ms. mmsuamam as THE m HAAS-VAN mm mm: BY ms sow mamas; 133 295 THS ”(has-2's :39? {he Segraa of M. S. ‘ «2 “LM' a? «glistzu ',5 g‘: raft ‘ EI-‘aiufizzsélw‘¢ 3mm; Efih‘ii’vERSmY TTTTTT ' Innlmnmmlimmmnmmrurmmml": _ 3 1293 01743 0012 ' LIBRARY Michigan State University ABSTRACT THE MEASUREMENT OF THE DE HAAS—VAN ALPHEN EFFECT BY THE GOUY METHOD by Richard Norman wagner A torsionptype, magnetic susceptibility, beam.balance capable of making differential weighings of 30 micrograms on samples weighing up to 50 grams was built. Force measurements were made on a single crys- tal of bismuth with magnetic fields up to 17,700 gauss at a temperature of h.2°K using the Gouy method. The results of these measurements illustrate the utility of the Gouy method in detecting De Haas-Van Alphen oscillations. THE MEASUREMENT OF THE DE HAAS—VAN ALPHEN EFFECT BY THE GOUY METHOD BY Richard Norman wagner A THESIS Submitted to Michigan State University In partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy l96h . \ V;C) will a“ L}‘ .N 4;“ “k (2: ACKNOWLEDGMENTS The author wishes to thank Dr. Meyer Garber for his guidance in the design of the apparatus and his suggestion to use the Gouy method to measure De Haas-Van Alphen oscillations. This work was made possi- ble through the financial support of the National Science Foundation and the Office of Ordnance Research. ii TABLE OF CONTENTS CHAPTER PAGE I INTRODUCTION . . . . . . o . o o o o o . o . . o . o o . o 0 II THEORX . . . o . . . . . . . . o o . . o o o o o o o o o o 0 III APPARATUS . . . . . . . . . o . o . . . o o . o o . o o o o MagnetandMagneticShield...............o c» -a -4 \u +4 Vacuum.Enclosure . . . . . . . . . . . . . . . . . . . . . BalanceandOpticalSystem................ 13 Electronics . . . . . . . . . . . . . . . . . o . . o . o 20 IV DESCRIPTION OF EXPERIMENTS . . . . . . . . . . . . . . . . . 26 Absolute Susceptibility . . . . . . . . . . . . . . . . . 28 Rotational Experiment . . . . . . . . o . o . . . . . . o 32 De Haas-van Alphen Effect . . . . . . . . . . . . . . . o 37 vDISCUSSION......................... 1:3 VIBIBLIOGRAPHY........................ uh VIIAPPENDIXOOOOOQQOO00000000000000... ’45 iii TABLE II III VI VII VIII XI XII LIST OF TABLES Electrical Components 0 o o o o o o o o o o o o o o o o o Magnet Calibration with Three Inch Gap Using Rawson Type 820 Fluxmeter o o o o o o o o o o o o o o o o o o o o o Calculations of the Absolute Susceptibility at Room _ Temperature..........ooooaooooooo Orientation of the Trigonal Axis Calculated from the Rotational Data 0 o o o o o o o o o o o o o o o o o o 0 Comparison of Susceptibility Peaks and Magnetothermal Peaks 0 o o o o o o o o o o o o o o o o o o o o o o o 0 Absolute Susceptibility Data, Taken at Room Temperature (22°C) with the Sample in a Vacuum . . . . . . o o . . Absolute Susceptibility Date, Taken at Room Temperature (22°C) with the Sample in One Atmosphere of Oxygen . o Rotational Data, at Room Temperature (22°C) and at a Field of h,2OO Gauss, Taken August 28, 1963 . . . . . . Rotational Data, at Room Temperature (23%) and at a Field of h,200 Gauss, Taken August 29, 1963 . . . . . Rotational.Data, at Room.Temperature (22°C) and at a Field of 11,270 Gauss, Taken August 28, 1963 . . . . . Rotational Data, at Room Temperature (23°C) and at a Field of 11,270 Gauss, Taken August 29, 1963 . . . . . De Haas-Van Alphen Effect.Data Taken at h.2°K . . . . . . iv PAGE 23 27 31 36 h2 h6 h7 h8 h9 SO 51 52 . . r r O a v a I a r n O ‘ r a a o a n a v a I O Q ’ c O 1 . a - o a. a r t 9 o . I a . v I c i n n a a I t a - a . c a. LIST OF FIGURES FIGURE PAGE 1 Magnet Shield . .’. . . . . . . . . . . . . . . . . o . o o 9 2 Effect of Magnet Shield . . . . . . . . . . . . . . . . . . 10 3 Vacuum Can . . . . . . . . . . . . . . . . . . . o . . . . . 11 h Balance . . . . . . o o . o . o . . . . . . o . . . . . . . 15 S OpticalSystem....................... 17 6 Sample Holder . . . e o o . o . o o o o o . . o . . o . o . 18 7 Improved Beam End Coupling . . . . . . . . . . . . o . . . . l9 8 Servosystem . . . . . . . . . . o o . . . . . . . . . . . . 21 9 Circuit 0 o o o o o . o o o o o o o o o o o o o o o o o o o 22 10 Effect of Surrounding the Sample with One Atmosphere of Oxygen on the Restoring Voltage, Room Temperature . . . . 30 11 Dependence of Restoring Voltage on Magnet Position, Room Temperature, h,200 Gauss . . . . . . . . o o . o . . . . o 33 12 Dependence of Restoring Voltage on Nhgnet Position, Room Temperature, 11,270 Gauss . . . . ... . . . . . . . . o o 3h 13 Crystal Orientation Coordinates . . . . . o . . o o o o . o 35 1h Dependence of Restoring Voltage on the Magnetic Field, h.2°K 39 15 Field Dependence of DHEVA Oscillations, h.2°K . . . . . . . hO l6 Dependence of DHeVA Oscillations on the Reciprocal of the Fiéld,h020Kooooooooooo000000000000 ’41 LIST OF ILLUSTRATIONS ILLUSTRATION PAGE I Balance 0 o o o o o o o o o o o o o o o o o o o o o o o o 0 1h II Balance and Optical System 0 o o o o o o o o o o o o o o o o 16 vi I INTRODUCTION A torsion beam balance was constructed to measure the magnetic susceptibility of metals