THE UNIVESZTY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGSINEERING MEASUREMENT OF THE g-FACTOR ANOMALY OF FREE, HIGH ENERGY ELECTRONS Arthr.l< Auguist SchP pp, Jr A dissertation sulbmitted in partial fu.fillment of %he requirements for the degree of Do-tor of Philosophy in the UiTverslit.y of 0Michigan 1959 Agigs.st, 1959 IP-377

Doctoral Committee: Professor H. Richard Crane, Chairman Professor Robert C. Bartels Professor Kenneth M. Case Assistant Professor Peter A. Franken Professor Robert W. Pidd Professor Marcellus L. Wiedenbeck

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ACKNOWLEDGEMENTS The author wishes to express his appreciation to the chairman of his committee, Dr. H. R. Crane, and to Dr. R. W. Pidd, for their encouragement throughout this work as well as their many helpful ideas and suggestions about the experiment. The author also wishes to thank Mr. H. A. Westrick for his cooperation and skill in the design and construction of much of the apparatus, to the graduate students who also helped in the construction and assembly of the apparatus, to his wife for her cooperation and patience, and to his father for his encouragement throughout the course of this work. The author wishes to express his appreciation to the United States Atomic Energy Commission for support of this work under Contract No. AT(11-1)-70, and to the Industry Program of the College of Engineering for the typing and printing of this thesis. iii

TABLE OF CONTENTS Page FRONTISPIECE.......................................... ii ACKNOWLEDGEMENTS............................................ iii LIST OF TABLES.............................vi LIST OF FIGURES.............................................. vii I. INTRODUCTION............................................. 1 Review of Electron g-Factor and Magnetic Moment Experiments................................., 1 Present Experimental Proposal............................ 7 Results and Conclusions................................. 11 II. APPARATUS............................................. 12 General Physical Layout........................ 12 High Voltage Supply................................12 Electron Gun............................. 13 Beam Deflecting and Focusing Coils..............13 Polarizer Assembly......................15 Solenoid................................................. 15 Earth Correction Coils................................... 16 Field Shaping Coil................................17 Trapping Assembly............................ 17 Analyzer Assembly............................. 18 Control Circuitry........................................ 21 Timing Circuit................. 23 Counting Circuits............ 25 III. APPARATUS STUDIES........................................ 27 Trapping Procedure................................27 Magnetic Field........................ 28 Trapping Duration................................30 Drift Frequency............................. 33 Spurious Asymmetries............................. 33 IV. EXPERIMENTAL RESULTS AND ANALYSIS........................ 41 Theory................ 41 Procedure for Measuring the g-Factor Anomaly............. 44 Interpretation of Results........ *..........o 63 Discussion of Errors.................................... 71 Results and Conclusions................................. 74 Suggestions for Making a More Precise Measurement........ 74 iv

TABLE OF CONTENTS (CONetD) Page APPENDIX I - TABLES OF EXPERIMENTAL RESULTS.................... 76 Part A: Gold - Aluminum Scattering............ 76 Part B: Gold - Gold Scattering....,....O.... 77 Part C: Continuous Run...................... 81 APPENDIX II - TABLES OF CALCULATED VALUES............... 82 Part A: Calculated Values of Magnetic Field Due to the Solenoid and Field Shaping Coil. 82 Part B: Calculated Values of Effective Potential Well............,..................... 83 BIBLIOGRAPHY....*.......................84..*........... 84 or

LIST OF TABLES Table Page I Phase of the Double Scattering Asymmetry Runs....... 46 II Time Average Magnetic Field........................ 61 III Experimental Values of./.o........................ 63 vi

LIST OF FIGURES Figure Page 1. Schematic Diagram of the Experiment.................. 9 2. Drawing of the Gun Assembly..................... 14 3. Drawing of the Trapping Assembly..................... 19 4. Photograph of the Trapping Assembly.................. 20 5. Block Diagram of Control Circuitry.................. 22 6. Block Diagram of Timing Circuit...................... 24 7. Axial Magnetic Field.......................... 31 8. Radial Magnetic Field......................... 32 9. Drift Frequency......................... 34 10. Spurious Asymmetries..................... 36 11. Spurious Asymmetries................................ 37 12. Spurious Asymmetries......................... 38 13. Spurious Asymmetries................................. 39 14. Spurious Asymmetries...............................40 15. Double Scattering Asymmetry......................... 48 16. Double Scattering Asymmetry.......................... 49 17. Double Scattering Asymmetry......................... 50 18. Double Scattering Asymmetry......................... 51 19. Double Scattering Asymmetry.......................... 52 20. Double Scattering Asymmetry.......................... 53 21. Double Scattering Asymmetry.......................... 54 22. Double Scattering Asymmetry....................... 55 23. Double Scattering Asymmetry....................... 56 24. Double Scattering Asymmetry.......................... 57 vii

LIST OF FIGURES (CONT'D) Figure Page 25. Effective Axial Potential Well...................... 59 26. Oscillation Period in Well.......................... 60 27. VD/wo Versus 1/B2 - Straight Line Fit............... 64 28. mD/wo Versus 1/B2 - Evaluation of "a"............... 68 29. OD/ Vo Versus 1/B2 - Evaluation of "f"............... 70 viii

I. INTRODUCTION Review of Electron g-Factor and Magnetic Moment Experiments The gyromagnetic ratio is defined as the ratio of the magnetic moment of a body to its angular momentum. Classically, for all bodies with the same charge to mass ratio throughout, the gyromagnetic ratio is G=e/z where e is the charge on the body and m is the mass of the body. For a quantum mechanical particle, in this case the electron, the gyromagnetic ratio has to be modified to account for relativistic and quantum electrodynamic effects and can be written Ge= Se e/ m = (i + / where g is a dimensionless quantity called the g-factor of the free electron and a is called the g-factor anomaly. It is the purpose of this dissertation to report on a precision determination of the electronic g-factor anomaly by a measurement on high energy, unbound electrons. This quantity provides one of the few experimental checks on the present theory of quantum electrodynamics which predicts an infinite series of radiative correction terms to the Dirac value of 2 for the g-factor of the free electron. The difficulty in checking the results of this theory arises from the fact that the radiative corrections are small compared to the results of semiclassical theory and thus have been checked only by the Lamb shift and magnetic moment measurements on low energy, bound electrons. In 1947, Lamb and Retherford(l) showed experimentally that the 22 S1/2 state of atomic hydrogen did not coincide exactly with the 22P1/2 -1

-2state as predicted by Dirac theory () This was the first conclusive evidence of a difference between experiment and the existing theory and signaled the introduction of quantum electrodynamics to resolve the differences. About a month after this discovery Bethe(3) reinterpreted the existing quantum electrodynamic theory and was able to explain the Lamb shift. By subtraction techniques he removed the divergences of the theory and found that the 22S1/2 state of hydrogen should be 1040 megacycles above the value due to Dirac theory. This value is in good agreement with the experimental value of 1000 megacycles and demonstrated the applicability of quantum electrodynamics. At about the same time Breit(4) suggested that the deviations of the level splitting of the hyperfine structure of the ground state of hydrogen and deuterium for zero magnetic field(5'6) could be accounted for by postulating the existence of an anomalous magnetic moment for the electron. He found that the theoretical predictions would agree with experimental evidence if the magnetic moment of the electron was slightly larger than one Bohr magneton as predicted by Dirac theory. In 1948, Schwinger(7'8) showed that the magnetic moment and g-factor of the free electron is modified by the quantization of the electromagnetic field. The g-factor is expressed as a series of radiative correction terms added to the Dirac value of 2. These terms are expressible in powers of the fine structure constant(9) o( C = e 7. z29 719/ x/-3 The first of these terms, the first order in a, was calculated by Schwinger and he found that the g-factor for the free electron should be ge= 2(1+ /zrrr) = 2 (1. oo//638)

-3The second order term in a, has been evaluated by Karplus and Kroll(lO) and by Sommerfieldfll). The results are 9e-= 2 ( + -/27 -.973 oc/t) = (1 00 11454) (Karplus - Kroll) 9e 2(1 +0/s1Tr-,0328cr/f1) =1(I.0011S96) (Sommerfield) Recently this difference has been resolved and the accepted value is 2(1.0011596). It is also in agreement with the bounds calculated by Petermann. (2) Since then a number of experiments have been completed that determine the magnetic moment and g-factor of an electron when bound in an atom. Kusch and Foley(l3) have shown that g = 2(1.00119 + 0.00005) by atomic beam measurements of the Zeeman effect in the ground state of Ga, In, and Na. A more precise determination of the g-factor can be achieved by performing two separate experiments. One experiment determines the magnetic moment of the proton in units of the Bohr magneton {(j/0i), and the other experiment determines the ratio of the magnetic moment of the electron to the magnetic moment of the proton (>.e/kp). The product of these two results gives the magnetic moment in units of Bohr magnetons or one-half the g-factor for the bound electron. By applying suitable relativistic corrections the magnetic moment of the free electron can be obtained. In 1949, Taub and Kusch(14) made the first determination of (le/ ip) by a molecular beam magnetic resonance method. They measured the frequency corresponding to a reorientation of the proton in the molecule NaOH and the frequency corresponding to a transition between

-4certain of the hyperfine structure levels of the ground state of both Cs133 and In115. By combining this result with measurements obtained in related experiments(13,15) they found that /e//lp(NcvOH) = 658.2/ 0. 03 By observing the ratio of the electronic-spin g-value of atomic hydro2 gen in the S1/2 state to the proton g-value in a sample of mineral oil or water the most precise values of (>e//p) for a bound state are obtained. A relativistic correction is applied to account for the binding energy of the electron in the hydrogen atom and then (4e/pp) for the free electron is obtained. Three results have been published by different groups(16-18) using the above methods and the results are all in agreement to within two parts per million. The average of their results yield le//lp(o ) 658.2 29 2 + 0.0 /2 A determination of ip(Oil)/o0 has been completed by Gardner and Purcell(l9) and independently by Franken and Liebes(20). In these experiments, the ratio of the nuclear magnetic resonance frequency of protons to the cyclotron frequency of free low energy electrons was determined. Both frequency determinations were performed, as nearly as possible, in the same location in the magnetic field so that a measurement of the magnetic field was not necessary. The results of these experiments were /(O l) = 657. L75 ~ 0Q008 (Gardner - Purcell),/10/Lp(Ol = 6S7.I62 ^ 0.003 (Franken - Liebes)

-5When these results are combined with the value for tle/pp(Oil) the following values for the g-factor are obtained qe = 2(1,0011/46 + 0.000012) (Gardner - Purcell) Be = 2(1.00/168 ~ 0.000005) (Franken - Liebes) The Gardner and Purcell measurement is considerably below the present theoretical value of 2(1.0011596) and has been recently remeasured by Hardy and Purcell(21) with the result that e 2^.00/!6!.' o 0.00000o/5o) (Hardy - Purcell) In all of the above experiments the g-factor of the free electron was obtained by a combination of two different experiments. This was necessary to eliminate a common factor, the magnetic moment of the proton in mineral oil, so that the magnetic moment of the electron in units of Bohr magnetons could be obtained. To find the g-factor for free electrons a correction term had to be considered that would account for the electron binding energy. For these reasons and others, some systematic differences between results on bound and on free electrons may be present, thus making a measurement of the g-factor directly on free electrons highly desirable. A measurement at high energies would also be desirable to further check the relativistic effects and the applicability of quantum electrodynamics. A number of proposals have been advanced for experimentally measuring the g-factor of free electrons. One such experiment, as proposed by Bloch(22), is based on magnetic resonance techniques on slow electrons in a magnetic field. Here electrons are sorted and trapped

-6by electric and magnetic fields such that only electrons in the lowest magnetic state are trapped. A weak oscillating electric and magnetic field is then applied to induce transitions to higher states and cause the electrons to leave the trap and be detected. Two resonant frequencies should be found, one corresponding to a spin transition and the other corresponding to an orbital transition. The g-factor of the free electron is equal to twice the ratio of these frequencies. Another group of experiments uses the double scattering process with a magnetic field between the first and second scatterer. The theory of the double scattering process has been worked out in detail by Mott(23) and shows that a beam of electrons is partially polarized, a partial alignment of the spins or magnetic moments, in a direction perpendicular to a plane formed by the incident beam and the beam after single nuclear scattering. If this beam is scattered again, the theory shows that there will be an azimuthal asymmetry in scattering due to the polarization of the beam. Many attempts have been made to observe this asymmetry and the most recent of these, by Nelson(24), gives excellent agreement with theory. Tolhoek and DeGroot(25) proposed an experiment using the double scattering process with a radio frequency field between the first and second scatterers. This field could be adjusted in frequency until the asymmetry after second scattering vanished indicating that the spin resonant frequency had been reached and the beam depolarized. This frequency and the magnetic field in which the beam was depolarized would then give the g-factor for the free electron. Another experiment using the double scattering process, proposed by Crane, Pidd and Louisell (2) has been completed as a thesis by Louisell (27). In this experiment a magnetic

