ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Report No. 7 MAGNETICALLY SENSITIVE ELECTRICAL RESISTOR MATERIAL January 1, 1956 to April 30, 1956 E. Katz H. Patterson W. Tantraporn H. Boyne Project 2136 DEPARTMENT OF THE ARMY LABORATORY PROCUREMENT OFFICE U. S. SIGNAL CORPS SUPPLY AGENCY CONTRACT DA-36-039-SC-52601 FORT MONMOUTH, NEW JERSEY June 1956

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE OF CONTENTS Page LIST OF IIJUSTRATIONS iii ABSTRACT iv OBJECTIVE iv A. INTRODUCTION 1 B. PREPARATION OF SINGLE CRYSTALS OF Bi AND CERTA Bi ALLOYS 2 1. Purpose 2 2. Materials 2 35 Preparation of the Raw Material Used for Growing Single Crystals 2 4, Preparation of Thin Bi Wires 3 5 Preparation of Bi Wires for Galvanomagnetic Work 3 6, Preparation of Bi Rods for the Absolute Determination of the Conductivity 4 7. Preparation of Bi Alloy Single Crystals 6 C. EQUIPMEINT 6 1. Magnets 6 20 Detection System T 3. Polar -Diagram Detection System 8 4 Mounting of Samples 10 Dv- MEASUREMENTS AND RESULTS 11 1. Measurements of Zero-Order Brackets at Room Temperature 11 2, Measurement of Zero-Order Brackets at Various Temperatures 14 3- Measurement of First-Order Brackets at Various Temperatures 16 4. Method of Determining Second-Order Brackets 19 5- Results of Measurements of Second-Order Brackets 26 6* Measurements on Alloys 29 70 Discussion 29 APPENDIX 30 ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN LIST.OF ILLUSTRATIONS Figure Page 1. Schematic diagram of amplifier. 9 2. Block diagram of resolver equipment: 9 3. Sample holder (semi-schematic) showing coordinates xx2x3s current, potential, and Hall electrodes, and 0, f, rotation axes. 12 4. Resistivity of Bi as a function of the angle Q between wire and trigonal axis. 13 5 Principal resistivities of bismuth as a function of temperature, 15 6. Measurements from which [100123 is found by formula (4)* 18 7. Measurements from which [00l]12 is found by forrmula (5). 20 8, Comparison of brackets of zero- and first-order by various authors * 22 9- Orientation of symmetzry coordinates klk2k3 relative to the laboratory coordinates xlx2x3 defined by the sample holder. 23 10 -Orientation of 'the laboratory coordinates relative to he downward vertical D and tihe horizontal magnetic field B. 23 11. Comparison of [200]23 and of [200]1l - [200122 by various authors. 28 Al*, Presence of fourth harmonic at -150TC. 31 A2. Relative amplitude of Co, C2, and C4. 32 Table I, Experimental Values of the Zero-Order Brackets as a FUnction of Temperature 14 IIo Experimental Values of the First-Order Brackets as a Function of Temperature 17 III, Comparison for Zero- and First-Order Brackets Among Different Workers 21 IV. Experimental Values of Second-Order Bracket Combinations as a Function of Temperature 27 V. Comparison for Second-Order Brackets Among Different Workers 27 iii