at low temperatures. The balance follows the 1’2 Samples weighing up to 50 general design of previous balances. grams weight can be used. The uncertainty in a given weighing is about 30 micrograms at present. Restoring force is supplied by a current carrying coil interacting with a permanent magnet. An optical-elec- trical servosystem is used to keep the beam in null position, minimiz- ing eddy current effects and increasing the effective stiffness. The balance is sensitive enough to use the Faraday method, but to avoid the necessity of obtaining properly shaped pole faces as are used to reduce sample positioning errors, the Gouy method was used. Further, it was decided to investigate the suggestion, to the author, by Dr. J. F. Cochran and Dr. M. Garber that De Haas-Van Alphen (here after referred to as DHeVA) oscillations might be studied using the Gouy method. Previous studies of DH-VA oscillations have been carried out with the sample suspended in a nearly uniform field. Shoenberg used the Far- aday method in his inital studies of the DH—VA effect in bismuth.3 This method measures some average value of the susceptibility when the 1M0 Garber, We Go H811”, and H. G. HOeve, Can. J. Phys. 2.9-) 1595, (1956). 2F. T. Hedgcock, Rev. Sci. Instr. 2A: 390, (1960). 3D. Shoenberg and M. Zakie Uddin, Proc. Roy. Soc. igé, 701, (1935). 1 susceptibility is field dependent, thereby making the Faraday meth- od useless in resolving closely spaced DH+VA oscillations. For the above reason Shoenberg used the torsion.method in a later experiment on bismuth.)4 The torsion.method is simple and sensitive, but it has the disadvantage that it measures the difference in the susceptibili- ties along the principal axes. Present day DHAVA studies are fre- quently made with pulsed very high fields using a pick-up method for measuring the susceptibility. The balance was constructed for use in investigating the suscepti- bility of alloys but it was decided to try and use it to detect‘DHeVA oscillations in bismuth. The absolute susceptibility of bismuth was not accurately determined because the orientation of the bismuth crys- tal was not well determined. The measurements, however, show that DHEVA oscillations can be detected easilye hD. Shoenberg, Proc. Roy. Soc. 119, 3h1, (1939). II THEORY The magnetic force on a long rod of material suspended in a.mag— netic field gradient can be calculated by considering the virtual work required to make a virtual displacement, 62, of the rod. This virtual work corresponds to the change in the magnetic free energy of the rod when the displacement is an isothermal reversible process. Calcula- tion of the virtual work can be simplified if the following assumptions, readily realizable with the Gouy method, are made: 1) The rod material is homogeneous. 2) The cross sectional area, A, of the rod in the plane perpendicun lar to the displacement, Oz, is a constant. 3) The ends of the rod are in regions of uniform field. Now, the virtual work required to displace the entire rod a dis- tancetéz can be considered as the work done in taking a small volume, AOz, from one end of the rod to the other. hath no essential loss of generality it will be further assumed that the permeability of the rod material is isotropic, i.e., a scalar. The medium surrounding the rod is a vacuum, and one end of the rod is in a region of zero field. The difference in the magnetic free energy of the rod is equal to the magnetic free energy of the small volume of material, A62, in the high field region. This energy is the difference between the energy of the field, due to fixed current sources, without the presence of the small piece of magnetic material, and, if the magnetic material is pres- ent, the amount of work done by the current sources necessary to re- 3 I; store the currents from zero to their initial value.5 The additional work necessary to build up the currents in the presence of the body is B U-= £3 [HIE-HBFHBi-ZSHdBJdV V 0 where V is the volume of the magnetic material, H and B are the fields in the presence of the magnetic material, and H1 and B1 are the fields in the absence of the magnetic material. By substituting BI=JJ.(HI+MI)=}J.HI 3 B=p.(H+M)= AWN-X) into the above equation we have (/1, -:..- permeability of free Space) * H U= $11.8 [X(H,—H)H+2$HdM-Jdv ' v o If the high field region is at the bottom of the sample and the mechanical force, 3‘, on the sample and the displacement, 82, are taken in the upward direction H 352 =12A62N.[‘X.(H,—H)H +28 HdM] o H 3- =él-A1J.[’X.