UNCLASSIFIED ORNL 744 * CY 9 t1 ** * * 4 . + 1 :- . ' . Dior.pl. 744 A REVIEW OF THE MAGNETIC FIELD SHAPING METHODS USED IN THE DEVELOPMENT OF THE 255° OAK RIDGE ISOTOPE SEPARATOR A Thesis Presented to the Graduate Council of The University of Tennessee In Partial Fuifillment of the Requirements for the Degree vaster of Science : 2 by William Kelly Dagenhart 1 .- December 1964 1.- 8 . A W S i! ACKNOWLEDGMENTS The author wishes to express his appreciation to the Oak Ridge National Laboratory* for making available the information concerning the Oak Ridge National Laboratory 255° isotope separator, which provided the nucleus for the development of this thesis. The separator development was the joint responsibility of Mr. T. W. Whitehead, Jr., and myself with the source, receiver, and liner designs by Mr. G. D. Alton under the supervision of Mr. W. A. Bell and Mr. L. 0. Love of the Electromagnetic Isotope Separations Department, Isotopes Division, Oak Ridge National Laboratory. Their continued support of this thesis program is deeply appreciated. The author also wishes to acknowledge the invaluable assistance of Dr. H. P. Carter and Mrs. Joan Rayburn of the Mathematics Division for preparing the orbit and overrelaxation computer programs for this separator, and for the persistent and constructive help of Miss Sandra Stambaugh in the typing of the manuscript. The patience, encouragement, and editorial assistance of my wife, Sally, has greatly aided in the formulation of this thesis. *Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. ii TABLE OF CONTENTS CHAPTER PAGE I. 1 INTRODUCTION ............ ......... History of the Oak Ridge National Laboratory isotope separations program ................... Presentation of the thesis problem ............ 4 Synopsis of Chapters II through VI ............ SOME THEORETICAL CONSIDERATIONS IN THE CHOICE OF A II. MAGNETIC FIELD SHAPE ................... 7 Some practical restrictions on the choice of a magnetic field shape .................. 7 Development of the vector potential and magnetic field expansions .................... Orbit differential equations ............... Zeroth order solution of the differential equations ... First order solution ..... Properties of the first order solutions .... Second order solutions ............. Third, fourth, and fifth order considerations .. III. THEORETICAL CALCULATION OF THE SHIM CONTOUR ...... Size limitations placed on the shims ........... First approximation of the shim contour neglecting the end boundary conditions .............. 31 iii Tv t SY CHAPTER Evaluation of the shim contour using the > - magnetostatic equipotential function .......... 34 F First order taper of the shims .............. 35 The use of relaxation methods for determining the shim contour .................... 40 Calculation of the shim contour ........ IV. MAGNETIC FIELD MEASUREMENTS ..... ........... - Selection of the field measuring technique . Transducer theory and circuitry ............. Probe box construction and component mounting Cabling system ..................... . * * Measuring and data logging system .* .... Input signals to the digital data system - ... 12 Digital data system ......... Probe positioning .......... " Syst em calibration .. M Call raion . . . . . . . . . . . . . . . . . . . Tiin Field measurement data handling procedures Data analyses ...................... Digital voltmeter stability ....... System stability check ............ Measurement data validation ......... . . -4 . ST * Sensitivity of the micplane focus to measuring errors .. SHIM CORRECTION PROGRAM ... V. 1 . .. 12 AT Program goals... . + - " 2.1" .." CHAPTER 99 I Shim correction method .................. Analysis of the results obtained from the . . 7 . second set of shims .................. 103 VI. EXPERIMENTAL TESTING OF THE SEPARATOR AND CONCLUSIONS .... Separator testing ving . . . . . . . . . . . . . . . . . Conclusions ....................... BIBLIOGRAPHY • • • 126 . . " t, * : APPENDIXES . A. DERIVATION OF THE VECTOR POTENTIAL FORM .......... 129 .. " B. DERIVATION OF TIE THREE DIMENSIONAL MAGNETIC FIELD AND Sim VECTOR POTENTIAL FIELD EXPANSIONS ......... ..... 133 Three dimensional magnetic field expansion ........ 133 - Three dimensional vector potential expansion ....... 136 C. MEASURING EQUIPMENT LIST AND PERTINENT SPECIFICATIONS .... - CLT i D. 255° ISCTOPE SEPARATOR CONSTRUCTION AND OPERATING PARAMETERS ..................... . . 146 * Magnet Construction .............. 146 4 1 - . . Vacuum System ............... . G Electrical Supplies ............. Ion Source ........................ 147 147 148 SU : IT Receiver . . . . . . . . . . . . . . . . . . . . . . . 3 . ; * ry 7 A + -. . 1 In *** " I *** . 1. .1 7. . * ; '- LIST OF TABLES TABLE : . * I. Coordinates of the Contours for the 255° Separator Linear, "Fifth Order", end First Relaxation Calculated Shims Which Were Used .................. 39 1 - 2 II. Sample Output Data from the Least Squares Calibration 78 . 2 L - III. Program ......................... Calibration Repeatability Analysis ............... Unprocessed Magnetic Field Measurement Data ......... IV. 83 Processed Magnetic Fieid Measurement Data ......... 255º Isotope Separator Magnet Shim Contour Data ..... VI. * . . . . . . . . S S . . . . . . YA . . . . . . . . . V 2 LY . - in... 1 i N. " > ...... . . W M , . . . FC i 9. 1 . .rt C !'. * ' N t I vi i is L - LIST OF FIGURES - FIGURE PAGE 35 $ 1. Installation of the Auxiliary Yoke Steel to Subdivide , the Beta Calutron Track in Building 9204-3 ........ 3 2. Schematic Outline of a Beta Calutron Isotope Separator ... 8 3. Polar Coordinate System and a 0-plane Schematic Cross-section of the 255° Separator Shims ................ 10 255° Separator Computer Calculated Optimum Three-dimensional Focal Pattern as Projected into the 0-plane, 254.5°, Using the Fifth-order Theoretical Field and a Plane Curved . . Y . Object .......................... 27 3 . 5. Schematic Outline of the 255º Isotope Separator Beam 2 : 29 and Beam Focusing System ................. A 255° Separator Shim Mounted on the Rail Carrier I - 6. . . . . t " ? Installation Rig ...................... E 11 : 7. Shims Installed in the 255° Isotope Separator Vacuum Tank .. 32 8. 255° Separator Shim Schematic Cross-section .......... . A 9. ta Relaxation Calculated Midplane Magnetic Field Shape Deviations of 255° Separator, Linear, Fifth-order, and Relaxation Calculated Contoured Shims ........... Calculated Midplane Focus Which Would Have Resulted if 2 . . 38 10. $ . TU ... 1 . Linear (Conical) Shims Had Been Used in the + A -- & . 255° Isotope Separator .................. 41 . 1 .4. vii . .- **************** **...********* ... SRL mer in viii FIGURE PAGE li. Calculated Midplane Focus Which Would Have Resulted if "Fifth-order" Shims, Contoured According to Equation (3-8), (Page 35), Had Been Used in the 2550 Isotope Separator ..................... 42 12. Midplane Magnetic Field Shape Deviations from the Theoretical for Shim Set Number One in Tank XBX-1 ..... 44 . . 13. Schematic of the 255° Separator Magnetostatic Boundaries .. 45 . 2. 14. A 255° Separator Shim Mounted on a Large Vertical n., '. ,- Lathe during Machining ................... . 15. Contour of the First 255° Separator Shim .......... 16. Hall Probe Schematic and Coordinate System ......... 17. Scanning Hall Probe Assembly with the lid Removed and Mounted on the Positioning Wheel ............ 18. Hall Probe Transducer Circuitry ............ 19. 20. 21. 22. Hall Probe Temperature Regulation ............. Magnetic Field Measuring Digital Data System ........ Magnetic Field Measuring Digital Data System ........ Schematic Layout of the Analog to Digital Conversion and Logging Data System Used in the 255° Separator Magnetic 67 Field Measurements . . . . . . . . . . . . . . . . . . . . 23. Analog Output from a Magnetic Field Measuring Azimuthal Scan ...................... Hall Probe Positioning Mechanism .............. 71 73 24. cioscorior, that the ini 41. ix FIGURE PAGE 25. Measured Deviation of the Midplane Magnetic Field Shape of the First Set of Shims as a Function of Radius for Tank XBX-1 at 0 = 128°. Also Plotted Are the Effective . i . Perturbations in the Radial Field Shape Deviations as Caused by Azinuthal Field Variations, and Thece no. Perturbations Must Be Added to the Deviation at ^ = 128° . .... to Get the Total Effective Radial Field Shape Deviation .. 85 26. Measured Percentage Deviation of the Midplane Magnetic .. . Field Shape of the First Set of Shims as a Function of . Es 86 -2 Azimuth for Tank 545 ................... Midplane Focus Which Was Calculated from the Measured 27. 271 Magnetic Field of Shim Set One in Tank XBX-1 for > 28. B. = 8,533 gauss ...................... XBX Yoke Magnetization Curve ................ Digital Data System Stability Check ............. 29. 90 30. Midplane Focus as Calculated from the Measured Magnetic 34 . ? 92 . Field of Shim Set, Number Two in Tank XBX-1 for the Virtual Object at r = 22.75 Inches ................ Midplane Focus as Calculated from the Measured Magnetic Field of Shim Set Number Two in Tank XBX-1 for the Virtual 31. 4 47 Object at r = 17.25 Inches ................ 93 32. Midplane Focus as Calculated from the Measured Magnetic Field ' .. of Shim Set Number Two in Tank XBX-l for the Virtual Object at r = 20 Inches and a = - 10° and for a Mass Corresponding .. . P2 to o m/m of + 0.0177 ................... 94 rice 7 2 : . .. FIGURE PAGE 33. Midplane Focus as Calculated from the measured Magnetic Field of Shim Set Number Two in Tank XBX-1 for the Virtual Object at r = 20 Inches and = 0° and for a Mass Corresponding to a 0 m/m of + 0.0177 ........... 96 34. Midplane Focus as Calculated from the Measured Magnetic Field of Shim Set Number One in Tank 653 with the Data Being Rounded off One Digit ................... 97 The Desired and A lished Facto nction for Shim Set Number Two in Tank XBX-1 I . . . . . . . . . . . . . 1 101 1 36. Midplane Magnetic Field Shape Deviations for the Original and Corrected Shim Sets in Tank XBX-1 ........... 104 37. Midplane Focus as Calculated from the measured Magnetic Field with a B. = 8,077 Gauss for the Corrected Shims in Tank XBX-). ....................... 105 38. Measured Midplane Magnetic Field Shape Deviations for the Correa 106 Corrected Shims at B = 3,620 Gauss in Tank XBX-l ..... Midplane ocus as Calculated from the Measured Magnetic 39. Field at B = 3,620 Gauss for the corrected Shims in Tank XBX-1 ....................... 107 40. Average Enhancement Factors for the Best Cadmium Isotope Separations Series in the Alpha Calutron and 255° Separators ...................... 108 41. Average Enhancement Factors for the Best Tungsten Isotope Separations Series in the Beta Calutron and 255° Separators ...................... 109 FIGURE PAGE 42. Isotope Contamination Factors for the 108ca Isotope Using Samples Collected with the corrected and Uncorrected Shims in Tanks XBX-1 and XBX-2 Respectively ...................... 111 43. Measured Midplane Magnetic Field Shape Deviations for Shim Set Number One in the Magnetic Gap of Tank XBX-2. Which Is 0.070 Irich Too Narrow .............. 112 44. Midplane Focus as Calculated from the Measured Magnetic Field of Shim Set One in the Narrow Gap of Tank XBX-2 ... 113 45. First Experimental Ion Source and Receiver for the 255° Separator Mounted on the Vacuum Tank Front ........ 116 46. 255º Isotope Separator Midplane Ion Beam Outline and Several $-plane Beam Cross-sections ............ 117 47. 255° Separator Straight Focal Pattern Receiver ....... 118 48. Isotopic Enrichments Which Compare the Best Single Sample Results of the Alpha, Beta, and 255° Separators ...... 120 49. Mercury Ion Beam Scan Using a 0.046 Inch by 4 Inch Receiver Beam Defining Slot in the 255° Separator ......... 121 255° Separator Receiver Beam Defining Faceplate (Using a 50. Straight Object) after a Long Experimental Collection ... 122 Isotopic Contamination Factors for the Best 108ca 51. Isotope Sample from Each of Three Best Different ORNL Isotope Separations ................... 123 CHAPTER I INTRODUCTION History of the Oak Ridge National Laboratory 1sotope separations program. Large scale isotope separations began in the United States in 1943 with the inception of the wartime program for obtaining uranium enriched in the fissionable 1sotope 2350. This program had to be pursued with much haste, due to its late start during World War II, and in this haste many different methods used for obtaining isotopically enriched samples were investigated. Of these, only the electromagnetic and gaseous diffusion processes were developed and put into large scale production. The gaseous diffusion process proved to be the most efficient one in the large scale separation of the isotopes of uranium, and almost all isotopically enriched uranium has been obtained from this process since the war. The large electromagnetic process equipment at ühe Y-12 plant in Oak Ridge, Tennessee, was put on a stand-by basis at the end of the war, and only four of the isotope separators were in use up until 1960. The increasing demnand for the enriched isotopes of almost all the elements, which are used for nuclear cross-section determinations, medical research and treatment, cyclotron and reactor transmutations, and other nuclear research studies, could most efficiently be satisfied by a reactivation of a large number of the electromagnetic isotope separators. One beta building which originally contained seventy-two of the 24-inch radius beta calutron isotope separators has been reactivated and now has .. thirty-two of these separators in round-the-clock use in addition to the two 48-inch radius alpha and two beta separators in the original pilot plant, which have been in use since the war. The beta separators, whose first working model was developed and tested at the Berkeley Laboratories of the University of California by Dr. E. 0. Lawrence, proved to be the most flexible and efficient tool for obtaining gram to kilogram quantities of enriched isotopes of most of the elements. The large beta building had two separate magnetic yoke systems, each originally containing thirty-six calutrons. Each of these yokes 18 now further divided by addig magnetic shorting bars to effectively divide the original yoke into three sections with eight calutrons each and one section with six calutrons (See Figure 1). Each section 18 excited separately and almost independently by a motor driven direct current generator, thereby enabling each section to be used for processing a different element. Presentation of the thesis problem. In the processing of elements with mass greater than 100 atomic mass units, serious limitations are imposed on the isotope separation process by the decreasing mass disper- sion of the beta calutron, which for example would be only one-tenth of an inch for a mass difference of one part in 240. These limitations are primarily: (1) the small space available for receiver pocket construc- tion, thus limiting the receiver pocket wall thickness and hence the ion bombardment time of a pocket due to sputtering; (2) the limitation on T . K 'N ...... ...... .............. .. ! 