-7field is inserted between the first and second scatterers parallel to the electron's velocity. This field causes the polarization vector of the beam to precess and since the plane of maximum asymmetry depends on the polarization direction, this also precesses. Louisell showed that the g-factor can be obtained by a measurement of the magnetic field, electron energy, distance between the first and second targets, and the angle of rotation of the asymmetry. If the experimental conditions are such that there are many rotations of the asymmetry then the measurement of the angle of rotation can be very precise. Since only five revolutions of the beam were used in this experiment, the precision was limited to + 1/2 percent and yielded 9e = 2.00 ~0.0,! This experiment was not accurate enough to show the radiative corrections but did prove the feasibility of making a direct measurement on free electrons. Present Experimental Proposal As seen from the previous discussion, a precision measurement of the g-factor of high energy, free electrons is highly desirable. The present experiment was, therefore, undertaken with a precision of a few parts per million in mind. The double scattering process is used with a beam energy of 100 kilovolts since the Mott asymmetry is near its maximum there. A magnetic field, produced by a solenoid, is interposed between the polarizer (first) and the analyzer (second) targets. This field is oriented in such a way that the electrons, before scattering, travel along a field line and scatter off the polarizer target so that the scattered beam is traveling at approximately ninety degrees to the

-8magnetic field. The polarization vector is initially normal to the plane through the incident and first scattered beam and, thus, pointing in the radial direction. Between the targets the beam will travel in a helical path and rotate at the cyclotron frequency while the polarization vector will precess slightly faster due to the anomalous magnetic moment of the electron. The difference frequency, the precession frequency minus the cyclotron frequency, is the quantity that is measured directly by studying the intensity asymmetry after second scattering as a function of the time spent between scatterers. A schematic diagram of this experiment is shown in Figure 1. Since the precession frequency is slightly greater than the orbital frequency of the electron, the polarization vector will precess about the magnetic field and eventually become parallel to the direction of motion of the electrons. In this state, the asymmetry after second scattering will vanish. At a later time, the polarization vector will be opposite to the initial direction and the asymmetry will reverse. In this manner, a cosine curve can be traced whose frequency is the difference frequency. Mendlowitz and Case(28) have theoretically investigated the motion of an electron in a homogeneous magnetic field using the Dirac equation with the addition of a Pauli type term to account for the anomalous magnetic moment. These calculations show that Us = ((/ + ai_ ) when the electron velocity is perpendicular to the magnetic field. Rearranging this equation to give the g-factor anomaly a, we find a= (c-)(A-c)9- f, )' = 4/)o

TRAPPING yltm^- * - i ^ ^C O IL POLARIZ ER TARGET ANALZERiment TARGET P. Figure 1. Schematic Diagram of the Experiment

-10where xD is the difference frequency (Ws - c%), Co is the zero energy cyclotron frequency (eB/mo), e is the electron charge, B is the magnetic field, and mo is the rest mass of the electron. Since the g-factor anomaly is approximately one-thousandth of the g-factor, a determination of the difference frequency and the magnetic field to one part in a thousand will give the g-factor to about one part in a million. In order to determine the difference frequency accurately, a large number of revolutions are required which necessitates keeping the electrons in the magnetic field for a long period of time. This is accomplished by giving the magnetic field a slight betatron shape by the addition of the external trapping coil shown in Figure 1. Before the electron pulse enters the system, the second set of concentric cylinders (those nearest the analyzer target) are pulsed 100 volts negative. As the electron pulse crosses the gap between the two sets of cylinders, the momentum is reduced and some of the electrons will be reflected back towards the first target by the shaped magnetic field. Before these reflected electrons return to the gap, the voltage is removed and these electrons are then permanently trapped in the betatron field. At some later time the first set of concentric cylinders are pulsed negative giving those electrons on the polarizer side of the gap an additional 100 volts of energy as they cross the. gap. These electrons will then leave the shaped field region and encounter the second target. In this way many cycles of the difference frequency can be observed and the period can be determined very accurately. The magnetic field must also be measured to a part in a thousand and this is accomplished by making the inhomogeneities small and measuring the field with a proton resonance device.

-11 - Results and Conclusions The procedure just described has been successful in trapping electrons for more than 300 microseconds which corresponds to approximately 100 cycles of the difference frequency. The difference frequency has been determined to about 0.1 percent for electron energies between 50 and 100 kilovolts. Combining these results with the corresponding magnetic fields yields the following value for the g-factor anomaly a = 0.0011609 ~0.0000021 = o/2 0 - (0.1 o. Lf)c'/1' where the error represents the standard deviation in the measurement. This value is in excellent agreement with the current theoretical value of a = o/z2r-0,328 /lfr = 0.001/sq6 The data have also been evaluated, by assuming the theoretical value for the g-factor anomaly, to give an upper limit on the electric dipole moment of the electron of 3 X JOI CT (times the electronic charge) The principal sources of error are the determination of the effective magnetic field in the trapping region, the determination of the difference frequency, and the evaluation of the stray electric fields due to charging of the concentric cylinders in the trapping region.

II. APPARATUS General Physical Layout The apparatus necessary for the measurement of the g-factor anomaly consists of the following elements: 1) a source of electrons, including a high voltage supply and an electron gun; 2) a polarizer target; 3) a magnetic field; 4) field trimming coils; 5) trapping apparatus; and 6) analyzing and detecting equipment. The construction of each of these elements along with the relevant circuitry will be described in this chapter. The solenoid, field shaping coil, gun, and earth correction coils can be seen in the Frontispiece. High Voltage Supply The high voltage supply consists of a voltage doubling circuit capable of giving D.C. voltages to 120 kilovolts. Regulation of the high voltage is desirable, since the orbits depend on the beam energy, and is achieved by taking a signal from the precision resistor used for the high voltage meter, converting this signal to an A.C. signal by an electronic chopper, amplifying this signal by a high gain A.C. amplifier, and reconverting back to a D.C. signal. This signal is then used to control the primary voltage of the high voltage transformer. This circuit is capable of regulating the high voltage to + 1 percent after allowing a suitable warm-up period. Since the regulating circuit is essentially a D.C. regulator, line voltage ripple and spikes in the high voltage due to corona discharges are not eliminated. These variations have been observed by placing an oscilloscope across the amplifier input and found to be + 2 percent. -12

-13Electron Gun The electron gun is designed to produce a pulsed beam of high peak current (approximately 100 milliamperes). It consists of a watercooled, copper, cup-shaped cathode, 1 inch in diameter and 1-1/4 inches deep. The filament, placed 1/2 inch behind the rim of the cathode cup, is a flat spiral of four turns of 0.02-inch tungsten wire. This assembly is placed inside and insulated from the case of the gun which is machined from a section of 2-inch diameter aluminum pipe. The gun case is supported on glass insulators at one end,and from the top of a 16-inch diameter glass cylinder at the other end. The top of the glass cylinder and the gun are connected to the high voltage supply so that the complete assembly is at -100 kilovolts. This assembly is shown in Figure 2. The cathode assembly is insulated from the gun case so that it can be biased 500 volts positive with respect to the case structure (acting like a grid) and, thus, keep the gun cut off. A negative 4000 volt pulse, 0.13 microseconds wide is generated by a thyratron pulser and applied to the cathode assembly. This causes a pulse of electrons to be accelerated through the grid structure into the 100 kilovolt field between the gun structure and ground. The beam current is measured by placing a collector cup in the beam path and calculating the peak current from the average value obtained. Beam Deflecting and Focusing Coils The electron beam diverges as it passes through the accelerating gap and must be focused on the polarizer target about 50 inches in front of the gun. Partial focusing is provided by the main solenoid field, but an additional focusing field is needed since the main solenoid

3/16 IN. DIA. COPPER TUBES FOR FILAMENT- \ L WATER COOLING SYSTEM LEAD nr',.- POSITION ADJUSTMENT BELLOWS-\ 9 a111 ft ASSEMBLY |CLAMPING SCREWS- INSULATED FROM OUTER COPPER DISK 1 IN. THICK \ - "11 Ki! PALA,- TEIlC 3/16 IN. DIA. COPPER TUBES FOR / MICA INSULATING WASHERS I -t~i? PLATE ^ WATER COOLING SYSTEM COPPER SCREEN l l l: I.....................-POLISHED ALUMINUM i|| I | CYLI DER s / / / / / POLSHED PI ^ CYLINDDS FLAMENT- 20 MIL TUNOSTEN 2 IN. DIA. ALUMINUM WATER JACKET FOR COOLING DETAIL OF FILAMENT ASSEMBLY I \^ ONTO GLASS AT EACH II 7/S IN. O.D. I/B IN. THICK ~DEFLECTION AND FOCUS I ALUMINUM PIPE IN THICK ALUMINUM PLATE \ LAMENT ASSEMLY \OLTAGE FIBER DRAW ^-GLASS PORTS ^1ALUMINUM RETAINING AGLASS SUPPORTING RING RODS Figure 2. Drawing of the Gun Assembly CYLINDER 2 IN. DIA. ALUMINUM WATER JACKET FOR COOLING LAVITE SPACER DETAIL OF FILAMENT ASSEMBLY, BRASS RING WAXED ONTO GLASS AT EACH -I 7/ IN. O.D. I/8 IN. THICKBS O N END OF CYLINDER ALUMINEM CYLINDER FOCUS,**..~*-* —-I IN. THICK ALUMINUM PLATE ~ ~ — FILAMENT ASSEMBLY BRASS BOX -GASPRBAUIU EANN GASSPOTN ~~~~~~~~RIN R O D S ~ - Figure 2. Drawing of the Gun~~~~~~~~~~~~~~~~~~~ssembly~~~)

-15is varied during the course of the g-factor anomaly measurement. This additional field is provided by a small solenoid 10 inches long and 2 inches in diameter, wound with 4000 turns of #28 magnet wire. In order to correct for any misalignment of the gun with respect to the polarizer target, horizontal and vertical deflection coils are also necessary. These coils consist of two sets of two coils 3-1/2 inches wide and 8 inches long, wound with 500 turns each of #26 wire. These three sets of coils are located in a vacuum-tight box just in front of the gun as shown in Figure 2, and are supplied by regulated current supplies. Polarizer Assembly The polarizer assembly consists of a slit to restrict the incident beam, a target wheel, and a second slit to restrict the scattered beam. The target wheel is movable from outside the vacuum system and has four positions. These four positions contain a gold polarizer target which produces the polarized beam, an aluminum target which produces an unpolarized beam necessary for studying possible spurious instrumental asymmetries, a blank target holder used to study background counting rates, and a collector cup used to monitor the beam intensity. The polarizer assembly can be seen in the photograph of the complete assembly shown in Figure 4. Solenoid The solenoid is wound on an aluminum form made from two 10foot sections of 12-inch inside diameter pipe, which also serves as the vacuum chamber. The winding consists of four layers of #10 heavy Formvar magnet wire, close-wound with 2160 turns per layer. For this

-16experiment, the four layers are connected in series and produce a field of approximately 107 gauss at the center with a current of 6 amperes. Regulation of the magnetic field is necessary since the anomalous gfactor is inversely proportional to the field and is accomplished by passing the solenoid current through a bank of 48 power triodes (6AS7's) connected in parallel. The triodes are controlled by the output of a high gain D.C. difference amplifier whose input voltages are a reference voltage and a voltage proportional to the solenoid current. The solenoid current is monitored by a diode that is sensitive to magnetic fields (G. E. 2B23) placed in an auxiliary coil in series with the solenoid. The plate (current) voltage of this diode is proportional to the solenoid current and is used as the input signal for the difference amplifier. The regulator is capable of eliminating ripple to better than 0.1 percent and has a drift rate of less than 0.05 percent per hour. The current in the solenoid was set by monitoring the voltage across a standard shunt with a precision potentiometer. In this manner the current could be set to about 0.1 percent. The potentiometer was calibrated by measuring the magnetic field with a proton resonance device. Corrections were applied to account for the component of the earth's magnetic field along the solenoid axis. Earth Correction Coils The horizontal and vertical components of the earth's magnetic field within the trapping region are cancelled by two sets of correction coils. These coils are wound with 30 turns each of #14 heavy Formvar magnet wire in wooden forms 28.25 feet long and 5 feet wide. They produce a uniform field of 0.16 gauss per ampere, and are supplied by a