(HI-H)H+23HOM] - , o 5J. A. Stratton, Electricity and Magnetism (McGraw—Hill, New York, 1911-1), p. 128 o The force is proportional to the free energy density at the high field end of the rod and to the cross sectional area of the rod. There- fore fluctuations in the force on the rod correspond directly to fluc- tuations of the free energy density at the high field end of the rod. It is often assumed that the presence of the magnetic material does not affect the current sources, in which case H1 = H, and the usu- al expression for the force on the rod is written H 3=ApongM . C) For materials in which M/ H -'-' 'X. Z constant 3‘ = 'Ep.A'X.H2. In cases where the magnetic susceptibility varies with the applied magnetic field the theoretical analysis is usually based on the free energy expression. This corresponds directly to the measured force on the sample. This simple and direct feature of the Gouy method seems hitherto to have been overlooked. In the case of the De Haas-Van Alphen effect the expression for the free energy may be written6 § _2m_<_Tm‘c F‘> flog AdOHfim—Sm an zommHm ZOHmmOu. .52 522 a: AHoo moHob E04: mafimmm .2 .m FIGURE h ammo}; ' ILLUSTRATION II BALANCE AND OPTICAL l7 i b?— \ L A \ LIGHT BULB “" T >— BAFFLES d LENS SLIT-IMAGE 4 LT / / / L A [1% L 1 / [ii—t L H \ PHOTO-CELL SECTION A-A / SLIT SP} TERI CAL r 7 E70 ECJ IOU) FULL SCALE FIGURE 5 OPTICAL srs'mu 18 PYREX TUBE 30mm WITH CERROSEAIr-BS ——--——-——J GERMAN SILVER YOKE -C. I -C ‘I -- m I‘ i / SOLDERED COPPER WIRE \ SAMPLE SCALE: ~ 381 FIGURE 6 SAMPLE HOIDER l9 BEAM END SOIDERED WIS]r 0.001 INCH THICK PHOSPHOR-BRONZE 0.002 INCH DIAMETER BERYLLIUM-COPPER WIRE _| / PHOSPHOR-BRONZE RIBBON SCALE: ~3:1 FIGURE 7 DIPROVED BEAM END COUPLING 20 DC power supply having a stabilization factor of 2,000. When the slit- image illuminates both photo-cells equally the beam is said to be in null position. The sample is suspended by a 1.5:mm diameter Pyrex tube. The sam- ple is attached to the bottom of the tube with a copper wire which goes through a small horizontal hole in the top of the sample and is solder- ed at both ends to a german silver yoke (see Figure 6). The Pyrex sample suspension tube and the voice coil are attached to the beam ends by 0.75" x 0.10" x 0.001" phosphor bronze ribbons, which are pinched in slots by tightening the #0-80 screws at the beam ends. This was later found to be unsatisfactory as the stiffness of the phosphor bronze ribbon increased the effective torsion constant of the beam to about 30,000 dyne-cm/radian. After these experiments the phosphor bronze ribbons were replaced by a yoke suspended by two 0.002 inch diameter wires (see Figure 7). This new suspension, less stiff than the old, and placed so as to raise the center of mass of the beam reduced the effective torsion constant of the beam to 1,000 dyne-cm/ radian. A vane on the bottom of the beam.is immersed in glycerin to pro- vide viscous damping of the beam motion. The beam and optical system are mounted on a brass plate which is isolated from vibration with four rubber vibration mountings (Lord # lO6PL—2). Electronics The general features of the electronics are illustrated in Figure 8 and the details in Figure 9 and Table I. The circuit is 21 DIFFERENTIAL IMPEDANCE NETWORK 4- MATCHING .....— AMPLIFIER VARIABLE 4‘ 4—— CURRENT I SUPPLY NULL CURRENT DETECTOR DETECTOR l ' v I A —“ l I V I é I BEAM <, , t/ VOICE COIL FIGURE 8 SERVOSYSTEM 22 g y _ — — — — _ _ — fl —— _— \ as" CIRCUIT FIGURE 9 Resistors R1, R2 £3,310 R 122 R7, R8, R9 1 R12 213 R 1h, 1 R16 5 Capacitors C1 C2, C3 Batteries Bl: B2 B3 Switches 51: S2 53: Sh Transistor Q Photo-cells P01, P02 Voice coil VC Ammeter A Null.Detector NDl Potentiometer P 23 TABLE I ELECTRICAL COMPONENTS l K ohm 1% 1/2 watt wire wound precision 100 ohm helipot 300 K ohm helipot 10 K ohm helipot 200 ohm helipot l K ohm 1% 1/8 watt. metal film 800 ohm 1% 1/8 watt metal film 100 ohm 1% 1/2 watt wire wound precision 1 K ohm 'wire wound potentiometer 39 K ohm 1% 1/8 watt metal film 22 K ohm 1% 1/8 watt metal film 300 mfd 10 WWDC SO mfd 10 WVDC Two series connected 6 VDC automobile batteries. 6 VDC automobile battery DPST toggle switch DPDT toggle switch RCA type 2N2h7 Clairex type hOhSL About 75 turns of #hO wire wound in one layer on an aluminum foil form. 5 - O - 5 ma (zero center) Leeds and Northrup DC microvolt amplifier, used on 1 - O - l mv (zero center) scale Leeds and Northrup type K-3, used with a Rubicon galvanometer, catologue no. 3hl7 sensitivity .005 a/mm 2h almost identical to one used by Henry.2 Null detection is achieved by means of a Leeds and Northrup DC microvolt amplifier attached across a Wheatstone bridge (called the in- put bridge) which has two photo-cells on adjacent arms. The DC component of the unbalance voltage goes directly across another Wheatstone bridge (called the output bridge) which contains the voice coil in one arm. This current through the voice coil in- teracts with a PM speaker magnet partially restoring the beam to null position. This restoring current adds a torsion constant of about 5,000 dyne-cm/radian to the effective mechanical torsion constant of 30,000 dyne-cm/radian; a negligible increase. An operational amplifier is now being tried as a stage of amplification between the input bridge and the output bridge in an effort to increase the electronic torsion constant. The unbalance voltage of the input bridge is also impressed across a differential network, 03 and R16. This velocity dependent signal is impressed on the base of the emitter follower connected transistor. The transistor has a current gain of about 35 and affords impedance matching between the differential network and the voice coil. Gain is adjusted by R13 for most effective damping. A too high gain setting results in an unstable system and the beam oscillates. The electronic damping is effective in reducing oscillations of the beam caused by me- chanical disturbances such as room vibrations and a varying magnetic force on the sample. 