ Ltur~e of' a Z'Verroel7e l~r~ic ~Ceromie, 8anpl~o 1. 8ainplec belweeon pu rallel pol~a F;. (Mag. ~< 20X

844* -80ARROW SHOWS DIRECTION OF APPLIED FIELD e) Outline of grains in photographs 5.16 a, b, c, and d. Figure 5.16. (Continued)

-81a. Discussion of Observed Phenomena The optical observation on ferroelectric, ceramic thin sections seems to indicate that their optical anisotropy is both a function of temperature and of an external DC field. Unfortunately, no experimental data has yet been published on the optical properties of good barium-strontium titanate crystals, containing more than 10% strontium titanate. It is therefore necessary to rely on the observations made on ceramic materials in the course of the present investigation. Because of the nature of the material, these observations can only be descriptive. Nevertheless, it is possible to make some general statements about the optical behavior of the mixed titanate ferroelectrics with respect to temperature and electric field. (1) Birefringence as a Function of Temperature The birefringence in mixed barium-strontium titanate compositions appears to be lower than in pure barium titanate (Figure 5.17). For barium titanate it increases from approximately.025 at 120~C, the tetragonal to cubic transition temperature, to.060 at 00C.(84) (Meyerhofer(l20) reports an increase from.040 to.075 for the same range.) The highest interference color observed on the ceramic Sample 4, at room temperature, is a first order green for a thin section 304 thick. The highest interference color observed at 83~, in the region of the tetragonal to cubic phase transition, is a first order white. Hence the change in birefringence is about.022 for a 60~ range. In Figure 5o17c the curve of Sample 4 has been shifted along the temperature axis so that its transition temperature at 830C coincides with the transition temperature of pure BaTiO3.

-820.08 i 0.07 0.06 - 0.05 - 0.04 - U 002 0.03 - 0.01 - TEMPERATURE, C -0020 0 20 40 60 80 0 i i i I I 0 20 40 60 80 100 120 ORTHORHOMBIC TETRAGONAL TETRAGONAL TEMPERATURE, ~C CUBIC TRANSITION TRANSITION Figure 5.17. Variation of Birefringence with Temperature in Tetragonal Barium Titanate and (Ba,Sr)Ti(5 Ceramic.

-83(2) Birefringence as a Function of DC Field Observations on ceramic barium titanate and Sample 4 indicate that the field has very little effect on the birefringence at temperatures substantially below the transition temperature; but within a range of about 20~C to 25~C of the peak incremental permittivity temperature, the field increases the optical anisotropy of the ceramic grainso At this point it should be mentioned that the electric field across the sample was placed at 30 to 45~ with respect to the analyzer and polarizero Any changes in the ceramic which the field might produce would in this way be readily observableo The increase in anisotropy results in a general brightening of the entire thin section, Closer examination of the anisotropic effect seems to indicate that this is partially due to an induced anisotropy in a few paraelectric areaso It is also due to an increase in birefringence in some ferroelectric grains, when a strong DC field (of about 11,000 volts/cm for Sample 4), is applied to the ceramic~ Higher interference colors appear with field than are observed at zero biaso In an aggregate of randomly oriented grains which show a first order grey, whites yellow' an.l rd at zero field, a second order blue cai. be detected when a field is appliedo The higher order colors are not due to a more favorable orientation of the polar axes in the field direction, since they represent a birefringence which is larger than that of the highest interfence color found among the ceramic grains at zero fieldo The estimated change in An.of o005 with field corresponds to a change of temperature of about 14~C (see Figure 5o17c)o

-84(3) The Effect of a Field on the Twinning Lamellae Some areas which show twinning lamellae become areas of uniform interference colors at a temperature close to that of the transitiono When a strong DC field is applied to the sample, the lamellae reappear. As Little(69) has pointed out, large mechanical as well as electrical stresses exist at 90~ domain boundaries, This is the result of twinning along the face diagonal of a pseudocubic, tetragonal crystal. The resulting mechanical distortion explains why a 90~ domain wall is readily visible in transmitted light. Close to the transition temperature, a barium-strontium titanate ferroelectric very nearly approaches the cubic structure. The stresses are greatly reduced and the twin lamellae fade away, The twinned area then appears as a uniform region of first order white. It is possible that a strong field applied in a favorable direction will greatly stress one axis and thereby restore the anisotropy and mechanical distortion which led to the original 90~ twin formationo In addition, field-induced twinning seems to be very temperature sensitiveo A heating of only 2 to 3~C will again change the area into one of uniform polarizationo (4) Increase of Light Intensity with Time In some samples, the initial field-induced change in anisotropy seems to increase with timeo After several seconds, the observed transmitted light intensity saturateso An interpretation here is difficult. Perhaps the initial stressing of the optical axis' is followed by a small degree of orientation in the field direction,

-85On the other hand, the observed phenomena may be somehow or other associated with space-charge effects. For example, it is well known from classical electrodynamics that in a dielectric medium the space charge will diffuse according to ot P = pe where p is the space charge density, po the initial density, a the conductivity, E the permittivity (e = er co, with er the relative permittivity and Ec the permittivity of a vacuum)o In BaTi03, -11 farads a is of the order of 10-8(ohm-m)-1 and er _ 103 and o 10 farads m. so that t (the time-constant for space-charge effects) becomes 0lxlO11 1Q-8 1 sec This is of the same order as the observed transient ~0e effects. In order to examine this and other possibilities a more extensive study of these materials would have to be conducted. (5) Domain Rotation Domain rotation is difficult to detect. An increase in intensity with field may be either due to a rotation of the optic axis into the field direction (assuming that the field direction lies at about 45~ to the privileged directions of the polars), or due to a field produced increase in the birefringence of the ceramic grain (Section V-B-2). On the other hand, a darkening of a grain with field, at constant temperature, seems to indicate that he field has rotated the optic axis r to coincide more nearly with the privileged directions of the polars (Figure 5.18a).

-86P PP' = privileged direction of polarizer S1 AA' = privileged direction of analyzer \/S' EE' = direction of DC field \2 S S' = polar axis of domain AlA 9 1-'- 1 1 at zero field a2\7 / a \S S2S2 = polar axis of domain s/^\X~~~ \.~ ~with field S1 Figure 5.18a, Schematic: Rotation of the Polar Axis in a Ferroelectric Domain Produced by an Applied Field. Merz(68) was able to show the existence of anti-parallel 180~ walls by applying a strong field perpendicular to the direction of the spontaneous polarization (Figure 5.18b). The small rotation of the optic axes into the field direction produced a strain anisotropy at the domain boundaries which made 180~ domains visible, I ferroelectric vector at 0 field \/- --- ferroelectric vector with field \ 1/ direction of applied field Figure 5.18b. Field Produced Rotation of the Polar Axes in Anti-parallel Domains.

-87It is felt, from observations on the ceramic grains, that the isolated occurrences of "domain rotation" are not the result of a field produced nucleation and growth of domains. They seem to arise from a small field-produced distortion of the optic axes, similar to the one produced in Merz's experimento(68) This is illustrated in Figure 5.19, grain 5o An area is shown which appears relatively bright on zero field, but is dark at 18,000 volts/cm. The grain has been almost stressed into an extinction position by the fieldo In Sample 2 and Sample 4, areas become light as well as dark when a field is applied. These samples are highly twinned and also have pronounced "domain movement." Rotation was not readily detected on any other compositions, (6) "Domain Motion" The movement observed in mixed titanate ceramics, again near the transition temperature, is confined to black, irregular lines which are sometimes distributed over the grains (Figure 5,19 grain 4). The lines do not necessarily adhere to grain boundaries. They have been observed only in a few compositions and seem to be affected by humidity, frequency of exposure to the field, and proximity to the electrodes. Under the influence of a DC field, they rearrange themselves into new patterns within a second or less, Sometimes they delineate bright areas which either increase or shrink as the field is switched on and off, This type of "motion" has constituted one of the most perplexing problems in the present study, mainly because of the total lack of reproducibility, It

, i:: I~,~i,~.. "M a) 50C, ze?,o applied f ield b) 35~C0 18000 vots/acm c) Orain 4. Light grain with irregular dark linee, located approximately in center of photographs Grain 5 - Position as indicated in sketch, upper right hand corner of grain 4. Figure 5.19. Extinction of an Anisotropic Area in a aFeroeleotric Grain of Sample 1, Upon Applying a Strongt DO ield. Sample between crooeed polars. (Mag. 320X)

-89is now felt, that space charges are responsible for the movemento Space charges build up at the electrodes and in the interior of the ferroelectric when DC fields are used. The charge accumulation is especially serious near the Curie point. It depends largely on the type of electrical contact made to the crystal. Meyerhofer(l20) has found that crystals with tin oxide electrodes have less space charges than those with evaporated metal electrodes. A permanent space charge layer in the cubic state has been described by Chynoweth.(121) He reports that a cubic barium titanate single crystal carries a small degree of surface polarization, although its spontaneous polarization is zero. At times the space charge is even large enough to induce birefringence in a cubic crystal. 2o Long Term Effects Optical and structural changes which are produced in a ferroelectric by the prolonged exposure to an external DC field, and which remain after the field is removed are described as long term, or "poling" effectso Sample 1 provides a good example of the poling effect. A series of photographs were taken of the composition between parallel and crossedpolars, at different temperatures, and different applied electric fields, Repeated heating and cooling of a ceramic through its peak incremental permittivity temperature does not essentially change the original domain patterns. In some ways these patterns conform to an existing structural mold. Perhaps domains in single crystals nucleate at impurities and imperfections, and they are produced by the stresses existing at these

-90imperfectionso Ceramic crystallites are, in addition, geometrically confined by their grain boundaries. The photographs of Figure 5.16 Sample 1, show that the repeated exposure of the ceramic to a field has produced some permanent changes, although the wedge was heated 20~C beyond its peak incremental permittivity temperature in the process. The areas in which poling can be detected are primarily in grains 1 and 2. Parallel twin lines are faintly visible, The grains are about to become isotropic, given another five degrees of heating. Twinning in grain 2 is perpendicular to the twinning in grain 1. When the field is applied, twinning completely disappears in grain 1 and is enhanced in grain 2. After exposing the sample intermittently to the field for a total of about 30 minutes, photographs Figure 5o16c were taken. Now grain 1 hardly shows any twin lamellae at zero field and the ones in grain 2 seem more pronounced. Grain 1 in photographs b and c have become almost identical. A high applied DC field does not alter the picture substantially, except that it increases the lines in grain 2, Figure 5.16d, No visible poling has occurred in grain 6. In grain 7, all twin lines seem to have been poled in one direction, Another example of poling is illustrated in the sketches drawn from Sample 1 before and after a 20 minute exposure to a field of 17,000 volts/cm at room temperature (Figure 5.20). Aside from changes in direction of the twin lamellae, an increase in the interference colors of the grains was observed. These, unfortunately, cannot be recognized from the black and white photographs of Figure 5.16. Poling on other samples has not been observed to this extent. Sample 4 in particular seems insensitive to the field in its ferroelectric state, The reason for the variation in behavior of the different samples is not known,

-91ELECTRODES ELECTRODES (a) O- FIELD (b) 17,000 VOLTS/CM. FOR 20 MINUTES INTERFERENCE COLORS INTERFERENCE COLORS YELLOW, WHITE, GREEN, RED, YELLOW, GREY. WHITE, GREY. Figure 5.20. Poling in Sample 1. Thickness 40p.

VIo CALCULATION OF THE LIGHT INTENSITIES TRANSMITTED BY (Ba, Sr) TITANATE CERAMIC THIN SECTIONS Several aspects of the percent light transmission versus temperature data taken on the ceramic barium-strontium titanate thin sectionsare examined more closely in this section on calculations. First, the following basic question is posed: Given a ferroelectric barium-strontium titanate thin section, what percentage of the incident light can be theoretically expected to emerge from the analyzer if the section is placed between crossed polars? The ceramic section is considered to consist of a single layer of randomly oriented crystallites, all of which are in the ferroelectric, anisotropic phase, and have the same value of maximum birefrigence. Secondly, the percentage of light transmission versus temperature of a barium-strontium titanate thin section is calculated from theoretical considerations. It seems areasonable assumption that the transition temperatures of a large aggregate of crystallites composing some barium-strontium titanate composition should be distributed about the peak permittivity temperature of the ceramic in a Gaussian distribution. The distribution of transition temperatures results from the compositional inhomogeneities in the ceramic as well as from internal strains in the material which affect the Curie temperature of the crystallites. The percent of light transmitted by a thin section of a bariumstrontium titanate ceramic through crossed polars is closely related to the number of anisotropic, ferroelectric grains in the thin section. The number of ferroelectric grains existing at some temperature, T, may be -92

derived by integrating over the distribution curve of transition tempera tures from the lower limit up to the desired temperature (Figure 6.6). Wh.en the integration is performed for all temperatures, one obtains a relationship of the percentage of anisotropic grains in the ceramic with respect to temperature. It will be subsequently shown that such a curve is of the form (l-erf), where erf is the error function (Figure 6,7). The shape of the experimental percent light transmission curves, as shown in Figures 5.2, 5~4, 5.6, 5o8, 5.10, 5.12, 5.13, closely seem to resemble the function (1-erf). In order to make an exact comparison of the predicted and experimental percent light transmission data, theoretical light transmission curves have been calculated for four large grained samples, samples 1-4 The shift of the transmission curves with field, as predicted theoretically and determined experimentally have been compared in Figures 6.8, 6.9, 6.10, 6.11. The agreement between the theoretical and experimental curves for both samples is good. The calculations in this section fall under the following three headings: Ao Fraction of light transmitted by a (Ba, Sr) TiO3 ceramic thin section at some temperature, T, in the ferroelectric regiono B. Change with temperature in the relative amount of light transmitted by a barium-strontium titanate thin section. C, Change with temperature and applied DC field in the relative amount of light transmitted by a barium-strontium titanate thin section. Do Comparison of the relative theoretical and experimental light transmission curveso

-94The temperature range of interest and the one considered here lies between 20~C and 1400~C This range includes the tetragonal-ferroelectric to cubic-paraelectric transition only. None of the ferroelectric transitions which occur at lower temperatures (for barium titanate, the orthorhombic to tetragonal occurs at about OC, and the rhombohedral to orthorhombic at -80~C) have been taken into account here. The calculations are based on a rough physical model which does not fully describe the structural complexities in the actual ceramic. However, the model is a starting point for an analysis of the optial anisotropy which exists in these ceramics, and should provide at least a prediction of their qualitative behavior. Ao Fraction of Light Transmitted by a (Ba, Sr) TiO3 Ceramic Thin Section at Some Temperature, T, in the Ferroelectric Region The intensity of an emerging ray passing through a crystal section between polars has been given by Johannsen(122) as I = I[cos cp-sin2(-cp)sin2@sin2 n2 )] (see Figure 61) (61) 0 A A \ V P' P U' VA Figure 6.1. Orientation of the Privileged Directions of a Crystal in the Optical Field of a Polarizing Microscope.

-95AA' PPI are the privileged directions of the analyzer and polarizer, UU1, VVI are the privileged directions of the crystal examined, @cp is the angle between the privileged direction of the analyzer and polarizer, and e is the acute angle between the privileged direction of the polarizer and one of the crystal directionso In Equation (6 1) I represents the intensity of the emerging light, and Io the intensity of the incident lighto n2 and nl are the indices of the crystal for the privileged directions U and V, m is the thickness of the crystal, and X the wave length in air of the incident light. When the polars are crossed, cp 90~, Then Equation (601) reduces to I = I n2sin 2(jsin2(- --- (6.2) The relative intensity R of a monochromatic ray passing through a crystal thus becomes R - sin 2Q2sin2 ) (6-5) 0o In order to estimate roughly the relative light intensity that is transmitted by a monolayer of randomly oriented grains, a simple physical model is substituted for the more complex ceramic thin sectiono The thin section is replaced by an "equivalent" crystallite with a maximum birefringence of E-c0 (nc-na) of about -o015, as experimentally determined from a large grained thin section, illustrated in Figure 5o17o The "equivalent" crystallite lies at some angle e with respect to the polars, which ideally

-96transmits a percentage of light equivalent to the amount which is transmitted by a multitude of randomly oriented crystallites. For purposes of calculation, the light passing through the "equivalent" crystallite is arbitrarily chosen o o to be composed of wavelengths between 5200 A and 6200 A only. This band approximately represents the range of maximum sensitivity of the photocell. It corresponds roughly to the area under the relative sensitivity curve of the photocell bounded by 4800 A to 6400 A (Figure 6.4). The relative transmitted light intensity is correlated with the thickness of the equivalent crystallite. Intensities for a crystal varying from 20 to 60 t have been calculated. The results are given in Table 6.1. o1 Calculation of 9 9 represents the acute angle of one of the privileged directions of the "equivalent" crystallite with respect to the privileged direction of the polarizer (see Figure 6.1). When @ = 0 or 90~, the privileged directions of the tetragonal crystal coincide with the privileged directions of the polarizer and analyzer, and the crystal is at extinction. When Q = 45~, a maximum of light is transmitted by the anisotropic crystal. To find an average value for the transmitted light intensity for the entire collection of randomly oriented anisotropic grains in the thin section, the relative transmitted light intensity is averaged over all angles Q from 0~ to 90~. From Equation (653), the relative transmitted light intensity R = = sin229 sin2()). Then the average relative transmitted light intensity R1 of the equivalent

-97crystallite, as a function of birefringence An, crystal thickness m, and wave length X of the incident light is represented by it/2 2f sin 29dG Rl(An,m,X) = sin2(6m) (6.4) f dG 0 Equation (6.4) can be reduced to 1 (29) - -1sin4Gj"i2 R (Zn,m,) = sin2(-nm) 2 2 -o X t/2 1 sin2 6nm (6.5) 2 X From Equation (6.5) it is seen that the relative intensity of the equivalent crystallite R1 can be represented by a sine squared function of x, where x = the phase difference angle. This is shown in Figure 6.2. It is interesting to note that the maximum value of the function is 0.5. This means that in a randomly oriented aggregate of anisotropic crystals the distribution of the crystallites alone limits the emerging light intensity to one-half of the incident light intensity. This is true for the most favorable values of birefringence, crystal thickness, and wavelengths of transmitted light. 1 o0 R1 05 0 Et 3Ct x Figure 6.02 Relative Intensity R1 of the Equivalent Crystallite with Respect to x, the Phase Difference Angle (see Equation 6,5)o

-982o Calculation of the Birefringence of the Equivalent Crystallite As previously described, the "equivalent" crystallite is substituted for the actual grains that compose a ferroelectric thin section in order to facilitate the calculations of optical light transmission. From experimental observations, the maximum birefringence of a barium-strontium titanate grain is known to be.015 at about 20~ below the peak incremental permittivity temperature (Figure 5.17). The birefringence of the equivalent crystallite is obtained by estimating the average of all observed values of birefringence in the randomly oriented crystallites of the ferroelectric thin section. The birefringence of the "equivalent" crystallite can also be obtained by averaging over allvalues of birefringence in the uniaxial indicatrix, having C - E or na-nc=.015. z oc Figure 635. Ellipsoid of Revolution. The equation of an ellipsoid of revolution or of a biaxial indicatrix is given in polar form, by 2 2 2 2 2 2 2 2 n cos y cos + n sin y cos n sin 1 (6.6) + n2 + 2 a nb n

-99where na, nb, and n are the principal indices of the indicatrix, and n is some index which is related to na, nb and nc by the angles y and p ( (see Figure 653)~ For na = nb nc as is the case for the uniaxial inditrix, the equation condenses to 2 cos sin2l /2 n(7 A) =, 2ros + n ) 1/2 (6-7) (l-k2cos2)l7/2 (6.8) where 2 2 nc k = 1 na If the birefringence An = na - n, then nc,An = n - -- c,/ (6.9) = na (1-k2cos2)1/2 By averaging over all values of n of the indicatrix from p = 0 to F = /2, the average index nay and hence the average birefringence An can be obtainedo:t/2 rncd a~ Bo (n.-' —1-k cos2,- (nav. = na - - c / (60) f dp P=0 This is graphically represented in Figure 6.5, which shows the birefringence An as a function of the angle (. By dividing the area under the curve by (, an average Anav of.00757 is obtained. The value used for X0 = na is 2~4 and for E = nc is 2.385.