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT The galvanomagnetic constants (brackets), as defined in our previous reportt, have been measured for Bi between room temperature and liquid-air temperature for orders zero and one and partially for order two. The results agree roughly with those of Okada and somewhat better with. those of Abeles and Meiboom. Also, preliminary measurements on some Bi alloys are reported. The growing and preparation of single crystals in the shape desired for our measurements are described in Section B. lie measuring equipment is described in Section C. The results are given in Section D, together with a particularly suitable method of determining the second-order constants. OBJECTITVE This project aims at developing the understanding of the magneto resistance effect (change of electrical resistivity in a magnetic field) by theoretical and experimental research, with the ultimate aim of developing materials with more favorable magneto resistance properties t.han are available at present. I —_ --- —_ ----__ -- iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A.T. R ODIXC't1ON It 'was stated in the previous report th-at the present report would; not be concerned with. further progress made meanwhile in regard to tthe theory of the galvanomagnetic effects. Accordingly, the present report covers tlhe experimental-work of the period MarcC 1, 1955 to March 351, 1956. _she length of thie period covered permits one to see some of thle wo:rUk, especially in th.e earlier- alf, in proper perspective and, consequently, some work of a preliminary nature will be omitted arid various inte1mediate stages in improving the techniques of measurement will tbe described only very 'briefly*. he main emph.asis will be on t~he techniques as developed presently and the results obtained. According to the previous theor-etical -report, the experimental task of measxring thie gal.vanomagnetic effects consists of the% folLowing steps 4 a, Preparation of sing.le crystal.s of given purity, dimensions, and orientation of crystal axes, b, Moimating of the sas ple with respect to electrodes for current, longitudinal voltage, transve:se voltage, magnetic field, and temperature,* cT T aking o exacio of measthe galanomagnetic material constants (brackets) from the measurements. d. Discussion of results. hrhese four steps will be descr:ibed p resently. The major part of t he work was done with' ismutha, later aug ent.ed wi.th some work on certain bismutl alloys. It is- recalled from. previous reports:how the galvanomagnetic constants are defined. If reference is made to a llsylmetz:y coordinate systems which is adapted to the symetry axes of the crystal. (k3 along th e trigonal axis, kl along a binary axis), then the conductivity components cij can be expressed in terms of a power series in the coQponents BL B2 BS of t;Le magnetic field and the coefficientss (brackets) are the galvano'magnetic constants. W n ma i = Z z [m - pptn - m]ij Blm P BBsn -m n=O m=O p.O.__________________________ 1 _________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For bismuth (group Di ) the independent brackets are, for n < 2-: n = 0 [o00]7i [000]33 n - 1 [100]23 [001]o2 -n = 2 [200] [2]22 [200200]33 [002].. [011] z [Ol]0 2 In the present report, work is described leading to the measurement of these brackets as a function of the temperature* The work on brackets with n = 0 is largely completed, that on brackets with n = 1 is completed to the point where results are available which seem satisfactory although some further improvements in accuracy are indicated, while that on brackets with n = 2 is just producing the first results. B. PREPARATION OF SINGLE CRYSTALS OF Bi AND CERTAIN Bi ALLOYS 1. PURPOSE It is the purpose to grow single crystals of pure Bi and of certain Bi alloys in a shape suitable for subsequent electrical measurements and with known orientation of crystal axes, 2, MATSERIALS The Bi and Te used in this work were obtained from the American Smelting Co. of South Plainfield, N. J. Both were of 99.99+- purity. The Sn was from the General Chemical Co. of New York, N. Y., and was of 99*97% purity. 5. PREPARATION OF TEE RAW MATERIAL USED FOR GROWING SINGLE CRYSTALS The raw material is made in the form of a rod of Bi (or alloy) in a glass tube. A 50-ml crucible containing about 50 gm of:Bi is placed in a vacuum bell jar: Several glass tubes about 35 cm long with one end closed and an inner diameter of from 2-5 mm are placed vertically into the crucible with the open ends down. The bell jar is then rinsed three times with He and evacuated to about 10- mm Hg, (The He rinse is done to prevent the formation of an oxide film on the Bi when heated. ) By means of a heater coil around the crucible the Bi is melted and forms a seal around the lower end of the _ --- —---------- 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - glass tubes. The glass tubes are also heated over their entire length by a vertical oven. Then He is let into the bell jar and pushes the Bi up into the glass tubes. The bell jar is then removed, the tubes are then drawn up at the rate of about 15 cm/hr out of the vertical oven past an air jet so that the Ri solidifies slowly from the upper end. This method prevents the breaking of the glass upon solidifying (Bi expands when solidifying). This method sometimeas gives single crystals; the trigonal axis is then usually normal to the length of the tube. The Bi rods so obtained serve as the raw material to be regrown by the methods described under Sections B.4, 5, 6 ' The material had a clean, shiny appearance. 4. PREPARATION OF THIN Bi WIRES A good deal of time was spent in attempts to prepare thin Bi wires of the order of 50 microns in diameter for magneto resistance measurements. Such wires, owing to their large surface-volume ratio, insure the best thermal homogeneity. However, the experimental difficulties encountered with these wires led us finally to abandon this procedure and to use thicker wires as described below. For the sake of putting our experience on record, the following details are reported. The Bi rods as described under Section B.3 are heated in their glass tube and then pulled. A section of about one inch is broken off from the long fiber so obtained and the cLeavage planes at both ends are investigated under the microscope. If they are parallel at both ends, which is the usual situation, we assume the fiber to be a single crystal. This assumption has been tested in a number of instances by means of X-rays and by cleaving a fiber at a dozen or so intermediate places. The normal to the cleavage plane was assumed to be the trigonal axis. Later experience with somewhat larger samples indicates, however, that often cleavage of a glassenclosed fiber will occur along a plane containing the binary axis (secondary or imperfect cleavage plane). This uncertainty and the relatively large amount of labor involved in X-ray determination of the trigonal axis direction was one of the reasons why the work with thin fibers was later abandoned. The glass at the ends of the fiber is then cracked off mechanically and the sample mounted in a special capsule with gallium electrodes which is mounted in a Cardan suspension-type holder for magneto resistance measurements. 5. PREPARATION OF Bi WIRES FOR GALVANOMAGNETIC WORK a. Seeded-Zone Melting of "Raw Material."1-A Bi rod, still in glass tubing (' 0.07 inch inner diameter), is held vertically. A heating coil is moved downward by a clock motor at the rate of about 1/64 inch/min. The melting zone in the operation is about 3/8 inch long. The operation is done in about 10-2 mm vacuum* (At lower -pressures the Bi melt would develop enough vapor pressure to kick itself out of the tube.) To start the operation the 3 --- —----— 5 --- —-------