2W. G. Henry, Private communication with the author, National Ilesearch Council, Ottawa, Canada. 25 A constant current is supplied through the voice coil by battery, 82, connected across the output bridge. This current is adjustable by means of the series connected potentiometers Rh, R5, and R6. The current direction can be reversed with switch S3. The purpose of the output bridge is to isolate the battery, B2, from the input bridge. Current through the voice coil is measured roughly with the ammeter, A, and accurately by measuring the voltage drop across the 100 ohm wire wound resistor, R12, with a Leeds and Northrup type K—3 potentimeter,P. The input bridge is balanced by replacing the photo-cells with two matched resistors and adjusting R.3 until the null detector, NDl, reads zero. This is done with 82 open and 31 closed. Likewise, the output bridge is balanced by adjusting R10 for a zero reading on NDl with 32 closed and S1 open. During an experiment the current through the voice coil was adjust- ed with Rh’ R5, and R6 so that NDl would read zero with 51 and $2 closed and the voltage across R12 measured with the potentiometer. IV DESCRIPTION OF EXPERIMENTS The experiments that will be described were carried out using the same bismuth crystal. The rotational and absolute susceptibility experiments were done to get the "feel" of the apparatus besides any useful data that might come from them. The DHEVA experiment was done to test the utility of the Gouy method in detecting DHFVA oscillation. Because the balance was not calibrated the data presented is the voltage drop across R12, called the restoring voltage, necessary to balance out the magnetic force on the sample (see Figure 9). This voltage is the difference between the voltage necessary to balance the gravitational torque on the beam when the magnetic field is zero, called the field-off voltage, and the voltage necessary to balance the beam when the magnetic field is on, called the field—on voltage. Be- cause the field—off voltage drifted at the rate of'about two microvolts per minute it was necessary either to take a field—off voltage reading between each field-on reading as done in the absolute susceptibility experiment or to take field-off voltage readings at the beginning and the end of the run and interpolate between by assuming a linear drift with time as done in the rotational and DH+VA experiments. The dial readings on the magnet current supply were calibrated against the magnetic field with a Rawson type 820 rotating coil flux- meter (see Table II). The magnet was cycled to about 18,000 gauss two times before the calibration and before each series of measure- 26 27 TABLE II MAGNET CALIBRATION WITH THREE INCH GAP USING RAWSON TYPE 820 FLUXMETER m magnet current control measured magnetic field dial readings (gauss) Course Fine (field increasing) (field decreasing) 0 0 58 68 0 300 12h 135 O 500 --- 177 0 800 235 2&1 6 500 305 317 10 500 390 th 23 500 682 700 35 500 951 967 us 500 1,175 1,196 100 500 ---- 2,h3o 135 500 3,196 3,21h 165 500 ----- 3,886 180 500 h,l97 h,220 2&0 500 5,550 5,561 330 500 7,559 7,577 390 500 8,881 8,912 500 500 11,268 11,296 650 500 1h,076 1h,100 800 500 16,061 16,090 1,000 500 17,685 ---- 1,000 1,000 17,723 -—-—- NOTE: The magnet was cycled to maximum field twice before making the measurements. 28 ments. The increasing field values were used in the following experi- ments 0 Absolute Susceptibility Measurements were made on the crystal at room temperature with the crystal in a vacuum and with the crystal in one atmosphere of oxygen. The crystal was suspended from the beam with a 0.002 inch diameter berylliumpcopper wire thereby allowing the sample to rotate about the vertical axis. The sample could be seen to rotate when a field of 100 gauss was applied and, therefore, the sample was assumed to align itself with the direction of minimum (minimum absolute value of $6) susceptibility in the plane of rotation along the direction of the magnetic field. One atmosphere of oxygen was admitted into the evacuated vacuum can directly from the storage bottle. The pressure was measured with a bellows-type gauge to an estimated accuracy of t0.5 inches of Hg. Data from these measurements are presented in Tables VI and VII. It can be seen from the voltage readings repeated at the same magnetic field that the readings are reproducible to 30 microvolts making the values‘for the restoring voltage with fields less than 2,000 gauss meaningless. However, the error in the restoring voltage values at 17,685 gauss due to voltage measuring errors is only 0.02%. The downward magnetic force, in dynes, acting on a cylindrical sample of uniform cross sectional area, A, in a medium is _ 2 IX E’I'sm" (Xses’ xmem) (H12- HZ ) 2' ’ * Refers to st - Vsm