-17regulated current. The current is regulated by circuits identical to the solenoid regulator and have an error of + 0.1 percent. Field Shaping Coil The uniform field of the solenoid must be shaped to produce a trapping field (betatron field) for the electrons. This is accomplished by the addition of an external coil 18 inches in diameter, 6 inches wide, with 593 turns of #30 magnet wire, wound in a single layer. The current in this coil is regulated by a chopper stabilized D.C. amplifier circuit, and holds the field to + 0.001 gauss at the center of the coil. The radial and axial components of the field were calculated by graphical integration of the field components due to a single loop. The calculated values are shown in Appendix II, Part A. These components were added to the calculated components of the solenoid field, and are shown in Figure 7. Trapping Assembly The magnetic field shown in Figure 7 is designed to trap any electrons that are reflected by the betatron-shaped field. Actually, all electrons entering the shaped field will gain enough energy as they approach the center of the betatron field so that none will be trapped unless they lose some axial momentum while in this region. The momentum of the electrons is decreased by applying an injection (decelerating) voltage pulse across a /4-inch gap at the center of the trapping region. Some of the electrons will now be reflected back towards the polarizer target. The injection pulse is removed before the electrons get back to the gap so that the loss of energy is permanent and the electrons are

-18trapped. After a predetermined time (trapping time), an ejection (accelerating) voltage pulse is applied across the gap and the energy of the electrons is restored. The electrons then leave the trap and strike the analyzer target. The mechanical construction of the entire trapping apparatus including targets and counters, is shown in the drawing in Figure 3 and in the photograph in Figure 4. This apparatus consists of three sets of two concentric cylinders of 9-1/4 inches in diameter, and 5-3/4 inches in diameter. The first two sets (the set nearest the polarizer target and the center set) are insulated from ground, and the third set is grounded. The second set is pulsed negative to decelerate the electrons and the first set is pulsed negative to accelerate and eject the electrons. The timing and control of these pulses will be described in a following section. Analyzer Assembly The analyzer assembly consists of a target holder and three geiger counters. The target holder contains two targets, a gold and an aluminum foil, that can be lowered into the path of the circulating beam from outside the vacuum. Slits before the target restrict the beam to a radial spread of 1/4 inch. The targets are supported at one point only from outside the beam radius so that there is no scattering off the target holder. Three geiger counters are used to detect the scattered beam, two at an 80~ scattering angle for measurement of the asymmetry, and one at 15~ as a monitor of the beam intensity. These counters are standard end window geiger counters, 2-1/2 inches long with an inside diameter of 1/2 inch. The central wire, a piece of 0.005-inch diameter

rBEAM FROM ELECTRON GUN POLARIZER TARGET GAP, 1/4 IN. WIDE /\ f Z' -34 37 89 NYLON SPACER SOLENOID FORM BRASS SHIELD / | 1 /./ > \ COUNTER C ZCOUNTER C, ANALYZER TARGET CYLINDRCAL BOX CONTAINING ONTO /_.,.-._ ANALTZ~tK lA~bDIFFERENCE CATHODE FOLLOWERS MON ITOR Figure 3. Drawing of the Trapping Assembly

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-21tungsten wire, is supported at one end by a glass insulator and is beaded at the other. The window is made from a piece of aluminized milar film 0.00025 inches thick. Control Circuitry The sequence of events is: 1) the injection pulse is applied to the second set of cylinders, 2) the gun emits a pulse of electrons, 3) the injection pulse is removed, 4) the geiger counters are pulsed into the operating region, 5) the ejection pulse is applied to the first set of cylinders, 6) the counting circuit gates are opened, and 7) the ejection pulse is removed. This cycle is repeated a thousand times each second. A block diagram of the system is shown in Figure 5. The timing circuit is a crystal-controlled circuit that generates two main pulses separated by an adjustable number of 1/2 microsecond steps. This circuit will be described in detail in the next section of this chapter. The first of these pulses (1) drives a thyratron pulser that transmits a signal up to circuits riding at -100 kilovolts. Here the pulse is reshaped and used to drive another thyratron pulser that generates a 0.13 microsecond, 4000 volt high pulse for the gun. This pulse is delayed and clippedr to 3000 volts before reaching the cathode of the gun. The time relationship between the gun pulse and the injection pulse is very critical,. and jitter is greatly reduced by triggering the injection pulse circuits directly from the gun pulse, thus, eliminating any jitter due to the transmission of the timing pulse and thyratron pulser. This is accomplished by returning the gun pulse, before the delay line, to ground potential through a 150 kilovolt condenser and using this pulse to trigger the injection pulse circuits. A 0-1 microsecond variable delay line

~ cT"'GE"0 i TIMING CIRCUIT ~ ~~ \J1 1~ S SCALER SCALER —t - - |-tHER~~ I | I I I, t I I I. t I I THYRATRON ^_______|_____ l l | l ~~~~~~~~~~~~~~~~~~~~~~PULSER DAMPLIFIERENC AMPL IFIERA | AMPLIFIER 1J ___ | COUNTAERPL ALIFIER | GT ^ |] ECIO INJECTION- MICRO~! —---' ------ -----' ----— ______ ______ SECOND GATE —--- ----— ~~~~~~~~~~~~~~~~~~~~~~ ~ DELAY LINE GATE -------— GATESHP O SHAPER 0 LY~~~~~~~L O DISCRIMINATOR DISCRIMINATOR ctp~~~ ~ AND AND0-1S 0-1 s. C- SHAPER SHAPER VARIABLE VARIABLE o DIFFERENCE DIFFERENCE T G I-J CATHODE - -- CATHODEO OU FOLLOWER FOLLOWERPULSER PULSER CATHODE I._ COUNTER CC 1^~ r=1 ----- ~ ~~GATE' — EJECTION INJECTION P GENERATOR I PULSE PULSE D.C. SUPPLY GEIGER G-EIGERI COUNTER COUNTERl cl _______ ^ _ _ __ _

-23is used to accurately adjust the time of the injection pulse with respect to the gun pulse. The injection pulse is a pulse 0.5 microseconds wide and 100 volts high with a rise time of 0.03 microseconds, generated by a delay line pulser. The second pulse from the timing circuit (2) can be adjusted in time with respect to the first from 0 to 999.5 microseconds, and controls the length of the trapping duration. This pulse turns on the counting circuits to be described later and also turns on the ejection pulser which is identical to the injection pulser. A variable delay line is inserted just before the pulser so that the timing of the counting circuits can be varied with respect to the ejection time of the electrons. Timing Circuit The timing circuit block diagram is shown in Figure 6. This circuit is a very accurate delay pulse generator controlled by a onemegacycle crystal oscillator with an accuracy of 0.01 percent. The oscillator is followed by a shaper that generates both a positive and a negative pulse 0.06 microseconds wide and 40 volts high. This pulse drives the first of three scales of ten circuits, using Burroughs Magnetic Beam Switching tubes (6700). This tube has ten outputs that can be selected by a switch so that a pulse is obtained when the beam is formed on the selected anode. The output from the tenth (zero) anode is fed into a coincident circuit along with the pulses from the master oscillator so that the output is a pulse in phase with the master oscillator, but delayed by ten microseconds. This pulse drives another scale of ten, identical with the first, which gives an output pulse in phase with the master oscillator but separated by 100 microseconds. A third

-24I IC 06C. SHAPER BEAM SWITCHING COINCIDENT TUBE CIRCUIT VARIABLE TAP FOUR - FOLD BEAM SITCHI CONCIDENT COINCIDENT IRUITBE CIRCUIT IRUTVARIABLE TAP BEAM SWITCHIN6 COINCIDENT AMPLIFER FLIP-FLOP TE IR VARIABLE TAP CATHODE SCOPE AMPLIFIER FOLLOWER TRIER DELAY PULSER FOLLOWER SHAPER I MIXER'SCOPE AMPUFIER ATH ECTON FOLLOWER J A) COUTON TU TE Figure 6. Block Diagram of Timing Circuit

-25circuit, identical to the first two, gives a total scale of 1000. The coincident circuits are necessary to eliminate any delay time due to the circuits. The adjustable output from each scale of ten is fed into a four-fold coincident circuit with the master oscillator pulses, and when there is a coincidence between all four pulses an output pulse is obtained. In this manner, any pulse from 0 to 999 microseconds delay with respect to the pulse from the tenth anode of the last scale of ten circuit is obtained. The half microsecond steps are obtained by switching in or out a fixed 1/2 microsecond delay line before the pulse is shaped. This delayed pulse controls the counter gate circuits and the ejection pulse applied to the first set of cylinders. A mixer circuit mixes the pulses from the master oscillator with the zero time pulse and the delayed pulse for monitoring with an oscilloscope. Counting Circuits The counting circuit block diagram is shown in Figure 5. The geiger counters are pulsed into the operating region about 0.2 microseconds before the beam is ejected. This time is controlled by the variable delay line just before the ejection pulser. Pulsing of the geiger counters is necessary because a large number of electrons enter the counter from the part of the injection beam that is not trapped. These electrons render the geiger counters inoperative during their dead time of about 100 microseconds, and would make it impossible to take data for trapping times less than the dead time. This trouble is eliminated by keeping the geiger counters just below their threshold voltage so that entering electrons do not cause a geiger discharge. The pulses from the geiger counters contain both the wanted geiger pulse plus the applied

-26pulse necessary to turn the counters on. This pulse is fed into a difference cathode follower along with the applied pulse and the difference, just the geiger pulse, is obtained between cathodes at a low output impedance. These circuits are situated as close as possible to the geiger counters so that any capacity to ground will be minimized. The outputs are fed into a discriminator circuit through a shielded balanced line and then shaped. Another gate circuit is added to insure counting only pulses that occur during a short period of time after the ejection pulse is applied. The output from the gate is amplified and then fed into the scalars.

III. APPARATUS STUDIES Trapping Procedure Trapping the electron beam was accomplished in two main steps. First, the trapping duration was set for approximately 5 microseconds since trapping durations of this length are insensitive to small changes in the solenoid current, field shaping coil current, and injection pulse voltage. This allowed adjustment of the beam on the polarizer target by changing currents in the deflecting and focusing coils until the counting rate was maximized. Rough adjustment of the solenoid current, shaping coil current, trapping voltage, and turn off time of the injection pulse with respect to the gun pulse could also be made at the same time. The second step was to slowly increase the trapping duration and make slight adjustments in the above parameters to maximize the counting rates. The main adjustments necessary were the field shaping coil current, the timing of the injection pulse, and placement of small shim magnets. Two small permanent magnets were used to shim the solenoid field and were placed before the trapping region near the polarizer target. The field of these magnets is essentially a dipole field, and was less than 10-5 webers per square meter in the beam path as determined by the proton resonant device. Since the electrons only passed the region of the magnets once as they were entering the trapping region, it is believed that the magnets perturbed the initial beam direction in such a way that successful trapping was possible. In the trapping region the field of these magnets has fallen off to approximately 10-8 webers per square meter, and is completely negligible compared to the solenoid field of about 10-2 webers per square meter. -27

-28By careful adjustment of all the parameters, a good beam can be trapped for periods of 600 microseconds. After obtaining a trapped beam, the field shaping coil current is slowly reduced, while adjustments are made on the other parameters until the minimum possible current is reached consistent with good trapping. This procedure reduces the field irhomogeneities to a minimum and, thus, simplifies the procedure for calculating the average magnetic field in the trap. Magnetic Field An accurate determination of the magnetic field within the trapping region is necessary since the g-factor anomaly is inversely proportional to the average magnetic field encountered by the beam as it passes between the two targets. The field of the solenoid was calculated by expanding the field on the axis in a power series of the radius. The axial and radial components are given by E (jL00 ) ( (3.1)'7=0 00B (a?) (!)l) o*,) p (3.2) (nwhere B( Oz) is the nth derivative of the field) on the axis of where Bn) (Oz) is the nth derivative of the field on the axis of the solenoid 3( ) N L_ ____ Z-Lt_ (