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - upper end of the Bi is heated to melt, and a seed crystal, mounted such that the trigonal axis of the seed is at the desired angle wilth the rod to be seeded, is moved down to touch the melt. After 10 min the connection is found to be in equilibrium, and the coil is moved downward as far as the desired length of the newly grown single crystal. This method of growing is very satisfactoy for the trigonal axis away from the rod and is 350% successful for the trigonal axis along the rod. Vibration in the room is found to be the disturbing factor. For best results, the seed crystal should be slightly larger than the rod. After removing the glass rod from tihe growing apparatus the glass around the Bi is taken off by IF in a Lucite container (r cite appears to be resistant to BF) and the Bi surface cleaned 'by a bath of about 0.05'iN nitric acid, then rinsed and dried. b, Determination of ~)he Crystalloga- c Axes,.- single crystal of Bi can be cleaved in four directions, thee containing the binary axes (inperfect planes) and one normal to thie trigonal axis (perfect plane). The planes can be distinguis!.ed visually~ All crystal axes are determined from. the perfect plane. ThLe trigonal axis is determined a cc;.rately 'by- means of a shadowgrahic method. The crystal is mounted in a light beri and a 5 "o 10 &times enlarged image is observed, drawn on paper, and the angie measured. The crystal is rotated around its own axis normal +o to the light path, ntil 'the focused sh~adow of the perfect plane is a s'traigh> line. This a..llows for the measurement of theie angle 1by an ordinary protracftor- to Ibe accurat e wit iin 1/2' or better: It is also requiaLed to know t>e orientation of t`.he binary axes* The sample is first mounted in its holderr '(ee Fig. 5 page 1i2 ) and the ang.e between one of the binary axes and the normal to the wire in tih.e plane of the sample holder (the x2-direction in Fig. 5, whiach is to be thle direction of Hall potential measturement) can be found as follows. On the perfect plane, one can see three sets of parallel lines intersecting at.60~~ T'hese lines are parallel to tie bin.zay axes. *.-rod is mointled so that tihe perfect plane and its line of intersection wit t^. e 'ode. poane &.re exactly horizontal. Tlhen under a vertical microscope one can see "he binary axis lines making some angles with. the cross-.air line in the eyepiece of the microscope permitting direct measurement of the angle between tbe binary axis and a horizontal line fixed on the sample holder by elementary manipul.ation, with an acuracy of better than 1/21'o 6. PREPARATION OF Bi RODS FOR, THE ABSO0:E. DETRMINATION OF TE CODUCTIVITY In order to obtain significant galvanomagnetic constants, it is necessary to know the absolute conductivities [K000]z and [OO] 33^ Special 1. See Y. Tanabe, TokSku Univ. Res. Inst. Sci. R.,l, 275 (1949). 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN samples were prepared for the measurement of these zero-order brackets as follows. Three V-grooves were cut into a 7/8-inch-diameter steel rod (three grooves were used for a higher rate of production of crystals). An oven was made out of Vycor tubing. Stopcocks were incorporated to allow a vacuum to be created within it. The inside diameter was large enough to insert the steel rod. A heater coil was wound on the outside of the Vycor tubing, tightly at one end, with the pitch gradually increasing toward the other end. This arrangement made it possible to obtain a thermal gradient along the steel when it was inserted. The glass is removed mechanically from rods of raw material (see Section B.3) and these are placed in the V-grooves of the steel rod, sometimes with thin sheets of mica between Bi and steel. (The purpose of the mica is to prevent any diffusion of steel into Bi; however, it was found that no observable difference results from omitting the mica as far as conductivity measurements are concerned.) After the Bi is inserted, the oven is evacuated and flushed with helium three times. After the last flush the oven is again evacuated to a pressure of approximately 10- 3 mm of mercury. The heater coil is then heated, allowing the Bi to melt from one end. After the rods of Bi are completely melted, the heating current is gradually decreased, allowing the steel rod to cool slowly but always maintaining a good thermal gradient. The Bi melt then starts to solidify along this thermal gradient and single crystals are produced. In this process no precautions need to be taken to insure that each melt starts crystallizing from one spot in the melt. It just happens. There seems to be no preferred direction of growth. Crystals of all orientations have so far been obtained, randomly. In order to obtain any desired axis orientation, the same method has been employed with a slight variation, using seed crystals. A seed crystal and a Bi rod are introduced into the V-groove. The Bi rod and part of the seed are melted to insure a good connection. The melt is then cooled from the seed as before. Single-crystal Bi rods of about 12 cm by 7 Ma2 obtained in this way were used for the measurement of the conductivity. The seed can be gotten either from a previously grown crystal or from a method devised for this purpose as follows. A metal strip in the shape of a quarter circle, one inch wide and 1/2 inch thick, has one continuous Vgroove cut along six consecutive sides of a regular 24-sided polygon, so that each straight section is about one inch long and makes an angle of 15~ with adjacent sections. A previously grown crystal of arbitrary orientation is bent to lie in the V-groove, preferably with its trigonal axis in the plane of the polygon. The metal strip is then heated at one end and the Bi is melted along a temperature gradient. However, the melting is stopped short of the A -------------------- 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN end of the Bi rod. This allows the end to remain solid and act as its own seed. Cooling is then started toward the opposite end. When finished there remains a single crystal of Bi in the shape of six connected straight sections. It is then cleared at each bend, thereby producing six given seed crystals all differing by 15~ from each preceding one. Any crystal can be used as a starting point. In this way seed crystals of any desired axis orientation can be obtained. The way in which the exact directions of the axes are determined for a Bi sample of this type was the same as that described in Section B.5. 7. PREPARATION OF Bi ALLOY SINGLE CRYSTALS It was intended to study the effect of impurities on the conductivity and on the galvanomagnetic constants of Bi, for substitutional impurities containing fewer or more electrons per atom than Bi, or the same number. Preliminary results on Bi-Sn alloys were given in an earlier report. The conductivity decreases as Sn is added. Presently some preliminary studies on Bi-Te and on Bi-Te-Sn alloys are reported. a. Bi-Te.-Our first alloys were made with 0.0212% and 0.122% tellurium by weight. The process of growing single crystals was the same as described for Bi in Section B.5, The electrical measurements indicated that more precautions are necessary to insure homogeneous composition' of alloy rods. A stirring device for the original melt in the crucible from which the raw material is made has since been added, but since crystals are grown along a temperature gradient, the effect of zone refining is superimposed upon the crystal-growing process. This means that the impurity, in all probability, will be distributed unevenly throughout the crystal. The crystals were harder and more brittle than pure Bi and showed a charcoal-grey color. The cleavage properties and the appearance of binary axis lines on the cleavage planes were similar to those of Bi..b Bi-Teh-Sn — One alloy was prepared with 0,05% Sn and 0o.05. Te by weight. The procedure and the difficulties were the same as for Bi.-Te alloys.:t The alloys are likewise similar, C. EQUIPME~I 1. MAGNETS In Report No. 5 the electromagnet is described which was used for thin wire samples. When the need for larger samples and different measurements on each sample came up, it was desirable to have a magnet with a larger gap and with a permanent field. The gap width should accommodate a Dewar for temperature studies. A C magnet was acquired. In each pole of this magnet - --------- 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN there are 28 rods of Alnico bars ( 1 inch diam., 4 inches length) closely packed behind a thin,soft, iron pole piece-. Energizing coils are mounted around the Alnico bars. IThe field can be set at any arbitrary value by running an energizing pulse through the coils, after -.twhich the Alnico bars will retain the value of the field "'permanently." The pole pieces are 6-5/4 inches in diameter at the tapered ends and are separated by a 3-1/2-inch gap. The uniformity of t-he field was tested by a sensitive flaumeter and testing coil (200 turns, 1 square inch area). The field was found to be constant within half a percent in a region 2-1/2 inches in diameter around the. center and decreasing slowly toward the edge, thLe decrease being less tnan 5'% at tkie very edge. The maxinr field attained wit. the present pulse source is approximately 1500 gauss. ThLe permanence of the magnetic field allows measurements to be taken on different samples, with varying termperat'e:e. The value of the field has been set to be 345 '- 3 gauss, meas-ured 'botoh by fluxmeter and proton-resonance equipment. The low field was chosen for the studying of the lower-order brackets of conductivity in order to minimize the influence of terms of higher than second order in B. A proton resonator has been built after the design of N. J. Hopkinss, with slight modification. For low fields of tie order of 250 gauss the resonance signal:requires considerable amplification, and the noise-to-signal ratio is not negligible. Consequently, the location of Lthe resonance peak requires more time for such weak fields than for strong fields. However, if the magnitude of the field is knomwn approximately, — the proton resonator permits determination to within 1 down to 250 gauss. 2. DETECT-IO. S.BSTEF4 Previous measuriements were:made with a galvanometer-Wheatstone bridge arrangement. In order to speed up measurements and to increase the s.ensitivity, use was made of a recorder and amplifier kindly loaned to us by the Leeds and Northrup Co. The sensitivity was thus about ltC0 times greater. This increase in sensitivity is desirable because it perriits a decrease in measuring current through the sample, tehus inimi zing any therma effects* A measuring current of approximately 0.15 amp was used. Associated, with thiis change were the following additions. a. Potentiometer Bridge. -Because of the planned use of samples of lower resistance than used previously, the Wheatstone bridge was replaced by a simple potentiometer bridge circuit. b. Vibrator. —A 120-cycle vib:rator has been used at the output of the potentiometer bridge to convert the bridge signal to ac. This signal is fed into a sharply tuned 120-cycle linear amplifier with an amplification factor of 500* This signal is then fed into the recorde r The input impedance 2. S ~ r* a 20. 401 (1950)^ _______________)_________