of Table III. 29 where the subscripts s and m refer to the sample and the medium sur- rounding the sample respectively,‘x.is the specific susceptibility, e the density, H1 the field at the bottom of the sample, and H2 the field at the top of the sample (values of Hz were estimated from Figure 2). The subscript v replaces m when the medium is a vacuum. In the case where the susceptibility is independent of the field a plot of 3'versus (H12 - H22) is a line of constant slope. However, the plot of restoring voltage versus (H12 - H22) is a line of decreas- ing slope indicating a non-linear relation between the restoring volt- age and the force on the sample (see Figure 10). Using the above equation for the force the absolute susceptibility of the sample may be calculated from 9m 33v x=x — ——-——- S mes 3‘5v"'3‘3m or if the restoring voltage is directly proportional to the magnetic force m The calculations of 7‘s are tabulated in Table III using, 7§m 3'106.2 x 10"6,l em 3 1.33 x 10'"3 g/cc,2 .98 = 9.86 g/cc.3 1Charles D. Hodgmen (ed.), Handbook of Chemistry and Physics (hlst ed.), (Chemical Rubber PubIlshing 00., Cleveland, 1959), p. 2651. 2RobertW. Vance and W. M; Duke (eds.), Applied Cryogenic Engi- ' nearing (John Wiley and Sons, New York, 1962), p.'hE0,IFig. A.6. 3H. M. Trent, D. E. Stone, and R. Bruce Lindsay (eds.), ”Density of Solids," American Institute of Physics Handbook (lst ed.), (MCGrawh Hill Book Company, New York, 1957), p. 2717. 3O mmm Aouoa x mmmammv mm: n ma: . oom mew 0mm mmm 08 m: oma mNH OOH ma. om mm o -. - 0.0 a . _ _ _ ~ _ A _ _ _ _ o \e r. mo.o 6:59; a 5.” macaw e \ r. commxo mo o\\\\. 40.0 ounfimoEpm 98 ca magnum 0 I. 00.0 I. mo.o m I. OH.o a I: mm.o I. a :~.o I. ed.o a 4/ cacao acwemcoo mo mafia I. ma.o a, _ i _ _ a _ _ _ a _ _ o~.o eBeitoa Butaoisag (sitoa) FIGURE 10 OF OXYGEN ON THE RESTORING VOLTAGE,ROOM TEMPERATURE EFFECT OF SURROUNDING THE SAMPLE WITH ONE ATMOSPHERE 31 TABLE :LII CALCULATIONS OF THE ABSOLUTE SUSCEPTIBILITY AT ROOM TEMPERATURE Restoring Restoring Susceptibility Field voltage voltage V v - V m X5 (gauss) st (volts) Vsm (volts) Ivoltsg st/(st - Vsm) (cgs x 10 390 0.00008 0.00011 -0.00003 -2.60 -0.03 951 0.0005h 0.00052 0.0000h 11.2 0.16 1,175 0.0008h 0.00081 0.00003 31.0 0.hh 2,h10 0.00333 0.00383 -0.00009 - 5.8 -0.51 3,196 0.00591 0.00601 -0.00010 -59.1 -0.85 3,860 0.00867 0.00883 -0.00015 -56.3 -0.81 8,197 0.01028 0.01036 -0.00012 -88.5 -l.27 5,550 0.0178h 0.01807 -0.00023 -78.3 -l.12 7.559 0.03302 0.03389 -0.000h7 -70.h -1.01 8,881 0.08567 0.0t632 -0.00065 -69.8 -1.00 12,230 0.08578 0.08715 -0.001h2 -60.5 -O.87 18,076 0.11066 0.11231 —0.00165 -67.1 -0.96 16,061 0.13972 0.1b167 —0.0019h -72.0 -1.03 17,685 0.16361 0.16592 -0.00232 -70.6 —1.01 aThe magnetic field at the bThe voltage drop across R1 high field end of the sample. required to balance the magnetic force on the sample with the sample in a vacuum. °The voltage drop across R12 required to balance the magnetic force on the sample with the sample in one atmosphere of oxygen. dThe specific susceptibility at room temperature calculated frog X8 = Xm emvsv/ 98(st - Vsm): where Xm em/e s = 1.143 x 10- chO 32 Because the percentage error, due to the 30 microvolt uncertainty in voltage measurements, in the value of st - V sm is the smallest (’w 2%) for the 17,685 gauss case; the susceptibility of the crystal for the previously described orientation is thought to be -l.01 x 10"6 cgs 112%. The values of st and Vsm are so close that the non-linear- ity of the balance has little effect on the above calculation. Rotational Experiment In an effort to determine the angle between the trigonal axis of the crystal and the vertical two series of room temperature measurements were made by rotating the magnet 190 degrees about a vertical axis in 15 degree steps. Measurements were made at fields of h,200 gauss and 11,270 gauss. The sample was suspended from the beam with a Pyrex tabe as described in the section, Balance and Optical System. The data from these measurements are listed in Tables VIII, IX, X, and XI and plotted in Figures 11 and 12. At room temperature bismuth has two principal susceptibilities, one parallel to the trigonal axis of X" = -1.053 x 10'“6 and the other perpendicular to the trigonal axis of ?g_= -l.b82 x 10'6.h If the trigonal axis, along 0P (see Figure 13), makes an angle 4) with the vertical and the magnetic field is in the x direction the observed susceptibility is5 (xusmzcb + xlcoszflcosze +xlsmze . 11L. F. Bates, Mbdern Magnetism (Cambridge Universtiy Press, London, 1961), p. 172. 51818., p. 173. 33 0°0116 I I I I I I I I I I I 0.01114 _ / /8- \O — '\ f, \ . DATA TAKEN 8-28-63 7, 0.0112 -— / \O\ 0 DATA TAKEN 8-29-63 ‘1 / \ 0 ’~ 0.0110_. / (a " 3 d \ 0 e ‘1 / > __ / ~I 0.0108 . “ (D \ / 8° \ )4 :3. 0.0106— — o \ > I , / 3° 0.010m- \ ..