-29where L = 2.896 meters is the half length, R is the winding radius, =o = 4t x 10-7, N = 1492 is the number of turns per meter, and I is the current in amperes. The first three terms of each of these expansions were evaluated for p = 0.0953 meters, the mean radius of the beam, and R = 0.169 meters, the mean radius of the windings. The error in neglecting the rest of the terms and in using the mean radius of the winding is less than 0.1 percent for both the radial and axial components. The field due to the shaping coil was calculated by first determining the field due to a single loop using the formula given in Smythe. (29) Curves were plotted for both the axial and radial components and graphically integrated over the external coil length. The magnetic field was measured experimentally by probing with a proton resonant device. The radio frequency coil and proton sample were fixed at the beam radius, but the azimuthal angle and the axial position could be varied. The experimental points shown in Figure 7 are an average of the values obtained at different azimuthal angles since the differences were less than 0.05 percent. The resonant frequency was determined by beating with a 500 kilocycle crystal oscillator, and measuring the beat frequency on a calibrated oscilloscope. All points were slightly below the field (1.1743 x 10-2 webers per square meter) corresponding to 500 kilocycles. The earth's magnetic field was cancelled by the earth correction coils except for the axial component of 1.6 x 10-5 webers per square meter which was opposed to the main solenoid field. This component of the earthts field was measured by reversing the solenoid field, measuring the field at a given point in the solenoid, subtracting this value from the value obtained with the normal field

-30direction, and taking one-half of the difference. Figure 7 shows the experimental points along with the calculated curve that fits the points best. The value of current for the solenoid and field shaping coil corresponding to the calculated curve are I(solenoid) = 6.2718 amperes and I(shaping coil) = 0.01762 amperes. The corresponding radial component is shown in Figure 8. The calculated values are tabulated in Appendix II, Part A. Trapping Duration The time of flight between the two targets must also be determined accurately. The timing circuit generates accurately spaced pulses but the zero point, the actual injection time with respect to the generated pulses, has to be determined. By operating the counters as straight geiger counters (not pulsed) and by keeping the beam intensity low, both the initial pulses and the delayed pulses can be observed simultaneously on an oscilloscope. The initial pulses correspond to electrons with too much axial momentum to be trapped, and they indicate the true injection time. By monitoring both the delayed and initial pulses on a calibrated scope, the zero time was determined to be 0.6 + 0.05 microseconds. This means that 0.6 microseconds should be subtracted from the times indicated on the timing circuit, and in the graphs indicating trapping time to obtain the true trapping time. The zero time can also be determined by a statistical analysis of the difference frequency curves for a number of different trapping times. This method will be treated in more detail in the next chapter.

CALCULATED TIME AVERAGE 1.174 MAGNETIC FIELD 2 Ic 3 1.1 732 I E N 4. u1.172','.j -- -T ZANALYZER 1.171 1.170 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Z (IN METERS FROM CENTER OF SOLENOID) Figure 7. Axial Mbgnetic Field

a)*~~~~~~ 6 W W~~~~~~~~ a. / 2" ANALYZER 4 (I, ce. > - \ ^ ^ ^ w w -08 -0.6 -0.4 -0.2 0.2 0 0.6 0.8 1.0 1.2 60 1 POLARIZER -2 z -4 N \ -6 Z (IN METERS FROM CENTER OF SOLENOID) Figure 8. Radial Magnetic Field

-33Drift Frequency The frequency with which the electrons oscillate in the shaped magnetic field is useful for calculating the average magnetic field and can be determined experimentally. For trapping durations up to about 35 microseconds the electrons remain bunched as they drift back and forth in the betatron field. This allows a measurement of the drift frequency by plotting the counting rate in either counter versus the trapping time, since more electrons will be ejected when the bunch is on the polarizer side of the gap than when it is on the analyzer side. The trapping time was varied in steps of 0.1 microseconds and the results are shown in Figure 9. The frequency determined from this graph is 1.14 megacycles which corresponds to a period of 0.88 microseconds. Spurious Asymmetries Spurious asymmetries may be present after second scattering even for the case of no Mott polarization of the beam. This is due to possible asymmetric scattering angles to the two counters, asymmetric magnetic fields, and misalignment in the slits, counters, and target holder. The scattering angle for the two counters is different due to the fact that the beam travels in a helix and, therefore, strikes the analyzer target at a slight angle. The two counting rates are, therefore, different due to the angle dependence of the scattering cross section. The angle of incidence is not fixed in time since it depends on the axial momentum given the beam on ejection which, in turn, depends on the position of the beam in the trap and on the trapping time. By using an aluminum analyzer target the spurious instrumental asymmetries

52 (4 MINUTES RUNS ) POLARIZER -GOLD a 48 ANALIZER-ALUMINUM z C) 0 44 z o 40 0 36 32 20.0 20.2 20.4 20.6 20.8 21.0 21.2 21.4 21.6 21.8 22.0 TRAPPING TIME (MICROSECONDS) Figure 9. Drift Frequency

-35can be measured since the Mott polarization for low Z elements is very small and any observed asymmetry will be due to spurious sources. Graphs of the spurious asymmetries for trapping times from 30 to 300 microseconds are shown in Figures 10 to 12, and the raw data is tabulated in Appendix I, Part A. The spurious asymmetries are considerably smaller than the Mott asymmetry, both are shown in the 300.0 to 305.5 microsecond graph shown in Figure 12 for comparison, and were neglected in the final analysis for the difference frequency. The error flags shown are due to just the statistical fluctuations in the number of electrons counted. Other spurious asymmetries can also arise from time variations in the high voltage used to accelerate the electrons, in the solenoid current, in the shaping coil current, and in the ejection pulse voltage. These asymmetries are much harder to evaluate and are minimized by highly regulating the above parameters. Graphs of the counting ratio for a gold polarizer and an aluminum analyzer target versus the solenoid current and high voltage are shown in Figurie 13, and Figure 14 shows the counting ratio versus the shaping coil current and the ejection pulse voltage. The regulating range indicated in each graph is the maximum drift observed over a period of a few hours. From these graphs it can be seen that spurious asymmetries of this type are very small (^0.5%) and are negligible compared to the Mott asymmetry. Drifts of this type are slow compared to the time necessary to measure one section of the difference frequency (5.5 microsecond period of trapping time) and, thus, cause a slow level shift of the counting ratio with only a negligible effect on the difference frequency.

-361.02 0.98 - i 4 L 0. I I96 I I I I II I I 30 31 32 33 34 35 TRAPPING TIME (MICROSECONDS) 1.02 0oo - < 0.98 z 96' 0.96, -I I I I- I I I I I' 60 61 62 63 64 65 i~Z ~ TRAPPING TIME (MICROSECONDS) 1.02 - 0.98 0.96 I I I I I I I I I I 90 911 92 93 94 95 TRAPPING TIME (MICROSECONDS) 1.02 1.00" n n i ~ ~ 0.98 _ 120 121 122 123 124 125 TRAPPING TIME (MICROSECONDS) Figure 10. Spurious Asymmetries

-371.02 1.00 0.98 0.96, I I I,. I I I 150 151 152 153 154 155 156 TRAPPING TIME (MICROSECONDS) Cu 1.02 1.00 o 0.98 D.96 180 i81 182 183 184 185 186 z TRAPPING TIME (MICROSECONDS) o 1.02 1.0 F I,,.oo~ ^ 0.98 210 211 212 213 214 215 216 TRAPPING TIME (MICROSECONDS) 1.02 1.00 v P P 0.98 0 96241 2 2 2 2 240 241 242 243 244 245 246 TRAPPING TIME (MICROSECONDS) Figure 11. Spurious Asymmetries

-381.04 1.02 1.00 a d 4 d I U) 0.980 0.96'''' l 0 270 271 272 273 274 275 276 TRAPPING TIME, (MICROSECONDS) z v3.- 104 0 o0 1.03 ~0.998\\ w 1.02 N 1.01 0 Z 1.00 0.99 E 0.98 0.97 300 301 30'2 303 304 305 306 TRAPPING TIME, (MICROSECONDS) Figure 12. Spurious Asymmetries

-391.08 1.06 1.04 1.02 1.00 0.98 0.96,- o 94 p t t rr- REGULATION 0.94 1 RANGE 9 0.92 a0.90 OPERATING POINT - 0.88 Z 6.0 6.1 6.2 6.3 6.4 6.5 6.6 0 SOLENOID CURRENT (AMPERES) a,, ].04 N J 1.03- REGULATION e RANGE I 0.97 1 0.96 I 94 96 98 100 102 104 106 ACCELERATING VOLTAGE (KILOVOLTS) Figure 13. Spurious Asymmetries

-401.14 1.10 1 1.06 - REGULATING 1.02 -i RANGE o 0.98 0~94 L OPERATING POINT 0 p t+YI - 0.90 14 15 16 17 18 19 20 (E SHAPING COIL CURRENT (MILLIAMPERES) z Iz 0 o 0 a 1.14 _ -1.10 r1.06 - Z 1.02 0.98 REGULATING 0.94- RANGE OPERATING POINT 0.90 0.86 72 86 100 114 128 142 156 EJECTION PULSE (VOLTS) Figure 14. Spurious Asymmetries

IV. EXPERIMENTAL RESULTS AND ANALYSIS Theory The theory of double scattering was investigated by Mott(23) in 1929. Using the Dirac wave equation, he calculated the double scattering cross section of a beam of electrons. The basic assumptions for the calculation were an unpolarized initial beam that undergoes single elastic scattering by an atomic coulomb field. The results of this calculation show that there is an azimuthal asymmetry in the double scattering cross section given by cr(e,e.t4)=,a)cr(~)[l+ 6(zieZaZ )c. os j (4.1) where a(Q) is the single scattering cross section for an unpolarized beam, 6(Z11QZ2Q2) is the Mott asymmetry factor, and 02 is the azimuthal angle about the second incident beam direction. This theory has been experimentally checked by Nelson(24) and excellent agreement is obtained for the energies and angles used in the present g-factor anomaly experiment. The addition of a magnetic field between the two scatterers is not included in this calculation. Mendlowitz and Case(28) investigated the theory of double scattering with a uniform magnetic field interposed between the scatterers. The Dirac relativistic wave equation was used with the addition of a Pauli type term to account for the anomalous magnetic moment of the electron. This Hamiltonian was transformed to the Foldy-Wouthuysen representation where the spin and space dependence can be separated to a very high degree of accuracy. In this form, both the spin and spatial -41

-42motion of the particle can be easily predicted and were found to agree with the classical model of a charged spinning top with a magnetic moment precessing in the magnetic field. The spin precession frequency obtained for the case of a uniform magnetic field is Ws = - ac[( Y)b -aL(7-1) s V? ^*(4.2) where xc = eB/ymo is the cyclotron frequency, a is the g-factor anomaly defined in terms of the magnetic moment by =p (l+a)eTi'/2m, b is a unit vector in the magnetic field direction, and v is the electron velocity vector. For the particular case of the electron velocity perpendicular to the magnetic field between scatterers, the precession frequency reduces to,3 + ( 7A )6 ^ (4-3) where dc is given by -eBb/ymO and y = (1 - v2/c2)-1/2. The difference frequency is defined by =[34S G| = I G)CAi = &j(4.4) where wo = eB/mo. Case and Mendlowitz also showed, for this particular case, that the double scattering cross section is cr(,, 0) - (Z Z COS i (4.5) ~r~~~~~e, e~ ~) ~! + ~(z, a z~)~os.~:o~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t6Z, 8,Z,,CO ~, O

-43where E = yaWT = wDT, T is the time spent in the magnetic field, 6(Q1z1Z2z2) is the Mott asymmetry factor, and 02 is the azimuthal angle. This equation shows that the double scattering asymmetry rotates at the difference frequency as the trapping time is changed. A measurement of this frequency and the magnetic field constitutes a direct measurement of the g-factor anomaly since O 7ncC/ Q mow/eB (4.6) A more general expression for the spin precession has been obtained by Ford(30) using the Dirac wave equation with the addition of terms to account for the anomalous magnetic moment and for a possible electric dipole moment of the electron. The result is 7rnL X+IV Ca ecz- 7I E] (4.\) -~[E- ~7T+VXBJ where f, defined in terms of the electric dipole moment vector, is - feMt /2 c o(4.8) and a, the anomalous g-factor defined in terms of the magnetic moment vector, is /?=fi+a~e hoy+2 (4.9) E is an arbitrary macroscopic electric field, and B is an arbitrary macroscopic magnetic field. This expression will be applied to the results of this experiment in a following section of this chapter.