ENGINEERING RESEARCH INSTITUTE e UNIVERSITY OF MICHIGAN of the amplifier is 50,000 ohms, hence no effective change in bridge balance results with the addition of this component. The 120-cycle-amplifier power supply has been replaced by a battery supply in order to decrease spurious 60-cycle distortion. The sensitivity of the recorder is such that a fullscale deflection of the recorder corresponds to 10 p.v, which represents a change of about 0.06% of the total sample resistance under average conditions of our measurements. 53- POLAR-DIAGRAM DETECTION SYSTEM The following equipment has been constructed in order to obtain quickly a polar diagram of the change in magneto resistance due to rotation of the crystal in the magnetic field such as are referred to in the literature and in previous reports. a. Sine-Cosine Resolver. -A sine-cosine resolver was built, consisting of a long solenoid primary and two secondary coils located at the midpoint of the solenoid. The secondary coils are displaced 900 from each other and rotate about a common axis. The dimensions of the primary solenoid are radiusz 2.75 inches (inner) length'ls 12 inches wound with 3500 turns of No. 28 "Nyclad" wire. The dimensions of the secondary coils are radius- 2.0 inches (inner) thickness 0.75 inches. Each coil is wound with 750 turns of No. 28 wire. The axis of the secondary coils is mechanically coupled to the axis of the sample holder. This axis can be driven by a motor at a slow rate so as to avoid eddy currents. b. Amplifier. —A three-stage linear amplifier has been assembled to amplify the signal from the bridge to the primary of the resolver 250 times (Fig. l). This amplifier has a battery power supply in order to eliminate 60-cycle interference which would be present if an electronic power supply were used. The amplifier tubes are rated for very low noise operation. c. Circuit.-The entire circuit (see Fig. 2) operates as follows. The d-c signal from the bridge is fed into the 120-cycle vibrator and then through the three-stage amplifier to the primary of the resolver* From the two secondary coils, the signal is fed through two identical 120-cycle sharply tuned amplifiers to the X and Y plates of a type 304-H oscilloscope. --------------- ---- 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN I mo' ' '''00 - '-A Fig. 1. Schematic diagram of amplifier. — r> ----- * --- a120-cycle d To x-axis Potentiomefer 120-cycle Linear Resolver - Amplifier o f scope Bridge Vibrator Amplifier Unitle o -o 120- cycle _ To y-axis Amplifier of scope Fig. 2. Block diagram of resolver equipment. With no magnetic field, a rotation of the sample through 3600 describes on the oscilloscope a rotating straight line whose envelope is a circle The radius of the circle is determined by the amount by which the bridge is unbalanced. If the bridge is balanced while the sample is outside of the magnetic field, and subsequently the sample is moved into the field and rotated, the length of the rotating line on the scope varies in general with the angle and the envelope is a polar diagram as expected. d. Choice of Frequency.-It has been apparent in the preceding discussion that the frequency for the operation of the resolver is 120 cycles* The choice was made because the two sharply tuned 120-cycle amplifiers had already been built and were available on loan from Dr. Enns of this laboratory. These amplifiers are quite satisfactory for our work. Therefore, the amplifiers were considered the basic unit to which the other equipment has been adapted. 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN e. Future Plans. —With the sine-cosine resolver ready for operation we expect to photograph the polar diagrams obtained from various samples, using different orientations of the crystal axes. The use of the resolver will enable us to check quickly what the dependence of the galvancxnagnetc effet'.t is terms of the field, i.e., the number of terms of the power expansion influencing the effect. It is also hoped that a detailed study can be made on crystal samples in order to analyze the rosette pattern in terms of the theory. 4. MOUNTING OF SAMPLES a. For Zero-Order Bracket Measurement.-Many crystals of varied orientations were grown by the procedure described in Section B.*6 A Leeds and Northrup Kelvin bridge was used to determine thle resistance of the crystals with an accuracy of two parts in the fourth significant digit. Direct current was used and balancing was accomplished with a sensitive galvanometer. The sample holder consisted of two knife edges for potential leads and two flat contacts for current leads, and could take samples from three to twenty inches in length. The base of the sample holder was made of Lucite with a long V-groove cut into it. This formed a perfect seat for the bismuth crystals since they were shaped from a V-groove. Another sample holder based on the same principle but holding six samples was also used for part of the measurements. Some measurements were taken at ambient temperature in air, whereas another part of the measurements was taken with the samples and holder immersed in a water bath, thermostatically controlled to within 0.l0F. b. For Measurements of Brackets of Orders 0,,2, *..etc., as a Function of Temperature.-The sample holder and sample to be described below were immersed in a liquid bath for temperature control. The liquid was contained in a Dewar and could be moved into and out of the magnetic field of the Alnico imagnet described under Section C.l. For slow, continuous variation in temperature, isopentane (mp<-1500C) is used. The sample holder is submerged in isopentane in the Dewar which is cooled by a controlled flow of liquid air through a copper coil submerged in the isopentane until freezing. Then the bath is warmed up slowly after the flow of liquid air is stopped. The measurement can thus be done continuously at all temperatures upward. The temperature is measured by a copper-constantan thermocouple (accuracy +- 1~C)........ --- —--- - 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For long isothermal measurements such as required for second-order brackets, different kinds of liquid will be used with the freezing points in steps from -150 to +20~C. Liquid-air cooling will again be used to freeze the liquids. It seems best to describe the sample holder together with the orientation of the crystal with respect to it. The crystal, in the form of a cylindrical rod 1-1/2 inches long and approximately 0.07 inch in diameter, grown by the seeded-zone melting method described in Section B.5, is placed in a V-shaped groove in a.l ucite disc, such that the trigonal axis and the rod form a plane perpendicular to the plane of the disc. Two Hall probes, in the form of brass wedges lying in another V-shaped groove exactly perpendicular to the rod, are held against the rod by two small bronze springs. Thus the "laboratory coordinate system" xlx2x3 as defined in the previous report is fixed with respect to the sample holder: x1 along the rod, x2 in the direction of the Hall probes, and x3 accordingly. The lucite disc is mounted on a brass disc such that the whole can rotate around the x2-axis (i.e., Hall probe direction), which is held by two brass prongs. The x2-axis and the prongs can be rotated in a horizontal plane, around a vertical or 0-axis. The rotations around the vertical 0-axis and about the horizontal x -axis or.-axis are accomplished by a gear arrangement and can be done from outside the Dewar. T6he magnetic field B is horizontal, hence perpendicular to the 0-axis. By manipulation of the gear system the sample holder can be placed in any desired orientation with respect to the magnetic field. The current leads (No. 28 Nyclad) are soldered to the ends of the crystal, using the bismuth itself as solder. The MR potential probes are sharp bronze pieces which are held against the rod by light spring pressure. The springed contacts of the MR potential probes and of the Hall probes are necessary to avoid stress in the sample and insure good contact during temperature variation. The details of the sample holder are shown in Fig. 3. D. MEASJREMENTS AND RESULTS 1. MEASUREMENTS OF ZERO-ORDER BRACKETS AT ROOM TEMPERATURE In order to measure the zero-order brackets according to the expression Vm 2 1 a + 1 I d12 V [m O= { ] [+^ + cos2 o ([OO]3 (1) o it is necessary to measure the resistance V/I, the mass per unit length m/f, the length A, the density d, for several samples with different angles 9 11 -