// ‘ ‘1 $4 .3 \ o e / ,9} 0.0102 - / I \ / 0.0100 - °\ /° “ \\ I /’ 0.0098 - \9‘" . —‘ 0.0096 1 l I J I l I All 1 J l to 20 0 -20 4.0 --60 ~80 ~100 -120 -180 -160 Magnet position (degrees) FIGURE 11 DEPENDENCE OF RESTORINC VOLTAGE 0N MAGNET POSITION, ROOM TEMPERATURE, 14,200 GAUSS 3h $0790" I I I I I I I I I I 0.0780 -' _ . DATA TAKEN 8-28-63 / 0.0770 - 0 DATA TAKEN 8-29-63 9’ _ / / 0.0760 *- / _ J / 0.0750 *- 55/ _ 0007110 — - l _. / / \ / 0.0710 — \ . f/ _I \ / 0.0700 L- ‘°' _ 0 0690 ‘ 1 l I 1 I I I I I ' - 1.0 20 -20 4.0 -60 -80 -100 -120 -lI.0 -160 Magnet position (degrees) FIGURE 12 DEPENDENCE OF RESTORING VOLTAGE 0N MAGNET POSITION, ROOM TEMPERATURE, 11,270 GAUSS 35 N I-e\/ '0 FIGURE 13 CRYSTAL ORIENTATION COORDINATES 36 TABLE IV ORIENTATION OF THE TRIGONAL AXIS CALCULATED FROM THE ROTATIONAL DATA Minimum Maximum 'X. min/x mat-x Angle a Date Field voltage voltage 0 (gauss) mim (volts) max (volts) Vmim/vmax (degrees) 8-28-63 8.200 0.00980 0.01126 0.870 hh 8-29-63 u,200 0.00978 0.01186 0.861 A5 8—28-63 11,270 0.06903 0.07788 0.887 36 8-29-63 11, 270 0.07015 0.07780 0.902 3).; a'The angle between the trigonal axis and the vertical calculated from the data. 37 As the crystal is rotated about its vertical axis a maximum sus- ceptibility, xmax ='-')(,‘L will be observed at 9 -'-' 90° and a minimum susceptibility, xmin = X“ sin2¢ + '>(._L c0524), at 9 = 0°. The ratio of these observed susceptibilities is 'X. X nun n 2 ' 2 “max :LJ. 4) ¢ By using the above values for'Xfl and'Xl and the experimental values of ' Xmin/Xmax : vmin/vmax listed in Table IV, 6 can be found from $9110.: 2 2 = —0.276 smzo . Xmax 0.724 Slh ¢+COS 4) l The value of ¢Iwas calculated to be h5° at h,200 gauss and 35° at 11,270 gauss (see Table IVL This calculation may be unreliable for the followb ing reason. In the process of cutting the sample from the larger crys- tal a small piece broke off the top of the sample. Crude measurements showed that the angle between the normal to the surface of the break and the vertical axis of the sample was 16° £20. If it is assumed that the sample cleaved along the basal plane,6 then the trigonal axis would make an angle of 16° with the vertical axis of the sample. This assumption is borne out moreover by results described in the next sec- tion. Ferromagnetic contamination of the sample would make the measured minimum susceptibility less (in absolute value) and would also decrease the measured rotationd. values of x’min/ x’max’ the low field values more than the high field values, thereby accounting qualitatively for this descrepancy. 61mm, p. 178 . 38 De Haas-Van Alphen Effect The crystal was suspended by the Pyrex tube as in the rotational measurements and the magnet was rotated to the direction of maximum room temperature susceptibility. In this orientation the magnetic field is perpendicular to the trigonal axis of the crystal, i.e.,‘X.=Xmax= Xi . The measurements were made at h.2°K. The data from these measurements are listed in Table XII. The low field at the top of the sample has been neglected as this decreases the restoring voltage at 17,685 gauss by only 2% and less at lower fields. The restoring voltage, V, versus magnetic field curve (see Figure 1h) clearly shows the quadratic rise in the restoring voltage. The oscilla- tions can be seen more clearly in the plot of V/H2 versus H (see Figure 15) and the plot of V/H2 versus l/H illustrates the periodic variation in l/H (see Figure 16). The double peak at 13,000 gauss is attributed to spin splitting of the Landau levels by Kunzler at al..7 Comparison of the position of the susceptibility peaks and the period of the oscillations are compared with magnetothermal oscillations8 with the field along the binary axis of the crystal in Table V. The close agreement shown in Table V indicates that the field was nearly parallel to the binary axis of the crystal when the susceptibility measurements referred to in the preceding section were made. 7J. E. Kunzler, F. S. L. Hsu, and W. S. Boyle, Phys. Rev. III, 13§: 109A (1962). 81bid., p. 1090. 39 Amloa x oesowv me i 2 2 fl 3.. So: 638mm: a m _ _ w a i _ _ _ 00.0 w0.0 0H.0 NH.0 :a.o onoo oH.0 0~.0 eBquOA Butaoqseu . FIGURE 11; DEPENDENCE OF RESTORING VOLTAGE ON THE MAGNETIC FIEID, h.2°x (9110A) ho Restoring voltage/field2 (volts/gaussz) FIGURE 15 EIEDD DEPENDENCE OE’DR-VA OSCILLATIONS, h.2°x I II I I I I I I I I I I n. O . O h- 0 I— . o I- - , cith ._ Q e——o— :% ——<>—— “" 0 ~""“""——O , O It_____L_____1 g) I I g) I I I I I I gi 53 53 a: 53 E3 «3 £3 E3 85 23 £8 53 a: no a: co co e~ c» c» r- r— xo ~o 12 13 1h 15 16 17 18 10 Magnetic field (gauss x 10'3) bl 8: A68 x 723.3 0.368 638mm; 0% 8m can can 8m 8m cow 2“ 6.3 com 9: 8H ofi 8H 8H _ _ d _ _ _ A _ _ _ _ _ _ _ . _ L 8 m Restoring voltage/field all 000 00» O N N 8 3 o- e— (volts/gaussz) 2 O Q P- 0 0 .3 N a: an § 08 FIGURE 16 82 TABLE V COMPARISON OF SUSCEPTIBILITY PEAKS AND MAGNETOTHERMAL PEAKS Peak Susceptibility pe ks magnetothermal peaks number Field H 10 /pH H A binary axis 10 /pH p (gauss) (gauss) 13.750 15,500 1 } 13,330 71.5 } 673 12,980 18,200 2 7,170 ~703 7,150 700 3 h, 800 693 14,720 703 b 3,610 693 3,750 700 5 2,850 701 ---- --- V 'DISCUSSION This work has successfully shown that the Gouy method is useful for detecting De Haas-Van Alphen oscillations. Moreover, the method has the advantage that the oscillations in free energy are measured rather than those in