-10035 Calculation of the Relative Intensity for All Wave Lengths from 0 0 5200A to 6200A As mentioned previously, the photocell employed in measuring the light intensity is essentially sensitive in the yellow portion of the spectrum. Hence, a constant amplitude band between 5200A and 6200 has been chosen to approximate the range of maximum sensitivity in the spectral intensity curve of the photocell (Figure 6.4). The relative intensity is given by (cf. Equation (6.5)) 1 i 2/nAnm R(An,m,X) = sin ) ( 2 Let c where c is the velocity of light and f the frequency. Then R(An,m,f) = sin2(nmf) (6.11) 2 (C6 Averaging between fl = and f2 = where X = 6200A and X2 = 5200A, Xc 2 we get J2 sin ( c ) fdf 2 f, R(An,m) = (6.12) f df fl Let tAnm A = -c C 3 and Af = x, dx = Adf.

-101Then (6012) can be written as 21 f2 sin2Af d(Af) R(An,m) = (6o13) f2 f df fl 1 (sin2Af2 - sin2Afl) "- 8A(f2-fl) 64 Equation (6.14) represents the relative light-intensity trans0o mitted by the equivalent crystallite, for wavelengths between 5200A and o 6200A. Some numerical values for the transmitted intensity can be obtained by assuming the previously calculated value of an average birefringence L^nav = o00757, and thicknesses varying from 20i to 60oj. The results of these calculations are listed in Table 6.1o 4. Discussion of the Calculated Relative Transmitted Intensities The object of the above calculations is to obtain an estimate of the transmitted light intensity that emerges from a thin section of a typical ferroelectric (Ba,Sr)TiO3 composition when the latter is placed between crossed polarso The model used is a rough one. Absorption losses have not been considered, nor has the effect of light scattering and multiple reflections on the emerging intensity been taken into account. These undoubtedly take place in the ceramic thin section, but are difficult to determine with any degree of accuracyo Two factors which emerge from the calculations seem to be worth noting here:

TABLE 6,1 C o RELATIVE LIGHT INTENSITY R (BETWEEN 5200A AND 6200A) TIANSMITTED BY A (BaSr)TiO3 CRYSTALLITE UNDER CROSSED POLARS, RELATIVE INTENSITY R VS THICKNESS OF SAMPLE 1 (sin 2Af2 - sin 2Afl) R (An,m) = - - 4 8A(f2- fl) Sample 2Af2= 2Af= sin 2Af2 - Thickness, m 2gAnnm 2rtAnm sin 2Af2 sin 2Afl sin 2Af2 - 8A(f2-fl) sin 2Afl/ R(Anm) (in microns) X2 Xl sin 2Af1 8A(f2-f1) 20.584gc,488-t.966.998 -.032 1.18 -.027.277 25 730rt.610it.755.940 -185 1.47 -.126.376 30.876.7325rt.345.743 -.368 1.77 -.208.458 35 1.020Ot.854rT -.070.438 -.508 2.06 -..247.497 40 1.165A.976it -.515.070 -.585 2.36 -.248.498 45 1.314it 1.098t -.834 -.301 -.555 2.65 -.201.451 50 1.46o0i 1.220T -.993 -636 -.357 2.95 -.121.371 55 1.6o61t 1.342i -.946 -.883 -.o63 3.24 -.019.269 60 1.752T 1.464,c -.707 -.994 +.287 3.54.o81.169

-103 - RANGE OF MAX. VALUE - - 100 90 80 70 6050 4030LI -J 20 I 00 0 2000 4000 6000 8000 9000 WAVELENGTH ANGSTROMS Figure 6.4. Spectral Sensitivity Curve of RCA Photocell 6957.

7r/2 ~0 16 ~ A nov 0 f --.00757 c /2'2.012 f d Z.008 z o i.004 - 0 0 10 20 30 40 50 60 70 80 90 ANGLE f IN DEGREES Figure 6.5. Birefringence An of a Barium-Strontium Titanate Crystal as a Function of Angle p.

-105First, Equation (6~5) and Figure 6.2 show that the light intensity transmitted by a two dimensional aggregate of randomly oriented anisotropic grains, between crossed polars, can at most be one-half of the incident intensity. The treatment neglects reflection and absorption losses in the ceramic which would further reduce this value, Secondly, Table 6o1 illustrates the dependency of the relative transmitted intensity on the sample thickness for the specified wavelengths and average birefringence used in Equation (6o14)o Between thicknesses of 30 to 45p, the calculated relative light intensity varies between 45 to almost 50%o Hence thin sections of 30 to 455 will transmit an optimum amount of light, The four large-grained ceramics used for light intensity measurements are 50 to 40k. thick. Their relative transmitted intensities vary between 12 and 22%o Bo Change with Temperature in the Relative Amount of Light Transmitted by a Barium-Strontium Titanate Thin Section Since the upper ferroelectric to paraelectric Curie transition in an unstressed barium-strontium titanate single crystal represents a phase transition from the tetragonal to the cubic state, the existence or the lack of an optical anisotropy in such a crystal becomes a means of determining the phase of the crystalo Similarly, for a ceramic, the optical anisotropy of a bariumstrontium titanate thin-section can give an indication of the crystallographic state of the ceramic, if the following factors are taken into account-

-1061) In a thin-section prepared from a commercial mixed titanate ceramic, the transition from a totally ferroelectric to totally paraelectric state does not take place at some definite temperature, as is the case for a single crystal. Instead, the transition in the ceramic occurs over a temperature range, owing to the variations in Curie temperatures of the grains composing the ceramic. (These small variations in transition temperature are essentially caused by compositional inhomogeneities (123) in the material. A complete reaction of the compositional end members of "mixed" barium-strontium titanate can be promoted by prolonged sintering at higher temperatures. However, materials thus reacted are generally too brittle and show too sharp a peak in permittivity to be of commercial value). 2) It seems a reasonable assumption that the Curie temperatures of the ceramic grains are distributed with temperature in the form of a Gaussian curve, with the maximum occurrences (Curie transitions) at the peak incremental permittivity temperature, see Figure.6.6 The area under the curve represents the total number of tetragonal to cubic transitions and therefore the total number of grains in the ceramic material. 3) Thie optical anisotropy in a ferroelectric ceramic thin-section can be empirically correlated with its tetragonal phase by measuring the intensity transmitted by the section between crossed polars. The observed intensity is the result of the light transmitted by the randomly oriented ferroelectric, tetragonal grains in the thin-section. When such intensity

-1070 z'- FERROELECTRIC/ PARAELECTRIC Tr 4TEMPERATURE Figure 6.6. Assumed Distribution of the Tetragonal to Cubic Phase Transitions of the Ceramic Grains About the Peak Incremental Permittivity Temperature Tr of the Ceramic. 0.8 0.6 0.4 0.2 0 0.5 1.0 1.5 2.0 igure 6.7. The Error Function erf(x) 2 2 Figure 6.7. The Error Function erf(x) L e.

-108measurements are taken on a barium-strontium titanate thinsection over a temperature range which includes the peak incremental permittivity temperature, the transmitted intensity shows the following behavior: It changes from some relatively high value of light intensity for the ferroelectric phase to almost complete extinction in the paraelectric phase. The relative transmitted light intensity of the thin-section can be directly related to that fraction of grains of the ceramic which are ferroelectric and tetragonal at any given temperature. At temperatures close to that of the peak incremental permittivity, the relative light intensity will experience its most significant changeso This follows from the substantial number of transitions which take place in the cermic at the peak incremental permittivity temperature, as seen in the schematic in Figure 606, It will be subsequently shown that the curve of the relative transmitted light intensity with temperature is of the form of (l-erf), where erf is the error function (Figure 6~7). Consider a (Ba,Sr)TiO3 ceramic composed of numerous grains.* The ceramic has a peak incremental permittivity at some temperature Tr ~ Let the probability of one grain changing phase from ferroelectric to paraelectric within the temperature range of dT be given by -(T-Tr 2 dP(T) P0 e dT (6.15) The assumption is made that it is valid to consider the phase transition temperatures of the ceramic grains to follow a normal distribution.

-109P is the probability, Po is a constant, T is the temperature variable, T is the temperature of the peak incremental permittivity for a particular composition, and a is a variance parameter which will determine how broad the distribution will be. Now let the total number of grains in a ceramic be N o Let dn/dT be the number of grains transforming (i.e. changing phase) in a range dT such that f (d) dT = N (6.16) 0 Let dn/dT be proportional to dP/dT as a reasonable assumption. Then (T-Tr)2 dn - _- —' dn a X (6.17) dT = n e (6 and,T-Tr) N = f no e dT. (6.18) 0 Thus the total number of grains in the ceramic are described as the number which change phase between the temperature limits of 0~K to oo It is assumed here for purposes of calculation that no other phases except the tetragonal and the cubic exist within this range. To calculate the value of no, let N n = a - --- (6.19) oo (T-Tr)2 J eT dT 0 Compute -Tr2 f e dT. (6.20) 0

-110T-Tr By substituting x - into Equation (6.20), we get 00 2 a eI dx (6.21) -Tr From the tables of integrals it is known that 00 -X 7 e dx = (6.22) 0 Equation (6,21) can be also written as 0 -x2 00 x2 a e dx + a f e dx, (6.23) -Tr 0 -x2 e dx + a 2 (6o24) -T 2 a Tr for one of the measured mixed titanate ceramics is about 330~K, a is about 15. The lower limit of integration in Equation (6.24) is therefore about -22. According to the tables of error functions(l24) the error integral, erf = f e dx = - - ex dx J -x O0 (6.25) 2 x -x2 - - f e dx IdO reaches 99.9% of its full value for x = -2.4 o At x = -22, the error integral can be considered to be unity (Figure 6.7). Hence, by extending

-111-x to -0o in Equation (6.25), the value of the error integral essentially remains the same. Likewise, the following approximations can be made for Equation (6.24) 0 2 0 -x2? a ex dx + a - a I e dx a - ar (6.26) -Tr 2 -00oo a -Tr Equation (6.19) therefore becomes ~n~o~~~~ -,~~N ^(6.27) Thus, Equation (6.18), which represents the total number of grains that change phase from tetragonal to isotropic within the temperature interval from 0 to Xo can be alternately written as T-Tr 2 N = fN e ) dT (6.28) 0 0 As previously indicated, the intensity of the light transmitted by a ferroelectric thin section between crossed polars can be assumed to be proportional to the number of anisotropic or tetragonal grains in the ceramic. The limits of integration in Equation (6.28) can be modified to give the function eT d( f(T) = e a dT (6.29) This expression represents the number of isotropic grains in the material at some temperature T, ioeo, the number of grains that have changed phase between 0~K to T~Ko The remaining anisotropic grains in the ceramic at T are therefore described by

-112Tr T~ (6~30) N - f(T) = N 1 - 1 e a dT (63.0) L1' 2 \fn 0 If the assumptions made in the present calculation are valid, there should be a direct relationship between the number of anisotropic grains in the ceramic, as a function of temperature, and the optical behavior of the experimental light transmission curves. Equation (6o30) can be evaluated in the following manner. Let the fraction of anisotropic grains in the ceramic be given by g(T) = 1 - = 1 e dT. (6.31) N a lTC o T-Tr Substitute x = -- into Equation(63.1). Then T-Tr 1 X 2 g(T) = 1 - 1 e- dx (6.32) QS/T -Tr Equation (6032) can also be written as T-Tr 0 ~ 2 ~ 2 g(T) 1 0 -X -X g(T) = 1 - dx + e dx (6.33) fTc -Tr 0 2 0 2 Tr The integral - e dx for -- < -27 is approximately unity Tr a (Figure 6o7) - Tr for the measured barium-strontium titanate thin sections is about -22. Equation (6.33) therefore becomes

-113g(T) 1 - 1 + J dx (64) 2 JI- 0 T-Tr 1 S x2 = 5 + _ e- dx. (6~35) Ad 0 T-Tr 2 x2 The integral - j e dx can be directly evaluated from the tables nT: O of integrals. As the temperature T ranges from 293~K to 393 ~K with Tc being about 330~K and a r 15, the integral goes from -1 to 1, and the function g(T) from 1 to 0. Hence g(T) describes the ceramic as being composed of exclusively anisotropic crystallites at the low end of the temperature scale and of exclusively isotropic crystallites at the high endo g(T) has been calculated for four large-grained thin sections, both for conditions of zero field, and for large biasing fields. The computed curves have been rescaled to allow comparison with the experimental percent light transmission data. Results are shown in Figures 6.8, 6o9, 6.10, and 6,11. CO Change with Temperature and Applied DC Field in the Relative Amount of Light Transmitted by a Barium-Strontium Titanate Thin Section Several investigators(78' 79, 100) have reported a shift in the Curie temperature of single crystal barium titanate under the influence of a fieldo The Curie temperature is shifted to higher values when the field is applied either parallel or perpendicular to the polar axis of the crystal, Measurements made by Merz(78) at temperatures above the Curie point of "good" BaTiO3 single crystals show that he was able to induce a ferroelectric state in the crystals on alternate cycles of a strong

-11460-cycle field. It has been also noted experimentally that a strong field causes a shift in the peak incremental permittivity temperatures of ceramic barium titanate and barium-strontium titanate ferroelectricso (Figures 2.4 and 2.5) According to Merz, (7 the Curie temperature shift AT of a crystal which is subjected to an electric field E can be expressed to the first approximaltion by the formula AT = C x |E. (6.36) AT represents the increase in the Curie temperature of the biased crystal in ~C, E is the field applied in volts/cm, and C is to a first approximation a constant = 1.43 x 10-3~C cm/volt. By adapting Equation (6.36) to the case of ceramics, the "new" peak incremental permittivity temperature, Tr Tr, of the crystallites composing an electrically biased ceramic thin section, can be expressed by Tr = Tr + 7 E (637) Tr is the peak incremental permittivity of the ceramic at zero electric field, y is a constant expressed in ~C cm/volts, and E is the applied *T-T* field in volts/cm By using Tr for Tr and x =-T in Equation By sig fr r (6.33) we obtain an expression for the anisotropy in a mixed titanate ceramic thin section as a function of temperature and electric field., - 0 X2 -x?2 g(T)* = 1 xI eTX dx + e dx (6.38) 00 L r Ce~J ~

-115Again, Equation (6.38) can be simplified to T-Tr* 2 g(T)* = 1 | 1 + 2 e 2 dxl (6~39) 22 \/JT 0 T-Tr 1 r jxv2 5 + - ex dx (6.40) Since the peak incremental permittivity temperature Tr of a biased ceramic is greater than the temperature T in the unbiased state, the function g(T)* will be located at higher values along the temperature axis than g(T). This is shown in Figures 6.8, 6.9, 6.10, 6.11. D. Comparison of the Relative Theoretical and Experimental Light Transmission Curves A meaningful evaluation of the percent light transmission vs temperature curves for a mixed titanate thin section using (6.35) and (6~40) depends on the correct choice of the constants Tr, Tr and a. Since good single crystal data on solid solutions of barium-strontium titanates are lacking, it is not possible to evaluate the Curie temperature and the shift in Curie temperature of these compositions directlyo The constants Tr, Tr,as well as the variance or dispersion parameter of the ceramic distribution a, have therefore been obtained from available experimental as well as theoretical(6) data on ceramic compositions. 1. Consideration of Tr., the Peak Incremental Permittivity Temperature of the Ceramic at Zero Field It is known(6) that a linear relationship exists between the peak incremental permittivity temperature of barium-strontium titanate ceramics and their composition. For pure barium titanate, the transition

26 I [ I 1 EXPERIMENTAL 22 /W ~~~~~~(b) LTHEORETICAL 22(b) 2L 4 10 0 | z\\ \ a) o VOLTS/CM., 8 ~ X b) 17,000 VOLTS/CM. 2 N 60 \\ LI) o I H I — LF 6.. C i of n j I a2 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, ~C Figure 6.8. Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 1.