4 IZ RT 1. . 1 A1 2 . . A a . A I . Lic . M RE Figure 1. Installation of the auxiliary yoke steel to subdivide the beta calutron track in Building 9204-3. PX . . * . . the amount of liquid cooling lines which can be applied to metal pockets to aid in the retention of low boiling point elements; and (3) the decreasing beam separation for the smaller relative mass differences. Various workers 1) in the beta and mass spectrometry fields have - - - proposed the use of axially invariant radially inhomogeneous magnetic fields to increase the mass dispersion and beam resolution of spectrom- eters. Several beta ray spectrometers have been successfully constructed using these types of fields, and have verified the second order focusing predictions of the existing theory. Artsimovich ") has constructed a E . 235° focal angle (See Figure 5 for a focal angle illustration) isotope . separator, which will transmit milliampere ion beams, but unfortunately not enough information was available to ascertain its merits. Very little information was available to ascertain the focusing properties of - these various magnetic lenses for ion trajectories outside of the . magnetic midplane. This information is vital since a practical isotope separator should accept 110° of source radial angular divergence and should not have serious image distortion for source ion slits with a length of 0.3 r, where r is the separator equilibrium orbit radius. TH . . .. Thus the basic problem was to modify the calutron magnet system to increase the beam resolution, the mass dispersion, and the final isotopic purities with a minimum of other calutron equipment modification and without a loss in ion beam throughput. - - 2 : wrn ES Synopsis of Chapters II through Chapter VI. Chapter II contains a summary of the general restrictions placed on the separator dimensions . 121" . . . A ri Ket :: and on the electrical and vacuum service equipment. An analysis of the magnetic fields, which satisfied these conditions, is presented, from which a choice of the magnetic field configuration was made. Three dimensional trajectories are then presented, which verified the focusing properties of the chosen field outside of the median plane. Chapter III gives the method and results of the theoretical shim shape calculations, which ignore the terminal boundary conditions placed upon the shims, and gives the limitation of this type of calculation. A review of the relaxation techniques used in solving the three dimensional magneto- static potential problem of determining the shim shape ectually used in the 255° separator is given, including a discussion of the boundary value approximations used. Chapter IV reviews the problems encountered in measuring the magnetic field of the constructed separator along with the reasons for the method chosen. A description of the measuring equipment used is given with a discussion of the data analyses and calibration methods used. The results of the field measurements are given with error analyses and with a specification of how they are used to obtain orbits in the midplane of the separator. Chapter V details the shim correction method used and gives the results of the first set of corrected shims. Chapter VI gives a brief description of the separa- tor equipment, which is used in the high purity isotope separations program and presents experimental data to verify the focal predictions of the theory and magnetic field measurement orbits. A comparison is made of the isotopic purities of the 255° isotope separator to those of other high output separators in existence. Chapter VI also reviews the basic approach used in achieving the magnetic field and gives suggestions as to how the process could be made more accurate and efficient. The thesis will emphasize Chapters III, IV, and V, which concern the shaping and measuring of inhomogeneous magnetic fields. 33 2 .:.. .- . CHAPTER II SOME THEORETICAL CONSIDERATIONS IN THE CHOICE OF A MAGNETIC FIELD SHAPE Some practical restrictions on the choice of a magnetic field shape. The most practical and useful new design for an isotope separa- tor for the Oak Ringe National Laboratory 1sotope separations program was one which could be achieved by installing a new set of magnetic shim plates into an existing vacuum tank without major alterations to the vacuum, electrical, and magnetic yoke systems. Figure 2 gives a schematic outline of a beta calutron isotope separator. Sector 1sotope separators with their source and receiver outside of the main magnetic field were not deemed practical for the present application due to their low throughput and to the difficulty of correcting for the ion beam defocusing effects of the large fringing magnetic fields of the beta yoke system. The choice of the magnetic field configuration was then narrowed to those which had the object and image inside the main field and which had equilibrium radii r less than 24 inches. This restriction on the radius was necessary in order to keep the ion beam out of the fringing fields near the gap boundary. With the foregoing restrictions in mind, the focal properties and practical possibilities of that class of magnetic fields which still remained were examined. . Development of the vector potential and magnetic field expansion. The following development deals only with magnetic fields which are Vacuum manifold Receiver Beam Gate valves 2 20-inch diffusion pumps Source << Linear shims Vacuum tank To roughing pump Figure 2. Schematic outline of a beta calutron isotope separator. circularly symmetric about the polar axis 2 and which are mirror symmetric with respect to the magnetic midplane, 2 = 0. The magnetic vector potential Ã' as a consequence can contain A and A only as constants (See Appendix A) which are defined to be zero leaving only A which is a function of r and z. From Maxwell's equations expressed -.-- in cylindrical coordinates (See Figure 3) and in a region free of current elements the magnetic induction vector Ę can be calculated from B = 8x : (2-1) which ieads to . a дА. B , and B, - (546), and B = = (2-2) - where A is a constant with respect to Ⓡ. Converting to the normalized -- ....... coordinate system defined by r - r (2-3) with r = r. (p + 1), the field components become B- and B. Ta + 1S {(0 + 1) As} · (2-4) The vector potential can then be expanded in a double power series about p = 0 = 0 as 0 (2-5) I sN 1=ó i=O j=o Expanding B and B in a double power series through terms of the fifth order gives Section "AA" Figure 3. Polar coordinate system and a 0-plane schematic cross-section of the 255° separator shims. ܙ ܙ ܂ - ܫ 1 ܪܽܟܕ ܂ ܪ . ; 11 - - ܂ B = ܝ . ܕܘܕܬܘ ܢ - ܕ,u - A { ( bo - + b . (... ܀ 64184 + bancs ܀ bean2 + ܡܝܝ (2 - 6) _ ) + bos + b ܀ ܘܦܕ besc2 + . . . ) cs + ( bos .. + 5:(... + since (o, - o) for mirror symmetry. Also B Bo 1 : 23ܘ21,ai J=o + ܘܘܢ܀ + ܘܘ azop2 + aao93 + a40e4 + agon5 (... + + (eo2 + alzp + aaap2 + aseos + ...)32 . . . . . ... + * :(... + ܩܦܨܦ + ao4) + (2-7) . . ܫܝܚ ••• • ܚܝܕ . ܂ since B (0,0) = Bule, - 0) for mirror symmetry. From Equation (2-4), using the Ap expression in Equation (2-5), B and B, are expressed as ܂ ܣ ܤ , and (2-8) o j=o !. - ܂ܐ ܠܗܘ , - ܙܘܪܙM ܐ 1 ܀ ܘ }g- ܙܘܘ ܬܘܙܨ . {ܠܐܙ ££ rolo + (2-9) : ܠ 3=o ܘ 12 The zeroth order vector potential coefficient Aoo is evaluated through the use of Stoke's theorem, $ 2.dö - [ {3- I BÃ. For the homogeneous magnetic field E = Bē, = Boē, , and À = (Aco + A100)ęc . Stoke's theorem then gives the following relationship between the vector potential coefficient Aco and the magnetic field as Apo = 1/2 Boro, where the integration has been performed in the midplane, z = 0, around the reference radius ro. Equating the coefficients of like terms in Equations (2-8) and (2-9) with those of (2-6) and (2-7) expresses the vector potential coefficients Aq, in terms of the field component expression coefficients az and bey Expressing the bar's and azi's in terms of the oº coefficients of B., regarded as a boundary condition at the magnetic midplane, by using Maxwell's equations, x = 0, and Ž · = 0, one can express the Asy's through terms of the sixth order (See Appendix B) as Aco – Boro , A10 = Boro , broa10 13 A20 = { Boro@10 , - } Boro(820 - 210), – Å Boroleao - $ 220 + $ 410) , - } Boroſeso - $ 230 + z 220 - & azo), Boro (250 - 260 + Ž 230 - 220 * 2:0), - - - 0210 , a Borolą 220 - 220 - 3430), + 4 2 230 + 4240 + 10a50) , A12 = - Boroazo , A2z = - Borokso, Ž Boroago , A32 = - 2Boroa 40 , мл oroa 50 ] Aqz -- { Boroaso, -- Borol-[ 270 + 220 - { 490 - 62.go) , and A24 = - ] Borož 220 - 220 + { 230 - 22.40 - 10850) - a 10 + a20 and - ano + - 2ado - 1 (2-10) These expressions for the vector potential coefficients agree with those of Bretcher (3), who has calculated them through terms of the fourth order. . . : 14 Orbit differential equations. The orbit differential equations have been derived using the principle of least action. Solutions to these differential equations have been obtained in the magnetic midplane through terms of the second order and outside the midplane through the first order in order to show the basic focusing properties of the inhomogeneous fields. The action is defined as the integral A-SE Pq da, (2-11) where q, and P, are the i th coordinate and its conjugate momenta and where the range of integration is taken between fixed spatial end points. Since the Hamiltonian of a charged particle is conserved in a magnetic field, the principle of least action gives os p4 dą, - -/-də = 0 , (2-12) where is the inechanical momentum of the ion and ds its vector displacement "7. For an ion in a magnetic field > is replaced by a canoni cal momentum, ; - qÃ, where a subsequently refers to the ion charge, m to the ion mass, and to its mechanical momentum 14). The variation in the action is then expressed as Apo-qÃ).aš = 0. (2-13) For the case of a static magnetic field and no electrostatic potentials the Hamiltonian and momentum are constants, leading to E . . 15 -" AS (pds - gã-də) = 0 , and - S (as oma ras) = 0 . (2-14) Using dố = (drvē + (rao jē+ (dz vē, and à = , (2-15) ds may be expressed as 2 do={(an)2 + (100)2 + (dz)2}/2 = {x^2 + y2 + zva }'ladi, (2-16) where primes indicate differentiation with respect to Ⓡ. Substituting Equation (2-16) into Equation (2-14) and using p = mv = (2qVm)/2, where V 18 the accelerating potential of the ion source and v the ion velocity, one can express the variation of the action as [ {(x®2 + y2 + 2+2;4/2 - ( 2 m )/2 for } a = 0 . } ao = 0 . (2-17) (2-17) Defining n = ( )/2 and using the normalized coordinates defined in Equation (2-3), with primes again denoting differentiation with respects to o, one can write Equation (2-17) as 1TR 2. LE SI 16 2 + (6 + 2)2 + o'z}/2 - 160 + 1)2. 10 = 0 . (2-18) Equation (2-18) may be abbreviated to A Fall = 0 , oc where F is defined as P= [ {^2 + ( + 1)2 + 0*2}}/2 - 160 + 1)A. ]. (2-19) The differential equations derived from the A variation of Equation (2-18), which use $ as the system parameter, have the form of the Euler- Lagrange differential equations "*, do = 0 , and son = 0 . (2-20) Expanding F in a power series, through terms of the fifth order, in , e', 0, and o expresses F as F = { 1 + + 22 t 2 to + Io p={+p+] (9*2 + 0*2) - 260*2 + 0*2) + p°(0*2 + 0*2) - }("+ 0*2) - Ž (0** + 2pº2,42 + 0*4) + plo** + 20*20*2 + 0*43} - n6e + 1) {400 + 420p + Azoo2 2 t . P a w + Agop® + A400* + A5005 + A0202 + A12PO2 + A220202 + Agzpo2 + A0 404 + A2 400* (2-21) Collecting the coefficients of equal powers of the coordinates makes the following definitions convenient. 2 نما P = { (o^2 + (-2) - (p2 + 0-2) + 02 (o^2 + 0*2) - 3 pP(*2 + *2) - 5 (* + 20+202 + *4) 10'4 + 20°20'2 + O'*)p + Foo + F100 + F2002 + F3003 + F 800* + F5005 + F0202 + F0404 + F12002 + F220202 + F320862 + F7 4004 , where = 1 - n10o = 1 - Boro , F10 = 1 - n(A20 + Aco) = 1 - 1 Boro , F20 = = n(A20 + A10) = - Boro (220 + 1) , -- (430 + Azo) = - } Boro (220 + 810) , = - (A40 + Ago) = - Boro(ago + azo) , . 