-44Procedure for Measuring the g-Factor Anomaly A measurement of the g-factor anomaly requires measuring both the difference frequency and the magnetic field. The difference frequency is obtained by measuring the counting rate ratio after double scattering as a function of trapping time. Thus, four experimental numbers are obtained, the two counting rates, the trapping time, and the magnetic field. The counting rate in either counter may be expressed by An= I(t)(Nsd.^A )( A,.d-) 6 % (e,) 42a(~,) (4.1o) x [ + 6(z., z2 e)COS 2 cos -,TJ A(T) where I(t) is the initial beam intensity which may depend on the time, NZ is the number of atoms per unit volume in the target, dZ is the effective target thickness, AQ is the solid angle, e is the counting system efficiency, a(Q) is the single scattering cross section b(zlQ1z2g2) is the Mott asymmetry factor, and A(T) is the spurious instrumental asymmetry which may depend on the trapping time, The ratio of the counting rates in the two counters Cl and C2, see Figure 1, is now given by m, _ A,(T) I \6(Z, eS Z )COS 2 COWDT (4.11 n A?(T) I +6(Z,e,Ze O)COs 02CO3G)T The factor C is a constant with respect to the trapping time and the ratio of the spurious instrumental asymmetries is negligible compared to the Mott asymmetry as shown in Chapter III. Since the Mott asymmetry factor is small compared to unity, this expression (4.11) may be

-45simplified by expanding to first order with the following result 1t C 12( Ze&,Z2)cs 60TC~oS 02/-cosA23 (4.12) The angles 021 and 022 are fixed at 021 = 19~, and 022 = 161~ for this experiment so that the normalized counting ratio becomes - = I + 1.9 6(eze COS G)T (4.13) The counting rate ratio was measured for various trapping times and energies. The trapping time was varied in steps of 0.5 microseconds over the range of 30 to 300 microseconds. Twenty-five groups of twelve points (5.5 microseconds) were taken and the raw data are tabulated in Appendix II, Part B. The magnetic field, shown in the tables as the voltage across the solenoid shunt, was varied with the energy to keep the beam radius fixed. The phase of each of these curves was determined by fitting a cosine by the method of least squares. An IBM 650 digital computer was used to determine the phase and the results of this calculation are shown in Table I. The phase indicated in Table I is the time corresponding to the maximum of the cosine curve. The difference frequency is found by determining the number of cycles of a single cosine curve between zero trapping time and the phase point indicated in Table I. The period is given by XT M-to (4.14) where Mi is the time corresponding to the maximum of the ith run, to is the time corresponding to zero trapping time (to = 0e6 + 0.05

-46TABLE I PHASE OF THE DOUBLE SCATTERING ASYMMETRY RUNS Run Number Phase Run Number Phase I 32.030 XIV 300.346 II 60.822 XV 302.657 III 92.307 XVI 203.042 IV 121.128 XVII 252.321 V 152.507 XVIII 268.062 VI 181.164 XIX 281.120 VII 210.189 XX 302.003 VIII 241.346 XXI 252.236 IX 270.422 XXII 301.776 X 301.703 XXIII 301.745 XI 301.941 XXIV 301.990 XII 301.913 XXV 303.266 XIII 301.961

-47microseconds, as shown in Chapter III), and Ni is the number of cycles th to the maximum of the i run. The errors in Mi and to are independent of the run number so the average period for a particular magnetic field is given by _ 2 2 A; (4.15) T = 4 ZN t The twenty-five runs are shown in Figures 15 through 23. The small arrows along the horizontal axis represent the maximum of the single cosine curve with a period given by Equation (4.15). The effect of an error in counting the number of cycles is shown in the first group of curves for the highest magnetic field used. The arrows with a plus sign represent the maximum points of a cosine curve with Ni = Ni + 1 where Ni is the number of cycles for the best fit cosine. The arrows with a minus sign have Ni = Ni - 1. In this way the correct period, corresponding to Ni, can easily be found. A continuous run from 30 to 130 microseconds was also taken and the number of cycles in this run counted. The raw data for this run are tabulated in Appendix I, Part C, and the points are plotted in Figure 24. The time average magnetic field seen by the trapped electrons must also be calculated. The axial force on the electrons is given by F = e B( Z) = tn (4.16) where v0 is the orbital velocity of the electrons and can be considered constant for this calculation since the magnetic field is essentially uniform and Bp(z) is the radial magnetic field shown in Figure 8.

-481.02 - 1.0 I N ~ I T,% 1.00 = 0.99 I 0.98 0.97RUN I + - |_j _______.. 30 31 32 33 34 35 T 1.03 - 1.02 - v 1.0 I0.99 0.98 - 0Q97 -RUN II + I I ~ I I X 60 61 62 63 64 65 1.03 1.02 CN 1.01 C Ii 1.0 1.00 0.99 0.98 0.97 RUN III I. I,. e I, I, I I I, ~ 1 90 91 92 93 94 95 T Figure 15. Double Scattering Asymmetry

-491.03 - 1.02 - 1.01,., 1.00c 0.99 6 0.98 - 0.97- RUN IV 1 + l,,*,,,*, i 4 i 120 121 122 123 124 125 T 1.03 I.O I 1.02, 1.01 I1.00 0.99 0.98 RUN V 0.97 + 150 151 152 153 154 155 T 1.03 - 1.02 i.02 - C 1.01'- 1.000.98 0.97 RUN VI _ T 180 181 182 183 184 185 Figure 16. Double Scattering Asymmetry

-501.03 1. 02 1.011'~ 000.99 1 I l 0.98 - 0.97- RUN VII + 1i I I I1 1 210 211 212 213 214 215 T 1.03 1.02 - 240 241 242 243 244 245 T T -N 1.01 1.00 - 0.99 - 0.98 -IX 0.97- RUN VIII I I 1 I I I I I 1 240 241 242 243 244 245 T 1.03 1.02 c* 1.01 E' 1.00 - 0.99 - 0.97 - - + 270 271 272 273 274 275 Figure 17. Double Scattering Asymmetry

-511.03 1.01 -1.00 0.99 0.98 0.978 3 RUN X O. 97 300 301 302 303 304 305 T 1.03 1.02 - Nc 1.01 1- -.00 0.99 - T 0.98 0.97 RUN XI I _ 1, I I 1 I I 300 301 302 303 304 305 T 1.03 1.02 1.01 - C 1.00 I 0.98 - 098 RUN XII 0.97 3 3 300 301 302 303 304 305 T Figure 18. Double Scattering Asymmetry

-521.03 1.02 I I I I I I I I I * I 300 301 302 303 304 305 1.00 Q0.99- 0.98 0.97 - RU N XIV { 0.98I I, I I I I, I 300 301 302 303 304 305 T 1.03 1.02 - 1.01I - 1.00 0.99 0.98 0.97 t I RUN XV I 1 I I I I I I, I, I 1 300 301 302 303 304 305 T 1.03 D'1.00 - 0.990.98 - O.97- - I RUN XV

-531.03 1.02 i.00 I 0.99 0.97 RUN XVI 1 I I I I I I 200 201 202 203 204 205 T 1.03 1.02 U 1*.00 I 0.99 - 0.98 0.97- RUN XVII I I II I I I I I 250 251 252 253 254 255 T 1.031.02 { *~ I II 1.00 0.99 0.98 0.97 RUN XVIII 265 266 267 268 269 270 Figure 20. Double Scattering Asymmetry

-541.03 1.02 1.01I I T ( % 1.00 0.99 0.98 0.97 RUN XIX. I II I I I I 1 I 285 286 287 288 289 290 T 1.03 - 1.02 N 1.01 0.99 { 0.98 - 0.97 - RUN XX 300 301 302 303 304 305 T 1.03 1.02 - N 1.01 0.99 A 0.98 0.97 RUN XXI 250 251 252 253 254 255 T Figure 21. Double Scattering Asymmetry

-551.03 1.02 - 1.00 C -.99 { 0.98 - 097 - RUN XXII I I I I I I I 300 301 302 303 304 305 T 1.03 1.02 - 1.01 - 1.00C 0.990.98 0.97 RUN XXIII I I I I I I 300 301 302 303 304 305 T 1.03 1.02 - 1.01 1 C 1.00 C-0.99 T I 0.98 - 0.97 RUN XXIV 300 301 302 303 304 305 T Figure 22. Double Scattering Asymmetry

-561.04 1.03 1.02 4 { 1.01<' 1.00 N_ 4.oo ~ 0.99 - 0.98, 0.97 RUN 0.96 I 300 301 302 303 304 305 306 T Figure 23. Double Scattering Asymmetry

-570.64 0.,62 {f ^ d d{'{{P( i?} {+ 0.58 0.560 0.54 I 1 30 35 40 45 50 55 0.66 ^,,,ciV. /, 1,. 0.56 4 i4 o0 55 60 70 75 80 Io.,, z 0.62 0.60 T 0.54 i ifi 0.52 4 4 80 85 90 95 100 105 0.66 0.64. 0.62- I i 0.60S 0.58-.i{I I{ l 1 CI 0.56^ 1 0.54 0.52 4 105 110 115 120 125 130 TRAPPING TIME, MICROSECONDS Figure 24. Double Scattering Asymmetry

-58Integrating Equation (4.16) once we obtain 7(~ =J vO ZdE (4-17) The radial magnetic field was graphically integrated to give the effective axial potential well and the results are tabulated in Appendix II, Part B, and plotted in Figure 25. Solving Equation (4.17) for the time we Obtain ~t =f,~ de-2, M(4.18) Another graphical integration will then give the period of oscillation for electrons in the well. The axial velocity of the trapped electrons goes to zero on each side of the potential well so that the integrand of Equation (4.18) becomes infinite at these points. The graphical integration is good only from center of the well out to some finite distance from the point where the electron velocity goes to zero. Assuming the force over this small final interval to be constant, the time necessary for the electrons to travel this final interval is calculated by using t- V ThL/F (4.19) where L is the length of the final interval and F is the average force over the interval. In this way the period of oscillation for trapped electrons with different amplitudes can be calculated. Figure 26 shows the results of this calculation. In Chapter III the measured period of oscillation was shown to be 0.88 microseconds, which corresponds to an amplitude of 0.487 meters. The time the electron spends in each interval

0 0 0 OD a -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 Z (IN METERS FROM CENTER) Figure 25. Effective Axial Potential Well n)\I Q W w - ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~\ o \c w 03 0 0 -1.6 12 08-0.4 0 0.4 0. 8 1.21. Z (IN MlrETERS FROM CENTER)) Figure 25. Effective Axial Potential Well

1.8 1.61.4 I z I / I Q/) 8 1.2 / 0 0 1.0 0.8 0.6 I -o 0.4 0.2 0'- I''I 0 0.1 0.2 0.3 0.4 0.5 0.6 AMPLITUDE, ( METERS ) Figure 26. Oscillation Period in Well

-61Az can be determined for this amplitude using the same procedure as used for the calculation of the oscillation period. The time average magnetic field can now be calculated using B5 T Bit t (4.20) L where T is the period of oscillation (0.88 microseconds), Bi is the magnetic field in the ith interval, and Ati is the time spent in the ith interval. For a solenoid shunt reading of 0.8313 volts the calculated time average axial magnetic field seen by the trapped electrons is 0.0117319 webers per square meter. Since the earth's magnetic field has a component of 16 x 10-6 webers per square meter opposed to the solenoid field, the actual solenoid field corresponding to a shunt reading of 0.8313 volts is 0.117479 webers per square meter. Table II shows the time average magnetic field seen by the electrons for the different values of field used in measuring the double scattering asymmetry. TABLE II TIME AVERAGE MAGNETIC FIELD Shunt Time Shunt Time Reading Average Reading Average (Volts) Magnetic (Volts) Magnetic Field Field 0.5819 0.0082074 0.7500 0.0105830 0.6286 0.0088674 0.8000 0.0113056 0.6500 0.0091698 0.8212 0.0115892 0.7000 0.0098764 0.8313 0.0117319

-62The average values of hD/%o may now be calculated for the various values of energy and magnetic field used. The value for each measurement of the difference frequency period may be expressed by (4.21) (cJD/o) = 2a-rmO/eB = k /B kNj/B(Mjt) (4.21) where k = 3.5712 x 10-5 kilograms per coulomb, B is the magnetic field and Tj is the difference frequency period for the jth run. The average value of D/wo is given by d(' ( " ) (4.22) Z (1)0 where s is the standard deviation for the measurement of (caD/%o)j and the sum is over all runs with the same magnetic field. Errors are present in the phase (M), the zero time (to), and in the magnetic field (B). These errors propagate into a standard deviation for the jth run as follows b2(tir t) 6^ ) ^ - t2 2 + B (Am + (4.23) where sM is the standard deviation in the phase, st is the standard deviation in the zero time, and sB is the standard deviation in the magnetic field. Table III shows the average value of TD/%o and the standard deviation for each magnetic field used. These values of AD/%O are plotted against 1/B2 as shown in Figure 27, and a straight line was fitted by the method of least squares. The intercept of this line at 1/B2 = 0 is 1.16064 x 10-3 + 1.21 x 10-6, and the slope is 2.309 x 10-10 + 1.24 x 1010.