14 _EXPERIMENTAL THEORETICAL - b) — _ a = 10,^ —a = 15 \12 "" ^^ ^ ^.\\ a) 0 VOLTS/CM. a-^ ~^ \ v;~ b) 10,900 VOLTS/CM. W 8 \ -, \\ \. z 6I0S- 4]_ 2Iw (-) w 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, 0C Figure 6.9. Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 2.

20 I I I I, I -EXPERIMENTAL (~b) -~: ~.~' ~ THEORETICAL 18 O — a * 0 _ \ \. \ -.-a =15 \ \ \ a) 0 VOLTS/CM. 16 \b) 10,700 VOLTS/CM. (a)'" \ 14 0u~TEMPERATURE, ~C\\' 12 z Figure 610. Comparison of Theoretl ad El P t z \\\ \v 3 I 20 30 40 50 60 70 80 90 100 110 120 TEMPERATURE, 0C Figure 6.10. Comparison of Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 5.

- EXPERIMENTAL THEORETICAL 15 - -- a - 20 -.-a = 15 13 -A -~~~ - - - - " s^^^.^ 5 a0) o VOLTS/CM. t ~~~~~~13 h' -"- b) 10,800 VOLTS/CM. Z - o IIC,): - \ \\ \(b) ( 9a _ LD I3 - z w 05 cr a. 20 30 40 50 60 TE 80 90 100 110 120 130I TEMPERATUREt, 0a Figure 6.11. Comparison o f Theoretical and Experimental Percent Light Transmission Curves for a Ceramic Thin Section, Sample 4.

-120occurs at about 120~C and decreases linearly with additions of strontium to absolute zero for pure strontium titanate. It should therefore be possible to determine the peak incremental permittivity of such a ceramic if its composition were known. Unfortunately, the peak permittivity temperatures of the mixed titanates are influenced not only by their composition, but by the completeness of the stoichiometric reaction, and by factors like impurities and internal strains. In general, the harder the ceramics are fired, the higher their peak incremental permittivity temperature. (Pure barium titanate ceramics prepared at the Electromagnetic Materials Laboratory of the University.of Michigan, Department of Electrical Engineering, have shown peaks in incremental permittivity varying from 120~C to 140~C. In every instance, the material was commercially acceptable, i.e,, it was not "underfired". The method of preparation of the "green body" was essentially the same, the firing cycles, however, varied somewhat in length.) Hence the linear relationship between the transition temperature and composition in mixed barium-strontium titanates can at best only be a guide. An important factor influencing the peak incremental permittivity temperatures of two of the four samples measured here are the small amounts of fluxing agents added to two compositions. (Table 4.1). The particular metal oxides added to the compositions tend to increase grain growth; they also increase the peak incremental permittivity temperature by about 20~C over the predicted temperature for the "pure" mixed titanate. For these reasons, Tr has been taken directly from the experimental zero field permittivity vs temperature data. Tr is defined as the peak incremental permittivity temperature of a particular barium-strontium

-121titanate ceramic with a specific history of preparation. It is also that temperature at which the largest number of crystallites change phase with in the ceramic. 2. Consideration of T*, the Peak Incremental Permittivity Temperature of a Ceramic Under the Influence of an Electric Field The shift in the peak incremental permittivity temperature with field is obtained in the following manner: the value 7 in the Equation Tr = Tr + yE is determined from the incremental permittivity vs temperature data of a specific mixed titanate sample. Knowing the value y and the electric field across the thin section end of the sample, the shift in the peak permittivity temperature at the thin section end can be evaluatedo Several y s are listed in Table 6.2 The values of y are lower than for single crystal barium titanate which has been reported as 12~0cm/Kv by Kaenzig and Maikoff,(79) l.04cm/Kv by Merz,(7 and 1.6~cm/Kv by Kawabe(100)* When a certain critical temperature is exceeded, a ferroelectric phase can no longer be induced in a single crystal, regardless of the applied field strength. Below the critical temperature, the polarization of the single crystal experiences a sudden increase as the crystal goes from paraelectric to ferroelectrico Above the critical temperature, the polarization changes continuously with increasing field (see Figure 6o12b). It should be pointed out that strictly speaking, y is not a constanto y, the shift in the transition temperature with field decreases with increasing fields. Since the values of Y on (Ba,Sr)TiO ceramics were determined at high fields, they are necessarily somewhat lower than the average y for the entire "inducable" temperature range.

-122WITH FIELD'T1 <WITH FIELD FIELD P Tc T0.... T - T (a) (b) Figure 6.12. Polarization as a Function of Temperature, With and Without a Biasing Field. (a) Near a Second Order Transition. (12) (b) Near a First Order Transition (After Devonshire)125) From Merz's(78) double hysteresis loop experiments on barium titanate it can be calculated that the critical temperature is about 120C above the Curie temperature of the unbiased single crystal. Kawabe reports observing a field produced shift in the transition temperature of about 150C. In mixed barium-strontium titanate ceramics it is difficult to determine the maximum field produced shift in the Curie temperatures of the grains composing the ceramic because of their distribution over a temperature range. Based on single crystal data of barium titanate, the maximum obtainable shift is assumed to be about 150C for the ceramic materials. Tr - T for the relative light intensity curves of sample 1 is therefore taken to be 15~C rather than the calculated 19~C. 3. Discussions of a, the Variance Parameter of the Ceramic Distribution The value for a in the expression of the theoretical light intensity is obtained by fitting the experimental permittivity vs temperature

-123curves to theoretical curves with known values of a 0 It has been shown by Diamond(62) that the zero field incremental permittivity of a bariumstrontium titanate ceramic at some temperature can be evaluated from the expression Tc-Tr 2 00 ( rd ) f E(T,Tc) e dTc E(T) = (6.41) Tc-Tr 2 00 ( ) f e dTc 0 where Eav~ for T < Tc e(T,Tc) = [^~ - T<c.(6.42) (T-To for c(T) is the average permittivity at temperature T of the randomly distributed barium-strontium titanate crystallites that compose the ceramic. In this model, the Curie temperatures Tc of the crystallites form a Gaussian distribution about some chosen temperature, Tr, which is primarily dependent on the composition of the mixed titanate. Furthermore, E(T,Tc), the incremental permittivity of the individual crystallite, assumes a similar dielectric behavior as observed on a barium titanate single crystal. At temperatures below the Curie temperature Tc, the permittivity takes on a value cav obtained by averaging over the ellipsoid associated with the permittivity tensor of the individual crystallite. (The lower temperature ferroelectric transitions have not been considered in this treatment.) At temperatures higher than Tc, the incremental permittivity of the crystal4jCl lite follows the Curie Weiss law, e — 1, where C is the Curie constant ~C lower than T Based on Equation (6.4 and T, for BaTiO5 is about 100C lower than Tc. Based on Equation (6.41),

-124Diamond computed the incremental permittivity vs temperature characteristics of some typical barium-strontium titanate compositions for different values of a. The curves in Figure 6.13 represent ces of 10, 15, 20, and 50. T= 5000K,Tc = To + 10(78) and the Curie constant C = 1.4 x 104. (78) When the experimental permittivity curves of samples 1, 2, 3, and 4 (Figures 5o1, 5-3, 5.5, 5.7) are compared with the theoretical curves shown in Figure 6.13 several differences immediately become apparent: First, the incremental permittivities of Samples 1-4 are considerably lower than the theoretical values. The theoretical model represents an ideal ceramic, in which the permittivities of the ceramic crystallites are equivalent to that of single crystals. High values of permittivity are obtained in a sufficiently large, well sintered and dense piece of ceramic. The samples used for the present percent light transmission experiments, however, are less than 1 cm long and.1 cm wide, with thicknesses ranging from about olcm at one end to 40o. at the other. The initial low porosity of the material was increased during the grinding and polishing process. Some tearing-out of the material could not be avoided and the holes introduced were not completely removed by subsequent polishing. This fact may in part explain the low measured capacitances of the samples,* Secondly, when the experimental permittivity curves are superimposed on the theoretical curves of Figure 6.15 by adjusting the experimental * Even small amounts of porosity would be very effective in reducing the permittivity if the basicr aterial was a high permittivity ceramic (cf. Lichteneckers Formula 1~2): InET = x1ine1 + x2.enE2 +....)

9000 i I i| a= 10 E ^(T) BASED ON 8000 -TcTr \CZ-/ E(T) = E,(T,Tc) e dT a=15 7000 FOR VARIOUS VALUES OF THE PARAMETER a 4Q~~~~ a'~a20 6000 ~ 5000 - sooo4000 - 5 I| 3000 - z w Z 2000 - =2 1-0 a= I5 a = 10 1000 0 I - I I I I I I L I II 270 290 310 330 350 370 390 TEMPERATURE, OK Figure 6.13. lneoretical Incremental Permittivity vs. Temperature Curves for Various Values of the Parameters a.

-126permittivity value and temperature of the peak incremental permittivity to fit that of the theoretical curve, a distortion in the experimental curves on both sides of the peak becomes visible (Figures 6.14 - 6.17). It is felt that the misfit results from the diffifulty encountered in determining small capacitances (about 30~pf) with the instrumentation used. Although precautions were taken to account for stray capacitances (see Section IV), it is possible that some systematic errors have been introduced during the measurement of the samples. Figures 6,14, 6.16, 6.17 show that the incremental permittivity vs temperature curves of 3 samples closely resemble the theoretical curves for which a is 10 and 15. In Figure 6.15 the experimental curves resemble the theoretical ones for which a = 15 and 20 o These values for a have been used to plot the theoretical vs experimental percent light transmission curves (Figures 6.8-6.11). )L. Comparison of the Experimental and Theoretical Transmitted Light Intensity Curves Figures 6.8 to 6o11 represent the experimental light transmission vs temperature measurements taken on four large grained (Ba,Sr)TiO3 ceramics, and their respective theoretical transmission curves, based on Equation (6.55) and (6.40). The two sets of curves shown in each figure illustrate the optical behavior of the ceramic at zero field and at the strongest field employed during the experiment. Fields higher than the latter caused sparking at the sample electrodes and breakdown of the ceramic. A comparison of the theoretical and experimental curves shows that both have a relatively flat portion at the lower side of the temperature scale, followed by a sharp drop in the percent light transmission over

9000,,,,,,, a 10 8000 - 0/ \ --- EXPERIMENTAL(NORMALIZED) Gl/ 0~15 THEORETICAL 7000 - > 5000 0 \ \ / 6000 - )' / 15120 / 5 —=2q/ y//a~lo I 20 / \ 15000 -, t270 290 310 330 350 370 39-0 TEMPERATURE, ~K Figure 6.14. Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs. Temperature Curves for Sample 1. I' 300 z~~~~~~~~~~EPRTR, * u ~ ~ ~ ~ ~~~Fgr.14Joprsno~TertcladNraie xeietlIce ~; ~ ~ ~ ~ ~ ~~~ ~mna Peattvt vs Tepeatr Cuve =o aml 1.,2

9000 -1 a a 10 8000 --— EXPERIMENTAL (NORMALIZED) 0.15/~ \~ -THEORETICAL a 015 7000- / \ 6 000 \ \'A I 6000 I QL /. % > 5000 / / I- - I I- -' Hy\,^ ^ ^ 4000' R)J^.^^ - -. 3000 z w Z 2000a 15 1000 - 100 0 ------ -J______ i ___ I ___ l ___ l ___ i ___ —--------- 270 290 310 330 350 370 390 TEMPERATURE, *K Figure 6.15. Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs. Temperature Curves for Sample 2.

9000 1 —- \ - - ~ --- i - l --- T ------ l --- i --- I --- I ---- 9000 f -- EXPERIMENTAL(NORMALIZED) a* 5,l THEORETICAL 7000 f *ooo-~~~~ \j\ \S~ S 4 as u20 I 5000 - f/1; \\ ^ 6 0 / I I I 43000- / I S 3 000 0Ii0 I' s / yy2~~~ a =10^^ -5,2~20~/I Z 2000 - a=20 y/ ^- I 13000 - z (I Z 2 000 Q: =20 a:15 Q: J O 1000 270 290 310 330 350 370 390 TEMPERATURE, OK Figure 6.16. Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs. Temperature Curves for Sample 3.

9000 1 1 1 1 a 10 8000 8000/ \ ___- EXPERIMENTAL (NORMALIZED) THEORETICAL an 7000 - 6000- 7 - 5000 / I - A l l../ I. W 4000 I Z 3000 z Lj// a = Ias = 15,20 w 0 Z 2000 - a=2 i 15 a = 10 1000 -- 270 290 310 330 350 370 390 TEMPERATURE, OK Figure 6.17. Comparison of Theoretical and Normalized Experimental Incremental Permittivity vs. Temperature Curves for Sample 4.

-131-h a narrow temperature range. At high temperatures, the slope of the curve is again relatively small. The steep portion of the curve describes the temperature range at which most ferroelectric to paraelectric transitions in the ceramic occur. The fit between the theoretical and experimental curves is not necessarily good in every instance. As indicated earlier, the accuracy with which Equations (6.34) and (6.39) can predict the ratio of anisotropic to isotropic grains at any given temperature depends on the choice of the constants T* T and a for any given sample. Above all, the model on which the calculations are based must be basically valid. The model rests on two assumptions: 1) Each crystallite in the ceramic, which is considered the smallest ferroelectric unit with a distinct polar vector, has a tetragonal to cubic transition at which the optical anisotropy drops from some constant, high average value of birefringence to a constant low value. The biasing field simply shifts the transition to higher temperatures. This is seen from Equation (6037) where T* = Tr + yE 2) The transition temperature of the individual crystallites are numerically distributed about some temperature Tr in a Gaussian distribution. The exact position and shape of the distribution is determined by the constants T or Tr and a as expressed in Equations (6o35) and (6.40), The validity of the first assumption can be partly tested against the optical transmission measurements performed by Kawabe on a single crystal of barium titanateo ( ) Figure 6.18 shows that at zero applied field the

-13240^ \ Cr.,30- -3 E E E201010 t 20 130 140 Temprture (~C) Figure 6.18. Temperature Dependence of the Optical Transmission of Barium Titanate with Various Applied Fields Near 120~C.

-133crystal goes from anisotropic to completely isotropic at the transition temperatureo Below the transition temperature, there is a gradual increase in transparency with temperature. This follows from the known increase in birefringence upon cooling the crystal from its cubic-tetragonal to tetragonalorthorhombic transition (see Figure 5~17) When increasing fields are applied to the single crystal, the transition temperature not only shifts to higher temperatures (Figure 6l18) but the difference in transparency between the induced ferroelectric and paraelectric states becomes increasingly smallero At sufficiently high temperatures a ferroelectric phase can no longer be induced by the field. Thus, in addition to a simple temperature dependence as observed for the birefringence of the crystal at zero field, the transparency of the biased crystal depends on the field in that it 1) shifts the transition temperature, and 2) produces a temperature dependent strain anisotropy in both the paraelectric and ferroelectric states, The present theoretical model of a ceramic does not take the temperature dependence of the optical anisotropy into accounto A strain anisotropy of constant value is indirectly introduced in both paraelectric and ferroelectric states by fitting the error curve within the end points of the experimentally determined transmission curves. These end points represent the maxima and minima of the percent light transmission curves. In general, they are somewhat higher for the biased ceramic (Figures 6.8 to 6o11, curves b) than for the unbiased ceramic (Figures 6.8 to 6.11, curves a)o The decrease of the percent light transmission with temperature, whether it be in the predominantly ferroelectric region, in the transition region,

-134or in the predominantly paraelectric region, merely follows from the distribution of transition temperatures of the crystallites composing the ceramic. As calculated, the decrease in intensity is not due to a temperature dependence of the anisotropy or of the field produced strain anisotropy. This simplification of the model explains the relatively sharp corners observed in the theoretical curves in Figures 6.8 to 6,11, especially for the transmission curves with field. If the temperature dependence of the anisotropy were incorporated into the theoretical calculations, a more accurate description of the relative transmitted light intensity would undoubtedly result. Such a refinement, however, is beyond the intent of the present study. The accuracy with which the constant Tr, Tr and a can be determined is limited by the uncertainties in the experimental data. Tr is obtained from the peak incremental permittivity vs temperature curves (Table -.1 and Figures 5.1, 5.3, 5.5, 5.7). It represents the average of the peak permittivity temperatures of the heating and cooling curves taken on four ceramic samples. Tr is therefore easily subject to an error of + 1 - 20C. The accuracy of Tr = T + rE, the peak incremental permittivity temperature produced by a field, depends on Tr and yE. As stated above, Tr can vary by + 1 - 2~C, y is obtained from the observed maximum shift of the incremental permittivity peak with field (Table 6.2). For the average crystallite with a Curie temperature at Tr, the determination of y from the ceramic permittivity curves is difficult because one cannot establish the limiting temperatures and fields up to which a transition to the ferroelectric state can be induced. The ceramic is not exclusively composed of

TABLE 6.2 SHIFT IN THE 7-AK INCREMENTAL PERMITTIVITY TEMPERATURE WITH FIELD FROM Tr TO Tr*, USED IN THE EVALUATION OF THE THEORETICAL PERCENT LIGHT TRANSIMSSION CURVES'FIG. 6.7-6.10) From cA vs T Data: Shift from Tr to Tr* for the % Light Transmission Curves, in ~C Observed Shift Calculated Max. Field Applied Max. Applied Field of Peak Permittivity y to Thin Section, Calculated Value Actually I Sample Kv/cm Temperature, ~C ~l/Kv/cm Kv/cm Shift, ~C Used, ~C 1 14.2 15 j 1.06 17.0 18.0 15 2 13.2 14 1.06 10.9 11.6 12 5 10.5 10.95 10.7 | 10.2 10 4 15.0 10.666 10.8 7.2 7. -.. ___ I I - __ _ -____. _ _______ i _ -

-136crystallites with Curie temperatures Tr, but of crystallites having a distribution of Curie temperatures. When, for example, a crystallite with transition temperature Tr has reached its maximum field-produced shift in the peak permittivity temperature, crystallites with Curie temperatures lower than T are already in a permanently paraelectric state, whereas those with Curie temperatures above Tr can still be subjected to an induced ferroelectric state. The temperature range in which some degree of ferroelectricity can be induced in a ceramic is therefore wider than in a single crystal. Furthermore, at high biasing fields, the "peaks" of the permittivity vs temperature curves are considerably reduced. The choice of the peak incremental permittivity temperature is consequently subject to greater error than for low fields. Hence values of T* as listed in Table 6.3 are approximate, with estimated errors up to + 4~C. Based on observations on single crystals of barium titanate the maximum shift of the peak incremental permittivity with field has been limited to 150C. The choice of a variance parameter from the incremental permittivity vs temperature curves becomes very difficult when the measured curves are adapted for comparison to the theoretical curves (see Figures 6.14-6.18). The figures show the experimental curves with the two values of a which most closely resemble the theoretical ones. These two parameters largely determine the shape of the theoretical light transmission curves as seen in Figures 6.8-6,11. When one considers that the evaluation of the theoretical relative transmission curves is subject to the many errors described above, the resemblance between the theoretical and experimental curves must be considered

-137TABLE 6 3 CONSTANTS USED TO EVALUATE THE THEORETICAL PERCENT LIGHT TRANSMISSION CURVES, BASED ON EQUATIONS 6 35) AND (6 40) Sample a Tr Tr* 1 10, 15 49 64 2 10, 15 51 63 3 10, 15 52 62 4 15, 20 84 91 = variance parameter of the ceramic distribution, ~C Tr = peak incremental permittivity temperature, at 0 field, ~C Tr* = peak incremental permittivity temperature, at maximum applied field, ~C

-158a good one. In most cases, a better coincidence of the curves can be achieved by shifting the theoretical Tr by one or two degrees, and/or by adjusting y within the limits of the possible error. Such a refinement, however, was regarded as neither necessary nor desirable in the present analysis. In general, the theoretical curves with a = 15 for Sample 1 (Figure 6.8), a = 15 for Sample 2 (Figure 6.9), a = 10 for Sample 3 (Figure 6.10), and a = 15 for Sample 4 (Figure 6.11), follow the experimental curves quite well. A better fit of the curves representing the biased sample, notably for samples 2 and 5 (Figures 6.9 and 6.10), would result after including the temperature dependence of the strain anisotropy in the calculations. Nevertheless, as represented, the data of the theoretical relative light transmission basically reflects the trend of the optical light transmission of a barium-strontium titanate thin section as a function of temperature and field. It can be therefore concluded that the assumptions made in the analysis of the relative transmitted intensity are essentially valid.