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN gXI I ' Fig. 35 Sample holder (semi-schematic) showing coordinates xlxx3, current, potential, and Hall electrodes, and 0, t, rotation axes. between the trigonal axis and the rod. The resistivity measured at room temperature near 200C is plotted for different samples vs cos2 @ in Fig..4 after correcting all data to 20~C, using 0.004 per ~C as the thermal coefficient of resistivity. From the slope and intercept of the resulting straight line, the zero-order brackets are determined for Bi at 20~C: [000] 1 = 9.06 x 103 ohm-' cm-l [000]33 = 7.21 x 103 ohm-1 cm-1 The lengths between the marks made by the potential probes on the samples were measured with a traveling microscope to an accuracy of 10-3 cm. The density of Bi was taken as 9.67 gm/cm3 throughout. The mass per unit length was determined by cutting the sample with a sharp razor exactly at the potential probe marks normal to its length, after the electrical measurements were taken, and then weighing. The main source of error is believed to be due to Q; however, this should have very little effect near the endpoints of the line. The error in the above results is estimated to be less than.0.7%, but a few points deviate more than this amount for reasons that are not understood presently * 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 OC hO oo I \l l I \ I I I 1' N ~ Ia b0 ______ \___________________ _____ C00 o 0 LC)n 4) \d 0 cCc '0 _.. __ __ _ _) d ~ \N Md EC Ct, 0 -e (LNV) 0HO) 901x d

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The results obtained from similar measurements with the various allo samples scattered too much to allow separate determination of the two brackets. Further work to insure homogeneity is first required, On the whole, the re — sistivity of Bi-Te was from 10 to 305 lower than pue Bi (the concentration of Te was Oo0212 and 0.122%), while the resistivity of Bi-Sn-Te as described was some 8% higher than that of pure Bi.| 2 MEASUREMENT OF ZER0-ORDER BRA0KETS AT VARIOUS EFIRATUEES Two crystals, one wit1h the trigonal axis norn.mial and tihe othe.r with the trigonal axis parallel to th.Le rod, are used to study the brackets [000]n and [000133, respectively, at various temperaturese. 1 he resistance of each sample is measured in an isopentane bath. from. -50tC to +20C. The resistance values at 20~C are then converted to resistivity. values of 110 and 139 x 103 for. /[ 00].. 1/[]0,33 rhespectively,. which 'were established.peviously. Ly resistivity as a function of t.emperatur-e is then plotted proportionately in Fig- 5. From the curve, tihe. values of t.he two brackets at 0I intervals: are tabulated in Table:I. TABLE I EXPERIMETAL VALUES. OF "THE ZERO -CODEKR T ES AS A.,n CTICON 0F RRArC T. (OC) [coo [ ooo33 20 9. 06 x lO3 (a cm)-' 7.21 x 103 (- cm) 10 9.36 7,44 0 9.65 7.73 -10 9,97 8 02 -20 10,33 8*56 -30 10.71 8-71 -40 11.10l 9:08 -50 11*55 9*52 -60 12*02 9.92 -70 12.55 10I45 -80 135*10 10.95 -90 15.70 11.*55 -100 14.40 12..16 -110 151L 12*.85 -120 15*92 15370 -130 16.83 l4.41 -140 17*8o 15.28 -150 _ 18.99 16.20 The thermal coefficients of resistivity (a) at 0~C can be read from Fig. 5. It is found that 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 N. L, o-i Z4) O 0 L tO0.rI- C ---------- - ^ - ---------------- _ _ o e0 4-! to ____________ X ___? — ^ __ ________ 0_ 0 -- 0- 0-0- 0 1~~~~~~~~~~~5~7> 5 C)..,....,.. a..O....,..., (IN INO 90 x ] -\5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN =~-~ - (> - 1 d[0001]= aS = LodT 4 [0 0]3 dT o,0059/24/~oc F-1 ~N1.1 dt 000111 3.1 No measurements have been taken, so far, for intermediate angles ~ that would permit the determination of- c as a function of P. Theoretically the dependence of a on 9 is given by ae = all + 33- ll) (2) i + (y - 1) cos2, with y = ([000 ]/[000]33)oGc = 1.25 for bismuth. 3. MEASUREMENT OF FIRST-ORDER BRACKETS AT VARIOUS TE^MPEATURES There are two first-order brackets in bismuth, [100123 and [001]12. These can be measured only by means of the Hall probes. The relation between the transverse voltage V across a cylindrical uniform rod of diameter d, with total current I through it, and the brackets up to the first power of B is given by J2 =. = c o [ 1o ] [ooo B [0oo001o]203 (cos (x3) - sin + 3 sin 9 [00 1112 where it is assumed that the sample is oriented, in the holder as previously described, with the principle cleavage plane containing the x -direction* The angle between the rod and the trigonal axis is Q, as before. The angle 0 is the angle between the magnetic field and the direction x of the Hall probes. For two special orientations or settings the expression simplifies considerably. In the first setting the sample is placed with the trigonal axis vertically (along the 0-axis). Then 73 = 0 and F2 jF2j = B [[1~00]23 1 cos 0 sin 0. (4) ( 1H (I) = B{[oo] [ ii[000l]3sJ ( ------------- — 16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Thus by turning the sample around the 0-axis and measuring the amplitude of the Hall voltage the bracket [10013 is obtained. In the second setting the sample is placed with the trigonal axis horizontally (normal to the 0-axis). Then cos (x3,H) = 3 sin 9- and ( _ -B [00]2 sin 0 sin 0 (5) \J1IH -o\3o [ j11]l Thus by turning the sample around the 0-axis and measuring the amplitude of the Hall voltage the bracket [001]12 is now obtained. The values of the first order bracket at various temperatures are given in Table II. TABLE II EXPERIMENTAL VALUES OF THE FIRST-ORDER.BRACKETS AS A FUNCTION 0F 'ITEMERATURE T (oC) [LOO.]. [00111? 20 0.94 (a cm gauss)-" -1.3 x 10- (f cm gauss)-" 10 1.07 -1.5 0 1.27 -1.8 -10 1.41 -2*1 -20 1.68 -2.5 -50 1.97 -3.1 -40 2.54 -538 -50 2*75 -4.4 -60 3.20 -5.1 -70 3*86 -6*. -80 4*58 -7.5 -90 5.44 -9.2 -100 6.53 -11 -110 7.89 -15 -120 9.60 -16 -150 11.5 -20 -140 14,1 -25 -150 17.6 -35 The values of [100]23 were obtained from the experimental curve of sample M, the trigonal axis of which is 50~ from the rod. The signal, in microvolts per ampere between the Hall probes with 0 = 90~, is plotted in Fig. 6 for both directions of the measuring current. Other samples were measured and agree with the tabulated values better than within 104. The systematic deviations of the measured points from the average curve are correlated 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 0i0 /. CM c~ o4 —~~~~~~~~~~~~~~~(C E^~~~~~~~~~~~~~~~ /~~~~~~~~~~~~~~ c'o oo d ____ _AL____ U r4 0 7 1 0 ( C _ _ c o o /. 18~~~~~~~~~~~~~~~~~~~~~~~~I