susceptibility. Oscillations in metals should readily be observable providing the sensitivity of the apparatus is increased by an order of magnitude. h3 IV BIBLIOGRAPHY Bates, L. F. Modern Magnetism hth ed. Chambridge University Press, Hodgmen, Charles D. (ed.). Handbook of Chemistry and Physics hlst ed. Chemical Rubber Publishing Co., CIeveland, 1959. Statton, J. A. Electricity and Magnetism. McGraw—Hill Book Company, Trent, H. M., D. E. Stone, and R. Bruce Lindsay (eds.). "Density of Solids," American Institute of Physics Handbook lst ed. McGrawa Hill Book Company, NEw'York, 1957. Vance, Robert W. and W. M. Duke (eds.). Applied Cryogenic Engineer- ing. John Wiley and Sons, New'York, 1962. Chambers, R. G. Can. J. Phys. 33, 1395,(1956). Garber, ML, W. G. Henry, and H. G. Hoeve. Can. J. Phys. 28, 1595, (1956). Hedgcock, F. T. Rev. Sci. Inst. 31, 390,(1960). Kunzler, J. E., F. S. L. Hsu, and W. S. Boyle. Phys. Rev. III, 128, 10814, (1962). Shoenberg, D., and Zakie Uddin. Proc. Roy. Soc. 159, 701, (1935). Shoenberg, D. Proc. Roy. Soc. 119, 3h1, (1939). VII APPENDIX 1:5 86 TABLE VI ABSOLUTE SUSOEPTIEILITT DATA, TAKEN AT ROOM TEMPERATURE (22°C) WITH THE SAMPLE IN A VACUUM Low fielda High fieldb H12 - H22 Voltagec Restoring H2 H1 6 (volts) voltaged (gauss) (gauss) (gauss2 x 10- ) (volts) 0 0 0 0.07hh6 ------ 0 0 0 0.07h38 ----- - 0 390 0.152 0.07830 0.00008 0 0 0 0.07h37 --_-__- 0 951 0.90h 0.07385 0.0005h O O 0 000.7th "'- """ 0 1,175 1.380 0.07357 0.0008h 0 0 0 0.07hhl - ----- 0 2,h10 5.808 0.07109 0.00333 0 0 0 0.07816 ~--------- 0 3,196 10.21 0.06852 0.00591 0 0 0 0.07883 ----- 0 3,860 1h.90 0.0657h 0.00867 0 0 0 0.07hh1 ------ 0 8,197 17.62 0.06h16 0.0102h 0 0 0 0.07839 ------- 20 5,550 30.80 0.05657 0.0178A 0 0 0 0.07hh3 ——-—~- 90 7,559 57.13 0.0Alho 0.03302 0 0 0 0.07hh2 -— ----- 200 8,881 78.83 0.02875 0.08567 0 0 0 0.07881 ------ 860 12,230 1&8.8 -0.01131 0.0857u 0 0 0 0.07hh3 ---—-- 1,320 1h,076 196.8 -0.0362h 0.11066 0 0 0 0.07hh1 --—--- 1,7A0 16,061 255.0 -0.06533 0.13972 0 0 0 0.07h37 ----—-- 2,220 17,685 307.8 -0.0892h 0.16362 2,220 17,685 307.8 -0.08921 0.16359 0 0 0 0.07838 ----- - 0 0 0 0.07h37 -----__ aThe magnetic field at the low field end of the sample. bThe magnetic field at the high field end of the sample. 0The voltage drop across R12 (see Figure 9 ) with the beam re- stored to null position. dThe voltage drop across R12 required to balance the magnetic force on the sample. 87 TABLE VII ABSOLUTE SUSCEPTIEILITT DATA, TAKEN AT ROOM TEMPERATURE (22°C) WITH THE SAMPLE IN ONE ATMOSPHERE OF OXYGEN Low field High field H12 - H22 Voltage Restoring H2 H1 -6 (volts) voltage (gauss) (gauss) (gauss2 x 10 ) (volts) 0 0 0 0.07186 ------ 0 390 0.152 0.07135 0.00011 0 0 0 0.07188 ----- 0 951 0.908 0.07092 0.00052 0 0 0 0.07188 ------ 0 1,175 1.380 0.07066 0.00081 0 0 0 0.07189 --—-- 0 0 0 0.07187 ---- 0 2,810 6.808 0.06807 0.00383 0 0 0 0.07153 ----- 0 0 0 0.07151 ----- 0 3,196 10.21 0.06550 0.00601 0 0 0 0.07150 ---- 0 3,860 18.90 0.06268 0.00883 0 0 0 0.07150 ----- - 0 8,197 17.62 0.06116 0.01038 0 8,197 17.62 0.06113 0.01038 0 0 0 0.07150 ----- 20 5,550 30.80 0.05388 0.01807 0 0 0 0.07152 ---- 90 7.559 57.13 0-03802 0.03389 0 0 0 0.07151 ----- 0 0 0 0.07186 ----- 200 8,881 78.83 0.02518 0.08631 200 8,881 78.83 0.02512 0.08633 0 0 0 0.07188 ----- 0 O 0 0.07188 ---- 860 12,230 188.8 -0.01570 0.08715 860 12,230 188.8 -0.01570 0.08715 0 0 0 0.07183 ---- 0 0 0 0.07186 ------ 1,320 18,076 196.8 -0.08085 0.11231 0 0 0 0.07186 ---- 0 0 0 0.07183 ----- 1,780 16,061 255.0 -0.07028 0.18167 0 0 0 0.07183 ----- 0 0 0 0.07188 ----- 2,220 17,685 307.8 -0.09888 0.16592 2,220 17,685 307.8 -0.09888 0.16592 0 O 0 0.07188 ---- NOTE: See TableVI footnotes for an explanation of the headings. 88 TABLE VIII ROTATIONAL'DATA, AT ROOM TEMPERATURE (22°C) AND AT A FIELD OF 8,200 GAUSS, TAKEN AUGUST 28, 1963 Magnet . c Field-o Fie1d~off Restoring Timea positionb Fleld voltage voltagee voltage (degrees) (gauss) Vl (volts) V2 (volts) V2 - V1 (volts) 7:82 80 0 -—--- 0.16158 ----- 7:87 80 0 ---- 0.16165 ---- 7:89 80 0 ---- 0.16168 - 7:56 80 0 ----- 0.16161 ---- 8:00 80 8,200 0.15096 0.16166 0.01070 --- 25 8,200 0.15061 0.16167 0.01106 8:08 25 8,200 0.15065 0.16168 0.01103 8:08 10 8,200 0.15087 0.16169 0.01122 8:10 10 8,200 0.15086 0.16169 0.01123 ---- -5 8,200 0.15089 0.16170 0.01121 8:18 -20 8,200 0.15068 0.16171 0.01103 8:25 —35 8.200 0.15108 0.16172 0.01068 --—- -50 8,200 0.15138 0.16173 0.01035 8:38 -50 8,200 0.15180 0.16178 0.01038 8:37 -65 8,200 0.15167 0.16175 0.01008 8:58 -80 8,200 0.15190 0.16179 0.00989 8:57 -95 8.200 0.15195 0.16180 0.00985 8:59 -95 8,200 0.15198 0.16180 0.00986 9:00 -110 8,200 0.15173 0,16180 0.01007 9:02 -110 8,200 0.15178 0,16181 0.01007 9:05 -125 8,200 0.15135 0.16181 0.01086 9:07 -125 8,200 0.15139 0.16182 0.01082 9:11 -180 8,200 0.15098 0.16183 0.01089 --- -155 8,200 0.15061 0.16183 0.01122 ---- -155 8,200 0.15060 0.16183 0.01123 ---- -155 0 - ----- 0.16183 ------ aThe time at which the reading of V1 or V2 was taken. bThe position of the magnet relative to an arbitrarily picked position. cThe magnetic field at the high field end of the sample. dThe voltage drop across R12 (see figuref?) with the field on and the beam restored to null position. 