18 (2-22) F5o = - n(A50 + A80) = - Boro (a 40 + ago) , Foz = - NA2 = { nBoP6210, PO4 = - 1404 = ła Borolē 10 - 220 - 32go) , = - n(A12 + A02) = nBoro(220 + {210) , F22 = - n(A22 + A12) = nBorol? ago + 220), F32 = - 1(A32 + A22) = nBoro (22 40 MU and F14 = - n(A14+ 404) = nBoro(- 280 - 3 290 + } 220 - 3 010) · Zeroth order solution of the differential equations. When terms through the first order are retained, Equation (2-20) becomes de do. (Foo + F10p) - (Foo + F100) = 0 , (2-23) which reduces to F10 = 1 - nBoro = 0 . This is equivalent to To = }e (2007)/2, (2-24) by use of the Equation (2-17) definition of n. Equation (2-24) gives the relationship between the source accelerating voltage V, the magnetic field Bo at the equilibrium orbit ro, and the ion mass mo, which will traverse this equilibrium orbit. First order solution. Retaining terms through the second order in F produces F = 10'? + Oʻ2) + Foo + F100 + F2002 + F0202, (2-25) which upon substitution into Equation (2-20) produces the second order differential equation of motion Ó'' - 2F200 = F10 , and o'' - 2F020 = 0. (2-26) These have the harmonic solutions p = Aqsin kjø + B,cos ky® - 3 Fue , and = A,sin k, + B, 2 F20 20 land o = Cocos kao + Casin ka , where kz2 = - 2F20, and ke2 = - 2F02. Expressing the integration constants in terms of the initial conditions produces BIG 1 Fles Dok sin kyo + { . 1 2 F20 oſ Co NIN gla (2-27) where po and pó are the initial normalized radial coordinate and initial value of n is the customary designation for the exponent in the equation B = Bo (2)", B = 1 20 which expresses the first order behavior of a more general field whose first order coefficient in its series expansion 18 - n. The value of n would be between zero and one for an isotope separator. ki is expressed for the reference mass mo as ky? = - 2F20 = (1 - n). The axial solution becomes 0 ** oo Cos ką tu o sin kao , (2-28) where ka? = - 2F02 = n, and oó is the initial value of . Equations (2-27) and (2-28) agree with the results of Tasman (5) und Bretcher 3, except that the term - is missing in the printing of Equation (2-27) in Bretcher's work. Properties of the first order solutions. These solutions are accurate only for paraxial trajectories, but do define the first order behavior of all orbits in a very useful manner. Equation (2-27) gives Per = - po at the radial focal angle, ofr= - 17 , with the image i ! - n) 72 being reversed and with unit magnification of the object. Equation (2-28) gives the axial focal angle as 60 = 1 , with the image again being reversed and having unit magnification. These focal properties are independent of the source radial and axial angular divergence through the first order. Double directional focusing occurs whenever the ratio of the radial to the axial periods of oscillation is a whole number defired by 2 = L , where L = 1, 2, 3, ... . Only the case for n = 1/2 was applicable to the present problem since the other cases (n = 0.8, etc.) would have required the beam to travel through more than 360° of azimuth for the source and receiver in the field. For n = 1/2 the image is reversed and has the same dimensions as the object. Another very useful property of the n = 1/2 field is the increased mass dispersion which it provides. The equation for the mass dispersion - - may be derived by differentiating Equation (2-27) with respect to m - which provides Im - mo 22 - n mo Thus, the mass dispersion of the n = 1/2 field is double that of a homogeneous field separator, which has n = 0. Second order solutions. Retaining terms through the third order in F expresses F as Pa (02 + F "2 + o'?) - bolo'? + Oʻ2) + Foo + F100 + F2002 + Fsop® + F0202 + F12002, (2-29) which upon substitution into Equation (2-20) leads to the second order radial differential equation, 22 b" - 2F20p - F10 = pp"' + NIN 10^2 + 3F3002 a + F1202 . (2-30) The right hand side of Equation (2-30) contains only second and higher order terms, and hence a good approximation to it is obtained by substituting in the first order solutions in Equations (2-27) and (2-28). When these are substituted into Equation (2-30), it becomes b" + kp?o = T,sinºk,* + ž T-sin 2kjø + Tzsin kjo + T4cosak,0 + Tscos kjø + To , (2-31) where Ta 1 F 20 P: v mce (EP - 2) + *** (CS) Te = pó (3 - kg ) 78 - pé To = 3 *3* - * * * (*)* * 820 (ah) Ts = *; 2 (CFO) - { Fao (), and To – F30 * { 'so (F) To = Fo+ 2 (2-32) Using the method of variation of parameters to find the general solution to Equation (2-31) produces the following equations " + kz? = R(0) , Az = sin kjº , and pz = cos kjø . (2-33) The general solution 18 p = Aj (0)22 + B2 (0)22 , where PORC do + E , 2 - Papa RO) - pipa) do + F , and B.(•) --Stoper ope) do +P , and R(« ) = 7481nºk+ Tzsın 25+ Tosin kjo HICU • + Tasi + T4cosºk,® + Tscos kjø + Te . (2-34) .::- Equations (2-34) reduce to pa sih, kº { } 1361n*;* - Tacos®kjø + į Tgsin®kjo + } Pesin kjø coBºx20 + $ Toin 130 + į Tskjø + ] Tysin 2kzo + Tosin kjø+ 8) + Covento { - 1} Tacos kj06in®ky • - - - 1, ' . 24 - T c06 $40 + } Təsin”,0 + ] Takjo - Tosin 2x30 T.C08%20 - {Tgco?kjo - Tocos k30+p} - TACOB K, + FY (2-35) When E and F are evaluated in terms of the initial conditions, they become 12 F = kx?p0 + póz ( k, 2 / Sie + Fic ökal F20. , and B = pók, (1 - ) 12-36) For ^ = "an and for the double directional focusing inhomogeneous field fr with n = 1/2, Equation (2-35) reduces to Per et ( )-80 · (2-37) Since the angular divergence from the present Oak Ridge National Laboratory calutron ion source is predominately radial, the coefficient of the term containing the radial divergence in Equation (2-37) will be set equal to zero producing 3 - = 0 . Using the expression for F30 in Equations (2-22) and the expression for ki? in Equation (2-26) leads to the following expression for a zo of 220 - - $ 210 - . For the case of current interest with 210 = - n=-5, 220 becomes 220 = } . With 210 = - į and 220 = į , the image 18 independent of the source radial angular divergence through terms of the second order. ". . -.- . -. Third, fourth, and fifth order considerations. It can be seen from a consideration of the second order cobits in the midplane that the higher order radial angular abberations in the midplane, which become significant as the radial angular acceptance from the source is increased, can be eliminated by obtaining higher order solutions for the midplane field contants. This follows from the fact that orbits with higher angular divergence travel part of the time in a portion of the magnetic field which is unique with respect to those orbits of lower radial angular divergence. Hence, the field in this unique portion can be altered to provide steering corrections, which will affect only those orbits of higher orders. This process can be carried out in theory for very large angular divergences. Tne very large magnetic field volumes, which these large values of B would necessitate, would make the job of magnetic field shaping more difficult, but, definitely not impossible. This process has been used, through terms of the fifth order, by Beiduk and Konopinski " for the n = 0, 180° inhomogeneous field spectrograph, and by Lehr (7) and Verster (8) for the n = 1/2, 255° .. WIM... . . - WO W 26 double directional focusing field. No calculations had been made higher than the second order for trajectories outside of the midplane and so a computer program was written for the IBM 7090 to calculate fifth order trajectories throughout the field volume of interest. The optimum theoretical focal pattern for a curved object, which has been projected into the o plane, 254.5°, 18 shown in Figure 4 as calculated on the CDC-1604 computer using an optimum fifth order three dimensional n = 1/2 field shape. The midplane field coefficients for this field, which focuses radially at 0 = 254.56° and which was used in the constructed separator are, aio = New 220 = aso = o = - o , and aso = 522 (2-38) 20.05. 20.00 L 40% 70 Radius (inches) 19.90 ¿ Distance from the midplane (inches) Figure 4. 255° separator computer calculated optimum three-dimensional focal pattern as projected into the o-plane, 254.5°, using the fifth-order theoretical field and a plane curved object. *The data for this drawing is from a computer program by T. W. Whitehead (unpublished, Oak Ridge National Laboratory, Oak Ridge, Tennessee, January, 1964), which was used in a study of the theoretical focusing properties of fifth-order three-dimensional magnetic fields. CHAPTER III THEORETICAL CALCULATION OF THE SHIM CONTOUR The methods used to shape the almost perfect focusing field presented in Chapter II will be discussed. The presentation will cover the limitations imposed on the shim contour by the vacuum tank structure and by the necessity of maintaining high voltage clearances for the ion source. The contour as calculated neglecting boundary terminations, and the relaxation methods actually used for calculating the first shim contour will be presented. Size limitations placed on the chims. Figure 5 is a schematic outline of the 255° separator and or the beam shape in the median plane. In order to use the existing iron-walled vacuum tank all shimming had to be done inside it. The tank and coils were designed for the linear shimmed field, which has rectangular symmetry, and this fact greatly complicated the shaping of a field with circular symmetry inside the tank. Magnetic field measurements presented in Chapter IV show azimuthal field variations resulting from the tank rectangular symmetry and from certain iron structural protrusions. Sixty -inch diameter shim platos were the largest size which would fit into the calutron vacuum tank, and they had to be truncated at the front. Figure 6 shows a shim plate mounted on a rail carrier prior to its installation into the vacuum tank. The basic tank gap was sixteen inches with the minimum gap set at ten inches in order to leave sufficient room 28 TW g= 12.5 ZUILIIN g=16 g=15 Receiver 68 Contour g=10 $=255° Beam Tro-20 Source Vacuum tank wall Tank iron wall and coil core outline Dimensions in inches Figure 5. Schematic outline of the 2,5° isotope separator beam and beam focusing system. 30 2. + 2. 1 . 1 27 A M . The one or per LIN how 2 . 2 e ...' . 7 . L. . 2 Y . 15 . 4 W ! i . r . i As se 2! 2 no - i W in * . Vi . 3 . . . ' : . " . 93 Figure 6. A 255º separator shim mounted on the rail carrier installation rig. 31 for a receiver and a high output electron bombardment ion source. The field shaping had to be performed by contouring identically two 3-inch thick, 60-inch diameter soft iron discs. A 1.25 x 7.25 x 34 inch piece of iron was cut from each side of the tank at the rear to allow clearance for the shim plates and to remove as much of this large source of magnetic field disturbance as possible. A shim was bolted to each tank wall using 16 one-half inch bolts made from iron of an analysis close to SAE 10-20 iron. The tapped holes were used which had held the linear shims in place. The shim iron contained 0.4 per cent manganese and 0.13 per cent carbon as determined by a spectrographic analysis. The steel discs were cut from rolled stock and were anealed for eight hours at 1800°F in a non-vented furnace with an air atmosphere. No tests were performed to determine the presence of voids or other structural defects in the shim stock since these were assumed to be relatively unimportant for the present magnetic configuration. This assumption has been justified by magnetic field measurements on a corrected set of shims, which were cut from steel supplied by a different vendor. These shim correction calculations will be described in detail in Chapter V. Figure 7 shows the two shim plates installed in the vacuum tank and shows the iron cutouts at the back of the tank. From the photograph it can be seen that several bolts are directly downfield from the beam, a fact which was undetectable from the magnetic field measurements in the midplane. First approximation of the shim contour neglecting the end boundary conditions. The magnetic field defined in Equations (2-6,7) may be I . i 74 " 32 Figure 7. Shims installed in the 255° isotope separator vacuum tank. E , 1 . - $ . . . . .- - A - - - aas ? IAI ܕܶ . 3 2 - 12 . - . . :.: . . 12 . w i . 2 . . SA . E . S : pr - : si: TY 1 . . . . . . . .; - : - ? , S . # . 1 is , *- i . Y 1 : II. - --- ..4. . - : M F E - . . '-- . R -- . - = - =- ES . . - 7 .-212 V .' 'S . R ::. - = - - - ht EU derived from a scalar potential field V ab B= - ūv. This scalar field obeys the partial differential Equation (9) ovoz u t u DEV = 0 , (3-1) where u 18 the absolute permeability of the medium at the spatial point in question. An iron core electromagnet is used to provide the magnetomotive force for each separator gap thus necessitating the treatment of two regions with very different permeabilities. The tran- sition between low permeability air gap and the high permeability iron must be included in the solution for V from the boundary restrictions placed on V. The law of refraction of magnetostatic field lines is that Tan da = ? Tan ay , (3-2) Mi where H2 and (y are the relative permeability and field line angle with the normal to the shim surface in air, and H. and a, those in the iron. Using a value of u of 1,500 at the operating flux levels leads one to the conclusion that tan (a is very lose to zero for all reasonable values of dj and hence that the field lines leave the iron surface approximately perpendicular to it. The shim contour can thus be treated as an equipotential surface thereby greatly simplifying the determinations of the shim contour. :72 Evaluation of the shim contour using the magnetostatic equipotential function. The magnetic induction field B, whose shape is known, may be derived from a scalar potential function V as Blo,0) = . v(,0) , (3-3) leading to the equation By(0,0) - Orlocos (3-4) By integrating Equation (3-4), using the expression for B in Equation - (2-8), V may be expressed as men. V(0,0) = - roBo { (1 + 2100 + a2op2 + 830p® + 240p4 + ...), + (802 + 2z2P + 2z2p2 + ...)