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN with the direction of the measuring current through the sample and are probably of the type discussed in an earlier report; hence, should be averaged. The values of [00112, also from sample M, cannot claim less than 15% error, since the experimental points represent signals of the order of 1 microvolt for the "safe" measuring current, 0.2 amp, while the noise is about 0.1 microvolt (Fig. 7). As of this writing, we have the complete data for [001]12 from only this one sample. Other samples were used; however, previous to the sample M, inexperience prevented the results from being complete. The value of [001]12 for four measured samples is found to be the same at room temperature, but different at lower temperatures. The differences were probably due to using the current in one direction only and/or too large a current density. But until more samples are measured, the values of [001]12 listed should be taken only as order of magnitude. It should be remarked here that the preparation of the samples does not guarantee that they are of the same purity. The variation in the bracket values, if due to imperfect setting of directions, would be less than 5% The observed variations from sample to sample must be due to other causes, of which the purity is at present the most suspected one. Table III and Fig. 8 compare values obtained for the zero- and firstorder brackets as a function of temperature by Abeles and Meiboom,5 Okada,4 and the present authors. The agreement of our data with those of Abeles and Meiboom is seen to be much better than with Okada's data. 4. METHOD OF DETERMINING SECOND-ORDER BRACKETS The following conventions will be adhered to regarding the signs and zero points of the various coordinates and angles involved (see Figs. 5, 9 l and 10). The laboratory coordinates are defined as follows. The +x3-direction is the outward normal to the plane of the sample holder. The +x1-direction is along the sample with the trigonal axis falling into the -+xl+x3-quadrant, and +x1making an angle +G < 90~ with the trigonal axis. The +x2-direction follows right handedly. The symmetry coordinates are oriented as follows. The +k3-axis is the trigonal axis in the +x1+x3-quadrant. Normal to k3, a kl-axis is chosen along one of the binary axes; k2 then follows. The angle r is measured from the +x2- to the +kl-axis in a righthand-screw sense around +ks. The direction cosines ~ia between the ki-axis and the x3-axes are now 5 Phys. Rev., 101, 544 (1956). 4. J. Phys. Soc. Japan, 11, 89 (1956)* 19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 0 0 0 r0 0 / rC) II / o 0 rd Q.. ~ 0 / dV)~~~~~~~~~~~~~~~~~~ W r CH4 0 0 0/ 0 0 0 O 0 ' —" 0 ~ t 0 0 0 i N.^_____' —. 0(N A4 NI I-) H() 20 c0' 0 oo.D. ~ * C oJ0 d~V^)"^ -----------------------— 1" H --- —— r --- —---

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE III COMPARISON FOR ZERO- AND FIRST-ORDER BRACKETS AMONG DIFFERENT WORKERS Our data: [000] 33 from sample I, [000]11 from sample J, [100]23 and [001]12 from sample M Gauss-n ~-l cm1 -Brackets T (~C) Ours Okada Abeles and Meiboom [000] 1 +45 - 7.94 x 103 +27 8.90 x 103 -- 855 x l03 0 9.65 x l0s 9.48 x 103 -70 12*53 x 103 13.16 x l03 -160 20.3 x 10s 22.5 x 103.. -197 25.9 x 03 _ 27.7 x 103 [000]3 +45 6.26 x l03 +27 7.02 x 103 - 6.65 x 103 0 7.75 x lO3 7.41 x l03 -70 10.43 x l03 10.05 x 103 --160 17.3 x 103 19.2 x 10 -- -197 22.2 x 103 - 26.6 x 103 [100]23 +45 =- 0.601 +27 0.9 0.76 0 1.27 0.839 1.25 -70 3.86 1.36 3.67 -160 22.5 2.93 -. -197 ____ - -- ____- 61.7 [001]12 +45 -- -3.78 x 10-2 +27 -1.5 x 10-2 1 - -33 x 10-2 0 -1.8 x 10-2 -3578 x 0" -70 -6.3 x 10-2 -5.67 x 10-2 -160 - - -10.7 x 10-2.___ ___ -197.-., 1. ---. -207. x 10-2 21