VI CONCLUSIONS The objective of the optical study of ceramic barium-strontium titanate thin-sections has been to correlate optical and electrical properties by direct observation of the ferroelectric to paraelectric phase transition. The effect of a strong electric field on the optical behavior of the ceramic grains at the transition temperature has been given particular consideration. The predominant question throughout this investigation has been: Are the observed changes with field of the incremental permittivity of the ceramics in the region of the ferroelectric to paraelectric transition mainly due to a reorientation of the ferroelectric polar vectors in the direction of the field? Or are they a result of an induced ferroelectric state? The data indicate that the field induced decrease in the incremental permittivity peak, as observed in Figures 5ol, 5o3, 5o5 and 5~7, mainly results from an induced ferroelectric state in the paraelectric grains. There exists a marked increase in the percent light transmission curves (Figures 6,7 - 6o11) and the photographs (Figure 5,14) taken on the large-grained (50 to 150t) ceramics. These changes in transmission are largest at a temperature somewhat beyond the peak incremental permittivity temperature of the ceramic. The unbiased ceramic is mostly paraelectric at that temperature. The incremental permittivity of a ferroelectric grain (near the transition temperature) is smaller than the permittivity of a paraelectric grain (near the Curie temperature), when measured either parallel or perpendicular to the polar axiso Therefore, the existence of an induced ferroelectric state in many grains in the -139

-140ceramic will necessarily lower its total incremental permittivity. In addition, the direction of the induced polar vector - which is also the direction of the induced uniaxial optic axis - lies in the direction of the applied field. In this position, it represents the lowest possible incremental permittivity value when measured along the field direction (Figure 2.5)o The optical data taken on three fine-grained (5 to 15t) samples show a trend similar to that of the large grained samples, although the transmitted intensities are lower. Thin sections of about 30 to 40p. prepared from these materials actually represent a three dimensional aggregate of grains. A larger percentage of the light that enters the thin section is scattered at the numerous grain boundaries of the small grained material than in the large grained material. This accounts for the low percentage in transmitted intensities seen in the optical data of the fine grained materials (Figures 5,10, 5.12 and 5o13). Contrary to the case of large grained ceramics, no domains can be distinguished in the fine-grained samples at magnifications of 500X to 700Xo Hence no analysis of domain movement can be made here. An anisotropy that exists well beyond the peak incremental permittivity temperature has been observed in the fine-grained ceramics. This anisotropy is found in the cubic phase of the ceramic, and in the electrically unbiased stateo It indicates the existence of some very highly strained areas in the ceramicso Such areas have been observed. in microcrystalline ceramics (grain size = 1) and have been attributed to strains at the grain boundaries (117)

-141The model of induced ferroelectricity assumes that the ceramic ferroelectric crystallites are "frozen in," that is, no field produced 90~ domain formation is considered possible although a field may produce a reversal of the polarization vector. 180~ domain switching does not affect the incremental permittivity because of the symmetry of the polarization axis for 180~ reversals. Neither can 180~ domain switching be ordinarily observed by optical means, since it does not involve any change in the orientation of the optical indicatrix 90~ domain alignment, however, would significantly change the incremental permittivity of the ceramic. This follows from the large dielectric antisotropy which exists in a barium-strontium titanate single crystal, 90~ domains are readily visible under crossed polars as a series of lamellae, separated by sharp lines which have been called 90~ domain wallso The optical observations on the large-grained ceramic thin sections show a frequent occurrence of 90~ twinning in the ferroelectric grains. However, no field produced changes have been observed in the position of the twinning striations which could be interpreted as 90~ wall motion. The changes are essentially restricted to the temperature region bordering the peak incremental permittivity temperature.The effect of the field gradually becomes less pronounced as the ferroelectric is increasingly cooled below or heated above the peak incremental permittivity temperature. It therefore appears that the observed small increase in anisotropy in the ferroelectric region* of the ceramic cannot * The ferroelectric region of the ceramic is broadly defined as that temperature range for which the majority of the ceramic grains are ferroelectric, It therefore includes all temperatures below Tr and above the temperature of the orthorhombic-tetragonal transition. Although the orthorhombic phase is also ferroelectric, the observations made here are restricted only to the tetragonal, ferroelectric phase,

-142be explained by an alignment of polar vectors in the field direction. Another explanation is given below: 1) The increase in the optical anisotropy with field arises in part from a certain percentage of grains in the ceramic whose Curie temperatures lie below the mean (peak) incremental permittivity temperatures Tr o When the ceramic is heated, these grains will become isotropic at lower temperatures than the rest of the material. Hence, they will also be susceptible to an induced anisotropy at temperatures below Tr o In this manner, they contribute to the field produced increase in anisotropy of the "ferroelectric" ceramic 2) The increased anisotropy in the ferroelectric grains is also based on the temperature dependence of their birefrigence. Observations on the grains in the thin sections have shown that the birefringence of the ferroelectric grains increases with decreasing temperatures (Figure 5o17)o In addition, the field produces a strain anisotropy in both ferroelectric and paraelectric grains. This is clearly seen in the transmittency data of a single crystal of barium titanate (Figure 6.18)o Optically it is not possible to distinguish between variations in anisotropy produced either by a change in temperature, or by a field, in the form of induced ferroelectricity and strain anisotropyo Detailed observations seem to indicate, however, that these three factors - in

-143different degrees of importance - essentially account for the observed changes in the transmitted intensities of mixed titanate ceramic thin sections A mathematical model has been proposed to describe the relative light intensity transmitted by a ceramic thin section as a function of temperature and electric field. The ceramic is considered to consist of a very large number of crystallites whose transition temperatures are spread in a normal distribution over a temperature range centered about Tr, The transmitted light intensity is taken to be proportional to the number of ferroelectric or induced ferroelectric grains in the ceramic. No allowance has been made for the temperature dependence of the birefringence or the field-produced strain anisotropyo Even though the model is a simple one, the calculated curves (Figures 608 to 6o11) show a good resemblance to the experimental data within the limits of error introduced during the evaluation of the theoretical curves It can therefore be concluded that, in a mixed bariumstrontium titanate ceramic, the field-produced changes in anisotropy in +the peak incremental permittivity region are primarily due to an induced ferroelectric state in the paraelectric grainso In comparison, the effect of the field on the ferroelectric grains is small. 90~ domain motion does not satisfactorily explain the observed dielectric non-linearity with field since the 90~ reorientation requires a relatively long exposure to a biasing field (20 minutes and up)o The lowering of the incremental permittivity, however, follows almost immediately after

-144the application of the field to the ceramico The fact that an anisotropic state is also instantaneously induced by a field in the paraelectric grains at the tetragonal to cubic transition region indicates that this mechanism is chiefly responsible for the dielectric non-linearity in mixed barium-strontium titanate ceramicso

APPENDIX A TABLES A-1 - A-6 CAPACITANCE AND PERCENT LIGHT TRANSMISSION DATA AS A FUNCTION OF TEMPERATURE AND APPLIED DC FIELDS, FOR 6 CERAMIC BARIUM-STRONTIUM TITANATE SAMPLES TABLE A-7 PERCENT LIGHT TRANSMISSION DATA AS A FUNCTION OF TEMPERATURE AND APPLIED DC FIELDS, FOR CERAMIC BARIUM-STRONTIUM TITANATE SAMPLE 7 TABLE A-8 VOLTAGES APPLIED TO CERAMIC WEDGES AND DIMENSIONS OF WEDGES -145

ABBREVIATIONS USED IN THE RECORDING OF THE DATA my = millivolt reading of an iron-constantan temperature thermocouple. T,~C = temperature at which the capacitance and photocell current data was taken, cX cy = capacitance* readings on the Q-meter co = capacitance* of the sample, as measured on the Q-meter at 1600 Kc. co = c - cy. c -= actual capacitance* of the sample as computed from Equation (3). Ch = capacitance* of the sample holder. for definition see Sec. C = incremental permittivity of the sample i II, P.12 for calculation see Appendix C, p 165 [ia photocell current measuring the percent of light transmitted by a ceramic thin section, L = percentage of the incident light transmitted by a thin section between crossed polars. Superscripts and Subscripts: 1 = data taken at zero biasing field 2 3 = data taken with samples exposed to increasing DC fields, 4 J (see Table A-8). *See Section IV, po 44. -146

-147TABLE A-i 8ample 1 mY T100 0: 0~ 0~ II Ilb ICI C 0- C1 ~2 ~ La2 L2 1.38 26.6 54.6 38.4 33.6 1500 21.0 18.6 55.8 37.2 32.4 1445 21.6 19.5 1.60 30.8 53.8 39.2 37.0 1650 20.9 18.2 54.3 37.7 33.4 1490 21.6 19.5 1.80 34.6 53.1 39.9 38.8 1730 20.6 17.8 54.7 38.3 35.0 1560 21.3 18.8 1.90 36.5 52.5 40.5 40.4 1800 20.25 17.4 54.5 38.5 35.4 1580 21.4 19.0 2.00 38.4 51.8 41.2 42.2 1880 20.25 17.4 54.0 39.0 36.6 1630 21.6 19.5 2.10 40.4 50.6 42.4 45.1 2020 20.00 17.0 53.7 39.3 37.3 1670 21.5 19.3 2.20 42.5 49.4 43.6 48.6 2170 19.70 16.6 53.3 39.7 38.3 1710 21.25 18.8 2.30 44.0 46.8 46.2 54.1 2420. 19.40 16.2 52.2 40.8 41.0 1830 21.20 18.62 2.40 46.0 44.2 48.8 63.9 2850 19.10 15.5 51.0 42.0 44.2 1970 20.75 18.50 2.50 48.0 43.0 50.0 67.9 3030 18.60 14.8 49.5 43.5 48.2 2150 20.30 17.4 2.60 50.0 43.3 49.7 66.8 2980 18.00 14.1 48.5 44.5 50.8 2270 20.10 17.0 2.70 52.1 43.6 49.4 65.8 2940 16.80 12.6 47.7 45.3 53.2 2370 19.20 15.9 2.80 54.1 44.3 48.7 63.6 2840 15.70 11.5 47.7 45.3 53.2 2370 18.35 14.6 2.90 55.6 45.0 48.0 61.3 2740 14.20 9.8 47.8 45.2 52.3 2360 17.20 13.2 3.00 57.4 45.8 47.2 58.7 2620 13.10 8.9 48.0 45.0 52.3 2330 16.20 12.0 3.20 61.2 46.8 46.2 55.8 2490 11.00 6.9 48.7 44.3 50.3 2250 13.70 9.33 3.40 64.9 47.9 45.1 52.7 2350 9.60 5.7 49.2 43.8 49.0 2180 11.80 7.59 3.60 68.7 49.3 43.7 48.7 2170 8.65 5.1 50.4 42.6 45.7 2040 10.10 6.17 3.80 72.3 50.5 52.5 45.5 2030 8.20 4.8 51.1 41.9 43.9 1960 9.15 5.50 4.00 76.2 51.3 41.7 43.3 1930 7.80 4.6 52.9 40.1 40.7 1810 8.60 5.13 4.30 82.1 52.8 40.2 41.0 1830 7.60 4.3 53.3 39.7 38.2 1710 8.10 4.8 4.60 87.2 53.1 38.9 36.3 1640 7.50 4.3 54.5 38.5 35.4 1580 7.80 4.6 5.00 94.8 55.3 37.7 33.6 1500 7.30 4.1 55.6 37.4 32.7 1460 7.60 4.47 5.50 104.0 57.0 36.0.29.7 1350 7.30 4.1 57.1 35.9 30.0 1340 7.50 4.3 6.00 113.0 58.0 35.0 27.3 1230 7.30 4.2 58.1 34.9 27.3 1220 7.40 4.3 6.50 122.1 58.8 34.2 25.9 1155 7.25 4.2 59.0 34.0 25.4 1130 7.30 4.2 6.20 116.7 58.3 34.7 26.8 1190 7.20 4.2 58.5 34.5 26.5 1l8o 7.20 4.2 5.80 110.5 57.5 35.5 28.6 1275 7.20 4.2 57.6 35.4 28.4 1260 7.20 4.2 5.30 100.4 56.2 36.8 31.4 1400 7.-20 4.3 56.2 26.8 31.4 1400 7.20 4.4 4.80 91.0 54.5 38.5 35.4 1580 7.10 4. 54.7 38.3 35.0 1560 7.25 4.6 4.4o 83.6 52.7 40.3 39.9 1780 7.20 4.4 53.3 39.7 38.2 1700 7.40 4.7 4.10 78.0 51.4 41.6 43.1 1930 7.30 4.5 52.0 41.0 41.0 1850 780 5.0 3.90 74.2 50.3 42.7 46.0 2050 7.60 4.8 51.2 42.8 46.3 2065 8.30 5.4 3.70 70.5 49.2 43.8 49.0 2180 7.80 4.9 50.1 42.9 46.5 2075 8.80 5.7 3.50 66.8 47.9 45.1 52.7 2350 8.30 5.4 49.5 43.5 48.2 2150 9.70 6.5 3.30 63.1 46.5 46.5 56.7 2530 9.25 6.2 48.5 44.5 50.8 2270 11.00 7.6 3.10 59.2 45.0 48.0 61.3 2740 10.60 7.2 47.7 45.3 53.2 2470 13.00 9.6 2.90 55.6 43.4 49.6 66.5 2970 13.00 9.6 47.2 45.8 54.6 2440 15.00 11.8 2.70 52.1 42.2 50.8 70.6 3150 15.40 12.2 47.3 45.7 54.3 2420 16.90 13.8 2.60 50.0 41.7 51.3 72.4 3230 16.6 13.5 48.0 45.0 52.3 2430 17.90 15.1 2.50 48.0 41.5 51.5 73.1 3260 17.6 14.8 49.0 44.0 49.6 2210 18.50 16.2 2.40 46.0 42.4 50.6 69.8 3110 18.1 15.5 50.1 42.9 46.5 2070 18.90 16.6 2.30 44.0 45.1 47.9 60.8 2710 18.5 16.0 52.1 40.9 41.2 1835 19.30 17.2 2.20 42.5 48.2 44.8 51.8 2310 18.9 16.6 52.8 40.2 39.5 1760 19.60 17.8 2.10 40.4 50.0 43.0 46.7 2080 19.2 17.0 53.3 39.7 38.3 1710 19.75 18.2 2.00 38.4 51.1 41.9 43.7 1950 19.4 17.4 53.8 39.2 37.1 1660 20.10 18.6 1.90 36.5 52.0 41.0 41.4 1840 19.6 17.8 54.2 38.8 36.1 1610 20.30 18.8 1.70 32.7 53.0 40.0 39.0 1740 20.0 18.2 54.8 38.2 34.6 1570 20.60 19.2 1.50 28.8 53.6 39.4 37.5. 1670 20.2 8.6 57.2 35.8 29.3 1310 20.60 19.2 Cx = 93.0 Coh 23.8 e 44.6. C 0 3 3 3 4.4 4* mv T,~C ~ Co Cco 63 P6_ 3 13 Cy _,C Cs 64 _P6 4 _ L4 1.38 26.6 56.8 36.2 30.1 1340 22.3 20.6 58.3 34.7 26.8 1190 23.2 22.2 1.60 30.8 56.3 36.7 31.2 1400 22.25 20.6 58.1 34.9 27.3 1215 23.4 22.4 1.80 34.6 56.1 36.9 31.7 1410 22.1 20.2 57.8 35.2 27.9 1245 23.3 22.4 1.90 36.5 55.8 37.2 32.4 1440 22.75 20.6 57.9 25.1 27.8 1240 23.3 22.4 2.00 38.4 55.6 37.4 32.8 1460 22.5 20.9 57.5 35.5 28.6 1275 23.4 22.4 2.10 40.4 55.4 37.6 33.3 1470 22.45 20.9 57.4 35.6 28.8 1280 23.3 22.4 2.20 42.5 54.9 38.1 34.5 1530 22.3 20.6 57.3 35.7 29.4 1320 23.1 21.9 2.30 44.o 54.2 38.8 36.1 1610 21.9 20.0 57.0 36.0 28.9 1290 22.8 21.4 2.40 46.0 53.5 39.5 37.9 1690 21.6 19.5 56.6 36.4 30.5 1360 22.5 20.9.2.50 48.o 52.7, 40.3 39.7 1770 21.2 18.6 56.1 36.9 31.7 1430 22.1 20.4 2.60 50.0 51.5 41.5 42.7 1900 20;8 18.2 55.5 37.5 33.1 1475 21.8 20.0 2.70 52.1 50.8 42.2 44.7 1990 20.2 17.4 55.0 38.0 34.2 1520 21.3 18.6 2.80 54.1 50.5 42.5 45.5 2030 19.6 16.2 54.5 38.5 35.4 1580 20.8 18.2 2.90 55.6 50.5 42.5 45.5 2030 18.4 14.6 54.2 38.8 36.1 1610 20.2 17.4 3.00 57.4 50.5 42.5 45.5 2030 17.5 13.5 54.0 39.0 36.7 1635 19.3 15.9 3.20 61.2 50.3 42.7 45.6 2040 15.5 11.2 53.7 39.3 37.3 1660 17.7 13.4 3.40 64.9 50.8 52.2 44.6 1990 13.7 9.3 53.7 39.3 37.3 1660 16.25 12.0 3.60 68.7 51.5 41.5 42.7 1910 11.6 7.4 53.8 39.2 37.1 1650 14.30 10.0 3.80 72.3 52.1 40.9 41.2 1830 10.4 6.45 53.1 39.9 38.8 1730 12.80 8.5 4.00 76.2 52.8 40.2 39.6 1760 9.7 5.9 54.5 38.5 35.4 1580 11.90 7.7 4.30 82.1 54.1 38.9 36.4 1640 8.8 5.3 55.3 37.7 33.5 1495 10.70 7.8 4.60 87.2 54.8 38.2 34.6 1540 8.3 4.9 55.9 37.1 32.1 1430 9.90 6.17 5.00 94.8 45.0 37.0 31.9 1420 8.0 4.7 56.8 36.2 30.1 1340 9.30 5.6 5.50 104.0 57.4 35.6 28.8 1280 7.75 4.5 57.8 35.2 27.9 1240 8.70 5.0 6.00 113.0 58.4 34.6 26.7 1190 7.60 4.4 58.6 34.4 26.2 1170 8.25 4.8 6.50 122.1 59.2 33.8 24.9 1110 7.50 4.4 59.4 33.6 24.6 1100 7.90 4.9 6.20 116.7 58.6 34.4 26.2 1170 7.40 4.3 59.0 34.0 25.4 1130 7.85 4.9 5.80 110.5 57.9 35.1 27.7 1230 7.30 4.2 58.3 34.7 26.8 1195 8.oo00 5.1 5.30 100.4 56.6 36.4 30.5 1360 7.50 4.8 57.2 35.8 29.3 1310 8.50 5.6 4.80 91.0 55.1 37.9 34.0 1520 7.70 4.9 55.8 37.2 32.4 1450 9.20 6.0 4.40 83.6 53.8 39.2 37.1 1650 8.10 5.2 54.9 38.1 34.5 1540 10.00 6.8 4.10 78.0 52.7 40.3 39.7 1770 8.70 5.6 54.2 38.8 36.3 1620 11.20 7.8 3.90 74.2 52.1 40.9 41.2 1830 9.40 6.3 53.9 39.1 36.8 1640 12.00 8.5 3.70 70.5 51.3 41.7 43.3 1930 10.20 7.0 53.5 39.5 37.8 1680 12.90 9.3 3.50 66.8 50.8 42.2 44.6 1990 11.30 8.0 53.5 39.5 37.8 1680 14.20 10.7 3.30 63.1 50.3 42.7 45.8' 2040 12.90 9.3 53.5 39.5 37.8 1680 15.60 12.3 3.10 59.2 50.1 42.9 46.5 2070 14.70 11.2 53.2 39.8 38.6 1720 17.00 14.1 2.90 55.6 49.9 43.1 47.1 2100 16.50 13.5 54.2 38.8 36.1 1610 18.30 15.9 2.70 52.1 50.5 42.5 45.5 2030 17.90 15.1 54.8 38.2 34.5 1535 19.25 17.0 2.60 50.0 51.3 41.7 43.3 1930 18.60 16.2 55.5 37.5 33.1 1475 19.80 18.2 2.50 48.0 52.5 40.5 40.3 1800 19.1 17.0 56.2 36.8 32.5 1450 20.10 18.6 2.40 46.0 53.2 39.8 38.5 1720 19.4 17.4 56.5 36.5 30.8 1370 20.3 19.1 2.30 44.0 54.3 38.7 35.7 1590 19.8 18.2 57.1 35.9 29.4 1330 20.6 19.3 2.20 42.5 54.8 38.2 34.7 1550 20.0 18.2 57.2 35.8 29.8 1300 20.7 19.5 2.10 40.4 55.1 37.9 33.9 1510 20.2 8.6 57.5 35.5 28.6 1275 20.8 19.5 2.00 38.4 55.3. 37.7 33.4 1490. 20.4 19.0 57.6 35.4 28.4 1265 21.0 20.0 1.90 36.5 55.7 37.3 32.6 1450 20.6 19.0 57.8 35.5 28.6 1280 21.25 20.4 1.70 32.7 56.0 37.0 31.9 1420 20.8 19.0 57.9 35.1 29.1 1300 21.40 20.6 1.50 28.8 56.3 36.7 31.2 1390 20.9 19.5 58.2 34.8 27.1 1210 21.40 20.6 Cx = 93.0 0h = 23.8 6 = 44.6 C0