* + 20405 + ... } ... = - roBot , (3-5) . .- thereby defining a family of equipotential surfaces, where the value of - - I will determine the value of the equipotential. At p = 0 , 13-5) where so is the gap at the reference radius. Thus t may be evaluated as (3-7) To solve Equation (3-5) for o as a function of p and T, an expansion of the following forin is assumed : 35 0 - (CO2 + C110 + C2202 + co2ps + C430* + ...)t + (cos + C23P + C23p2 + ...)78 + C0575 + ... . (3-8) Substituting Equation (3-8) into Equation (3-5), equating the coefficients of like terms equal to zero, and using the relationships of the field constants in Appendix B, one may express the c's as Coi = 1 , C12 = - 810 , C21 = 2 102 - 820 , 031 = - 230 + 2220220 - 2103 , C41 = - 8 40 + 2210230 + azo2 - 3210-220 + 2204 , 202 = 220 + 420 , C13 = 42022 10 - 212 = -4220810 - 28102 + 3a30 + azo C23 = 4202820 - 2080 28102 + 4212810 - 222 = - (azo + IN 4220 · 102102) - 12a30210 - 4220220 + 2010* + 6245 + { 230 - 220 +220 , and Cos = 34 - 220 - 210)2 - 440 - Ž 430 + Žā 220 - Il 1910 · (3-9) First order taper of the shims. Differentiating Equation (3-8) with respect to e, through terms of the first order, leads to -- fi.. * IS7 X f (3-10) do - Ch17 -aso recente , and 0 = tan-- nemo, (3-11) where e is the shim taper angle as shown in Figure 8. do 18 positive. BT = Solving for the classical field inhomogeneity index, n, gives n = 20 tane. (3-12) For the 255° separator the gap width go was twelve inches with a taper angle of 0 = 8.5°, Figure 9 shows the deviation of the midplane magnetic field from that theoretically desired according to Equation (2-7) 1f the conical shim predicted by Equation (3-11) had been used in the 255° separator. Shown also in Figure 9 are the field deviations which would have been produced by the shim contour as calculated from Equation (3-8) and the field deviations expected from the first set of shims actually used. Table I gives the contour data for the linear, fifth order, and actual shims used. An Algol computer program was used to compute the linear and fifth order contours and is available from the author upon request. Inputs which must be specified are the reference radius ro, ... . the gap width go, the maximum and minimum values of r for the range of r for which the contours are desired, the increments of r for which output is desired, and the midplane field constants. The expected field deviations were calculated by the use of relaxation techniques to be .. 2 : discussed in the next section. The expected field deviation of the first 37 Shim Shim murado . -.. . .. Figure 8. 255° separator shim schematic cross-section. Fifth order contour Relaxation contour Linear contour ---- Percentage deviation 38 14 16 18 20 22 24 26 Radius (inches) Figure 9. Relaxation calculated midplane magnetic field shape deviations of 255° separator, linear, fifth-order, and relaxation calculated contoured shims. TABLE I COORDINATES OF THE CONTOURS FOR THE 255° SEPARATOR LINEAR, "FIFTH ORDER", AND FIRST RELAXATION CALCULATED SHIMS WHICH WEPE USED Contour 2 fifth order (inches) Coordinates Z linear (inches) 2 actual Contour 1 (inches) (inches) 0.0 5.0 5.0 5.0 12.0 12.5 13.0 23.5 5.0 5.0 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 5.0 5.0 5.0236 5.0913 5.1604 5.2307 5.3023 5.3751 5.4489 5.5238 5.5998 5.6766 5.7544 5.8329 5.9123 5.9925 6.0734 6.1549 6.2371 6.3199 6.4033 6.4873 6.5718 6.6568 6.7423 0.8283 6.9147 7.0016 7.0889 7.1766 7.2648 7.3534 7.4424 7.5 5.0 5.0 5.0100 5.0859 5.1618 5.2377 5.3136 5.3895 5.4654 5.5413 5.6172 5.6932 5.7691 5.8450 5.9209 5.9968 6.0727 6.1486 6.2245 67.7004 6.3763 6.4522 6.5282 6.6041 6.6800 6.7559 6.8318 6.9077 6.9836 7.0595 7.1354 7.2113 7.2872 7.3631 7.4391 7.5 7.5 5.0939 5.2788 5.4229 5.5232 5.5520 5.6055 5.6444 5.7285 5.8052 5.8813 5.9651 6.0747 6.1856 6.2647 6.3010 6.3351 6.3731 6.4104 6.14682 6.6090 6.7797 7.0287 7.4099 7.5 7.5 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.6 28.5 29.0 . 29.5 30.6 set of shims agreed with that produced by the first contour to within 0.045 per cent. This discrepancy has the proper sign and shape so that it could be explained as an iron saturation affect, since the nagnetic field changes by several thousand gauss over the field region used by the beam. Figures 10 and 11 show the midplane ion beam focus which would have been produced by the conical and fifth order contours (not actually used). The use of relaxation methods for determining the shim contour. .. Due to the large ratio of be of 0.6, it was not possible to ignore the effects of terminating the theoretical equipotential surface defined by Equation (3-8) at the tank wall and at the minimum gap of ten inches. The actual outside boundary contained azimuthal variations, and these were approximated and included in the relaxation calculation of the first shim contour, which assumed circular symmetry. The inside circular boundary was, of course, accurately defined. It was deemed impractical to try to include in the contour calculation the effects of shim iron saturation, tank structural tolerances, shim iron inhomogeneities, shim iron voids, and other commonly published field shaping problems. In the contour calculation an attempt was made to approximate the fixed magnetostatic boundaries and then to numerically solve the resulting magnetostatic problem accurately by numerical methods for various shim contours until the midplane magnetic field, evaluated from the gradient of the scalar potential, agreed with the theoretically desired field shape over as large a volume as possible. Fortunately the possible . Wu 1. 11! . > hu My . 1L DR EA In BE W . . UNCLASSIFIED ORNL 2 VEL 11, * . 1 Yr . VI - 20 XX ik Si yan VA LV U 2 V .PL - ' . KY + 744 20F4 GU . ! . . I .. - ? - w . Il . S 1. .' ' . . . .. .. .. . 2 : min e .**.* .*' !! . . . o ochronie . . Radius (inches ) 240 245 250 255 $( degrees) Figure 10. Calculated midplane focus which would have resulted if linear (conical) shims had been used in the 255° isotope separator. Radius (inches) ZH 1-10° 245 260 250 255 01 degrees) Figure ll. Calculated midplane focus which would have resulted if "fifth-orer" shims, contoured according to Equation (3-8), (Page 35), had been used in the 255° isotope separator. 43 volume coincided closely with that which was desired. It was assuned that the errors introduced into the field shape by the approximations of the contour calculation could be eliminated by the correction procedures outlined in Chapter V using the magnetic field measurements described in Chapter IV. The fielä deviation plots of Figure 12 illustrate the disturbing effects of improper boundary terminations. Plotted in this figure are the field deviations actuall; measured from the first set of shins, the deviations as expected from the relaxation calculations, and the deviations which would have been obtained from the first contour if the correct inside termination had been used. The incorrect termination resulted from a machining error which took 0.030 inch too much material off of the flats of the center of the shims resulting in a minimum gap of 10.160 inches instead of 10 inches: With these powerful numerica.l aids it was possible to get an accurate field shape with the second set of shims, thus bypassing the expensive and time consuming successive approximation shaping, which has usually been used in the past on other separators and spect roneters. The large 1, 400-pound discs make such procedures impractical and would probably not have provided the accurate field shape attained. -- - - Calculation of the shim contour. Figure 13 schematically shows the boundaries of the magnetostatic problem and the coordinate system used for its solution. Expanding the potential function in a two- dimensional Taylor series about r, and 21 gives - . F . metalen.. ..........don't A Shim • Shim set one, measured contour one, calculated from measurements contour one, relaxation calculated — Shim Percentage deviation 777 13.5 15.5 17.5 23.5 25.5 27.5 19.5 21.5 Radius (inches) Figure 12. Midplane magnetic field shape deviations from the theoretical for shim set number one in Tank XBX-1. 36" 75"- -- vir, -h, z, ) Virith, 2,) veri,2,) Viroozen) 175" E " Infinity approximation used Shim equipotential First approximation to infinity 11.5" lit e 16" 9" + Midplane V= 0 Figure 13. Schematic of the 255º separator magnetostatic boundaries. . . . ! v(rı + Ar, 21 + A2) = V(r1,21) V(21 + 5, 22 + 2) = V(92,2x) + (ar + az ) V(82,21) + šip (or get az ola) v(72,22) + ... *(ar šif + se fa)*V(+2,25) + ... . (5-13) + A2 Through terms of the fourth order using a cylindrical coordinate net and using ar = h, and az = 1 hy, Equation (3-13) becomes V(r2 + ,22) = V(r2 + 22) + do V(71,22) + V(12,22) + 3 de V (82,29) + * V (12,22) --------- + ... , (3-14) ? V631 - ,123) = V(81,81) - Ver.1,29) + TO V(72,25) Šat V22,23) (8 1,83) h DI . + ... , (3-15) -...--.-.-- V(11,22 + 4) = V(11,72) + ne de V(71,22) + } , a V(1922) .- . . *3 n, der V(81,923) + šte na 2h 3 23 ha da V(99,21) "z dz tine met sowat in indiano * s chri 1 . I. VOLW WN 47 mi je region '. ' of met 1 . 41 1 6 V i ne sa interesse # 11 ? 1,111, 1. . .' . gi + ... , and (3-16) V(r1,21 - 9 ) = V(r1,21) - h, v ) + 3 h? İzz V(r1,71) me (22,83) + at any ty V(82,23) + ... . (3-17) Laplace's equation in cylindrical coordinates for axially symmetrical potential fields is d2v, 1 av toy = 0. Šra + år + Izz = (3-18) Solving Equations (3-14,15,16,17) for the terms in Equation (3-18) through terms of the fourth order produces . 37 . (3-19) 1 V(82,22) - {V(92 + 529) + V(72 – .22) - 2V(71921) my Vera,a}, ty (29,22) - {V (49,99 + bg) + V(29,351 - ,) - 2019,83) Bemutat V(72,71)} , and 2 + 2 - N (3-20) warned. so many divertime t k * common .. * ** 48 el et V(49,89) - center {vers + hy 23) - Usa = hyves) Phry (3-21) Substituting Equations (3-19,20,21) into Equation (3-18) produces nz mo21) + V(rı - h, 21) - 2V(r1,22) - h 2rih ŽE SI V(71321)} + 2rzh, {V(+ _323) – V(32 = ,23) BP VC29,22)} + kes {067,48 + n) + V(29,123 - A) - 20(82929) - B me G V (92922)} = 0 . b303 h 21 + h) + V(r1,22 - h) Setting h = h = h, and solving for V(81,22) produces V(r1,25 1 + 1,21) + Vérı - 9,21) + V(r1,21 + h) + V (92,983 - 1) - } * V (1,31) + V(72922) } + {v(32 + ,23) - V(*a - 1,21) - to set (79,211} · (3-22) 2 ! i 4 vi www up24 em lis uminen . not bankowe A value of n = 0.5 inch was used for the shim calculations, producing the third and fourth order terms as third order = third order = 7,680 mm V(11,72) , and , and fourth order = Tek AV19,20) + (11,2;} . (3-23) From Equation (3-5) the partiai derivatives may be approximately evaluated as e ty V(12,2x) = - Borooz (6230 + 2402) , V opet V171,27) = - 24 Boroon , and de V(F1,2) = - 120 a 40Boroon, (3-24) which when used in Equation (3-23) provides a correction to the potential smaller than one per cent. Calculations made using potential values from the two dimensional relaxed net show that the sum of the third and fourth order corrections is in approximate agreement with the corrections as calculated from Equations (3-23,24). The magnitude of these correction terms is fairly small over the field region in which the beam travels and only becomes large in the regions where the contour is terminated in flat surfaces. By decreasing the mesh size from one-half to one-fourth of an inch, the magnitude of these third and fourth order corrections . 2. . . ! AV AP . . - ¿ {. 1. 1. . WW. TTT 4. 241 ky W 14. AT im! HITTIM" WWW.TWITTER . - 50 could be reduced by a factor of 16, with the work being increased by a factor of 4. The magnetic field deviations plotted in Figure 12 show T that a satisfactory magnetic field was produced by the set of shims whose contour was evaluated neglecting these third and fourth order terms. The successful shim correction procedure described in Chapter V also uses this approximation. The inclusion of these derivative terme would also lengthen and complicate the relaxation calculation. Thus, for the smoothly varying potential functions used, the third and higher orier terms may be dropped, producing V(r1,21) 1 + ,21) + vérı - h,21) + V(r1,21 + h) (3-25) ht, + V(12,21 - n)} - {0673 + 1,92) - V(ra - ,27)} . . Thus, the potential at any point can be computed in terms of the potentials of its four nearest neighboring points. A program has been written for the IBM 7090 computer which uses Equation (3-25) to evaluate the magnetic field produced by various shim contours. Fixed arbitrary potential assignments were given to the two :.:.:. fixed boundaries, with the midplane being zero and the shỉm contour and iron tank wall being ten. The boundary outside the gap was assigned values calculated to approximate infinity and were held constant for all . . ... shim contours as they had negligible influence on the field inside the . gap. The rest of the one-half inch spaced net points were given initial guessed potentials by a linear interpolation from the midplane to the : N AN. EN NA 2 . . 4 14 MU . .." 1 L ! WA FLP ) Wh !! ! - . . . JE 2 51 shim contour Equation (3-25) was then applied at each of the net points to compute the field residuale, defined as - RES = V calc - Poids - .