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN r 0t P, - - - - -- — 71 0 E.0. O c - o 0 OG).,.0 ~ o r / l/ i 0 0 -~ u o N- - _ _ _a 8 O,? t 0 0 ---_ T, ~-0 /I / /S NOX[OOV ~~oo co /n ^^ rr>, o 1-(;m U s 7,X( ssnv(o) ) NI x a[.ooo-J QNV [000]o 22

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN k3 X3 eJ rig. 9. Orientation of symmetry coordinates klkak3 relative to the laboratory coordinates xx2x.3 defined by the sample holder D X3 Fig, 10* Orientation of the laboratory coordinates relative to the downward vertical D and the horizontal magnetic field Bo. 23 -

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN l 1 = - sin. sin K ~21 = - sin 9 cos K s31 = cos 2' 12 = + cos K ~22 = - sin 32 = 0 (6) 3s a= + cos Q sin K 123 = + cos 9 cos s 33 = sin 0 The angle 0 is measured from B to the x2-axis in a right-hand-screw sense around the downward vertical. The angle r is measured from the downward vertical to the x1-axis in a right-hand-screw sense around x2 'The direction cosines 7i between the magnetic field B and the ki-axis are now, in terms of the Euler angles 9, 0, $: 71 - cos 0 cos K - sin 0 sin K cos (t - C) S 72 = - cos 0 sin K - sin 0 cos r cos (j - ) (7) 7s = sin-0 sin (* - ), For sufficiently low fields, according to equation (27) of the previous report, p o Rkl kYk 7 Z Ap Po where Po [o000o]11 [00033 + COs2 ([000]J - [000O]33) and the R"k are tabulated in ternm of brackets and 1, 12,31 in 'TabLe XV I of the previous report SBy suRastituting from -thi tahble in,the rigat-Ihamd side of the equation (8), making use of "u:e aL0oe expressions (6) and.(7) for ~ia and,k, an expression of Ap/p in terms of bracke-ts,, 9, 0 and, ea n be obtained. The brackets are material constan'ts Also, K and Q are constanrs for -any given sample but-may vary from s ample to sample, Thus if, is set at any fixed value, then Ap/p depends on 0. Ti.s dependence, which does not enter through Rkj. but only through-.7k ^, is obviousiy of the form* =r ^ Co+ cos-2 ( - 0 (9) bLP J-u.4r = const. where the tlree constants Co, C2, and 02 depend on t-he brackets and on, 9, as well as on 1r, for given B. The experimental determination of the brackets is now carried out by varying 0 with constant.r Of course, the varying of 0 only allows the * See, however, the Appendix. 24

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN determination of the three constants CO, C2, and 02, swhile there are eight independent brackets. It might seem that by evaluating the three constants for various values of r sufficient material -would be available to derive all tJae brackets. This, however, is not so. From one sample alone, at most six constants Rk" or six equivalent bracket combinations can be obtained. To do this, several settings of the angle 0 must be used and 0 must be varied for each setting. In order to determine the eight independent brackets of second order, some measurements must be made on at least two ssaples with different axis orientations relative to the length of the sample (different Q)o Setting 1 The trigonal axis is set along the 0-axis downward (r - ~ =;0') | This is the same setting as used for the measurements of [Z100]23. We denote the result by -P =- lo.+ 4C cor 2 (0- ~). (Le.) Po Setting 2 The trigonal axi is s et horizontally (r - ~ = 90), This is the same setting as used for tthe measurements of [001]i3.he esult is denot[ed by i P co 2 + 2C co soo 2 0 - 2y). <I1) Po For further necessary data we take the difference between two settings.: Setting 3 - e 45 Setting 4 -' 45 -The difference will have the form P45 "-45 =- CA - C0o cos 2 0o + A2 sin 2 0 (12). Po where 'the equality of the first two coefficients is a direct consequence of the relations developed in previous reports. Other required identities are 'Co + -c2 cos 2 02 - Co+ C2 cos 2 (15) and 202 sin 2 020 = CG2/%I (14) 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The brackets can now be expressed as follows in terms of these measured quantities, using the abbreviations Po = [00]1 {[000] 33 + cos2 9 ([000]11 - [000]33) (5) A = -2 sin2 @ sin 20 sin K (l + 2 cos 2,) (16) 2P 0C2 [200]23[00011 = - R~- sin2 9 sin 2 02 (17) 2 -4P CZ it ([200]82 - [200]22)[000]33 - [100132 = Si 2s sin(2 1 + 30) (1 2P 0 [Qll]11[o00]33 =:21 sin 28 %/i (19) [011]23[000]11 - [l0]23[001]L2 = 32- s s in C3 - sin t C cos 35) (20) 2([00 +200] 2 [ )[00 0]33 + [100]23] 2 ( -. cos2 0) + r 2~- Po "40 2{[200]33[0001o + [l]00]3 }[000]l cos2 O/[0oo]3 = -- (21.) ([002]1 -4- [o00L ]2/[C0o0])[000]33 (1 cos2 0) + [002]33[000]11 [000]33 Cos i0 (2 - (Ca cos 2 ).(22) The first four equations express, in Iterms of measured quantlties, th.e.rackets [200]23. [O1L11, [011123, and the cominationr [20013 - [200]2o Tiese can.*terefore be found from each. sample individuaallyy. The last two equations ex.press, in t;erms of.measured quaxnUtities, e he brackets and com bination.$s [200]j3 + [200]22, [200]2 and [002], [002] y masing several (a least tas o) samples with different 0e and plotting the rig.t-bas d side vs cos2.' a straig t line should result^ pe;rmiatting the dete.rmination of a ll f our quantities. FinG.ly, by comparing the- results for [2001 + 2022 and [200] - [200]22 one can arrive at [200] L and.[200]22 individually, While a number of other procedures may be followeds the above procedure is probably the simplest and most direct. It is interesting to note that the brackets that can be determined from one sample require knowledge of 0 a id K, while the brackets that come from several samples only require knowledge of., 5. RESULTS OF MEASIREBI NiS OF SECONDi-OKRDER BRACKETS The following (incomplete) results.ave so far dbeen obtained for tshe brackets of second order for pure bismlth. Tihey should be considered as sutbjf 26