-63TABLE III EXPERIMENTAL VALUES OF / Magnetic 2 Average Standard (B) 1/B D/0 Deviation 0.0082074 1.48453 x 104 1.16448 x 10-3 1.23695 x 10-6 0.0088674 1.27177 x 104 1.16255 x 10-3 1.22853 x 10-6 0.0091698 1.18927 x 104 1.16308 x 10-3 7.43456 x 10-7 0.0098764 1.02519 x 104 1.16346 x 10-3 6.15290 x 10-7 0.0105830 0.89285 x l04 1.16224 x 10-3 1.20615 x 10-6 0.0115892 0.74455 x 104 1.16168 x 10-3 1.19597 x 10-6 0.0117319 0.72655 x 104 1.16232 x 10-3 4.53150 x 10-7 Interpretation of Results According to Equation (4.6) the difference frequency divided by 0o is equal to the g-factor anomaly and should be independent of the energy and magnetic field. Equation (4.6) assumes only an axial component of the magnetic field and an electron velocity exactly perpendicular to the magnetic field. The systematic shift of wD/wo versus 1/B2 shown in Figure 27 must be due to fields not included in this equation. A radial magnetic field is present in the system due to the shaped magnetic field and an axial electric field is present during the injection and ejection of the beam. Since the electrons are trapped in the magnetic well, the average axial force must be zero. This requires the average radial magnetic field to be zero as can be seen from Equation (4.16). Thus, the average precession of the magnetic moment vector of the electron

0.001166 0.001165 0.001164 0.001ool163 1 0.001162 0.001161 _3 0.001160 0.001159 0.001158 0.001157 1 I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 I/B (IN 10 METERS PER WEBER ) Figure 27. acD/o Versus 1/B2 - Straight Line Fit

-65around the radial direction is also zero so that the radial component of the magnetic field will have negligible effect on the difference frequency and can be neglected. The axial electric field pulses also cause a precession of the magnetic moment vector about the radial direction due to the (v x E) term in Equation (4.7). These pulses are of the order of 200 volts per centimeter at the gap and are much smaller at the beam radius. An electron that is trapped will only pass this field once during injection and once during ejection, and the time spent in the field is approximately 10 seconds. This leads to a rotation of the spin vector of only about 10-2 radians. The rotations are in the opposite directions for injection and ejection since the electric pulse is reversed while the beam velocity (vt) remains unchanged. These two rotations tend to cancel each other, and the effect of the trapping pulses can be neglected in the measurement of the g-factor anomaly. The systematic shift of the data must be due to other fields than the two mentioned above. Three other effects may be present, one due to the fact that the beam has a finite pitch as it drifts back and forth in the well, another due to any stray electric fields present in the trapping region due to charging of the surfaces of the concentric trapping cylinders and a third due to a possible electric dipole moment for the electron. Equation (4.7) is a general expression for the precession of the electron and includes the three effects just mentioned. Upon examination of the terms in (4.7), it is found that the effect of a radial electric field is at least 100 times greater than that due to an axial electric field or an azimuthal electric field. A radial electric field is also the most probable field present, due to the geometry of the

-66trapping system. The effect of the radial magnetic field was negligible and due to cylindrical symmetry, there is no azimuthal magnetic field. Specializing to the case of axial magnetic field and a radial electric field, Equation (4.7) becomes s = -< j + y +(at- sI} bCCOS~c( -CL —y' s/ln o+ fI + os o0 where a is the slight pitch angle of the helical trajectory of the beam and cO = eB/mo. The precession of the magnetic moment vector about the 0 direction averages to zero over one complete cycle of the drift frequency since sina is an antisymmetric function of the motion. The only residual effect is due to one pass through the field from the first target to the second target. This precession amounts to only 0.001 radians when the values of the various trapping parameters are substituted into Equation (4.24) and can be neglected. One other term can be neglected. This is the term proportional to sin2a in the z component of the spin precession frequency. This term amounts to less than 0.1 percent of the g-factor anomaly, and since a 0.1 percent measurement is being made, this term is negligible. The cyclotron frequency for this type of motion is given by =0,- = o + " (4.25) ^c=~o~y rBjCOcCI JT

-67The difference frequency divided by CD is now obtained by subtracting Equation (4.25) from Equation (4.24), and then taking the absolute value of the resultant vector. This yields G)D v c8 B M'+r + _ + "-1 (4.26) The cosC factors have been made equal to unity since Cx is about 10 and the error introduced is negligible. Expression (4.26) contains three unknown quantities, 1) the g-factor anomaly a, 2) the electric dipole factor f, and 3) the stray electric field E. These quantities are evaluated by requiring this expression to pass through the weighted average of the experimental points shown in Figure 27, and have the same slope at this point as the straight line fitted by least squares. Theoretically, the electron possesses no electric dipole moment and the experimental evidence(31) to date is in agreement with this prediction. If the electric dipole moment factor, f, is put to zero, Expression (4.26) reduces to COAD.^+ E (~t / (4.27) C3D a + cSa P W. a+, This expression is shown in Figure 28 and represents the best fit to the experimental points. In the limit as P-+l, Expression (4.27) approaches the g-factor anomaly so that the intercept of the curve shown in Figure 28 gives the best experimental value of the anomalous g-factor. Using Equation(4.27), the g-factor anomaly, a, and the stray electric field can be found in terms of the weighted average of the experimental

0.001166 0.001165 0.001164 0.001163 0.001162 13 0.001161 1 0.001160 -* —THEORY 0.001159 0.001158 0.001157 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 1/B1 (IN 104 METERS4 PER WEBER2 ) Figure 28. cw/wo Versus 1/B2 - Evaluation of "a"

-69points and the slope of the least square straight line. This calculation yields a=N,+ER/c E=-cBN2/F (4.28) where N1 = weighted average of the experimental points 1.16278 x 10-3 N2 = slope of the least square straight line 2.309 x 10-10 R = 1/K K + 1 F = B2R K = epB/m c p = beam radius of 0.953 meters B = magnetic field at weighted average point 1.0389 x 10-2 webers per square meter. Substitution of these values in (4.28) yields a = 0.0011609 E = -4.0 volts per meter The data may also be interpreted to yield information about the electric dipole moment of the electron by assuming the theoretical value of 0.0011596 for the g-factor anomaly of the electron. A plot of Expression (4.26) for this case is shown in Figure 29. The f-factor and the electric field for the best fit are given by j _ (,-aT)F-NR E = c (-a)G cBN I JFI+R GR -IF GR (4.30)

0.001166 0.001165 0.001164 0.001163 0.001162 0.001161 4 0.001160 0.001159 0.001158 0.00115T7. - l i —. ------- I.. I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 4 4 2 I/B (IN 10 METERS PER WEBER) Figure 29. k/jo Versus 1/B2 - Evaluation of "f"

-71where aT = theoretical value of the g-factor anomaly 0.0011596 G = B2I I = K2/2aT(K2 + 1) the other factors were defined previously. Substitution of these values in (4.30) yields f = 8.4 x 10-5 E = -4.8 volts per meter. Discussion of Errors The quantities which may affect the final results are, 1) the mapping of the magnetic field in the trapping region, 2) the solenoid and shaping coil currents, 3) the beam energy, 4) the trapping time, 5) the stray electric field, and 6) the statistical fluctuations in the counting rates. The errors in each of these, except the last, contain a random and a systematic part. The combined effect of the random parts of all of the above errors was determined through an experimental study of the fluctuations in the phase of the double scattering asymmetry. Fifteen measurements of the phase of the double scattering asymmetry for the same experimental parameters were taken over a period of three weeks. The scatter in the results was such as to indicate a measurement of the standard deviation in the phase of 0.3 microseconds. This is the quantity labeled sM in Equation (4.23). The standard deviation in the zero time (sto) was determined to be 0.05 microseconds as shown in Chapter III. The standard deviation in the magnetic field was determined experimentally from the variations of the proton resonance curve observed on the oscilloscope. This included the variations in the solenoid current, the shaping coil current, and in the proton resonance device itself.

-72When these standard deviations are substituted in (4.23) and the square root of the sum of the squares over all runs of the same magnetic field is taken, the standard deviation for each measurement of WD/0o, shown in Table III, is obtained. These errors then propagate into the following standard deviations for the experimental results in the two cases mentioned above. a = 0.0011609 + 0.0000011 E = -4.0 + 2.1 volts per meter and f = (8.4 + 3.5) x 10-5 E = -4.8 + 1.3 volts per meter. These errors are just the random errors present in the experimental parameters. Systematic errors are also present in the absolute value of the magnetic field, in the trapping time, in the beam energy, and in the stray electric field. A systematic error of 0.1 percent in setting the magnetic field was found by resetting the field many times, and making an accurate measurement of the field with the proton resonance device. The voltage across the solenoid shunt was accurately reset each time with a potentiometer calibrated against a standard cell, but the magnetic field did not return to exactly the same value each time. This effect could be caused by a difference in contact potential each time the potentiometer is connected to the shunt, and by a difference in some small leakage current around the solenoid. The duration of trapping was controlled by a crystal oscillator having an absolute accuracy of 0.01 percent. The accuracy was obtained by checking the crystal against WWV. The beam energy enters into the final result only if there are stray radial electric fields in the apparatus, and then only in an insensitive

-73way. For the stray electric field indicated by the results, namely, about four volts per meter, a 5 percent variation in the energy would produce a change of about one part in 107 in the g-factor anomaly. The control of the high voltage was at least this good so that variations in the beam energy result in only about 0.01 percent variations in the measurement of the g-factor anomaly. Stray radial electric fields in the trap are an important possible source of error. They may be present due to charging of insulating films on the concentric cylinders. This field was determined, as shown earlier, by fitting the theoretical expression to the experimental points. This is an uncertain procedure, however, because it assumes that the average radial electric field was about the same for the runs at different magnetic fields. Since the stray fields represent only a few volts across the shields, a change from 100 kilovolts to about 50 kilovolts should not appreciably change the number of electrons reaching the surfaces. A day-to-day shift of the stray electric fields, due to changing of the insulating films, was ruled out because no progressive shift of the difference frequency for a given set of experimental parameters was found over a three-week period. The systematic errors may be listed as follows: Magnetic Field 0.10% Energy 0.01% Time 0.01% These errors are combined with the random errors by taking the square root of the sum of the squares. The final results with the combined errors are given in the next section.

-74Results and Conclusions The evaluation of the difference frequency and magnetic field data yield the following value for the g-factor anomaly of free, high energy electrons. a = 0.0011609 + 0.0000021 = a/2t - (0.1 + 0.4)0/ 2 where the error represents the standard deviation in the measurement. This value is in excellent agreement with the current theoretical value of a = a/2T - 0.328c2/3r2 = 0.0011596. The data have also been evaluated to give an upper limit on the electric dipole moment of the electron. The experiment yields the following value of the f-factor of the electron f = (8.4 + 6.7) x 10-5. This places an upper limit on the electric dipole moment of 3 x 10 15 cm (times the electronic charge). The principal sources of error are the determination of the effective magnetic field in the trapping region, the determination of the difference frequency, and the evaluation of stray fields due to charging of the concentric cylinders in the trapping region. Suggestions for Making a More Precise Measurement The precision of the experiment can be increased by reducing the errors mentioned above. The error in the effective value of the magnetic field is due to two different sources. The first is the absolute value of the solenoid and shaping coil fields. By using a proton resonance device to control the field, the absolute value can be held

-75to about one part in 105 which is a factor of 100 better than the value for the present experiment. The second is the determination of the time average magnetic field seen by the trapped electrons. It was found during the course of the experiment that the field inhomogeneities could be reduced to about 0.1 percent before the beam could not be trapped. By reducing this, another factor of three, the inhomogeneities would be small enough so that a measurement of the g-factor anomaly to ten times better accuracy could be made without averaging over the magnetic field. In the present experiment, no special precautions were taken to eliminate the stray electric fields. By plating the concentric cylinders and baking them in a vacuum, the stray fields can probably be reduced considerably. Also, methods are available to actually measure the electric fields. The accuracy of the determination of the difference frequency can be extended by increasing the trapping time and reducing the random and statistical errors. In this way, accuracies of one part in 10' or better should be possible.