8The voltage drop across R12 with the field off and the beam restored to null position. As the "field-off" readings are taken on- ly at the beginning and the end of a run; the values listed when the field is on are obtained by assuming a linear drift with time. fThe voltage drop across R12 required to balance the magnetic force on the sample. 89 TABLE IX ROTATIONAL DATA, AT ROOM TEMPERATURE (23°C) AND AT A FIELD OF 8,200 GAUSS, TAKEN AUGUST 29, 1963 Magnet . Field-on Field—off Restoring Time position Field voltage voltage voltage (degrees) (gauss) V1 (volts) V2 (volts) V2 - V1 (volts) 6:01 -155 0 ---- 0.16339 ----- 6:06 -155 8,200 0.15231 0.16339 0.01108 6:07 -155 8,200 0.15236 0.16339 0.01103 6:09 -155 8,200 0.15238 0.16380 0.01102 6:11 -155 8,200 0.15288 0.16380 0.01096 6:15 -180 8,200 0.15277 0.16380 0.01063 6:19 -125 8,200 0.15313 0.16381 0.01028 ---- -110 8,200 0.15381 0.16381 0.01000 ---- -95 8,200 0.15357 0.16381 0.00988 --- -80 8,200 0.15360 0.16381 0.00981 6:27 -80 8,200 0.15358 0.16382 0.00988 ---- -80 8,200 0.15359 0.16382 0.00983 6:29 -65 8,200 0.15381 0.16382 0.01001 --- -50 8,200 0.15293 0.16382 0.01089 --- -35 8,200 0.15255 0.16383 0.01088 6:81 -20 8,200 0.15227 0.16383 0.01116 7:00 -5 8.200 0.15210 0.16387 0.01137 7:01 -5 8,200 0.15205 0.16387 0.01182 7:03 -5 8,200 0.15205 0.16387 0.01182 7:05 10 8,200 0.15201 0.16388 0.01187 7:07 10 8,200 0.15205 0.16388 0.01183 7:09 10 8,200 0.15208 0.16388 0.01188 7:11 25 8.200 0.15218 0.16388 0.01130 -- 25 8,200 0.15218 0.16389 0.01131 7:16 80 8,200 0.15258 0.16389 0.01091 ---- 80 0 ----- 0.16389 ------ 7:21 80 0 ----- 0.16851 ----- NOTE: See TableVDIfootnotes for an explanation of the headings. 50 TABLE X ROTATIONAL DATA, AT ROOM TEMPERATURE (22°C) AND AT A FIELD OF 11,270 GAUSS, TAKEN AUGUST 28, 1963 magnet Field Field—on Field-off Restoring Time position (gauss) voltage voltage voltage (degrees) V1 (volts) V2 (volts) V2 - V1 (volts) 11:01 -155 0 ----- 0.16199 -—--~- 11:03 -155 0 ------ 0.16201 ----- 11:07 -155 11,270 0.08888 0.16201 0.07758 11:12 -180 11,270 0.08570 0.16202 0.07632 11:13 -180 11,270 0.08572 0.16202 0.07630 ----- -125 11,270 0.08783 0.16202 0.07819 11:19 -110 11,270 0.09003 0.16203 0.07200 11:22 -95 11,270 0.09159 0.16203 0.07088 11:28 -80 11,270 0.09210 0.16208 0.06998 11:27 -65 11,270 0.09183 0.16208 0.07061 11:31 -50 11,270 0.08990 0.16205 0.07215 ---- -35 11,270 0.08789 0.16205 0.07816 ---- -20 11,270 0.08608 0.16206 0.07598 ----- -5 11,270 0.08878 0.16206 0.07732 11:83 10 11,270 0.08821 0.16207 0.07786, 11:88 10 11,270 0.08820 0.16207 0.07787 11:87 25 11,270 0.08866 0.16208 0.07782 11:53 80 11,270 0.08600 0.16209 0.07609 12:00 80 11,270 0.08616 0.16210 0.07598 12:08 80 0 ---- 0.16213 ----- 12:06 80 0 ------ 0.16209 -—-- NOTE: See TableVUI footnotes for an explanation of the headings. 51 TABLE XI ROTATIONAL DATA, AT ROOM TEMPERATURE (23°C) AND AT A FIELD OF 11,270 GAUSS, TAKEN AUGUST 29, 1963 Magnet Field Field—on Field-off Restoring position ( ) voltage voltage voltage (degrees) gauss Vi (volts) V2 (volts) V2 - Vi (volts) 8:85 80 0 ---- 0.16358 ---- 8:50 80 0 ------ 0.16353 ----- 8:56 80 11,270 0.08810 0.16360 0.07550 8:59 80 11,270 0.08808 0.16361 0.07557 5:01 25 11,270 0.08632 0.16362 0.07730 5:08 10 11,270 0.08583 0.16363 0.07780 5:07 -50 11,270 0.08632 0.16365 0.07733 5:10 -20 11,270 0.08762 0.16366 0.07608 ---- -20 11,270 0.08758 0.16367 0.07609 5:15 -35 11,270 0.08951 0.16368 0.07817 5:19 -50 11,270 0.09136 0.16369 0.07233 5:21 -65 11,270 0.09292 0.16370 0.07078 5:23 -80 11,270 0.09351 0.16371 0.07020 5:25 -95 11,270 0.09298 0.16371 0.07073 5:27 -110 11,270 0.09138 0.16372 0.07238 5:31 -125 11,270 0.08918 0.16378 0.07856 5:38 -180 11,270 0.08703 0.16375 0.07672 -—-— -155 11,270 0.08625 0.16376 0.07751 5:83 -155 0 ----- 0.16378 NOTE: See TableVm footnotes for an explanation of the headings. 52 awn MOm mum men men we: mma mom M6m 6m6 ape 6mm owHH 000a 000a 000a ommm 6mm :me :He now ~00 see 000 :H» ~00 OHN mew 06> Hmm mmm pm: HmmI owm0 :.ma m.oa ma.a em.6 ~m.6 om.m ~6.e mm.m 66.m Hm.m 66.H 6H.H ~a~.o mmm.6 mwm.6 mwm.6 6mH.6 00000 6060 pxoc co eoseapcoo c 0000 HHr—IHHNNNNMMM COONOCDr-iMJNwON omme—quxoooomm O ae6.6 6:6H6.6 mmeoo.o 65666.6 me6oo.o m6466.6 mmeoo.6 6mm66.6 66N66.6 46666.6 :6H66.6 66H66.6 Nm666.6 66666.6 6m666.6 NH666.6 >m000.0l mm000.0 '0'..-" ‘l'll". ‘l'lll' :U'i ii'.’ 6mm6N.6 mmmwm.6 ammwm.6 mmmmm.o mmmwm.o Hmmmm.6 Hmmwm.6 6mmmm.o mmmmm.o 6Nm6N.6 emmmm.o 6Nmmm.6 mma6N.6 mmmmm.6 mam6N.6 mmamm.6 N~a6~.6 eaemm.6 6mm6N.6 NNm6N.6 am66~.6 ammmm.6 6mwem.o 6mamm.6 6m~6~.6 66~6N.6 a666N.6 Naemm.6 mm66N.6 6m66m.6 mmamm.6 46amm.6 ~mamm.6 am66N.6 mm66N.6 6666N.6 66m6N.6 66amm.6 4N66N.6 Hme6 mae6 pae6 maa6 maa6 HHA6 moa6 66.6 e6.6 No.6 66.6 6m.m 6m.m emam mmam amam meam Ream meem meam 6eam hmam A00H x Mlmmsmmv Ammmsmw\mpmo>v NAMIOH x mesmmv AmIOH x mmswwv %mpao>v A0¢Ho>v m> Ampaobv H> .moeaa e m H Nm\AH> I mm m e I we oommeao>_ pommeaoe NBoE 6 30$ 66636: 36:36am 8-36am «popmom M9034 .3. 2529. $3.9 eoEmm zmmqu 24>? ma HHx mqm