-. where Vcale is computed from Equation (3-25) and Voza is the current value of the potential at the net point. A new pctential assignment was - - - - made at the net point as computed from 'new = Poza + C RES, where C was an over-relaxation constant taken equal to 1.75 to obtain faster convergence. A new potential was calculated for every point in the net in this manner, and this process was repeated until the residual at each net point was less than 0.0001. This criterion required approximately 55 iterations to be fulfilled and indicated that the field had now been relaxed and satisfied Laplace's equation to the approximation previously indicated. The coarse net used contributed a negligible error due to the smooth variation of the potentials. The magnetic induction field B was then computed from this scaler magnetic potential field as B'= - ūv. (3-26) The value of B, from Equation (3-26) was then compared to the theoretically desired midplane field shape, and from this comparison a shim shape correction was calculated. The potential field was then recalculated, and a new B was calculated to be used in calculating a new shim en hy het baie maak, 296.4: **** 1. ! ! ! . S11 om 52 correction. This shim contouring was continued until the midplane field agreed with that which was desired over as large a range of r as possible. The z coordinates of the equipotential ten, which were those of the shim contour, were calculated using a quadratic interpola ilon between the net points. The numerical tape controlled machira group at the Y-12 shops filled in the shim contour between the supplied points, which were specified at one-half inch intervals of r, with the aid of a spline curve fitting routine. This routine passed a smooth curve through the supplied data points such that the curve tangents were matched at each data point and the curve curvature was minimized. A steel template was cut from the filled-in-curve on a numerical tape controlled grinder, and - this template controlled the large vertical lathe shown in Figure 14, - which is shown machining the shim. The finished shim tolerance was $ 0.002 inch, which is sufficient to shape an 0.01 per cent field with the large gap used. The first shim contour is shown in Figure 15. . *See Chapter V for the details of calculating the shim correctiors. - - -- - - . , ' 21 . " . . . . 1 Lys in AT . * y W EN RE S " ri SM . . " IA ".. 1. " : . 11 17 . - € ' WI th Vi :. . * 11 . 14. . KO TV .. i . W KA 2 Tit S 1.. . J ? - . . -- t TWO. . andere . 1 Ur you . : LU. . . . 4 ?! 13 . . . . co 1/ . 6 " . i . 19 , . S . . : : V . . : I . Y W NA w band Android mme the i . EN LA Figure 14. A 255° separator shim mounted on a large vertical lathe during machining. . . I . . LY . - . ... It 2 . . 1 I'W L1 Shim contour 3.5"-to 3" Tank wall LIIIIIIIIIIIIII 6 8 10 2 14 16 18 20 22 24 26 28 30 32 34 36 Radius (inches) Figure 15. Contour of the first 2550 separator shim. . .... 'rt ' . . ' CHAPTER IV MAGNETIC FIELD MEASUREMENTS A thorough study was made of the sensitivity of the focus of the 255° magnetic field to field shaping errors and this study demonstrated a need for fifth order field shaping if the optimum theoretical focal pattern were to be achieved. A program was initiated to obtain + 0.01 per cent field measurements in the separator magnetic midplane, which were used in the calculation of a second corrected shim contour and for obtaining ion trajectories in the actual separator field. The second set of shims was designed to give a nearly perfect midplane focus; and the shim contour compensated for the defocusing effects of azimuthal field variations, shim alignment tolerances, and radial field shaping errors due to shim saturation effects. Through the use of the field measurements, it was possible to reduce the midplane focal width from 0.110 inch to 0.025 inch for B = $ 10°, with the second set of shims. The predicted focal width for the second set of shims was 0.018 inch! Computer orbits obtained using the theoretical three dimensional field show that the midplane residual focal width adds as a constant to the optimin three dimensional focal pattern. Thus, the optimum focal pattern will be obtained when the midplane focal width is reduced to zero. - Selection of the field measuring technique. Hall probes were .. . - - - - chosen as the magnetic field transducers due to the ease with which they -- - - - - . . 56 could be positioned in the large midplane field area and for their adaptability to accurate analog to digital conversion techniques. The large field areas to be measured and the large fringing fields of the beta yoke system would have made it difficult to apply rotating coil techniques. The adaptability of Hall probes to the measurement of high - - gradient magnetic fields made them especially useful for the accurate measurement of the inhomogeneous 255º separator magnetic field. Two Hall probes were used for measuring the magnetic field, and the voltages from each of these probes was measured by a common analog to digital conversion data system. A munitor probe was held stationary at the center of the shims, and a scanning probe was mounted on a positioning wheel, which scanned it throughout a polar grid net to completely map out the midplane field. Hall voltage measuring errors due to magnetic field regulation, amplifier drifts, Hall probe drive M. .... current drifts, and digital voltmeter reference drifts appear symmet- . rically in both the monitor and scanning probe voltages. By taking a . proper ratio of these two signals, it was possible to completely . eliminate these sources of error. It will be shown that the residual E- T- ... field measuring errors may be attributed solely to digital voltmeter quantizing errors, which would be one part in 20,000 at a full scale - - - ... reading. The field measurements thus provided two voltages for each of .. . ... the 2,256 polar grid points, whose ratio was proportional to the magnetic .. field at those points. An absolute correspondence was set up between the scanning Hall voltages and the magnetic field strength through the use of a nuclear 57 magnetic resonance flux meter. This correspondence was expressed in terms of a quadratic equation for the field strength B as a function of the corrected scanning Hall voltages. The quadratic coefficients were computed with the aid of a least squares curve fitting routine, which used the Hall voltages and nuclear resonance determined field strength calibration data. This calibration data was taken in a homogeneous magnetic field. The 4,512 Hall voltages from the field measurements were converted to magnetic field values and used on the IBM 7090 to compute midplane ion trajectories. This magnetic field data was also used in the shim correction program as explained in Chapter V. Pertinent specifications of the magnetic field measuring equipment are given in Appendix C. Transducer theory and circuitry. The Hall probe develops a voltage across its width due to the deflection of the longitudinal drive current by the Lorentz force of the magnetic field component perpendicular to its sensing area. Figure 16 shows schematically the relationship between these parameters. The Hall voltage V is given in a first approximation by where Ry is the Hall coefficient, t the probe thickness, I, the longitudinal current component, and B, the magnetic field component normal to the sensing area. In Figure 16 the drive current is fed through terminals 1 and 2, and the Hall voltage appears between terminals po w erpoint f o the phone that it has becoming T . N . . IT WA RA L Current - Ako 2 Current electrode Hall voltage electrode Figure 16. Hall probe schematic and coordinate system. 3 and 4. The FC-34, Siemens and Halske indium arsenide-phosphide Hall probes used have a mean temperature coefficient of - 0.044/°C. A temperature compensation network was connected to the Hall probe output which reduced the temperature coefficient to 0.01%/°C at 33°C. The output from the probe networks was parabolic with temperature and reached a minimum at 28°C where the temperature coefficient was zero. The probes had to be operated above the highest ambient temperature since the temperature regulator used could only supply heat. The temperature regulation was better than $ 0.25ºC, and so it was not necessary to change the network minimum to the operating temperature, Probe box construction and component mounting. Figure 17 shows the scanning probe with the lid removed. The overall box dimensions were 0.7 x 1.5 x 3.75 inches with all walls being made of one-eighth inch brass. A screw adjustable one-eighth inch thick tray was mounted. in the center of the box to adjust the position of the Hall probe which was mounted on it. A sheet of fish paper the size of the tray was glued to it, and then a 0.010 inch thick sheet of copper was glued to the fish paper. The FC-34 Hall probe was glued to the copper and was insulated from it by a thin sheet of mica. Duco cement was used throughout. A Veeco 2307 thermi stor, Rg in Figure 18, was soldered to the copper sheet one-half inch from the probe, and this thermistor was in the temperature compensation network. Two thermistors were soldered to the inside bottom of the box at the end where the cables entered the - . TRIO NINY OS $ TI . Niti *** **KROHVITTER Prot hike in the B i H: 1. TE (1 ? . . . . ------ - amori ... ... mi ... . . **2:39 25 " 17 SALAT of ! AT La . . H LU . a 25 1 17 1 ' . . .-=- . . c s: POWEN 2 24 . At 42 AO 23 - M . a pu I ' . 1 INCHES 22 . . . 2. . 16 . 2 + . . 17 1 A : 21 Y D. VN 1. 22 M . 1. . R . . 2 20 ASE . * 1 le 7 S Payment IT - . . . 19 L . i - At ot wer *" . . are look . . . . . JA . : 2 . . . RA Figure 17. Scanning Hall probe assembly with the lid removed and mounted on the positioning wheel. . S Y /:. . +/h. 9 . *). 7.' LE .. ,..,...W.......... Fusioon on " O . . 1, WPA 1. . WWW. " . LR OLY TAN C . NU ON . Nhi R N LO UM TIL V1 IN HIER W 14 . W . ".. Current signal - Monitor probo Scanning probe L *. - - Current source Thormocouple po FC-34 Hall probo R, -93 ohms Rz*1300 ohms R270 ohms thermistor R4+2 ohms Ice reference junction - TO K-3 and digital data system Figure 18. Hall probe transducer circuitry. *** LS • 62 box. One of these, a 2307 thermistor, was the sensing element for the temperature regulator, and the other one, a 12E2 thermistor, was the power element. The 2307 was one leg of a resistance bridge and a Helipot potentiometer was the opposite leg in the temperature controller. The resistance bridge was fed by a constant voltage zener diode supply and provided an output voltage whenever the thermistor resistance at the probe temperature was different from the Helipot resistance. The bridge output voltage was amplified and used to control a power transistor which regulated the current to the 12E2 power thermistor. By changing the Helipot setting, the probe temperature setting could be changed, and the controller would regulate in the proportional mode about this point. The temperature in the building where the separator and measuring equipment was located varied from 65°F to 95°F, depending on the time of day and the weather. The probe operating temperature was set at 95°F, and the ambient temperature in the separator gap was kept near this value through the use of a 3 kw hot air heater. The ambient air temperature was adjusted so that the regulator would oscillate the temperature about the set point. With the large thermal mass of the brass probe box and the low thermal coupling to the copper foil on which the Hall probe was mounted, the copper foil temperature remained within + 0.2°C of the set temperature. The temperature was monitored with a copper-constantan thermocouple, which was mounted on the copper foil. See Figure 19 for a temperature fluctuation plot over an eight hour measurement. , A Scanning probe Monitor probe fluctuations (°C) of YA Temperature -0.2+ LIIIIIIIII 17.5 21.5 Scanning probe position (inches ) 13.5 25.5 Figure 19. Hall probe temperature regulation. 64 Cabling system. Figure 18 gives a schematic of the transducer cabling system, including the probe current source and the thermocouple system. All components and equipment, except for the monitor and scanning probes, were located in a temperature regulated screen room away from the large magnetic fringing fields of the beta yoke. Shielded cables 100 feet long connected the measuring equipment to the probes. The probe box was electrically and thermally insulated from the metal positioning carriage, and so all grounds for the probe were supplied inside the screen room. The common wire of the temperature regulator -=::.. . ---•, - . thermi stors provided the probe box ground. All other cable shields were insulated from and terminated just inside the probe box. A thin sheet of mica electrically insulated the thermocouple from the copper foil on which the Hall probe was mounted to reduce ground loop emf's. This was also a safety measure to prevent the grounding of the amplidyne magnetic field regulator system whose shunt input signal was connected to the K-3 input switch for monitoring. All of the Hall probe cable shields were tied to a common tie point which was located symmetrically in the drive current circuit between the monitor and scanning Hall probes. The ground for the Hall probe transducer system was provided at the curient source. No ground loop or noise pickup problems were experienced with this system. Measuring and data logging system. Figure 20 shows the following components of the field measuring system from left to right: (1) digital data system cabinet containing the digital voltmeter, amplifier; scanner, TW . . L . ith the w 123 PON . Www * W W 61 TL . . 2 . . '' V X no 1 "T S 12 YE * . P 3. AYINY . . . 7 NE Sibiu Wind 2. M . WIMBO 2 . . 3 . E- Yi TW 1 , It Wh thank and the . 2 ✓ 12 M. > Zig ta ' M . . -in. . . - > . I SORTS I " NO . . !! I w . , 11 ! XV Y LE W II - . JA. . * 1.*** . LE . B ' " ! andamentos M * 4. Figure 20. Magnetic field measuring digital data system. 