-148TABLE A-2 Sample 2 ~~ ~ ~ ~'~~isC2 c,2 c2 a2 L my T,0C C-1 C1 y a 2 1.42 27.4 55.2 37.8 33.9 2500 23.2 11.22 56.0 37.0 32.5 216o 23.4 11.4 1.60 30.8 54.8 38.2 34.8 2310 22.8 10.96 55.7 37.3 32.7 2170 23.8 11.2 1.80 34.6 54.2 38.8 35.9 2390 22.3 10.30 51.3 37.7 33.7 2240 22.6 10.7 2.00 38.4 53.0 40.0 37.1 2600 21.9 10.00 54.2 38.8 36.4 2420 22.2 10.4 2.20 42.5 51.5 41.5 42.8 2850 21.4 9.60 53.2 39.8 38.7 2570 21.6 9.8 2.30 44.0 49.6 43.4 48.1 3190 21.2 9.40 52.7 40.3 39.9 2650 21.6 9.8 2.40 46.0 47.4 45.6 54.1 3600 21.0 9.33 51.8 41.2 42.1 2800 21.4 9.7 2.50 48.o 95.1 47.9 59.7 3970 20.5 8.91 50.1 42.9 46.8 3110 21.1 9.5 2.60 50.0 43.9 49.1 65.1 4330 19.7 8,32 48.2 44.8 52.2 3470 20.3 8.8 2.74 52.5 44.0 49.0 65.8 4380 18.8 7.76 47.7 46.3 56.5 3750 19.6 8.3 2.78 53.7 44.2 48.8 64. 4260 18.3 7T 47.5 45.5 54.0 3590 19.2 8.0 2.90 3551 44.7 48.3 62.5 416o 17.3 6,76 47.5 45.5 54.0 3590 18.2 7.3 3.00 57.4 45.5 47.5 59.8 3970 16.2 545 47.5 45.5 54.0 3590 17.2 6.6 3.10 5.9P 46.. 4750 8 3B80 15.0 5.30 48.0 45.0 52 3500 15-8 5-7 3.20 61.2 46.4 46.6 57.3 3810 14.1 4.90 48.0 45.0 52.6 3430 14.8 5.2 3.40 64.9 47.8 45.2 53.1 3530 12.0 3.84 48.9 44.1 51.4 3415 12.6 4.1 3.60 68.7 49.1 43.9 49.4 3280 10.8 3.31 49.9 43.1 47.1 3130 11.2 3.4 3.80 72.3 50.1 42.9 46.8 3110 9.9 2.95 50.8 42.2 44.8 2980 10.1 3.0 4.00 76.2 51.0 42.0 44.3 2940 9.4 2.75 51.6 41.4 42.8 250 9.4 2.7 4.20 80.0 52.2 40.8 41.3 2740 8.7 2.51 52.5 40.5 40.6 2770 8.8 2.5 4.42 84.0 53.1 39.9 38.9 2580 8.4 2.34 53.3 39.7 38.4 2555 8.5 2.4 4.60 87.2 53.7 39.3 36.9 2450 8.3 2.34 54.0 39.0 36.7 2440 8.3 2.4 4.80 91.0 54.7 38.2 35.2 2340 8.1 2.29 55.0 38.0 34.4 2290 8.2 2.3 5.00 94.8 55.5 37.5 33.2 2210 8.0 2.29 55.7 37.3 32.9 2190 8.1 2.3 5.30 100.4 56.3 36.7 31.4 2090 7.9 2.24 54.5 36.5 31.0 2060 7.9 2.2 5.60 106.0 57.2 35.8 29.4 1950 7.8 2.19 57.4 35.6 29.0 1930 7.8 2.2 610o 114.9 57.9 35.1 27.9 1850 7.8 2.19 58.0 35.0 27.7 1845 7.8 2.2 6.50 122.1 58.8 34.2 26.3 1760 7.8 2.19 58.8 35.0 27.7 1845 7.8 2.2 5.70 108.3 57.1 35.9 29.7 1970 7.8 2.19 57.2 35.8 29.4 1955 7.8 2.2 5.40 102.2 56.1 36.9 31.9 2120 7.8 2.19 56.3 36.7 31.4 2090 7.8 2.2 5.10 96.7 55.4 37.6 33.4 2220 7.9 2.24 55.6 37.4 33.0 2190 7.9 2.2 4.90 92.9 55.0 38.0 34.5 2290 8.0 2.29 55.0 38.0 34.4 2290 8.0 2.3 4.70 89.2 54.2 58.8 6.3 2410 8.0 2.29 543.5 58.7 6.o 2400 8.1 2.5 4.50 85.4 53.3 39.7 39.5 2620 8.1 2.29 53.5 39.5 39.0 2590 8.2 2.3 4.30 82.1 52.3 40.7 41.0 2720 8.3 2.34 52.7 40.3 40.0 2660 8.4 2.4 4.10 78.0 51.4 41.6 43.3 2880 8.5 2.49 51.9 41.1 42.5 2830 8.6 2.45 3.90 74.2 50.2 42.8 46.4 3080 8.9 2.57 50.8 42.2 44.8 2980 9.1 2.70 3.70 70.5 49.2 43.8 49.2 3270 9.4 2.88 50.0 43.0 47.0 3130 9.8 2.90 3.50 66.8 47.8 45.2 53.1 3530 10.5 3.16 49.0 44.0 44.7 3300 10.8 3.30 3.20 61.2 45.6 47.4 59.7 3960 13.2 4.36 47.7 45.3 53.3 3540 13.5 4.50 3.10 59.2 45.1 47.9 61.2 4o6o 13.9 4.68 47.0 46.0 55.4 3680 14.7 5.1 3.00 57.4 44.5 48.5 63.1 4190 15.0 5.37 46.9 46.1 55.7 3700 16.0 5.8 2.90 55.6 44.0 49.0 64.7 4300 15.8 5.75 46.8 46.2 55.8 3715 16.7 6.3 2.80 54.1 43.3. 49.7 66.9 4440 16.8 6.46 46.7 47.3 59.3 3940 17.8 7.0 2.70 52.1 42.7 50.3 68.9 4580 17.7 6.91 46.9 46.1 55.7 3700 18.6 7.6 2.60 50.0 42.5 50.5 69.7 4630 18.6 7.59 47.9 45.1 52.8 3510 19.5 8.2 2.50 48.0 43.1 49.9 67.6 4490 19.4 8.13 49.5 43.5 48.4 3220 20.2 8.8 2.40 46.0 45.0 48.0 61.7 4100 20.0 8.51 51.3 41.7 43.7 2910 20.6 9.0 2.30 44.0 48.1 44.9 52.2 3470 20.4 8.91 52.4 40.6 40.6 2700 20.9 9.3 2.20 42.5 50.7 42.3 45.3 3010 20.7 9.12 53.3 39.7 38.6 2570 21.3 9.6 1.94 36.9 53.1 39.9 38.9 2580 21.5 9.77 54.4 38.6 35.9 2390 21.8 10.0 1.80 34.6 53.6 39.4 37.7 2500 21.7 10.00 54.8 38.2 34.8 2320 22.1 10.3 1.60 30.8 53.3 38.7 36.0 2390 22.0 10.00 55.5 37.5 33.4 2220 22.3 10.5 C, = 93.0 C, =23.6 =66.5 C. my T,~C c co C 63 a3 L3 Cy Co L4 y 30a 1.42 27.4 57.1 35.9 29.7 1975 23.7 11.7 58.7 34.3 26.0 1760 24.2 12.3 1.60 30.8 56.9 36.1 30.2 2010 23.5 11.5 58.4 34.6 26.9 1790 24.0 12.0 1.80 34.6 54.4 36.6 31.1 2070 22.9 11.0 58.3 34.7 27.1 1805 23.5 11.5 2.00 38.4 55.9 37.1 32.3 2150 22.5 10.6 57.8 35.2 28.1 1870 23.1 11.0 2.20 42.5 55.2 37.8 33.9 2250 22.0 10.2 57.5 35.5 28.6 1900 22.7 10.7 2.30 44.0 54.9 38.1 34.7 2310 22.2 10.4 57.1 35.9 29.7 1975 22.6 10.6 2.40 46.0 54.1 38.9 36.5 2430 21.8 10.0 57.0 360 29.9 1990 22.6 10.6 2.50 48.0 53.2 39.8 38.8 2580 21.5 9.8 56.5 36.5 30.9 2055 22.2 10.2 2.60 50.0 52.1 40.9 41.5 2760 20.9 9.3 55.7 37.3 32.7 2175 21.8 10.0 2.74 52.5 51.2 41.8 43.8 2895 20.2 8.9 55.2 37.8 33.9 2250 21.4 9.8 2.78 53.7 51.0 42.0 44.3 2945 20.0 8.6 55.1 37.9 34.2 2280 21.1 9.55 2.90 55.6 50.8 42.2 44.7 2970 19.0 7.8 54.6 38.4 35.3 2350 20.4 8.91 3.00 57.4 50.5 42.5 45.4 3020 18.2 7.3 54.3 38.7 36.2 2410 19.7 8.32 3.10 59.2 50.3 42.7 46.0 3060 16.8 6.4 54.0 39.0 36.8 2450 18.6 7.59 3.20 61.2 50.5 42.5 45.5 3030 15.8 5.7 54.0 39.0 36.8 2450 18.0 7.24 3.40 64.9 50.8 42.2 44.7 2970 13.3 4.4 53.6 39.4 37.7 2510 16.2 6.03 3.60 68.7 51.4 41.6 43.8 2910 11.8 3.7 53.9 39.1 37.0 2460 13.8 4.68 3.80 72.3 52.0 41.0 41.6 2770 10.5 3.2 54.0 39.0 36.6 2435 12.2 3.89 4.00 76.2 52.7 40.3 39.9 2650 9.6 2.8 54.4 38.6 35.7 2370 10.9 3.35 4.20 80.0 53.2 39.8 38.8 2580 9.0 2.6 54.6 38.4 33.5 2230 10.0 3.02 4.42 84.0 54.0 39.0 36.8 2450 8.6 2.45 55.2 37.8 33.9 2250 9.3 2.69 4.60 87.2 54,6 38.4 35.3 2350 8.4 2.4 55.7 37.3 32.7 2170 8.9 2.57 4.80 91.0 55.3 37.7 33.7 2240 8.2 2.3 56.3 36.7 31.4 2090 8.6 2.46 5.00 94.8 56.0 37.0 32.1 2140 8.1 2.3 56.7 36.3 30.5 2030 8.4 2.40 5.30 100.4 56.9 36.1 30.2 2010 7.9 2.2 57.6 35.4 28.5 1895 8.2 2.34 5.60 io6.o 57.7 35.3 28.4 1890 7.8 2.2 58.1 34.9 27.4 1850 8.1 2.29 6.o 114.9 58.3 34.7 26.9 1790 7.8 2.2 58.7 34.3 26.1 1770 7.9 2.24 6.50 122.1 59.1 33.9 25.2 1675 7.8 2.2 59.4 33.6 24.7 164o 7.8 2.19 5.70 108.3 57.3 35.7 29.0 1930 7.8 2.2 57.7 35.3 28.4 1890 8.1 2.29 5.40 102.2 56.5 36.5 29.6 1970 7.8 2.2 57.1 35.9 29.7 1975 8.2 2.29 5.10 96.7 55.8 37.2 32.5 216o 7.9 2.2 56.5 36.5 31.0 2060 8.2 2.34 4.90 92.9 55.4 37.6 33.4 2220 8.0 2.3 56.2 36.8 31.7 2110 8.5 2.51 4.70 89.2 54.6 38.4 35.3 2350 8.2 2.3 55.7 37.3 32.6 2170 8.7 2.51 4.50 85.4 54.0 39.0 36.8 2450 8.2 2.3 55.1 37.9 34.1 2270 9.0 2.60 4.30 82.1 53.4 39.6 38.1 2540 8.5 2.4 54.6 38.4 35.3 2350 9.7 2.88 4.10 78.0 52.7 40.3 39.9 2650 8.8 2.6 54.3 38.7 36.1 2400 10.4 5.16 3.90 74.2 51.8 41.2 42.1 2800 9.4 2.8 53.9 39.1 36.8 2450 11.8 3.72 3.70 70.5 51.4 41.6 43.2 2870 10.3 3.1 53.8 39.2 37.2 2470 13.3 4.40 3.50 66.8 50.7 42.3 45.0 2990 11.8 3.7 53.5 39.5 37.9 2520 15.4 5.50 3.20 61.2 50.1 42.9 46.8 3110 14.7 5.1 53.7 39.3 37.5 2495 18.5 7.59 3.10 59.2 50.0 43.0 47.0 3130 l16.2 6.0 54.0 39.0 37.2 2470 19.5 8.13 3.00 57.4 50.1 42.9 46.8 3110 17.2 6.6 54.2 38.2 36.3 2410 19.9 8.51 2.90 55.6 50.1 42.9 46.3 3110 l18.0 7.2 54.5 38.5 35.5 2360o 20.7 9.12 2.80 54.1 50.5 42.5 45.5 3030 18.9 7.8 54.9 38.1 34.5 2300 21.3 9.55 2.70 52.1 50.9 42.1 44.5 4960 19.6 8.3 55.3 37.7 33.6 2240 21.6 9.78 2.60 50.0 52.2 40.8 41.3 2750 20.3 8.8 56.2 36.8 31.7 2170 22.3 10.23 2.50 48.0 53.1 39.9 38.9 2590 20.9 9.3 56.7 36.3 30.5 2030 23.3 10.23 2.40 46.0 54.1 38.9 36.6 2430 21.4 9.7 57.2 35.8 29.4 I1960 23.0 10.96 2.30 44.0 54.8 38.2 34.8 2310 21.6 9.9 57.5 35.5 28.7 1910 23.0 10.96 2.20 42.5 55.4 37.6 33.6 2230 22.0 10.0 57.8 35.2 28.1 1870 23.4 11.30 1.94 36.9 56.3 36.7 31.4 2080 22.3 10.5 58.3 34.7 27.0 1795 23.5 11.50 1.80 34.6 56.4 36.6 31.1 2070 22.5 10.7 58.3 34.7 27.0 1795 23.6 11.50 1.60 30.8 56.7 36.3 30.5 2030 22.6 10.7 58.6 34.4 26.3 1750 23.6 11.50 CX =93.0 C0 -23.6 c -66.5 Cg