APPENDIX I TABLES OF EXPERIMENTAL RESULTS Part A: Gold - Aluminum Scattering For all runs: Energy 100 Kilovolts Magnetic Field 0.8313 volts Counts in C2 50,000 Normalized Normalized Normalized Normalized Normalized Trapping C1 Counting Trapping C Counting Trapping C1 Counting Trapping C1 Counting Trapping C, Counting Time Ratio Time 1 SRatio Time Ratio Time Ratio Time Ratio 30.0 39,081 1.0035 90.0 40,934 1.0046 150.0 4o,6o4 1.0067 210.0 4o,530 0.9997 270.0 39,285 0.9782 30.5 38,365 0.9851 90.5 4o,854 1.0026 150.5 40,753 1.0104 210.5 4o,417 0.9970 270.5 4o,422 1.0065 31.0 39,452 1.0130 91.0 4o,324 0.9896 151.0 40,653 1.0079 211.0 40,515 0.9994 271.0 40,575 1.0103 31.5 38,422 0.9866 91.5 40,893 1.0036 151.5 40,007 0.9919 211.5 4o,563 1.0006 271.5 39,763 0.9901 32.0 38,787 0.9960 92.0 41,034 1.0070 152.0 39,969 0.9909 212.0 4o,761 1.0054 272.0 4o,418 1.0064 32.5 38,547 0.9898 92.5 4o,836 1.0022 152.5 40,256 0.9981 212.5 4o,196 0.9915 272.5 4o,262 1.0025 33.0 38,638 0.9921 93.0 40,535 0.9948 153.0 40,758 1.0105 213.0 4o,742 1.0050 273.0 4o,274 1.0028 33.5 38,436 0.9870 93.5 4o,705 0.9990 153.5 4o,241 0.9977 213.5 4o,781 1.0059 273.5 40,365 1.0051 34.0 39,667 1.0186 94.o 4o,519 0.9944 154.0 39,859 0.9882 214.o 4o,202 0.9917 274.0 4o,309 1.0037 34.5 39,607 1.0170 94.5 4o,321 0.9895 154.5 40,085 0.9938 214.5 40,967 1.0105 274.5 4o,381 1.0055 35.0 39,550 1.0156 95.0 4o,951 1.0050 155.0 40,313 0.9995 215.0 4o,715 1.0043 275.0 39,999 0.9960 35.5 39,781 1.0215 95.5 4i,o53 1.0075 155.5 40,510 1.0044 215.5 4o,094 0.9890 275.5 39,871 0.9928 6o.o 39,771 1.0007 120.0 39,953 0.9977 180.0 4o,428 0.9937 240.0 4o,147 0.9945 300.0 39,502 1.0002 6o.5 39,692 0.9987 120.5 39,735 0.9923 18o.5 4o,861 1.0043 24o.5 40,747 1.0094 300.5 39,489 0.9999 61.o 39,543 0.9949 121.0 39,962 0.9979 l81.o 40,890 1.0050 241.o 4o,864 1.0123 301.0 39,512 1.0005 61.5 39,728 0.9996 121.5 4o,1o4 1.0015 181.5 40,796 1.0027 241.5 4o,072 0.9926 301.5 39,417 0.9981 62.0 39,788 1.0011 122.0 39,867 0.9956 182.0 41,o45 1.0088 242.0 4o,689 1.0079 302.0 39,370 0.9969 62.5 39,662 0.9979 122.5 4o,182 1.0034 182.5 40,883 1.oo48 242.5 40,507 1.0034 302.5 39,531 1.0010 63.0 39,745 1.0000 123.0 4o,039 0.9998 183.0 40,720 1.0008 243.0 4o,026 0.9915 303.0 39,560 1.0017 63.5 39,781 1.0009 123.5 4o,201 1.0039 183.5 4o,251 0.9893 243.5 39,896 0.9883 303.5 39,354 0.9965 64.o 39,673 0.9982 124.0 4o,075 1.0007 184.o 40,783 1.0024 244.0 4o,5o4 1.0033 304.o 39,546 1.0014 64.5 39,904 1.oo4o 124.5 39,782 0.9934 184.5 40,337 0.9914 244.5 4o,411 1.0010 304.5 39,537 1.0011 65.0 39,784 1.0010 125.0 4o,278 1.0058 185.0 4o,651 0.9991 245.0 4o,462 1.0023 305.0 39,520 1.0007 65.5 39,852 1.0027 125.5 40,362 1.0079 185.5 4o,590 0.9976 245.5 4o,lo4 0.9934 305.5 39,543 1.0013

-77Part B: Gold - Gold Scattering Energy - 100 Kilovolts Magnetic Field - 0.8313 Volts Normalized Normalized Normalized Trapping C1 Counting Trapping C1 Counting Trapping C1 Counting Time Ratio Time Ratio Time Ratio Run I Run III Run V C2 = 20,000 C2 = 20,000 C2 = 20,000 30.0 17,232 1.0311 90.0 17,543 1.0071 150.0 17,678 1.0091 30.5 16,828 1.0069 90.5 17,289 0.9925 150.5 17,949 1.0245 31.0 16,658 0.9968 91.0 16,746 0.9614 151.0 16,935 0.9667 31.5 17,113 1.0240 91.5 17,431 1.0007 151.5 17,465 0.9969 32.0 16,933 1.0132 92.0 17,629 1.0121 152.0 17,427 0.9947 32.5 16,780 1.0041 92.5 17,719 1.0172 152.5 17,904 1.0220 33.0 16,143 0.9660 93.0 17,382 0.9978 153.0 17,673 1.0088 33.5 16,113 0.9642 93.5 17,138 0.9839 153.5 17,000 0.9704 34.0 16,499 0.9873 94.0 17,039 0.9782 154.0 17,012 0.9711 34.5 17,107 1.0236 94.5 17,587 1.0096 154.5 17,693 1.0099 35.0 16,688 0.9986 95.0 18,087 1.0383 155.0 17,797 1.0159 35.5 16,447 0.9841 95.5 17,442 1.0013 155.5 17,695 1.0100 Run II Run IV Run VI 6o.o 17,095 0.9919 120.0 17,278 0.9875 180.0 17,411 0.9863 6o.5 17,340 1.0061 120.5 17,512 1.0009 180.5 17,640 0.9992 61.0 17,634 1.0232 121.0 17,686 1.0108 181.0 18,077 1.0240 61.5 17,250 1.0009 121.5 17,521 1.0014 181.5 18,008 1.0201 62.0 16,398 0.9514 122.0 17,443 0.9969 182.0 17,676 1.0013 62.5 17,394 1.0092 122.5 17,163 0.9809 182.5 17,289 0.9793 63.0 17,135 0.9942 123.0 17,643 1.0083 183.0 17,608 0.9974 63.5 17,483 1.0144 123.5 17,767 1.0154 183.5 18,016 1.0205 64.o 17,513 1.0161 124.0 17,954 1.0261 184.0 18,081 1.0242 64.5 17,096 0.9919 124.5 17,446 0.9971 184.5 17,551 0.9942 65.0 16,996 0.9861 125.0 17,269 0.9870 185.0 17,044 0.9655 65.5 17,486 1.0146 125.5 17,281 0.9877 185.5 17,443 0.9881

Part B (Continued) Energy - 100 Kilovolts Magnetic Field - 0.8313 Volts Normalized Normalized Normalized Normalized Trapping C1 Counting Trapping C1 Counting Trapping C1 Counting Trapping C1 Counting Time Ratio Time Ratio Time Ratio Time Ratio Run VII Run IX Run XI Run XIII C2 = 20,000 C2 = 20,000 C2 = 20,000 C2 = 30,000 210.0 17,782 1.0188 270.0 18,538 1.0193 300.0 17,324 0.9979 300.0 25,888 1.0090 210.5 17,522 1.0039 270.5 18,277 1.0049 300.5 16,973 0.9777 300.5 25,206 0.9824 211.0 17,213 0.9862 271.0 17,850 0.9814 301.0 17,358 0.9999 301.0 25,360 0.9884 211.5 16,958 0.9716 271.5 17,719 0.9742 301.5 17,372 1.0007 301.5 25,863 1.0080 212.0 17,135 0.9817 272.0 17,681 0.9722 302.0 17,889 1.0305 302.0 26,350 1.0270 212.5 17,514 1.0035 272.5 18,146 0.9977 302.5 17,542 1.0105 302.5 25,938 1.0110 213.0 18,072 1.0354 273.0 18,822 1.0349 303.0 17,281 0.9955 303.0 25,123 0.9792 213.5 17,496 1.0024 273.5 l8,4o4 1.0119 303.5 16,988 0.9786 303.5 25,114 0.9788 214.o 16,945 0.9709 274.0 18,261 l.oo4o 304.0 17,388 l.0oo6 304.0 25,510 0.9943 214.5 17,447 0.9996 274.5 17,780o 0.9776 304.5 17,936 1.0332 304.5 26,179 1.0204 215.0 17,711 1.0147 275.0 18,102 0.9953 305.0 17,471 1.0064 305.0 25,904 1.0096 215.5 17,650 1.0112 275.5 18,669 1.0265 305.5 16,794 0.9674 305.5 25,446 0.9918 Run VIII Run X Run XII 240.0 17,090 0.9627 300.0 17,563 0.9799 300.0 17,381 0.9994 240.5 17,472 0.9842 300.5 17,462 0.9743 300.5 17,074 0.9817 241.0 18,282 1.0299 301.0 18,255 1.0185 301.0 17,369 0.9987 241.5 18,678 1.0522 301.5 18,030 1.0059 301.5 17,481 1.0051 242.0 17,684 0.9962 302.0 18,574 1.0313 302.0 17,992 1.0345 242.5 17,390 0.9796 302.5 17,941 1.0010 302.5 17,516 1.0071 243.0 17,293 0.9742 303.0 l8,o66 1.0080 303.0 17,275 0.9933 243.5 18,190 1.0247 303.5 17,652 0.9849 303.5 17,022 0.9737 244.0 18,153 1.0226 304.0 18,251 1.0183 304.0 17,448 1.0032 244.5 17,883 1.0074 304.5 18,622 1.0390 304.5 17,906 1.0295 245.0 17,530 0.9875 305.0 17,725 0.9886 305.0 17,375 0.9990 245.5 17,375 0.9788 305.5 16,939 0.9451 305.5 16,864 0.9696

Part B (Continued) Normalized Normalized Normalized Normalized Trapping Cl Counting Trapping C, Counting Trapping C1 Counting Trapping C1 Counting Time Ratio Time Ratio Time Ratio Time Ratio Run XIV Run XVI Run XVIII Run XX C= 40,0002 2 0,000C0 = 40,000 C2 40,000 Energy - 98 Kilovolts Energy - 70 Kilovolts Energy - 70 Kilovolts Energy - 70 Kilovolts Magnetic Field - 0.8212 Volts Magnetic Field - 0.7000 Volts Magnetic Field - 0.7000 Volts Magnetic Field - 0.7000 Volts 300.0 35,355 1.0102 200.0 34,300 1.0062 265.0 39,238 1.0124 300.0 38,405 0.9892 300.5 35,842 1.0241 200.5 34,193 1.0031 265.5 39,259 1.0130 300.5 37,931 0.9770 301.0 34,838 0.9954 201.0 33,975 0.9967 266.0 38,696 0.9984 301.0 38,422 0.9896 301.5 34,101 0.9743 201.5 33,606 0.9859 266.5 38,o6o 0.9821 301.5 38,793 0.9992 302.0 34,134 0.9753 202.0 34,123 1.0010 267.0 38,371 0.9901 302.0 39,285 1.0118 302.5 35,270 1.0077 202.5 34,269 1.0053 267.5 38,718 0.9990 302.5 39,063 1.0o61 303.0 35,532 1.0152 203.0 34,747 1.0193 268.0 39,288 1.0137 303.0 38,964 1.0036 303.5 35,016 1.0005 203.5 34,504 1.0122 268.5 39,033 1.0072 303.5 38,581 0.9937 304.0 34,639 0.9897 204.0 33,959 0.9962 269.0 39,079 1.0083 304.0 38,548 0.9929 304.5 34,384 0.9824 204.5 33,574 0.9849 269.5 38,395 0.9907 304.5 39,226 1.0103 305.0 34,991 0.9998 205.0 33,632 o.9866 270.0 38,321 0.9888 305.0 39,338 1.0132305.5 35,885 1.0253 205.5 34,179 1.0027 270.5 38,609 0.9962 305.5 39,349 1.0135 Run XV Energy - 81 Kilovolts Run XVII Run XIX Magnetic Field. - 0.7500 Volts________________________ 300.0 37,299 1.0190 250.0 38,893 1.0017 285.0 34,138 0.9937 300.5 35,873 0.9800 250.5 38,361 0.9880 285.5 33,874 0.9860 301.0 35,662 0.9743 251.0 38,356 0.9879 286.0 34,762 1.0119 301.5 35,750 0.9767 251.5 38,804 0.9994 286.5 34,332 0.9994 302.0 36,690 1.0024 252.0 39,463 1.o164 287.0 33,902 0.9868 302.5 37,005 1.0110 252.5 39,397 1.0147 287.5 33,622 0.9787 303.0 37,209 1.0165 253.0 38,838 1.0003 288.0 34,340 0.9996 303.5 36,528 0.9979 253.5 38,475 0.9909 288.5 34,245 0.9968 304.0 36,151 0.9876 254.0 38,254 0.9852 289.0 35,335 1.0286 304.5 36,709 1.0029 254.5 38,714 0.9971 289.5 34,808 1.0132 305.0 36,978 1.0102 255.0 38,965 1.0036 290.0 34,689 1.0097 305.5 37,388 1.0214 255.5 39,403 1.0148 290.5 34,190 0.9952