66 paper tape printer, and programmer-buffer; (2) paper tape punch; (3) paper tape reel take-up; (4) nuclear magnetic resonance fluxmeter on top of the associated frequency counter; and (5) the x-y a.alog plotter. Figure 21 shows some more components of the field measuring system from left to right as: (1) cabinet containing the drive current source at the bottom, the wheel position controller and analog generator, and the temperature regulators; (2) K-3 galvanometer and potentiometer; (3) K-3 switch box; and again (4) the digital data system cabinet. The K-3 was used to monitor the magnet current shunt voltage, the Hall voltages, and the drive current voltage drop from R4 in Figure 18. The se . -. measurements were used only to monitor the system and w?re not included - -. -. in the final field measurement data. - - -- - - - v - - - - Input signals to the digital data system. During a measuring - - - . : - c. run, “nly the scanning probe and monitor probe voltages were measured. - .. The poler. data grid was scanned through o with the radius being incremented at the end of each azimuthal scan. At the end of each voltmeter reading of a 1.5000 volt auxiliary K-3 output voltage, the 0.096 volt drive current voltage amplified by gain 20, the amplifier zero offset, the digital voltmeter zero, and the amplified monitor and scanning probe Hall voltages. The monitor Hall probe voltage was kept near 0.100 volts, and the scanning probe voltage ranged from 0.950 to 0.100 volts, and both were amplified by gain 20 before being measured by the digital voltmeter. . 1 ML, . 4 " NO I il 361 23 ry 1. . . - 3 . = ./ 11 5 . 4 . . : WY 2 . IN " ! 14. 1908 . " 7 L . M S . 39 . . W . ! . 11 7 . 1 , 2 . 1 . W " " il h bi . . . ¢ 71 . TT lị. K IN 4. M AZ ?. . " - online yang 1 . V Super . . u.S w CA T an WM Sn ...1 MY! . 2 wy ., . ) 1 2 ( W ! L . . M 1 2 . * > . . . 3 S :.. ötu. . VWS . 07 2 s.. Figure 21. Magnetic field measuring digital data system, Di, Ital data system. Figure 22 gives a schematic layout of the analog to digital conversion and logging system. A mechanical stepping switch scanner sequentially connected the input signals to the digital voltmeter. The signal contacts of the stepping switches ană the input connector pins were goid plated to reduce thermal emf's. The scanner automatically switched in a differential amplifier with a dc gain of 20 to amplify the Hall voltages and the control current voltage. The amplifier was a full differential dc amplifier with guarded, floating, and isolated input and output circuitry. It had a 1.4 cycle bandwidth and better than t 0.01 per cent short term gain stability. See Appendix C for more detailed specifications on all the measuring equipment. The digital voltmeter input range was from 0 to 2 volts with a 100 microvolt least digit resolution or a full scale resolution of 1/20,000. A measuring and logging time of approximately four seconds per channel of data was required. The digital voltmeter had a visual digital readout and a parallel decimal output which was fed into the programmer buffer. The programmer buffer converted the parallel decimal signals into parallel staircase output to feed the Hewlett-Packard 560A printer and also performed a parallel to 1-2-4-8 binary coded decimal serial output conversion to feed the paper tape punch. Each 5-digit measurement provided by the digital voltmeter was checked as it was being punched to ensure that each word punched had five characters plus a space character. This was accomplished by counting the number of feed pulses supplied to the punch using a 1-2-4 transistor counter with an associated decision circuit, which locked up the system if the wrong 0 -0.1 Volt signals 0-2 Volt signals Amplifier Scanner ! ac converter ! Visual readout digital voltmeter Scanning wheel control Programmer buffer Tape punch Printer X-Y recorder Figure 22. Schematic layout of the analog to digital conversion and logging data system used in the 255° separator magnetic field measurements. - XL 47.7 number of characters had been punched in a word. This character counter overdid its job and indicated errors when other parts of the system were out of tolerance but still functioning properly. This counter prevented many tape punching errors. The Western Electric BRPE-2 tape perforator punched only one bad number out of approximately 100,000 and was very reliable. The monitor probe output was constant to within 30 digits and the scanning probe to within 600 digits over an azimuthal data scan. These variations were smooth and ideally should have been absent. It proved very beneficial to monitor the data with an x-y analog plotter, which plotted the variations of the last two digits of the five digit monitor and scanning probe voltages. See Figure 23 for the analog output from an azimuthal scan. The Hewlett-Packard printer analog output was adjusted so that a full scele ordinate deflection on the 11 x 16.5 inch graph paper corresponded to a 100 digit change in the voltages. Both of these signals were plotted on the same paper for each azimuthal scan (See Figure 23), and changes larger than 99 digits in the last two digits were reflected in a folded scale on the paper. An abscissa voltage was generated with a o analog generator, which provided a voltage proportional to the azimuthal angle at which the scanning probe was located. The generator consisted of a solenoid driven rachet potentiometer fed by a . . . constant voltage source, with the solenoid being driven in synchronism with the scanning probe wheel. With the voltages being analoged and A * plotted as they were measured, it was possible to spot immediately any 14 inconsistences in the data and take appropriate corrective action. All - . . . F B . . ." 19 I P I I n ' T IT ! . . f " TUUDII UUUUU CIUDUUUUU UDOTI UUU Q0000 UISLUIDO JDIUIOL ATIT UONON VII.2 ITC T 71 000000000000000UDIULOUUUUU LLLLLLL UUUUUUUUUUUUUUUUUUU 11 TTTTTTTTTTTT UUUUU Figure 23. Analog output from a magnetic field measuring azimuthal scan. is: 19 72 ? voltages were of the same polarity and of known magnitude and so no .. . decimal or polarity information was logged. The paper tape was converted to punched cards for processing on the IBM 7090 and the printed paper tape was used for cross checking the data. Probe positioning. Figure 24 shows a picture of the probe positioning micarta wheel. The monitor probe is shown just above the wheel axle at the end of the cable with the spiral wrap. The black cable is connected onto the nuclear magnetic resonance fluxmeter probe, which 18 also located in the homogeneous center portion of the magnetic field. The scanning probe is shown at the upper front of the wheel attached to the other spiral wrap data cable. Figure 17 shows the scanning probe fastened to its radial track. Its radial position 18 adjusted in one inch increments by locating the cart dowel pin in accurately positioned holes along the radial track. The radial positioning of the cart from the wheel center 18 accurate to within + 0.003 inch. Another pin hole is located at the other end of the probe cart, symmetrically about the Hall probe. By reversing the direction of the cart in the track and by measuring the radial gradient of the field at the probe position it was possible to calculate the position of the Hall probe magnetic center to within 1 0.002 inch from the track dowel pin holes. The radial position of the probe was adjusted manually after each azimuthal scan. An automatic air operated twin plugger device stepped the whee.. azimuthally in 2° increments. Azimuthal positioning -.- - = was not critical due to the low azimuthal field gradients. 1.. ., t. ! S1 . " UA. . ' I Det 1. IT ust be tau . M I . . point . ' . . . "PALU 72 * Su 1. : . 2 CA 6 . - -- . . - . Figure 24. Hall probe positioning mechanism. T '. '. . . . . . ," 1 ** * W WA In : 1 1 V .. . P , . . . ! . GA 'M 5 M : 1 .. 1 , 7 1 Wh - 1. Llor." * . * . MI . VI 2 VE Diese Master control for the wheel was located in the data scanner of Piet the digital system. After the monitor and scanning probe measurements had been made, the scanner provided a contact closure which actuated a wheel control relay. The relay actuated the wheel air positioner, and the wheel rotated to the next data point. While rotating, the wheel tripped a switch, which reset the data scanner for a new set of measurements. A switch at the last azimuthal position locked the system until it was re set for a new azimuthal scan. The wheel center was located within 0.050 inch of the shim magnetic 17 center of rotation. The effect of this displacement on the midplane orbits 18 completely negligible and amounts to moving the son exit slit by the same amount. Orbits will be analyzed in the data error analysis section to verify this statement. The wheel plane was located at the median plane to within 0.100 inch. An examination of Equation (2-8) shows that this will cause an error in the field shape determination of less than 1/100,000. * System calibration. The nuclear magnetic resonance probe and scanning probe were relocated to the homogeneous portion of the magnetic field at the center of the shims. The Hall probe sensing area was aligned normal to the magnetic field lines, and the value of the flux at the Hall probe was within 0.005 per cent of that at the nuclear resonance probe. Calibration points were taken every 150 gauss between 1 kilogauss and 10 kilogauss. Two separate nuclear resonance samples had to be used . : to cover this range. A frequency proportional to the nuclear resonance 411 . TA . .. IL Y PRE - - - - frequency was produced by the Varian F-8 nuclear fluxmeter and measured by a Hewlett-Packard 524C/D counter for each calibration value of the magnetic field. A simple constant supplied with the fluxmeter was used to convert these frequencies into absolute magnetic field values. Three voltages were measured by the digital data system at each calibration point. These were the amplified control current voltage Dr, the amplifier offset En, and the amplified Hall probe output voltage Bo®, where the subscript n 18 à sequential data point index and where the superscript 8 refers to the scanning probe. All calibration Hall voltages were normalized with respect to the value of the control current and amplifier gain at the first calibration point by the equation (4-1) where F8 is the normalized amplified scanning probe Hall voltage. The coefficients of the calibration equation, - . - : ! .. 1,8 = A3 (F@)2 + A2 (F.nº) + Ag , (4-2) 7 . were evaluated with the use of a least squares curve fitting routine, using the normal' zed scanning probe voltages in and the nuclear resonance magnetic field values Bas input data 10). This quadratic equation fitted the experimental data to within the digital voltmeter resolution. A typical value of the mean of the sum of the residuals squared would be 0.2 as calculated from . - .- 3 ta hi MSRS WE (Hmº - Buna (4-3) where N, the number of data points, is typically 60 and where Hº - B 18 the difference between the least squares equation value # of the magnetic field at the data point and the observed value Bni H would range from five to ten kilogauss typically. Two separate least squares fits had to be performed for each calibration, since the callbration flux range from one to ten kilogauss was divided and covered by two different resonance samples. The low field proton resonance sample, which covered the flux range from one to eight kilogauss, had a frequency conversion equation of ES * . ADET 4 B = 7.2577 · and the high field deuteron sample, which covered the range 6.5 to 10 kilogauss had a conversion equation of B = 0.6536 · The least squares curves for these two samples did not match up exactly . . in their overlap region, and so the curve corresponding to the 0.6536 constant was normalized to the other by a multiplicative constant. The accuracy of 1/42,577 quoted for the conversion constant has no affect on . . . . . . the determination of the midplane field shape. A small error in this ..com yok.....si mi constant, if linear, would only correspond to a small change in Bo in GA - EX F ... " . . $W WAK 1. KT S. . . ATTI 1. 1.7V .. 90 M . INT "* ": *" . 77 The scanning probe was calibrated for each group of Equation (2-7). field measurements made, and these calibration curves changed by less than 0.02 per cent over three months time. For field measurements to $ 0.01 per cent accuracy, a new calibration curve was required whenever the probe current, the amplifier gain, or the probe operating temperature were changed. Also a calibration was made for every group of measurements where the digital system was shut down for extended periods between measurements. Table II shows a partial listing of the ouput data from the least squares program applied to a low field calibration. BNS is the amplified scanning probe voltage, CN the fluxmeter output frequency, DN the amplified drive current voltage developed across a standard resistor, EN the amplifier offset, DELEN the change in EN from the first callbration point, DELDN the percentage change in DN from the first calibration point, YOBS the measured values of the magnetic induction in gauss, YCALC the calculated value of the magnetic induction at the calibration point using the least squares equation, FNS the normalized value of the scanning probe voltage, and the RESIDUALS which are equal to YOBS - YCALC. All voltages are in units of volts. Table III shows the absolute field measuring errors which would be made if a calibration was not made with each set of measurements. R is the radius of the magnetic center of the probe. A single measurement for each radius was taken, and the resulting FNS Hall voltage was applied to five different calibration equations. Calibration 5 is the one which applies to the data. Columns 2-6 give the percentage absolute deviation of the resulting field values calculated 103kisarankan von anderswo...