-149TABLE A-3 Sample 3 my OT,0c & c1 o1 e~ pa1 r~ o2 2 2 mv T, Cl C 1 h. ^ J C^ C^ 1.40 27.0 60.7 32.3 22.3 2107 20.5 15.0 61.1 31.9 21.5 2032 21.5 16.5 1.60 30.8 60.3 32.7 23.1 2183 20.2 14.8 60.8 32.2 22.1 2088 21.5 16.5 1.80 34.6 59.8 33.2 24.1 2277 19.8 14.2 60.5 32.5 22.7 2145 21.0 15.8 2.00 38.4 59.0 34.0 25.8 2438 18.1 12.2 60.0 33.0 23.6 2230 20.3 14.8 2.20 42.5 58.0 35.0 28.0 2646 18.3 12.4 59.5 33.5 24.8 2344 20.0 14.5 2.40 46.0 54.9 38.1 34.9 3298 17.0 11.0 57.5 35.5 29.1 2750 18.5 12.6 2.50 48.0 50.0 43.0 47.3 4470 16.2 10.2 56.0 37.0 32.4 3062 18.0 12.2 2.60 50.0 47.1 45.9 54.6 5160 14.3 8.2 53.6 39.4 38.0 3591 17.1 11.0 2.70 52.1 46.6 46.4 56.8 5368 12.7 7.0 53.1 39.9 39.3 3714 16.3 10.2 2.80 54.1 47.6 45.4 53.9 5094 o10.4 5.2 52.6 40.4 40.4 3818 14.6 8.6 2.90 55.6 48.0 45.0 52.8 4990 9.5 4.8 52.5 40.5 40.7 3846 13.7 7.8 3.00 57.4 49.6 43.4 48.3 4564 8.7 4.2 53.0 40.0 39.5 3733 12.0 6.4 3.10 59.2 50.1 42.9 46.8 4423 8.2 4.0 53.4 39.6 38.5 3638 10.6 5.5 3.20 61.2 50.7 41.3 42.7 4035 8.0 3.9 53.7 39.3 37.8 3572 10.1 5.2 3.40 64.9 53.7 39.3 37.8 3572 7.7 3.7 54.8 38.2 35.1 3317 9.0 4.4 3.60 68.7 54.2 38.8 36.6 3459 7.6 3.6 55.7 37.3 33.0 3119 8.4 4.1 3.80 72.3 55.3 37.7 34.0 3213 7.6 3.6 55.6 36.4 31.1 2939 8.1 4.0 4.00 76.2 56.6 36.4 31.0 2930 7.6 3.6 57.7 35.3 28.6 2703 8.0 3.9 4.30 82.1 57.9 35.1 28.2 2665 7.6 3.6 58.6 34.4 26.7 2523 6,7 3.8 4.60 87.2 59.2 33.8 25.4 2400 7.6 3.6 59.7 33.3 24.4 2306 7.7 3.7 5.00 94.8 6o.o 33.0 23.7 2240 7.6 3.6 60.4 32.6 23.9 2259 7.7 3.7 5.50 104.0 61.3 31.7 21.1 1994 7.6 3.6 61.5 31.5 20.8 1966 7.7 3.7 5.80 110.5 61.8 31.2 20.1 1899 7.6 3.6 62.1 30.9 19.6 1852 7.7 3.7 5.30 100.4 60.9 32.1 21.9 2070 7.6 3.6 61.2 31.8 21.4 2022 7.7 4.0 4.80 91.0 59.4 33.6 25.0 2363 7.6 3.6 59.8 33.2 24.1 2277 7.8 4.1 4.40 83.6 58.0 35.0 28.0 2646 7.6 4.0 58.7 34.3 26.4 2495 7.9 4.2 4.10 78.0 56.8 36.2 30.5 2882 7.6 4.0 57.5 35.5 29.1 2750 8.0 4.2 3.90 74.2 55.6 37.4 38.3 3147 7.6 4.0 56.5 36.5 31.2 2948 8.1 4.3 3.70 70.5 54.7 38.3 35.3 3336 7.6 4.0 56.1 36.9 32.1 3033 8.3 4.4 3.50 66.8 54.0 39.0 37.1 3506 7.6 4.0 55.0 38.0 34.6 3270 8.7 4.7 3.30 63.1 50.9 42.1 44.7 4224 7.7 4.0 53.7 39.3 3.8 3572 9.6 5.2 3.10 59.2 49.6 43.4 48.3 4564 8.0 4.2 53.2 39.8 3.9 3676 10.8 6. 3.00 57.4 48.4 44.6 51.6 4876 8.3 4.4 52.9 40.1 39.7 3752 11.9 6.8 2.90 55.6 47.5 45.5 54.2 5122 8.8 4.7 52.3 40.7 41.2 3893 13.2 7.8 2.80 54.1 46.6 46.4 56.9 5377 9.6 5.2 51.6 41.4 43.0 4064 13.2 7.8 2.70 52.1 45.8 47.2 59.3 5604 11.7 6.6 53.0 40o0 39.5 3733 15.8 10.1 2.60 50.1 46.3 46;7 57.7 5427 13.7 8.2 54.1 38. 36.8 3478 16.9 1.4 2.50 48.0 48.9 44.1 50.3 4753 15.5 9.8 56.5 36.5 31.3 2958 17.8 12.4 2.30 44.0 57.9 35.1 28.2 2665 17.0 11.4 58.9 34.1 26.0 2457 18.6 13.2 2.10 40.4 58.6 34.4 26.7 2523 18.0 12.5 59.9 33.1 24.0 2268 19.4 14.1 1.90 36.5 59.6 33.4 24.6 2325 18.7 13.4 60.8 32.2 22.1 2088 20.0 14.4 1.70 32.7 60.3 32.7 23.1 2183 19.4 14.2 61.0 32.0 21.8 2060 20.3 15.2 1.50 28.8 60.7 32.3 22.4 2117 19..8 14.6 61.3 31.7 25.1 1994 20.6 15.6 Cx = 93.0 Ch = 23.4 = 94.5 Cs v T, ~ yC C3 C8 E3 ~ a3 -L C0y 0 a 4 L84 4 1.40 27.0 61.7 31.3 20.3 1918 22,3 17.6 62.2 30.8 19.4 1833 23,1 19.0 1.60 30.8 61.5 31.5 20.7 1956 22.2 17.5 62.2 30.8 19.4 1833 23.2 19.0 1.80 34.6 61.1 31.9 21.5 2032 21.8 17.0 61.9 31.1 19.9 1880 23.0 19.0 2.00 38.4 60.8 32.2 22.1 2088 21.3 16.2 61.6 31.4 20.5 1937 22.5 18.0 2.20 42.5 60.1 32.9 23.5 2221 20.6 15.2 61.2 31.8 21.4 2022 22.0 17.2 2.40 46.0 59.0 34.0 25.8 2438 19.8 14.2 60.5 32.5 22.8 2155 21.2 16.0 2.50 48.0 58.1 34.9 27.7 2618 19.4 13.8 60.0 33.0 23.7 2240 21.0 15.8 2.60 50.0 56.2 36.8 31.9 3015 18.6 12.8 58.8 34.2 26.2 2476 20.5 15.0 2.70 52.1 55.7 37.3 33.1 3128 18.2 12.4 18.5 34.5 26.9 2542 20.3 14.8 2.80 54.1 54.9 38.1 34.9 3298 17.2 10.2 57.8 35.2 28.4 2684 19.6 14.0 2.90 55.6 54.7 38.3 35.4 3345 16.5 10.4 57.5 35.3 28.6 2703 19.3 13.6 3.00 57.4 54.8 38.2 35.1 3317 15.0 9.0 57.2 35.8 29.7 2807 18.4 12.5 3.10 59.2 55.1 37.9 34.5 3260 13.6 7.7 57.2 35.8 29.7 2807 17.3 11.6 3.20 61.2 55.1 37.9 34.5 3260 17.7 7.0 57.3 36.7 31.7 2996 16.6 10.6 3.40 64.9 55.8 37.2 32.8 3100 10.8 5.6 57.5 36.5 31.3 2958 14.7 9.7 3.60 68.7 56.7 36.3 30.9 2920 9.7 4.9 57.9 36.1 30.4 2873 13.1 7.3 3.80 72.3 57.4 35.6 29.2 2759 9.0 4.4 58.4 34.6 27.1 2561 11.7 6.2 4.00 76.2 58.2 34.8 27.5 2599 8.5 4.2 58.9 34.1 26.0 2457 10.4 5.4 4.30 82.1 59.1 33.9 25.6 2419 8.2 4.0 59.7 33.3 24.3 2296 9.4 4.6 4.60 87.2 60.0 33.0 23.7 2240 8.2 3.9 60.4 32.6 22.9 2164 8.8 4.3 5.00 94.8 60.7 32.3 22.3 2107 7.9 3.8 61.0 32.0 21.8 2060 8.5 4.2 5.50 104.0 61.8 31.2 20.1 1899 7.85 3.8 62.0 31.0 19.7 1862 8.2 3.9 5.80 110.5 62.1 30.9 19.6 1852 7.8 4.0 62.4 30.6 19.0 1796 8.2 4.0 5.30 100.4 61.4 31.6 20.9 1975 7.9 4.1 61.6 31.4 20.6 1947 8.3 4.2 4.80 1.0o 60.1 31.9 21.5 2032 8.0 4.2 60.4 32.6 22.9 2164 8.9 4.7 4.40 83.6 59.1 33.9 25.6 2419 8.3 4.4 59.6 33.4 24.6 2325 9.5 5.1 4.10 78.0 58.0 35.0 27.9 2637 8.6 4.6 58.8 34.2 26.2 2476 10.6 5.8 3.90 74.2 57.2 35.8 29.7 2807 9.1 4.9 58.3 34.7 27.3 2580 11.8 6.7 3.70 70.5 56.9 36.1 30.4 2873 9.4 5.1 58.1 34.9 27.7 2618 12.6 7.3 3.50 66.8 56.0 37.0 32.3 3052 10.4 5.7 57.5 35.5 29.0 2741 14.3 8.8 3.30 63.1 55.2 37.8 34.3 3241 12.2 7.0 57.3 35.7 29.5 2788 16.4 10.8 3.10 59.2 54.3 38.7 36.3 3430 13.9 8.4 57.3 35.7 29.5 2788 17.8 12.4 3.00 57.4 54.7 38.3 38.3 3336 15.2 9.6 57.5 35.5 29.1 2750 18.6 13.2 2.90 55.6 54.6 38.4 35.6 3364 16.2 10.5 57.6 34.4 27.2 2570 19.2 14.o 2.80 54.1 55.0 38.0 34.7 3279 17.0 11.4 58.0 35.0 27.9 (2637) 19.7 14.5 2.70 52.1 55.7 37.3 33.0 3185 18.0 12.6 58.8 34.2 26.2 2476 20.3 15.2 2.60 50.1 56.9 36.1 30.3 2863 18.6 13.2 59.3 33.7 25.2 2381 20.7 15.8 2.50 48.0 58.5 34.5 26.9 2542 19.2 13.8 60.5 32.5 22.7 2145 31.0 16.2 2.30 44.0 60.2 32.8 23.4 2211 19.8 14.6 61.4 31.6 20.9 1975 21.3 16.6 2.10 40.4 60.7 32.3 22.3 2107 20.3 15.2 61.9 31.1 19.9 1881 21.6 16.6 1.90 36.5 61.4 31.6 20.9 1975 20.7 15.8 62.1 30.9 19.6 1852 22.0 17.6 1.70 32.7 61.7 31.3 20.3 1918 21.0 16.2 62.5 30.5 18.2 1720 22.2 18.0 1.50 28.8 61.8 31.2 20.1 1899 21.3 16.6 62.5 30.5 18.2 1720 22.3 18.1 Cx = 93.0 Ch = 23.4 = 94.5 C,