-80Part B (Continued) Normalized Normalized Normalized Trapping C1 Counting Trapping C1 Counting Trapping C1 Counting Time Ratio Time Ratio Time Ratio Run XXI Run XXIII Run XXV C2 = 40,000ooo C2 = 40,000 C2 = 40,00 Energy - 60 Kilovolts Energy - 60 Kilovolts Energy - 50 Kilovolts Magnetic Field - 0.6500 Volts Magnetic Field - 0.6500 Volts Magnetic Field - 0.5819 Volts 250.0 37,962 0.9943 300.0 38,585 0.9865 300.0 36,543 J.0141 250.5 38,118 0.9984 300.5 38,818 0.9925 300.5 36,256 1.0061 251.0 37,699 0.9875 301.0 39,683 1.0146 301.0 35,736 0.9917 251.5 37,805 0.9902 301.5 39,527 1.0106 301.5 35,406 0.9825 252.0 38,492 1.0082 302.0 39,553 1.0113 302.0 36,367 1.0092 252.5 38,667 1.0128 302.5 39,244 1.0034 302.5 36,041 1.0001 253.0 38,112 0.9983 303.0 38,926 0.9953 303.0 36,038 1.0000 253.5 37,745 0.9887 303.5 38,713 0.9898 303.5 36,672 1.0176 254.0 37,891 0.9925 304.0 38,701 0.9895 304.0 36,056 1.0005 254.5 38,592 1.0108 304.5 39,044 0.9983 304.5 35,821 0.9940 255.0 38,684 1.0133 305.0 39,297 1.0048 305.0 35,512 0.9855 255.5 38,368 1.0050 305.5 39,242 1.0033 305.5 35,986 0.9986 Run XXIV Run XXII Energy - 58 Kilovolts Magnetic Field - 0.6286 Volts 300.0 34,708 0.9915 300.0 40,563 0.9946 300.5 34,786 0.9937 300.5 40,118 0.9836 301.0 34,713 0.9917 301.0 41,283 1.0122 301.5 35,542 1.0153 301.5 41,074 1.0071 302.0 34,977 0.9992 302.0 41,102 1.0078 302.5 35,227 1.0063 302.5 40,802 1.0004 303.0 34,830 0.9950 303.0 40,353 0.9894 303.5 34,286 0.9795 303.5 40,201 0.9857 304.0 35,159 1.0044 304.0 39,979 0.9802 304.5 35,197 1.0055 304.5 41,379 1.0147 305.0 35,181 1.0050 305.0 40,340 0.9801 305.5 35,451 1.0127 305.5 40,425 0.9912

-81Part C: Continuous Run from 30 to 130 Microseconds Energy - 95 Kilovolts Magnetic Field - 0.8000 Volts Trapping C1 C2 Trapping C1 C2 Trapping C1 C2 Trapping C1 C2 Trapping C1 C2 Tie Time Time 2 Time Time Time 2 30.0 28,100 50,000 50.0 6,154 10,000 70.0 6,002 lo,000 90.0 5,828 lo,000 110.0 5,842 10,000 30.5 28,559 50,000 50.5 6,083 10,000 70.5 6,254 10,000 90.5 5,873 10,000 110.5 5,734 10,000 31.0 28,400 50,000 51.0 5,823 10,000 71.0 6,342 10,000 91.0 5,579 10,000 111.0 5,572 10,000 31.5 27,753 50,000 51.5 5,779 10,000 71.5 6,131 10,000 91.5 5,644 10,000 111.5 5,810 10,000 32.0 27,822 50,000 52.0 6,052 10,000 72.0 6,023 10,000 92.0 5,590 10,000 112.0 5,912 10,000 32.5 28,210 50,000 52.5 6,199 10,000 72.5 5,936 10,000 92.5 5,570 10,000 112.5 6,028 10,000 33.0 29,130 50,000 53.0 6,o81 10,000 73.0 5.959 10,000 93.0 5,721 10,000 113.0 5,822 10,000 33.5 28,914 50,000 53.5 5,931 10,000 73.5 6,059 10,000 93.5 5,721 10,000 113.5 5,649 10,000 34.0 28,217 50,000 54.o 6,178 10,000 74.0 6,102 10,000 94.0 5,492 10,000 114.0 5,750 10,000 34.5 5,696 10,000 54.5 6,140 o10,000 74.5 6,169 10,000 94.5 5,381 10,000 114.5 6,148 10,000 35.0 5,626 10,000 55.0 6,o84 10,000 75.0 6,091 10,000 95.0 5,898 10,000 115.0 5,941 10,000 35.5 5,800 10,000 55.5 6,099 10,000 75.5 5,735 10,000 95.5 6,039 10,000 115.5 6,003 10,000 36.0 5,887 10,000 56.0 6,163 10,000 76.0 5,983 10,000 96.o 6,097 10,000 116.o 5,959 10,000 36.5 5,959 10,000 56.5 5,909 10,000 76.5 6,093 10,000 96.5 5,926 10,000 116.5 5,824 10,000 37.0 5,771 10,000 57.0 6,141 10,000 77.0 5,981 10,000 97.0 5,666 10,000 117.0 5,820 10,000 37.5 5,683 10,000 57.5 6,007 10,000 77.5 6,002 10,000 97.5 5,876 10,000 117.5 6,117 10,000 38.0 5,781 10,000 58.0 6,383 10,000 78.0 5,878 10,000 98.0 5,939 10,000 118.0 5,952 10,000 38.5 6,033 10,000 58.5 6,142 10,000 78.5 5,845 10,000 98.5 5,771 10,000 118.5 5,887 10,000 39.0 6,014 10,000 59.0 5,880 10,000 79.0 5,997 10,000 99.0 5,920 10,000 119.0 5,732 10,000 39.5 5,885 10,000 59.5 6,052 10,000 79.5 5,815 10,000 99.5 5,674 10,000 119.5 5,996 10,000 40.o 5,825 10,000 60.o 6,126 10,00 80.0 5,938 10,000 100.0 5,480 10,000 120.0 6,055 10,000 40.5 5,829 10,000 60.5 6,274 10,000 80.5 5,703 10,000 100.5 5,652 10,000 120.5 6,o6o0 10,000 41.0 5,900 10,000 61.0 6,282 10,000 81.0 5,644 10,000 101.0 5,852 10,000 121.0 5,947 10,000 41.5 5,863 10,000 61.5 6,219 10,000 81.5 5,890 10,o00 101.5 5,837 10,000 121.5 5,929 10,000 42.0 5,976 lo,000 62.0 6,187 lo,000 82.0 5,799 10,000 102.0 5,701 10,000 122.0 6,032 10,000 42.5 5,835 10,000 62.5 6,181 10,000 82.5 5,796 10,000 102.5 5,511 10,000 122.5 6,078 10,000 43.0 5,810 10,000 63.0 6,148 10,000 83.0 5,760 10,o00 103.0 5,672 10,000 123.0 6,311 10,000 43.5 5,863 10,000 63.5 6,217 10,000 83.5 5,503 10,000 103.5 5,672 10,000 123.5 6,221 10,000 44.0 6,o01o 10,000 64.o 6,375 10,000 84.0 5,503 10,000 104.0 5,759 10,000 124.0 5,958 10,000 44.5 6,238 10,o00 64.5 6,o01o 10,00 84.5 5,563 10,000 104.5 5,545 10,000 124.5 5,962 10,000 45.0 6,o6o l,000 65.0o 5,961 10,000 85.0 5,731 10,000 105.0 5,660 lo,o00 125.0 6,000ooo 10,000 45.5 5,759 10,000 65.5 6,221 10,000 85.5 5,552 10,000 105.5 5,345 10,000 125.5 6,300 10,000 46.o 5,8o6 1o,ooo 66.o 6,328 10,000 86.o 5,373 10,000 106.0 5,580 10,000 126.0 31,161 50,000 46.5 5,966 10o,ooo 66.5 6,184 10,000 86.5 5,511 10,000 106.5 5,728 10,000 126.5 30,492 50,000 47.0 6,232 10,000 67.0 6,o84 10,000 87.0 5,543 10,000 107.0 5,799 10,000 127.0 30,424 50,000 47.5 5,983 10,000 67.5 5,995 10,000 87.5 5,528 10,000 107.5 5,623 10,000 127.5 29,922 50,000 48.0 5,852 10,000 68.o 6,029 10,000 88.0 5,785 l0,000 108.0 5,593 10,000 128.0 30,478 50,000 48.5 5,916 10,000 68.5 6,212 10,000 88.5 5,496 10,000 108.5 5,677 10,000 128.5 31,023 50,000 49.0 5,942 10,000 69.0 6,131 10,000 89.0 5,380 10,000 109.0 5,773 10,000 129.0 31,355 50,000 49.5 6,018 10,000 69.5 6,117 10,000 89.5 5,345 10,000 109.5 5,915 10,000 129.5 30,734 50,000 130.0 30,114 50,000

-82APPENDIX II TABLES OF CALCULATED VALUES Part A: Calculated Values of Magnetic Field Due to the Solenoid and Field Shaping Coil Solenoid Dimensions: 5.792 Meters Long 0.338 Meters Diameter 1492 Turns per Meter Field Shaping Coil Dimensions: 0.154 Meters Long 0.464 Meters Diameter 593 Turns Total Fields Calculated for a Beam Radius of 0.095 Meters Z Solenoid Shaping Coil From Axial Field Radial Field Axial Field Radial Field Center 10-4 W/M2 10-6 W/M2 10-4 W/M2 10-6 W/M2 0.000 117.3900 0.00000 0.32000 0.00000 0.025 --- -- -- 1.70324 0.050 117.3893 0.03061 0.28387 3.40638 0.075 --- -- --- 4.54194 0.100 117.3886 O.06124 0.22967 5.26450 0.125 --- -- -- 5.41927 0.150 117.3880 0.09185 0.16516 5.05800 0.200 117.3874 0.12397 0.12129 3.87091 0.250 117.3855 0.15687 0.08774 2.73545 0.300 117.3836 0.18979 O.06193 1.96127 0.350 117.3808 — 0.0 4387 1.39359 0.400 117.3780 0.26041 0.03200 1.03221 0.450 117.3746 - 0.02479 0.72256 0.500 117.3711 0.33784 0.01909 0.48510 0.550 117.3665 - 0.01549 0.34069 o. 600 117.3617 0.45584 0.01238 0.20640 o.650 117.3570 -- 0.00878 0.15488 0.700 117.3523 0.52324 0.00722 0.10325. 800 117.3385 0.63784 0.00516 0.05163 0.900 117.3228 0.77315 _ --- 1.000 117.3027 0.93544 1.100 117.2789 1.22691 --- - 1.200 117.2507 1.37855 1.300 117.2174 1.68771.. 1.400 117.1741 2.08455

-83Part B: Calculated Values of Effective Potential Well Electron Electron Z Volts Z Volts 1.45 141.24 0.70 3.63 1.40 124.16 0.65 1.49 1.35 108.65 o.6o 0.41 1.30 94.63 0.55 0.00 1.25 81.84 0.50 1.90 1.20 70.13 0.45 5.78 1.15 59.57 0.4o 12.13 1.10 50.08 0.35 22.19 1.05 41.50 0.30 37.04 1.00 33.83 0.25 59.32 0.95 26.98 0.20 90.09 0.90 20.87 0.15 131.18 0.85 15.43 0.10 174.74 0.80 10.73 0.05 204.52 0.75 6.77 0.00 217.93

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