-.. .***** . . " M P 42 ........ or................... WS . " TABLE II SAMPLE OUTPUT DATA FROM THE LEAST SQUARES CALIBRATION PROGRAM o o o o o DAGENHART ROCILOMPOR ROMI 18 TANK XBX2 255 ORIS PARAMETER STD ERROR Al •10.054896•01 15,10193E-01 A2 50.54511E02 50.51679E01 A3 10.58126.00 11.75682E01 VARF 13.148176.02 VARPI 20.218156-04 SUM RESIDUALS SQUARED 55.22230E-01 ONS CN ON EN DELEN DELON YOAS YCALC 1.9987 6365.220 1.9932 .00090 0 .738716013 9.73806E«n3 1.9880 6332.470 1.9933 .00090 5.018826-03 1.680606.n3 9.68702E03 1.6872 6326.690 169932 .00060 1.000006•00 5.01082E-83 9,01435F.03 9.684566•03 1.9594 6245.000 1.9932 .00090 1.000006.00 -5.018826-039,594676•03 9,55507E03 1.9590 6244,090 1.9932 .00090 0 9.553356.03 0.553216.03 1.5318 6161.080 1.9932 .00070 O 9,426306.03 5.426378.03 1,5310 6158.560 1.9932 .00070 0 9.42252E03 9.42263.03 1.9092 6092,130 1.9932.00090 O 9.32000F.03 0.32086E03 1.9040 6076,190 1.9932 .00090 O 9.296906•03 9.296578.03 1.8794 6001,120 1.9932 .00090 0 9111646 ms 9.101586•03 1.0759 5990.050 1.9932 .00090 0 9,164705.039.16521E03 1.6493 5909.000 1.9932 .00090 0 9.040705.039.04072E.03 1.4470 5901.900 1.9932 .00070 0 9.029656•039.029956*03 1.2100 5012.950 1.9932 .00070 8.893746.73 6.89405€.03 1,6191 5810.130 1.9932 .00090 o 8,049436•03 8.819638.03 1.7089 5723,600 1.9932.00070 0 8.75704E+03 8.757526•03 1,9065 5716,490 1,9932 .00090 0.746166.03 .74625E03 1.7548 5619.360 1.9932 ..00070 O 8,50755€.n3 6.59730E+03 1.7537 5615.070 1.9932.00070 0 8,501916 13 6,59213E03 1.9282 5537.020 1.9933 .00070 0 5.018025-030,47137€•^3 8.491766•03 1.9273 5534,490 1.9932 .00000 1.00000F.no •5.01082E-93 8.467705.13 8.46787€ 03 1.7002 5451.560 1.9932 .00070 0 0.340826•n3 8.340296•03 1.6981 5444.300 1.9932 .00010 0 8.330025.038.33039E03 o o o PNS RESIDUALS 1.99000E.00 6.2592416.01 1,90720600 7.761626E-01 1,986500.00 2.150944601 1,950708.00 •2.019000E-01 1.996300.00 1.3976126.05 1.931104-00 1.133350802 1.930306.00 1.110349E-Oi 1.908506..0 2.5694376.02 1.903308.00 9.10703JE-02 109870600 5.012373E-02 1.875208.00 5.0912716-01 1.840608.00 2.4327046-02 1,846308.00 2.999556E-01 1.117306.00 +3.1071205.01 1,816406.00 •4.050665E-01 1.78820E.00 4.9045038-01 1,785808.00 -8.830019E-02 1.954106.00 2.468973E-Q11 1,75300E-00 -2.2469456-01 1,927416:00 •1,8410956.01 1726596.00 •1.6379996-01 1,699505.00 5.3306066-01 1.69740E 00 .7142496-01 o o o o o o o o o o o 1 TP N1 VITA ***** vamlancem.com.... . .... TABLE III CALIBRATION REPEATABILITY ANALYSIS . v NORMALIZED CALIBRATION REPEATABILITY, WOULD GIVE ERRORS IN FIELD SHAPE PAVEVIJ NADEV2J PADEV3J PADEVAJ PADEVEJ PASDEVIJ PRSDEV2J PRSDEV3J PRSUEVAJ PRSUEVS 12.428 2.1645 .2175 . 1952 .0974 C .0431 .0116 0139 . OUOU с 13,428 2.1016 .2165 .1942 0972 C .0402 0109 .0129 .0084 с 14.428 2.1574 .2153 . 1928 0965 C 0361 .0096 .0115 .0074 с 15.428 2,1973 .2131 . 1911 .0952 C .0311 .0081 ,0098 , 0063 UUUU 10.428 2.1465 2121 .1892 .0939 C ,0254 .0064 .0079 OUSU UUUU 17.428 2,1405 .2104 .1873 .0927 C 0193 .0048 0060 . 0038 .WOUU 18.428 2.1330 .2080 .1853 0914 C .0130 .0031 . 0040 .00 25 c 19.420 2,1273 . 2072 . 1833 0901 C 0166 .0015 .0020 ,0012 . VOUV Ĉu.428 2.1205 .2056 .1813 .0889 C ..000U c 21.420 2.1130 2042 .1793 .0877 C •.0068 -.0015 ..0020 c 2,1064 .2027 . 1772 .0864 C ..0138 *, 0040 22.420 23.428 .V0OU ..0012 ..0024 -60030 -.0029 0.0043 2.0491 .2013 . 1752 0853 ..0209 -,0061 c 64.428 2.0917 .2000 .1732 .0841 c ..0282 -.0057 ".0081 ..0948 c 25.428 2.0844 . 1981 .1713 .0830 c 0.0353 -.0469 -,0100 ..Ouby .UUUU 20.428 2.0777 .1977 .1695 .0820 c •.0419 -,0080 ..0118 ..0069 .UUUU 27,428 2.0718 , 1968 .1680 ,0812 -.0477 -.0089 ..0133 •.0077 14 . AL . . 2 80 with each of the five different calibration equations from the correct field values. Columns 7-11 give the resulting percentage error in the determination of the field shape. Field measurement data handling procedures. For each polar grid point between 0 = - 10° and 9 = 270º, and between r = 12.587 and r = 27.587 inches two amplified Hall voltages were measured. These were B. , the amplified scanning probe voltage, and B.", the amplified moritor probe voltage. The five digit voltages were punched in paper tape for later conversion to punched cards for processing on the IBM 7090 computer. The initial value of E. and Dm the amplifier offset and amplified control current voltage, were required for the first data point. No other measurements were required for processing the field measurements. Values of Dr, En, the K-3 reference voltage, the amplifier offset, and the digital voltmeter zero were taken after each azimuthal scan to keep a check on the voltmeter operation. No values of Bos or B" outside of the range from 0.0000 to 1.9999 were accepted by the computer. For values outside of this interval, which would indicate a faulty data logging, a linearly interpolated value from the two adjacent data points was used. No such error was ever made. All of the scanning probe amplified Hall voltages were normalized back to the values of the amplifier gain, the amplifier offset, and the control current which existed at the first calibration point by the equation (B." - E) (D.. - E. . F = = (Bº - Eb) - m. - - (4-4) - - - tham - - - it." red U - M . febr 19 . . P . 2 TO ii obert i R . 1 r .. 1 * Y . N . ! le - . . where Doo and E., are the amplified control current voltage and amplifier offset at the first calibration point, and D. and E, are the values of these parameters at the first measurement data poirt. B" is the value 7 - of the amplified monitor Hall probe voltage at the first data point, and B at subsequent data points. The values F were then consistent with their corresponding calibration equation. The value of F at each polar grid point was then converted into a magnetic field value using Equation (4-2). All polar grid coordinate information was computer generated. The measured magnetic field values H were then compared with the theoretically desired field defined by Vio Equations (2-7 and 2-38) to get a percentage error deviation. The theoretical field was normalized to the measured field at $ = 128° and . r r = 20 inches to compute the constant B. in Equation (2-7). The H array was written out on magnetic tape by the data processing progren for use in the ion orbit computer program. Midplane orbits were numerically computed using the H. array. A cubic Lagrange interpolation was used to determine field values between the polar grid points as the ion orbit progressed through the data grid. Table IV shows magnetic field measurement data at one position, and Table V shows the processed magnetic field measurement data at one º position. BNS, BNM, EN, and FNS are all in volts. HS is the measured * i - - : - magnetic field value in gauss, and HT is the theoretically desired value of the field. FDEV is the deviation of HS from HT, and FACTOR(R) is an " it 51 1 azimuthally invariant correction factor which will be explained in t. Chapter V. NII K . . ? . . e. 14 31 ' 1 * ca . . 12 4 . ' ., w - - -- -- TABLE IV UNPROCESSED MAGNETIC FIELD MEASUREMENT DATA RPC- 9 91). Il 11-21-43 NHL: 128 UNS 29.5870 26.5870 25.5871) 24.5870 23.5870 2?.5870 21,58711 21.5870 19.5870 18.58710 17.987 16.5870 15.5870 14.5870 13.5871 12.58711 HNM 1.849A 1,8497 1.8495 1.8497 1. À 495 1.8497 1.8496 1.8495 1.849.3 1.489 1.84AA 1.8489 1.84699 1.8461 1.8459 1,847.? 1.3448 1.3739 1.40 7.3 1.4431 1.4319) 1.5197 1.5595 1.61105 1.6419 1.6847 1.7207 1.7704 1.8134 1.8505 TH907 INS 1.3207 1.3448 1.3734 1.4173 1.4431 1.4910 1.5197 1.5595 1.61105 1.6414 1.6.947 1.7287 1.7704 1.8134 To 5135 1.8807 EN .pnog on09 0109 .019119 .000 . Onn . Onn Ono 9 OPOR .nl .0009 .no 0909 On09 .onin .Ona oriente... La mo........ . 11* TABLE V PROCESSED MAGNETIC FIELD MEASUREMENT DATA RDC 9 RDv• | | | : 1.84 8 0 40 : 8076.82 R2: 20.0 01 PHI: 128 FAS HS HS FACTOR(R) HT 27,587 1.3.19178976-001 67.7773952F+002 67.7773952E+002 67,0474897€+102 26.587 13,433 4 712E-601 68.97783526+002 68.9778.352E+002 68.66412326+002 25.587 1.3.72576656-001 70.4283656F+C02 70.4283654E+02 70.34 1 8 4245 102 24.587 14.1581 425E-001 72.1746731E00?72.1746731E002 72.0768819E+ņ0? 23.587 14.475531E-001 73.6517580E+02 73, A51758860n2 73.870759 16+102 22.587 14.79503496-001 75.7157023E+02 75.7152123E+02 75.7219254F+10? 21.587 15.182674 96-001 77.62559996+002 77,92559996+002 77.62941716+002 20.587 15.58119425-001 79.5862552F+002 62552F+002 79.58625525002 79.5917042F+002 1 9.587 15.9929326F-001 81.608.3559F + ņ02 81,60 A3559 Eon 2 A1.6163516F102 18.587 16,4103784E-001 83.65476576002 83.6547457E on 2 $3,6696500E+n02 17.587 1839 1846E-001 A5.7529659F +002 85,7529659E+On 2 A5.77628696+n0? 16.587 17.27834557-001 87.49773615+002 87,8977361Eon2 A7.91898 74E00? 15,587 17.72309956-001 90.0655956E+002 90.9655956E+002 90.0881675F-n02 14.587 TR.15279856-001 92.1569340E+002 92.1560.34 nE002 02.2715849E+n02 13.587 1.5263787E-001 93.9702318F+002 93.976231AE+002 04.4539898E10? 12.587 1.145949E-001 95.3678360F+NO2 95.3678361E002 96.6167763E+102 . . . . ? ." . 1' * . .. WRIT ULIK W . 7 . 11 12V 84 Data analyses. Figure 25 shows the measured deviation of the magnetic field of the first set of shims from the theoretical field at © = 128°, the position at which the beam is widest radially. For an azimuthally invariant magnetic field, the optimum focus would be obtained when this deviation 18 reduced to zero. The actual field has significant azimuthal variations whose effect 18 negligible for r less than 22 inches. Particle trajectories which lie partly in the field region with r greater than 22 inches are influenced by a field which is effectively weakened by the amount of the azimuthal perturbations plotted in Figure 25. The optimum field shape B.(r.) can be expressed in terms of the measured field, Bm (8,), PD (r,128°), the percentage deviation of BM (1,) at © = 128° from the theoretical field shape (See Equations 2-7,38), and AP(r), the azimuthal perturbation as B.(3,0) = B., (7,0) {1 - 10-2 * PD(7,128°) - 10-2 x AP(r)} , AP(r) 18 adjusted so that the midplane focal wiath is reduced to zero for particle trajectories calculated in the field B. (8,8). Figure 26 shows the field deviation of the first set of shims as a function of o for various r positions. Figure 27 shows the midplane focus for shim set Number 1. Figure 28 shows a magnetization curve for the XBX yoke in ... . . Building 9731. , . Digital voltmeter stability. The Kintel digital voltmeter measured . * the K-3 auxiliary output voltage of 1.5 volts after every azimuthal scan 7 during measurements. It read this voltage constant to within £ 0.0001 's . . . 1. 1 . 85 A Azimuthal perturbation • Deviation at $ = 128° batdaginn Percentage deviation 0.4 LIIIIIIIIIII 14.5 16.5 18.5 20.5 22.5 24.5 26.5 Radius (inches ) Figure 25. Measured deviation of the midplane magnetic field shape of the first set of shims as a function of radius for Tank XBX-l at ^ = 128º. Also plotted are the effective perturbations in the radial field shape deviations as caused by azimuthal field variations, and these perturbations must be added onto the deviation at = 128° to get the total effective radial field shape deviation. . . . . . . 11. 3. . . . -..- - : t UNCLASSIFIED ORNL 1R D USA.. ! IL UN TX . en , 4 . . N .... , ** * t - . : 7 P - --- ' , jcn 744 30F4 - 24 S VIA . Radius (inches) 25.568 2.568 19.568 *16.568 Percentage deviation 6 Lt ....... ......... .. ... 13.568 0 20 40 60 80 100 160 180 200 220 240 260 120 140 $ (degrees) Figure 26. Measured percentage deviation of the midplane magnetic field shape of the first set of shims as a function of azimuth for Tank 545. L 0 TTTTT Radius (inches) 245 260 250 255 $( degrees ) Figure 27. Midplane focus which was calculated from the measured magnetic field of shim set one in Tank XBX-) for B = 8,533 gauss. ior, 2 . * - Magnetic field (kilogauss ) IIIIIIIII 0 200 400 600 Coil current (amperes) Figure 28. XBX yoke magnetization curve. . volts at all times, which corresponds to the voltmeter quantizing error of + 1 digit. System stability check. Figure 29 shows the variations which occurred in the normalized scanning probe voltage F when the drive current, amplifier gain, and magnet current regulation were intentionally changed by several per cent during a measuring system stability check. The scanning probe was held stationary in the field, so that the final S normalized Hall probe voltage F should have been a constant, i.e., F should only be a function of position in the shaped magnetic field. Plotted in this figure are the variations in the normalized values F , which are within + 0.01 per cent. Any probe temperature changes, voltmeter measuring errors, or fast amplifier gain changes between the measurement of B and B", would show up as deviations in Figure 29. This check included all system electrical measuring errors and showed that the data would correspond to the calibration curve to within + 0.01 per cent. The standard deviation from 1.8532 volts for the stability check measurements was 0.000066 volts. Typical system drifts during actual measurements, which tha data normalization eliminated were less than 0.5 per cent. This stability check also shows that the final measured magnetic field is accurate to + 0.01 per cent, a determination which includes all errors except for probe positioning errors. Measurement data validation. Due to the large magnetic field gap and to the smooth shim contour, the midplane magnetic field should vary quite slowly as a function of $. A computer program was written for the +0.03 RDM-11 DOS STABILITY CHECK +0.02 +0.01 .. m -. . • * . o occ000 . PERCENTAGE DEVIATION voor Woor wood wood TV - ... -- -- · - ... .. -0.01 -0.02 60 90 120 150 150190 180 -0.03 v 30 130 Mesh Antworte_20 MEASUREMENT NUMBER Figure 29. Digital data system stability check. 