-150TAJI A-4 duple 4 my T,~C ^ 0' gl I. cog 2 P 2I 1.4C 97.0 55.3 37.7 33.7 634 19.2 13.9 55.6 37.4 33.0 620 19.2 13.8 1.65 31.8 55.6 37.4 33.0 621 19.2 13.9 56.0 37.0 32.1 602 19.1 13.8 1.9c 36.5 55.9 37.1 32.3 608 19.2 13.9 56.2 36.8 31.6 594 19.2 13.8 2.20,2.5 55.8 37.2 32.5 612 19.0 13.8 56.2 36.8 31.6 594 19.0 13.6 2.50 48.0 55.7 37.3 32.8 617 18.9 13.6 56.2 36.8 31.6 594 18.9 13.5 2.80 54.1 55.2 37.8 33.9 637 18.8 13.4 56.0 37.0 32.2 605 18.8 13.4 3.10 59.2 54.9 38.1 34.6 650 18.6 13.0 55.6 37.4 33.0 620 18.6 13.0 3.40 64.9 54.3 38.7 36.0 677 18.6 13.0 55.1 37.9 34.2 642 18.6 13.0 3.60 68.7 53.3 39.7 38.4 722 18.4 12.8 54.7 38.3 35.1 660 18.5 13.0 3.80 72.3 52.5 40.5 40.3 798 18.2 12.6 53.9 39.1 37.0 695 18.3 12.7 3.90 74.2 57.1 41.9 44.0 828 18.1 12.4 53.2 39.8 38.6 725 18.2 12.6 4.00 76.2 49.6 43.4 48.0 902 17.9 12.2 52.2 40.8 41.2 75 18.1 12.4 4.10 78.0 48.0 45.0 52.5 986 17.6 11.8 51.2 41.8 43.8 824 17.9 12.2 4.20 o.o 46.3 46 55.3 990 17.2 11.4 50.8 42.2 44.7 840 17.6 11.8 4.30 82.1 46.4 46,6 57.1 1071 16.6 10.6 49.7 43.3 47.8 900 17.2 11.0 4.40 83.6 46.1 46.9 58.1 1092 15.7 9.8 49.1 43.9 49.4 930 16.4 10.2 4.50 85.4 46.3 46.7 57.4 1080 14.4 8.4 48.9 44.1 50.0 940 1.3 9.4 4.60 87.2 46.3 46.7 57.4 1080 13.5 7.6 49.0 44.0 49.6 932 14.5 8.4 4.80 91.0 47.1 45.9 55.1 1036 10.3 5.4 49.1 43.9 49.5 930 12.1 6.5 5.00 94.8 47.6 45.4 53.6 1010 10.4 5.4 49.7 43.3 47.8 900 10.9 5.8 5.30 100.4 49.0 44.0 49.6 931 9.5 4.8 50.4 42.6 46.0 865 9.8 5.0 5.60 106.o 49.7 43.3 47.8 900 9.4 4.7 50.8 42.2 44.8 842 9.7 5.0 6.00 113.0 59.9 42.1 44.6 838 9.3 4.7 51.9 41.1 42.0 790 9.4 4.7 6.50 122.1 52.3 40.7 40.8 768 9.3 4.7 53.0 40.0 39.2 738 9.3 4.6 6,30 118.5 51.7 41.3 42.4 798 9*2 4.6 52.5 40.5 40.3 758 9.25 4.5 5.80 110.5 50.3 42.7 46.2 868 9.4 4.7 51.1 41.9 44.1 830 9.50 4.8 5.50 104.0 49.2 43.8 49.2 924 9.4 4.7 50.2 42.8 47.3 890 9.50 4.8 5.20 98.5 48.2 44.8 51.9 976 9.7 4.9 49.5 43.5 48.5 912 10.00 5.1 4.90 92.9 46.8 46.2 55.9 1050 10.5 5.4 48.7 44.3 50.5 950 11.2 6.0 4.70 89.2 46.0 47.0 58.4 1100 11.3 6.0 48.1 44.9 52.1 980 12.2 6.6 4.50 854 45.7 47.3 59.3 1120 12.8 7.0 48.2 44.8 52.0 976 14.0 8.1 4.40 83.6 45.6 47.4 59.6 1120 14.2 8.3 48.2 44.8 52.0 976 15.3 9.4 4.30 82.1 45.6 47.4 59.6 1120 15.0 9.0 48.4 44.6 51.3 96 17.1 11.2 4.20 80.0 46.1 46.9 58.1 1091 16.1 10.2 48.8 44.2 50.1 941 16.8 11.0 4.10 78.0 46.2 46.8 57.7 1084 16.7 10.8 49.5 43.5 48.5 912 17.2 11.4 4.00 76.2 47.2 45.8 54.8 1030 17.2 11.4 50.6 42.4 45.3 850 17.5 11.8 3.90 74.2 49.6 43.4 48.0 903 17.6 11.9 52.2 40.8 41.2 774 17.6 11.8 3.70 705 52.3. 40.7 41.0 770 17.8 12.1 53.5 39.5 38.0 714 18.0 12.3 3.50 66.8 53.4 39.6 38.2 718 18.0 12.4 54.3 38.7 36.1 678 18.1 12.4 3.20 61.2 54.2 38.8 36.3 683 18.2 12.6 55.2 37.8 33.9 638 18.2 12.6 2.90 55.6 55.1 37.9 34.2 643 18.4 12.9 55.7 37.3 32.8 616 18.4 12.9 2.60 50.0 55.5 37.5 33.2 624 18.5 13.0 56.2 36.8 31.7 596 18.5 13.0 2.30 44.0 55.9 37.1 32.4 608 18.5 13.0 56.3 36.7 31.5 593 18.5 13.0 2.00 38.4 56.2 26.8 31.6 594 18.5 13.0 56.6 36.4 30.8 579 18.5 13.0 1.70 32.7 56.3 36.7 31.5 592 18.5 13.0 56.6 36.4 30.8 579 18.5 13.0 1.40 27.0 56." 37.0 32.1 603 18.5 13.0 56.5 36.5 30.9 581 18.6 13.0 0= 93.0 Ch 23.6 a 18.8 CO my T,0C C3 3 CL3 C40 C4 C4.y 0 3 L3 y 0 4 s 1.40 27.0 56.2 36.8 31.6 594 19.2 13.8 56.8 36.2 30.2 568 19.2 14.0 1.65 31.8 56.6 36.4 30.7 578 19.1 13.8 57.2 35.8 29.4 553 19.1-2 14.0 1.90 36.5 56,9 36.1 30.1 573 19.2 13.8 57.5 35.5 28.8 542 19.25 14.0 2.20 42.5 56.8 36.2 31.2 586 19.1 13.8 57.5 35.5 28.8 542 19.25 14.0 2.50 48.0 56.9 36.1 30.1 566 19.0 13.6 57.4 35.6 29.0 545 19.2 14.0 2.80 54.1 56.7 36.3 30.5 573 19.0 13.6 57.3 35.7 29.2 549 19.2 14.0 3.10 59.2 56.7 36.3 30.5 573 18.9 13.5 57.1 35.9 29.7 558 19.1 13.8 3.40 64.9 56.0 37.0 32.2 605 18.9 13.5 56.7 36.3 30.5 572 19.0 13.6 3.60 68.7 55.7 37.3 32.8 616 18.8 13.4 56.3 36.7 31.5 592 19.0 13.6 3.80 72.3 55.0 38.0 34.3 645 18.6 13.0 55.9 37.1 32.4 610 19.0 13.6 3.90 74.2 54.4 38.6 35.8 673 18.5 13.0 55.5 37.5 33.2 628 18.9 13.6 4.00 76.2 53.8 39.2 35.8 673 18.4 12.9 5.4.8 38.2 34.9 656 18.9 13.6 4.10 78.0 53.1 39.9 38.9 732 18.3 12.7 54.3 38.7 36.0 678 18.9 13.6 4.20 80.0 52.2 40.8 41.2 775 18.2 12.6 53.9 39.1 37.0 695 18.7 13.4 4.30 82.1 51.4 41.6 43.3 815 17.8 12.1 53.9 39.1 37.0 695 18.5 13.0 4.40 83.6 51.1 41.9 44.0 827 17.3 11.5 52.4 40.6 40.7 765 18.2 12.6 4.50 85.4 50.8 42.2 44.7 840 16.5 9.4 52.2 40.8 41.2 775 17.7 12.0 4.60 87.2 50.5 42.5 45.6 856 14.8 8.8 51.8 41.2 42.2 793 17.2 11.4 4.80 91.0 50.4 42.6 45.86 861 13.4 7.6 51.6 41.4 42.6 800 15.0 9.0 5.00 94.8 50.8 42.2 44.8 844 11.9 6.4 51.8 41.2 42.2 795 13.4 7.6 5.30 100.4 51.3 41.7 43.5 818 10.3 5.4 51.9 41.1 42.0 790 11.2 5.9 5.60 106.0 51.7 41.3 42.4 797 10.0.5.1 52.3 40.7 40.8 768 10.4 5.4 6.00 113.0 52.4 40.6 40.7 766 9.4 4.7 52.9 40.1 39.2 736 9.6 4.8 6.50 122.1 53.2 39.8 38.7 727 9.3 4.6 53.5 39.5 38.0 715 9.2 4.6 6.30 118.5 52.7 40.3 40.0 752 9.2 4.4 53.1 39.9 39.6 745 9.2 4.6 5.80 110.5 51.8 41.2 42.2 794 9.5 4.8 53.2 39.8 39.5 743 9-7 4.9 5.50 104.0 51.0 42.0 44.2 830 9.7 5.0 51.9 41.1 41.9 787 10.2 5.2 5.20 98.5 50.2 42.8 46.4 872 10.5 5.5 51.2 41.8 43.8 824 11.4 6.0 4.90 92.9 49.8 43.2 47.5 883 12.2 6.6 50.7 42.3 45.1 848 13.5 7.6 4.70 89.2 49.5 43.5 48.3 910 13.5 7.8 50.9 42.1 44.6 838 15.2 9.2 4.50 85.4 49.7 43.3 47.8 900 15.4 9.4 51.1 41.9 44.1 828 16.8 11.2 4.40 83.6 50.0 43.0 47.1 885 16.5 10.6 51.5 41.5 43.1 810 17.4 11.6 4.30 82.1 50.1 42.9 46.7 878 17.0 11.1 51.8 41.2 42.1 790 17.7 12.0 4.20 80.0 50.9 42.1 44.6 838 17.5 11.8 52.1 40.9 41.5 780 18.0 12.3 4.10 78.0 51.3 41.7 43.6 820 17.6 11.8 52.5 40.5 40.4 760 18.1 12.4 4.00 76.2 52.1 40.4 41.6 782 17.9 12.2 53.5 39.5 37.9 713 18.2 12.6 3.90 74.2 53.7 39.3 37.5 705 18.0 12.3 54.5 38.5 35.7 671 18.3 12.7 3.70 70.5 54.5 38.5 35.6 670 18.1 12.4 55.5 37.5 33.2 624 18.4 12.8 3.50 66.8 55.3 37.7 33.7 634 18.3 12.7 56.1 36.9 31.9 600 18.5 13.0 3.20 61.2 55.9 37.1 32.3 607 18.3 12.7 56.6 36.4 30.8 579 18.5 13.0 2.90 55.6 56.4 36.6 31.2 586 18.4 12.9 57.1 35.9 29.6 556 18.6 13.2 2.60 50.0 57.1 35.9 29.6 556 18.6 13.0 57.4 35.6 29.0 545 18.6 13.2 2.30 44.0 57.0 36.0 29.9 562 18.5 13.0 57.6 35.4 28.6 538 18.6 13.2 2.00 38.4 57.2 35.8 29.5 554 18.5 13.0 57.8 35.2 28.2 530 18.6 13.2 1.70 32.7 57.1 35.9 29.6 557 18.6 13.0 57.8 35.2 28.2 530 18.8 13.2 1.40 27.0 57.0 36.0 29.9 562 18.6 13.0 57.6 35.4 28.5 536 18.6 13.2 Cx = 93'0 Ch = 23.6 ~ 18.8 Cg

-151TABLE A-5 Sample 5 my T,0C 0C l oa1 i~ o2 c2 6 Cs2 1a2 L2 1.60 30.8 62.9 30.1 17.7 1930 13.7 2.80 65.0 28.0 13.9 1520 15.3 3.3 1.80 34.6 52.9 30.1 17.7 1930 13.8 2.85 64.9 28.1 14.0 1525 15.5 3.3 2.00 38.4 62.8 30.2 17.9 1950 13.6 2.75 64.8 28.2 14.4 1570 15.3 3.3 2.50 48.0 63.0 30.0 17.5 1960 13.1 2.65 64.6 28.4 14.6 1590 14.5 3.0 3.00 57.4 63.1 29.9 17.4 1945 12.7 2.55 64.6 28.4 14.6 1590 13.8 2.8 3.50 66.8 62.5 29.5 16.7 1840 12.6 2.50 64.7 28.3 14.4 1570 13.7 2.8 4.00 76.2 62.7 29.3 16.2 1770 12.3 2.45 64.8 28.2 14.2 1550 12.9 2.6 4.50 85.4 64.2 28.8 15.3 1670 12-1 2.40 65.0 28.0 13.9 1520 12.6 2.5 5.00 94.8 64.8 28.2 14.4 1570 12.2 2.40 65.2 27.8 13.5 1470 12.5 2.5 5.50 104.0 65.0 28.0 13.9 1520 12.1 2.40 65.5 27.5 13.0 1415 12.3 2.45 6.00 113.0 65.4 27.6 13.1 1430 12.1 2.40 65.2 27.4 12.9 1410 12.2 2.4 6.5o 122.1 65.9 27.1 12.4 1350 12.1 2.40 65.9 27.1 12.3 1340 12.2 2.4 7.00 131.5 66.0 27.0 12.1 1320 12.2 2.40 66.0 27.0 12.0 1310 12.3 2.45 6.80 128.0 65.9 27.1 12.4 1350 12.25 2.45 66.0 27.0 12.0 1310 12.3 2.45 6.30 118.5 65.6 27.4 12.8 1395 12.25 2.45 65.8 27.2 12.4 1350 12.4 2.45 5.80 110.5 65.4 27.6 18.1 1430 12.3 2.45 65.5 27.5 13.0 1415 12.5 2.5 5.30 10oo.4 65.0 28.0 13.9 1520 12.3 2.45 65.3 27.7 13.4 146o0 12.5 2.5 4.80 91.0 64.6 28.4 14.6 1590 12.4 2.45 65.0 28.0 13.9 1490 12.6 2.5 4.30 82.1 64.1 28.9 15.6 1700 12.4 2.45 64.9 28.1 14.0 1525 12.8 2.6 3.80 72.3 63.8 29.2 16.1 1760 12.6 2.50 64.7 28.3 14.4 1570 13.1 2.7 3.30 63.1 63.4 29.6 16.8 1830 12.7 2.50 64.7 28.3 14.4 1570 13.5 2.8 2.80 54.1 63.1 29.9 17.4 1890 13.0 2.60 64.6 28.4 14.6 1590 13.9 2.8 2.50 48.0 63.0 30.0 17.5 1920 13.2 2.65 64.5 28.5 14.7 1600 14.6 3.1 2.30 44.0 62.9 30.1 17.8 1940 13.4 2.70 64.6 28.4 14.6 1590 14.9 3.2 2.10 40.4 62.8 30.2 17.9 1950 13.7 2.81 64.8 28.2 14.2 1550 15.2 3.25 1.90 36.5 62.8 30.2 17.9 1950 14.0 2.90 64.8 28.2 14.2 1550 16.0 3.5 1.73 33.3 62.8 30.2 17.9 1950 14.2 2.95 64.8 28.2 14.2 1550 16.4 3.6 1.53 29.4 62.8 30.2 17.9 1950 14.3 3.00 64.9 28.1 i.4.0 1525 16.5 3.7 Cx = 93.0 Ch= 23.7 = 109 Cs my T, Oc c3 0 6 33 3 4 64 44 y 03 3 L3 Y S4 L 1.60 30.8 65.8 27.2 12.4 1350 16.7 3.7 66.5 26.5 11.3 1230 18.7 4.5 1.80 34.6 65.7 27.3 12.6 1370 16.8 3.8 66.4 26.6 11.4 1240 18.7 4.5 2.00 38.4 65.6 27.4 12.8 1390 16.7 3.7 66.4 26.6 11.4 1240 18.6 4.45 2.50 48.0 65.4 27.6 13.1 1430 15.8 3.4 66.2 26.8 11.8 1285 17.6 4.05 3.00 57.4 65.4 27.6 13.1 1430 15.0 3.2 66.0 27.0 12.1 1320 16.6 3.70 3.50 66.8 65.3 27.7 13.2 1440 14.7 3.1 65.9 27.1 12.3 1340 16.2 3.6 4.00 76.2 65.2 27.8 13.5 1470 13.7 2.8 65.9 27.1 12.3 1340 14.8 3.06 4.50 85.4 65.3 27.7 13.2 1440 13.0 2.6 65.9 27.1 12,3 1340 13.4 2.75 5.00 94.8 65..5 27.5 13.0 1415 12.8 2.6 66.0 27.0 12.1 1320 13.6 2.80 5.50 104.0 65.6 27.4 12.8 1395 12.6 2.5 66.1 26.9 11.9 1290 13.2 2.70 6.00 113.0 65.9 27.1 12.3 1340 12.4 2.5 66.2 26.8 11.8 1280 12.8 2.60 6.50 122.1 66.0 27.0 12.1 1320 12.4 2.45 66.2 26.8 11.8 1280 12.8 2.60 7.00 131.5 66.1 26.9 11.9 1300 12.4 2.45 66.4 26.6 11.4 1240 12.7 2.55 6.80 128.0 66.1 26.9 11.9 1300 12.5 2.5 66.3 26.7 11.5 1250 12.8 2.60 6.30 118.5 65.9 27.1 12.3 1340 12.6 2.5 66.2 26.8 11.8 1285 12.9 2.60 5.80 110.5 65.8 27.2 12.4 1350 12.7 2.55 66.1 26.9 11.9 1300 13.2 2.70 5.30 100.4 65.6 27.4 12.8 1410 12.7 2.55 66.0 27.0 12.1 1320 13.3 2.70 4.80 91.0 65.5 27.5 13.0 1420 12.8 2.6 66.0 27.0 12.1 1320 13.6 2.80 4.30 82.1 65.4 27.6 13.1 1430 13.2 2.7 65.9 27.1 12.3 1340 14.1 2.90 3.80 72.3 65.3 27.7 13.4 1460 13.6 2.8 65.9 27.1 12.3 1340 14.7 3.10 3.30 63.1 65.3 27.7 13.4 1460 14.3 3.0 66.0 27.0 12.1 1320 15.7 3.40 2.80 54.1 65.4 27.6 13.1 1430 14.9 3.15 66.2 26.8 11.8 1285 16.6 3.70 2.50 48.0 65.3 27.7 13.4 1460 15.6 3.4 66.2 26.8 11.8 1285 17.2 3.90 2.30 44.0 65.5 27.5 13.0 1420 16.2 3.6 66.3 26.7 11.5 1250 17.8 4.15 2.10 40.4 65.5 27.5 13.0 1420 16.6 3.7 66.3 26.7 11.5 1250 18.2 4.30 1.90 36.5 65.6 27.4 12.8 1395 17.3 3.9 66.4 26.6 11.4 1240 19.1 4.65 1.73 33.3 65.6 27.4 12.8 1395 17.8 4.1 66.4 26.6 11.4 1240 19.5 4.80 1.53 29.4 65.7 27.3 12.6 1370 17.8 4.1 66.4 26.6 11.4 1240 19.5 4.80 Cx = 93.0 Ch= 23.7 = 109 Cs