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN to change when further work is done, but the general order-of-magnitude agreement itth Abeles and Meiboom's data indicates that they are most likely reliable. Since the data on Setting 3-Setting 4 were not obtained for the samples reported on, only the data listed below are as yet available., Table IV gives [200123 and [200]11 - [200]22 for the temperature range of 20~C to -70OC (see Appendix) as derived by equations (17) and (18). By combining these results with those of another sample, we arrive at the preliminary results for the second-order brackets at 20~C given in Table V, where also a comparison with similar data of Okada and of Abeles and Meiboom is shown. Except for the [200]2s value, the agreement is very satisfactory. Figure 11 shows that the agrieement with Okada at lower temperatures is not so good as at room temperature TABLE IV EXPERIMhENTAL VALUES OF SECO ND-ORDER -BRACKET CO1 INIATIONS AS A FUNCTION OF TEMPERATURE Data from samples E T [2O.00 [200fl1 - [200 2 -(cc) ^ Gauss-2 Ql cmO Gauss"2 "1 cm 20 -4,21 x 10-5 +11o7 x 10-5 10 -6.0 +14o88 0 -8.6 +20.2 -10 -12o7 +23.6 -20 -18,9 +3517 -30 -27.2 +41.1 -40 -37.8 +55. -50 -51.6 +71.3 -60 -65 08 +92.1 -70 -100... +129 TABLE V COMPARISON FOR SECOND-OEDER BRACK.ETS AMONG DIFFEERET.WORKERS Gauss -2 "-' Cm'1 Bracket Ours, 20'C Okada* _20C Abelees and Meiboom, 27' [200]Li -7.4 x 10-5 -7,4 x 10-5 -6o5 x 10-5 [200]22 -19.1 x 105 -20 x 10-5 -1953 x 10~5 [200]23 -4,21 x 10-5 -0,51 x 10-5 not given [002] L -3*56 x 10-5 -5,2 x 10-5 -2.25 x 10 -5 * Interpolated; Okada's data given at 0~ and 450~C 27

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 120 o Our sample E o * Okada's result 100 Abeles and Meiboom N 0 S(/) C) < 80 00 z 60 o 40 0 0 o 40 0 o 0 0 N 0 20- o o A 0 0 0 — \ --- T (OC) -80 ~ -60 -40 -20 0 ) 40 o 0 'S -20- o cQ 0 n _ (.) ZI I0 N -80 0 -60.. 0 -100- o Fig. 11. Comparison of [200]23 and of [200]11 - [200]22 by various authors. 28

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 6. MEASUREMNTS ON ALLOYS Only very preliminary measurements have been made on alloy samples, only at room temperature. For the Te-doped samples the magneto resistance.was an order of magnitude smaller than for pure Bi, For the sample of Sn-Te.-doped Bi the magneto resistance was about half that of pure Bi, but further and more systematic study is necessary along these lines* 7- DISCUSSION The results reported are just a beginnirg. It has taken considerable effort and time to develop the methods of growing suitable single crystals and of overcoming difficulties in the experimental teebmiques of measurement, While there is still room for considerable improvement, we are at present. in. a position to turn out bracket values for different imatersials and high.er order if this project is continued, as tpe resultns p ested show. For the near future it is intended to take fuxrther measuremente s tin order to corroborate the present results and, iLmprove theiraccu. Also, a more systematic study of alloys of BLi sand. r-iI compouads is intended. Axny detailed discussion will be deferr ed till these measurements ~e - aiailiee A general discussion for- Bi can be found in the paper. of Abel3es aid. Meieboom (Reference 35) While a certain amount of electron theoretical insterpretation is common nowadays in the literature for th.e,brackets up to order 2, the interpretation of t;e values of brackets of higher order is a matter" of fa:r/ther study. 29

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPEIDIX The Form of Equation (9) The result of a run as described in Section D.4 is in general of the form Ap =- Co.- cos 2 (c - 02) + C4 cos 4 0 - 04) + o(l) which is compatible with equation (9) only if C4, which is proportional to B34 is negligible compared to Co and C2, wilch are essentially 'proportional to,B2 Since Co = C02 B2 +.'Co4 4 +. C2 = C22 B2. +024 34 +. (A22) C4 = C44A.4 + o-r u this can always be achieved by making B sufficiently smalio. TMherefore, a first check run is made for each sample and temperature to make sue' tat;Ap/p as a function of 0 does not contain any appreciable amotnt of fourth or- ligher nharmnies. If fourth or higher hbarmonics are present in eq.aion (Al), it is not correct to subtract them out and use the Co and C2 terms only. since tese terms contain also fowuth-power contributions. The" best procedure for elimina ting the influence of fourth npower terms from Co and 2 if they are not.egliIt is preferred not to -vary B for, intercompalison of further mneasurements * T.t clearly visible, for example, in FigW Al, representing a curve of,Ap/p rs 0 for sample Eo Setting 1 at.-l5C.s The rel.ati.ve- m.agni tude of Co0 02. and C4 as a function of te'mperature is shown in Fig sA2, It is seen that the quad4 ratic approximation can be used successful-y only for temperatures above -100 C ' 50

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 700'..I 600 oo - 0> 0 _~O0 o 400 2 300X 0 z 00 - Sample E Setting I Ap/^pE-{478 + 85 sin 2( +29) poo00 _ ___ +50 sin 4(#+44 ___ /,V x 1.336 x 10'%/zV 100 L.. -,- - - - 0 20 40 60 80 100 120 140 160 180 200 220 Fig. Al. Presence of fourth harmonic at -150~C. 51

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 500 50 C ________ ________ Sample E Setting I A p =1.336x 04 400. o / 30C 0 0 90 -0 - 1 FigsA2. Relative amplitude of Co, C2 and C4. <3 200 100 --- C4. ---~ -90 -100 -110 -120 -130 - 140 -150 -160 T(~C) Fig. A2, Relative amplitude of Co; C2, and C4. 52