-152TABLE A-6 Sample 6 mv T, C C C C; B1 C C2 C e2 La2 L2 1.40 27.0 62.3 30.7 189 945 13.8 3.01 63.0 30.0 176 880 14.8 3.35 1.60 30.8 62.2 30.8 192 960 13.5 2.95 63.0 30.0 176 880 14.8 3.35 1.80 34.6 62.0 31.0 196 980 13.2 2.80 62.9 30.1 178 890 14.7 3.35 2.00 38.4 61.8 31.2 199 995 12.7 2.75 62.8 30.2 181 905 14.2 3.20 2.10 40.4 61.0 31.3 201 1005 2.20 42.5 61.7 31.3 201 1005 12.2 2.60 62.8 30.2 181 905 13.3 2.95 2.40 46.o 61.6 31.4 204 1020 11.7 2.50 62.7 30.3 182 910 13.1 2.85 2.50 48.o 61.6 31.4 204 1020 2.60 50.0 61.6 31.4 204 1020 11.4 2.40 62.7 30.3 18.2 910 12.7 2.75 2.80 54.1 61.6 31.4 204 1020 10.9 2.30 62.5 30.5 186 930 12.2 2.65 3.00 57.4 61.8 31.2 199 995 10.6 2.40 62.6 30.4 184 920 11.6 2.45 3.20 61.2 61.9 31.1 198 990 10.4 2.15 62.7 30.3 182 910 11.2 2.35 3.40 64.9 62.2 30.8 192 960 10.2 2.10 62.8 30.2 181 905 10.8 2.25 3.80 72.3 62.4 30.6 186 930 9.8 2.00 62.8 30.2 181 905 10.3 2.15 4.20 80.0 62.7 30.3 182 910 9.8 2.00 63.0 30.0 176 880 10.0 2.05 4.60 87.2 63.0 30.0 176 880 9.7 2.00 63.3 29.7 172 860 9.8 2.00 5,00 94.8 63.2 29.8 172 860 9.7 2.00 63.4 29.6 169 845 9.8 2.00 5.50 104.0 63.5 29.5 168 840 9.5 1.95 63.7 29.3 164 820 9.6 1.95 6.oo 113.0 63.8 29.2 161 805 9.5 1.95 63.9 29.1 161 805 9.5 1.95 5.30 100.4 63.4 29.6 169 845 9.5 1.95 63.6 29.4 165 825 9.5 1.95 4.80 91.0 63.1 29.9 174 870 9.5 2.00 63.4 29.6 169 845 9.6 2.05 4.30 82.1 62.8 30.2 181 905 9.5 2.00 63.2 29.8 174 870 9.7 2.10 4.00 76.2 62.6 30.4 183 915 9.7 2.10 63.0 30.0 176 880 10.0 2.15 3.70 70.5 62.4 30.6 186 930 9.8 2.10 62.9 30.1 178 890 10.3 2.25 3.40 64.9 62.2 30.8 192 960 10.0 2.15 62.7 30.3 182 91o 10.7 2.35 3.10 59.2 62.0 31.0 196 980 10.2 2.20 62.7 30.3 182 91o 11.2 2.50 2.90 55.6 61.8 31.2 199 995 10.4 2.30 62.6 30.4 184 920 11.7 2.65 2.70 52.1 61.6 31.4 204 1020 10.7 2.35 62.7 30.3 182 91o 12.2 2.75 2.50 48.0 61.6 31.4 204 1020 11.2 2.50 62.8 30.2 181 905 12.7 2.85 2.30 44.o 61.6 31.4 204 1020 11.5 2.55 62.9 30.1 178 890 13.4 3.10 2.10 40.4 61.7 31.3 201 1005 12.1 2.75 63.0 30.0 176 880 14.0 3.30 1.90 36.5 62.0 31.0 196 980 12.6 2.85 63.1 29.9 174 870 14.6 3.50 1.70 32.7 62.0 31.0 196 980 13.1 3.05 63.1 29.9 174 870 15.25 3.70 1.50 28.8 62.2 30.8 192 960 13.5 3.15 63.3 29.7 171 855 15.7 3.85 Cx = 93.0 ch = 23.6 = 50 C rv T,0C C3 C s 3 3 C4 C4 C4 4 L4 1.40 27.0 63.5 29.5 166 830 16.1 3.80 63.9 29.1 159 795 17.0 4.10 1.60 30.8 63.4 29.6 169 845 16.0 3.75 63.8 29.2 162 81o 17.1 4.20 1.80 34.6 63.4 29.6 169 845 15.7 3.55 63.8 29.2 162 810 16.7 4.00 2.00 38.4 63.4 29.6 169 845 15.2 3.50 63.7 29.3 163 815 16.2 3.80 2.10 40.4 2.20 42.5 63.2 29.8 1T3 865 14.8 3.38 63.6 29.4 165 825 15.8 3.70 2.40 46.0 63.1 29.9 174 870 14.1 3.15 63.5 29.5 166 830 15.3 3.55 2.50 48.0 2.60 50.0 63.0 30.0 176 880 13.8 3.05 63.5 29.5 166 830 14.9 3.40 2.80 54.1 63.1 29.9 174 870 13.4 2.95 63.4 29.6 169 845 14.5 3.30 3.00 57.4 63.1 29.9 174 870 12.7 2.75 63.5 29.5 166 830 13.8 3.10 3.20 61.2 63.1 29.9 174 870 12.2 2.65 63.5 29.5 166 830 13.4 2.95 3.40 64.9 63.1 29.9 174 870 11.8 2.55 63.5 29.5 166 830 12.9 2.80 3.80 72.3 63.2 29.8 173 865 11.0 2.30 63.5 29.5 166 830 11.9 2.55 4.20 80.0 63.4 29.6 169 845 10.6 2.40 63.5 29.5 166 830 10.4 2.40 4.60 87.2 63.5 29.5 166 830 10.2 2.10 63.7 29.3 163 815 10.7 2.25 5.00 94.8 63.6 29.4 165 825 10.0 2.10 63.7 29.3 163 815 10.4 2.15 5.50 104.0 63.8 29.2 162 810 9.7 2.10 63.9 29.1 159 795 10.0 2.05 6.00 113.0 64.0 29.0 158 790 9.6 64.0 29.0 158 790 9-7 2.00 5.30 100.4 63.7 29.3 164 820 9.7 2.10 63.8 29.2 162 81o 10.1 2.10 4.80 91.0 63.5 29.5 166 830 9.9 2.15 63.6 29.4 165 825 10.5 2.25 4.30 82.1 63.3 29.7 172 860 10.2 2.20 63.6 29.4 165 825 11.0 2.40 4.00 76.2 63.3 29.7 172 860 10.7 2.35 63.5 29.5 166 830 11.7 2.60 3.70 70.5 63.2 29.8 173 865 11.1 2.45 63.5 29.5 166 830 12.4 2.70 3.40 64.9 63.2 29.8 173 865 11.8 2.65 63.5 29.5 166 830 13.1 3.05 3.10 59.2 63.1 29.9 174 870 12.5 2.85 63.5 29.5 166 830 13.9 3.25 2.90 55.6 63.2 29.8 173 865 13.1 3.05 63.5 29.5 166 830 14.4 3.55 2.70 52.1 63.2 29.8 173 865 13.6 3.20 63r7 29.3 163 815 15.0 3.65 2.50 48.0 63.2 29.8 173 865 14.2 3.35 63.7 29.3 163 815 15.5 3.80 2.30 44.0 63.4 29.6 169 845 14.8 3.55 63.8 29.2 162 8o1 16.2 4.00 2.10 40.4 63.5 29.5 166 830 15.3 3.72 64.0 29.0 158 790 16.7 4.20 1.90 36.5 63.6 29.4 165 825 15.8 3.90 17.3 4.40 1.70 32.7 63.6 29.4 165 825 15.5 3.80 64.1 28.9 157 785 17.5 4.50 1.50 28.8 63.7 29.3 164 820 16.7 4.20 64.2 28.8 154 770 17.8 4.60 Cx = 93.0 ch = 23.6 E = 50 c

-153TABLE A-7 Sample 7 mv T,~C L a2 L2 a3 L3 1.48 28.4 19.1 3.40 19.5 3.55 20.0 3.70 1.60 30.8 19.1 3.40 19.4 3.50 20.5 3.85 1.80 34.6 19.1 3.40 19.3 3.45 22.0 3.70 2.00 38.4 19.0 3.40 19.2 3.45 19.9 3.65 2.20 42.5 18.9 3.35 19.2 3.45 19.6 3.60 2.40 46.0 18.8 3.35 19.1 3.40 19.4 3.50 2.60 50.0 18.75 3.30 19.0 3.40 19.3 3.45 2.80 54.1 18.8 3.35 19.0 3.40 19.3 3.45 3.00 57.4 18.8 3.35 19.0 3.40 19.25 3.45 350 66.8 18.8 3.35 19.0 3.40 19.2 3.45 4.oQ 76.2 18.5 3.25 18.8 3.35 19.0 3.40 4.50 85.4 18.5 3.25 18.8 3.35 19.0 3.40 5.00 94.8 18.5 3.25 18.7 3.30 19.0 3.40 5.50 104.0 18.5 3.25 18.7 3.30 19.0 3.40 5.20 98.5 4.70 89.2 18.5 3.25 18.7 3.30 19.0 3.40 4.30 82.1 18.6 3.25 18.7 3.30 19.1 3.40 3.80 72.3 3.30 63.1 18.6 3.25 18.7 3.30 19.1 3.40 2.80 54.1 18.6 3.25 18.8 3.35 19.3 3.45 2.70 52.1 2.50 48.0 18.75 3.30 18.9 3.35 19.5 3.55 2.30 44.0 2.10 40.4 18.80 3.35 19.1 3.40 19.7 3.60 1.90 36.5 1.70 32.7 19.1 3.40 19.4 3.50 20.2 3.75 1.50 28.8

TABLE A-8 VOLTAGES APPLIED TO CERAMIC WEDGES AND DIMENSIONS* OF WEDGES DC Voltages DC Field Sample Length of Average Width Width of Wedge Average Thick- Thickness at Applied to Field Applied Applied to (see Table 4.1 Wedge of Wedge at Thin Section ness of Wedge Thin Section Sample to Wedge Thin ection for compositions) (cm)cm) End (cm) (cm) End (cm) (Volts) (Volts/cm) (Volts/cm) 390 5700 6900 1.97.o69.058.018.004 6oo 8700 9500 980 14200 17000 500 i 6600 5400 2.96.076.0.02.016.0035 700 9200 7600 1000 120 10900 320 4500 4500 3.90.072.071.010.0035 500 7000 7000 760 10500 10700 o - 220 6000 4200 4.83.037.052.036.003 390 10500 7500 560 15000 10800 500 12200 16600 5.61.0o41.050.007.003 700 17100 23300 1000 j24400 33000 500 10000 10900 6.670.050.046.017.004 720 14400oo 15600 950 l! i 1 20600 I — 950 — 19000 20600 - 400 - 7000o 7 - -.057 - - - 960 1 16800 * Dimensions measured with an eye piece micrometer.

APPENDIX B -155

26 24 2220 // z 0 U 18 - C) 6 -16 (/) 4- 14 m 12 20 22 24 I — PHOTOCELL CURRENT I a Figure B-1. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 1. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density. Filter. Z 6 LJ 1 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, / L O Figure B-1. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sale 1.'Photocell Current vs. Percent Light Transmitted. by a Calibrated Diffuse Density Filter.

'I I I I' I I 1' I I I I I I I i! 14 12 10 0 (0 ~11 4 - IPHOTOCELL CURRENT, a Figure B-2. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 2. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter.

24 22 z 20 0 () W) 18 6,z n, 14 - 12o3 10 -J 8 p < w 6 -EF J 4' I 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, jo Figure B-3. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 3. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter.

26 24 2220 te // 16 u) z I 14- 8 - BEFORE RUN AFTER RUN I- 6 - 12 I.-I1 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, C ) Figure B-4. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 4. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter. Photocell Current vs. Percent Light Transmitted. by a Calibrated Diffuse Density Filter.

10 - (0 ^0 ) 6 z e ^5 I1- 4 -J 3Q- 02 a..,,,I I I I I I I I I I I i I I I I,I,, I 1 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, pL a Figure B-5. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 5. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter.

// 5 / I 3 0oL,,,,,,,,,,,,,,,,, I I I I I 0 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, Figure B-6. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 6. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter.

! I! I I I I I I I I I I I I 6 Z 0 ) 4 r2,/ I I 2 Iz.1) QI.^^ 0 L I I I I. I I I I I, I I I I I I I I I 1 2 4 6 8 10 12 14 16 18 20 22 24 PHOTOCELL CURRENT, u. Figure B-7. Calibration Curve for the Percent Light Transmission Measurements of Ceramic Sample 7. Photocell Current vs. Percent Light Transmitted by a Calibrated Diffuse Density Filter.

APPENDIX C -163

-164SAMPLE CALCULATION AND CALCULATION OF ERRORS The data presented in the preceding section on the incremental permittivity of the barium-strontium titanate compositions is subject to two types of errors; systematic errors, which are inherent in the measuring equipment, and random errors, which are introduced by the observer. The capacitance data which were taken on a Bontoon Q-meter, are accurate to within t 2% for the conditions of measurement,* A t 1.5% error is due to the limitations of the instrument itself, Another + 1% error is introduced during the course of the measurements by the observer, The compounded errors of instrument and observer are therefore somewhat less than + 2% for each capacitance measurements. In measuring the dimensions of the ceramic sample, a calibrated Bausch and Lomb mechanical microscope stage, an eye-piece micrometer, and a micrometer slide were used. The scales of these measuring devices were judged to be accurate, Certainly any deviations in their dimensions from the absolute length were insignificant in comparison with the accidental errors introduced during the measuring of the sample dimensions, Because of the small irregularities in the geometry of the wedges, an error of +.02cm is estimated for the length of the wedges, and an error of - 01cm is estimated for each measurement of the width and thickness (at the thick end) of the wedges, Ao Sample Calculation of the Incremental Permittivity of a Wedge of Ceramic Barium-Strontium Titanate, It is shown in Section IV, that the capacitance of a ceramic sample can be calculated from the capacitance data by the following * The samples all had a capacitance below 10fpf and were measured at 1600 KC,

-165equation: Co(CsT + CH) - CSTCH Cs = ----,\(C-l) CST - Co where Co is the capacitance of the sample as measured on the Q-meter, CST is the standard capacitance of the circuit, CH is the capacitance of the sampler holder, CS is the actual capacitance of the sample (see Figure 4.4, po43 ). The capacitance data of Sample 3, at zero field and 27~C (see data on Sample 3, Appendix A), is given as: Co = 3235p4f, CST = 110 oO f, H = 23o41 f. Therefore C = 32e3(110 + 23.4) - (110)(23.4) f 110 - 32o3 22.3 o The incremental permittivity is defined as Cs x t EA = - (C-2) A x,0885 where eA is incremental the permittivity of a sample, t is the distance between the electrodes, A is the area occupied by one electroded surface, CS is the capacitance of the sample.

-166A schematic of a wedge-shaped ceramic sample is shown in Figure 4.1, page 40, For Sample 3 the dimensions are as follows: Length of wedge =.90 cm, Width of wedge =.072 cm, Average thickness of wedge =.0095 cm, Hence the incremental permittivity of Sample 3 at 270~C and zero bias is: CA = 22.3 x.072.90 x o0095 x.0885 2120 B, Calculation of Probable Errors in the Permittivity of a Ceramic BariumStrontium Titanate Wedge. The general formula for the calculation of errors states that if R = f(rl, r2), (C-3) the probable error in R, E-, can be written as -i [( E)2 2] 12 (C-4) where E1 is the error in rl, and EFr is the error in r2 ~ Fl 2 2 The errors associated with the calculated incremental permittivity of Sample 5 at 27~C and zero bias are due to 1) Measurements of capacitance which are each subject to an error of + 2%. 2) Measurements of the sample dimensions which are subject to errors varying between +.01 and +,02cm.

-167The data for Sample 3, including probable errors, are given below. Capacitance data: Co = 93,0 + 2.0 - 60.7 t 1.2 = 32.3 + 2.35pf. 0 + CST = 110.0 + 2.24if. CH = 23.4 t 0.54lef. Dimensions: A = (.095 +.01)(.9 t.02) cm2. t =.072 +.017 cm. 1o Probable Error in Sample Capacitance, ECs Co (CST + CH) - CSTCH ( sample -1 -ST - Co Let C (CST + CH) CSTCH = CA CSTCH = CB CST -C CC Then C -CA - CB sample - a, Error. in CA, E CA Equation (C-4) is used to solve for ECA. Let CA = f[Co, (CST + CH)] [(A)2 E 2 (^CAA )2 E 2]l/2 Then EC A )2 Ec 2 + ( A +CH) E(CsT+CH) CA = C (C + CA = C (CST + CH) 32.3 + 2.3 (110.0 t 2.2 + 23.4 +.05) f2 3 2.3 - 2.3 (133.4 + 2.3)5f2' 4309 + ECA |f2

-168ECA = [(133.4)2 (2.3)2 + (323.)2 (2.3)2]1/2 = 31644f2 Therefore CA = 4309 + 316 f2 b Error in CgB EC - B CB STH' = (1o.o t 2.2)(23.4 t 0.5) 44f2 = 2574 ~ Ec 4f2. CB According to Equation (C-4), if CB = f(CST, CH) then E E 2 + E then ECB [( B )2 E 2 + ()2 E1 2]2 CB CT CST'C CH [(110.0)2 (0.5)2 + (253.4)2 (2.2)2]1/2 2 = 75t Lf Therefore CB = 2574 + 75f2. c, Error in CC, ECC CC = CST - Co = (110. + 2.2)-(32.3 + 2.3)f, =77.7 + 3.24f. d. Probable Error in CS, ECs S C. C = CC

-169(4309 + 316)-(2574 + 75) 77.7 + 3.2 1735 325 77.7 53.2 22533 + ECs According to Equation (C-4), If CS = f[(CA-CB), CC] then ES [(CS C 2 C 2 SC2 ECs2]l/2 CQ~~~S~ E ((CABC)B) + (C) ECs] (7.)2 (325)2 + (- 77 )2 53.2]/2 = 4.28 Therefore Cs = 22.3 + 4.3 f. Probable Error in the Incremental Permittivity of the Sample, EEA The incremental permittivity has been defined in the early part of this appendix as CS x t UES x t-~ ---.(C-2) A x.0885 Inserting the capacitance of the sample CS in Equation (C-2) and the dimensions of thickness (t) and the area (A) of the sample, the incremental permittivity of the sample is given by (22,3 + 4.3)(,072 +.017) E A. (.095 +.01)(.9 t.02) x.0885 2120 + E _ t

-1702. Probable Error in E, EE where c x t (c-2) A x o0885 Probable error in numerator, EN If numerator N = f(C,t) Then EN = [( 3)2 EC2 + () Et21/2 = [(22.3)2(o0017)2 + (o0072)2(4.3)2]1/2 =.003 Probable error in denominator, ED If denomenator D = f(Q,w) where Q = length, and w = width, then ED = [(N)2 Eg2 + ( N)2 E 2]1/2 x.0885 ) w = [(.095)2(.01)2 + (.9)2(oO2)2]1/2 x.0885 = o.0016 Probable error in EA, EEA If eA = f(ND), [ EEA)2 E,2]1/2 then EEA = +[( E + ( )2 1/ 1 [(3)2 + 161 )2(0016)2]1/2 [(00756)2(~3)+ (O.007562 44 2 oo for Sample 3 = 2120 + 44

APPENDIX D PREPARATION OF (Ba,Sr)TiO3 CERAMICS The mixed titanate ceramics, with the exception of Sample 5, a commercial ceramic, were all prepared at the Electromagnetic Materials Laboratory, Electrical Engineering Department, University of Michigan. Stoichiometric proportions of a high purity grade BaTiO3 and SrTiO3, sometimes with small additions of metal oxide fluxing agents (see Table 4.1), are combined in a porcelain ball mill to which water and a binder such as methyl cellulose or the commercial "Hyform" preparation is added. The powders are ball-milled with porcelain balls for one hour. The slurry is then drained into pans and oven-dried at about 80~C. After drying, the soft cake is granulated and passed through a 20 mesh screen. The powder is pressed into a thin rectangle at about 5000 lb. per sq. in. The slabs are placed on a platinum sheet, dusted with some of the prepared powder, and fired in a Globar furnace. The temperature of the furnace is increased at a rate of 250~C/hr to 1350~C, and is held at that temperature for two hours. The furnace is then allowed to cool naturally. This procedure produces dense ceramics which can be easily ground into wedges (see Section IV).

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UNIVERSITY OF MICHIGAN 3 9015 03695 6095