91 CDC 1604 computer which checked the smoothness of the midplane measurement data. This check permitted the rejection of bad data, which had obviously resulted from logging or gross measuring errors. Each data point measurement was compared to a value at that point determined from an interpolation in a cubic equation whose coefficients were determined from a least squares fit of the data at the eight nearest neighboring points in at the same radius. The data point measurements differed from the interpolated values by less than $ 0.02 per cent with 98 per cent of the differences less than 0.01 per cent. modern Sensitivity of the midplane focus to measuring errors. The normal errors made in determining the Bnº normalization constant, Se Doo - Eool from Equation (4-4) nave no affect on the midplane focus. Hall probe field averaging errors due to the finite size of the probe sensing area cause errors in the measurements less than 0.001 per cent and have no affect on the focus. The magnetic center of the Hail probe was determined to within 1 0.002 inch. Errors in getting the wheel center exactly aligned with the shim center are completely negligible and would correspond to relocating the ion source virtual object in the magnetic field. + . Figures 30 and 31 show the midplane focus obtainable in the actual magnetic field for a source virtual object at p = $ 0.1375, instead of . . . - - - . the equilibrium orbit position of p = 0. Figure 32 shows the midplane - - . . . . . ---- - . .. - - - ... - .-.- . _ __ _ _ _ TTTTTTTTTT 17 Radius (inches) 244 III 248 252 256 $( degrees ) 260 Figure 30. Midplane focus as calculated from the measured magnetic field of shim set number two in Tank XBX-l for the virtual object at r = 22.75 inches. Radius (inches) -8° -10° 244 260 248 252 256 $(degrees ) Figure 32. Midplane focus as calculated from the measured magnetic field of shim set number two in Tank XBX-1 for the virtual object at r = 17.25 inches. 1 Radius ( inches ) 1-100'-8° IIIIIII 238 242 246 250 254 $(degrees) Figure 32. Midplane focus as calculated from the measured magnetic field of shim set number two in Tank XBX-l for the virtual object at r = 20 inches and . = - 10° and for a mass corresponding to a m/m of + 0.0177. ZO. ATT. Linn W . 2. . 95 focus obtained in the actual magnetic field if the source virtual object were displaced to = - 10° from the symmetric theoretical position and with = + 0.0177. Figure 33 shows the focus ootained for the mo conditions in Figure 30, except that the virtual object is at ^ = 0°. Thus the ordinary small + 1/8 inch displacements of the source virtual object due to mechanical construction tolerances and other uncertainties in the position of the source virtual object will cause no defocusing of the ion beam. Sufficient beam receiver movement must be allowed for if the focal point is to be followed for large source displacements. Figure 34 shows the midplane focus obtained when all of the measurement data was rounded off to one less significant figure. This rounding off increased the focal width to 0.090 inch from an original width of 0.080 inch. - ei - ! W . . TILL .. . . . *. 1 4.1 . Radius (inches) -10° IIIIIIIIIII 248 252 256 260 264 $ ( degrees ) Figure 33. Midplane focus as calculated from the measured magnetic field of shim set number two in Tank XBX-l for the virtual object at r = 20 inches and * = 0° and for a mass corresponding to a A m/m of + 0.0177. . 1--::. 1 : . - , . . : 1 . 1 .. : . : - - - 97 Radius (inches ) 245 250 255 $(degrees) 260 Figure 34. Midplane focus as calculated from the measured magnetic field of shim set number one in Tank 653 with the data being rounded off one digit. MUNI- Wate! Austfritt , CHAPTER V SHIM CORRECTION PROGRAM Program goals. Although a large amount of work went into calculating the first shim contour it was recognized from the beginning that at least two successive contours would be required in order to arrive at the desired field shape. It was calculated, using computer determined orbits, that it would be necessary to shape the magnetic field with an overall effective error of less than + 0.02 per cent from the theoretical field shape in order to achieve the optimum results from the separator. It was further assumed that by reducing the field shaping errors from + 0.4 per cent to + 0.02 per cent that the isotopic mass abundance assays from the isotope separation runs would show a significant improvement. This assumption was based on the following reasoning: A. That the various image broadening and isotopic contaminating processes are basically additive. B. That some of the processes mentioned in (A) are: 1. Object broadening caused by electrostatic lens system aberrations. 2. Energy spread in the ions emerging from the source. 3. Ion source arc plasma oscillations. 4. Residual gas scattering along the ion flight path. 5. Magnetic lens aberrations. 6. Ion beam plasma instabilities. . + 98 os . 54 . R . * *** * Fo? * * S. W 29 " T A . 99 C. That the smaller receiver pocket opening used on the 255° separator to collect the ion beam would reduce the contamination from neutral thermal charge vapor. D. That the pocket would present a smaller area for the entry of beam ions which had charge exchanged with the neutral residual gas in the flight chamber. E. That the midplane magnetic lens aberrations could be reduced from an image broadening of 0.110 inch to approximately 0.020 inch by the improved magnetic field shape. Shim correction method. The shim correction process proceeded through the following four steps: 1. Measurement of the first shim magnetic field and, the calculation of its midplane focal properties. 2. Computation of an azimuthally invariant radial field correction, which reduced the midplane focal width to 0.005 . . inch in the altered measured field. - - - -. Computation of an azimuthally invariant radial shim contour correction for the first shim contour which would produce the field correction of (2) through the use oỉ the relaxation program. 4. Magnetic field measurements on the corrected shim field, and the determination of its midplane focal properties. The first step has been explained in Chapter IV. The computation of the radial field correction function in Sümp (2) was arbitrarily ES 100 calculated as that function necessary to alter the radial field shape at 0 - 128° to equal the average of the radial field shapes at $ = 118° and 138°. This radial function reduced the midplane focal width from 0.110 inch to 0.005 inch and is the correction function used in the contour calculation in Step (3). The corrected field may be expressed in terms of the measured field as B,(1,0) = B (7,0) {1 - 10-2 x PD(7,128°) - 10-2 x AP(x)} , S where . r * B.(r.) = calculated corrected midplane field, B (r,) = measured midplane fielä, Triin. w..:mirem. PD (r, 128°) = percentage derivation at © = 128° for the measured field, and AP(r) = azimuthal perturbation caused by the strong azimuthal *:*:' Etat etmiäez lite tid v field variations. The Factor(r) function is then expressed as ariww.city terms Factor(r) = 1 - 10-2 x PD(r,128°) - 10-2 x AP(r) . Step (3) is accomplished by empirically adjusting the azimuthally invariant shim contour in the relaxation calculation until the Factor(r) change is performed on üño midplane field over as large a range of r as the available shimming range will permit. Figure 35 shows a plot of the desired Factor(r) function and the one which was calculated for the second shim contour. Table VI gives the corrected shim contour data. 1.014 1.012 ----- Desired - Accomplished 1.010 1.008+ 1.006 1.004 101 Factor (r) 1.002 1.000 0.998+ 0.996 LLLLLLLLLL 14.5 16.5 18.5 20.5 22.5 24.5 26.5 Radius (inches) Figure 35. The desired and accomplished Factor (r) function for shim set number two in Tank XBX-1. 102 TABLE VI 255° SEPARATOR MAGNET SHIM CONTOUR DATA Radius (inches) First shim contour (inches) Second shim contour (inches) Difference (inches) 0.0 5.0800 5.0000 0.0800 5.6444 13.0000 13.5000 14.0000 14.5000 15.0000 15.5000 16.0000 16.5000 17.0000 17.5000 18.0000 18.5000 19.0000 19.5000 20.0000 20.5000 21.0000 21.5000 22.0000 22.5000 23.0000 23.5000 24.0000 24.5000 25.0000 25.5000 26.0000 5.0800 5.0800 5.0800 5.0939 5.2788 5.4229 5.5232 5.5520 5.6055 5.7285 5.8052 5.8813 5.9651 6.0747 6.1856 6.2647 6.3010 6.3351 6.3731 6.4104 6.4682 6.6090 6.7797 7.0287 7.4099 7.5000 5.0000 5.0000 5.0000 5.1675 5.3502 5.5045 5.5888 5.5843 3.6093 5.6253 5.7075 5.7809 5.8626 5.9646 6.0727 6.1897 6.28109 6.3205 6.3566 6.3931 6.4133 6.4375 6.5728 6.7506 7.1139 7.5000 7.5000 0.0000 0.0800 0.0800 -0.0736 -0.0714 -0.0816 -0.00156 -0.0323 -1.103A 0.0191 0.0210 0.0243 0.0187 0.0005 0.0020 -0.0041 -0.0202 -0.019; -(1.021 -1).0200) -0.0029 0.0307 0.0362 0.6291 -0.0852 -0.1901 0.0 30.000C 7.5000 7.5000 0.0 103 Analysis of the results obtained from the second set of shims. Magnetic field measurements performed on the set of corrected shims show that a factor of 10 reduction in the effective magnetic field deviation has been obtained. Figure 36 gives a plot of the deviation from theoretical of the magnetic field of the first and second set of shims at 0 = 128°. Figure 37 shows the focus of the corrected shims at a B. field of 8,077 gauss. Figure 38 shows the field deviation from theoretical plots at $ - 128° for the corrected shims for B. field strengths of 3,620 and 8,077 gauss. These plots clearly show the field shape changes which occur when the iron at sinaller radii and at higher field strengths begins to saturate. Figures 39 and 37 show the midplane focus obtainable at the B. field strengths of 3,620 and 8,077 gaus s respectively. Figures 40 and 41 are graphs of the average enhancement factors for the isotopic separations of cadmium and tungsten. Other elements have been processed, but not enough mass abundance assays were available to make reliable comparisons. Enhancement factors for the corrected and uncorrected shims are shown along with the best average enhancement factors obtained from either the 48-inch alpha or 24-inch beta calutron isotope separators. The enhancement factor is calculated from 100 - CF E.s. 100mm Too E.F. =- -- CE 100 - where Ce = the percentage concentration of the isotope in question in the feed material, • Corrected shims o Shim set one 0 Percentage deviation TTTTTTTTTT 104 14.58 17.58 20. 58 2 3.58 26.58 Radius (inches) Figure 36. Midplane magnetic field shape deviations for the original and corrected shim sets in Tank XBX-1. SEX 105 0 - - 5 - --- - - - - - - Radius (inches) :- Y . - - - . - - - . - - 01 .8- .. 250 258 260 Eni 252 254 256 $ ( degrees) - 1. - Figure 37. Midplane focus as calculated from the measured magnetic field with a B = 8,077 gauss for the corrected shime in Tank XBX-1. - -- I - + - ::--.€ . - - . •2.. - * -- Fe o Bo - 3620 gauss • B. - 8077 gauss Percentage deviation 106 * 12.587 16.587 24.587 27.587 20.587 Radius (inches) Figure 38. Measured midplane magnetic field shape deviations for the corrected shims at B * 3,620 gauss in Tank XBX-1. 107 Radius (irches ) \-10• -8. -6° IIIIIIIIIII- 250 252 254 256 258 260 $ ( degrees ) Figure 39. Midplane focus as calculated from the measured magnetic field at B = 3,620 gauss for the corrected shims in Tank XBX-1. 800 n Corrected shims Shim set one Alpha calutron e 600 108 Enrichment factor 11300 108cd Hicd Isotope Figure 40. Average enhancement factors for the best cadmium isotope separations series in the alpha calutron and 255° separators. 109 200 corrected shims Shim set one Bota calutron @ Enrichment factor Allllll \\\\ 180 W 182 184W 186w 183 W Isotope Figure 41. Average enhancement factors for the best tungsten isotope separations series in the beta calcutron and 255° separators. 110 Co = the percentage concentration of the 1sotope in question in the separated sample. Figure 42 shows a plot of the average contamination factors for the 108ca 1sotope as obtained by using all data from the corrected and uncorrected shim isotope collections. Approximately the same results hold true for the 111ca and 113ca isotopes, which were extensively assayed. The contamination factor 18 calculated from where A, = mass abundance assay of the 1° contamination in the 108ca pocket sample, A = mass abundance assay of the 108cd in the 108ca pocket, N = mass abundance of 108ca in the feed material, and N, = mass abundance of the i th contamination in the feed material. This contamination factor gives the contamination from neighboring isctopes, within the assay uncertainties, if all were of equal natural abwidance. Figures 43 and 44 show the midplane field deviation from theoretical at 0 = 128º and the focus obtained for shim set one installed in Tank XBX - 2, which has a gap width approximately 0.070 inch less than that of Tank 1. This gap width error produced an image broadening to 0.135 inch from the 0.110 inch determined in Tank 1, which has the theoretical 16 inch gap. All assay data presented for shim set one was 111 A Corrected shims • Shim set ono Contamination factor 0.01L IIIIII 108 110 112 114 106 116 Mass - . it 3 - Figure 42. Isotope contamination factors for the 108ca isotope using samples collected with the corrected and uncorrected shims in Tanks XBX-1 and XBX-2 respectively. . Percentage deviation ATT 14.43 23.43 26.43 17.43 20.43 Radius (inches) Figure 43. Measured midplane magnetic field shape deviations for shim set number one in the magnetic gap of Tank XBX-2 which is 0.070 inch too narrow. 113 UNCLASSIFIED ORNL OWG. 64-10615 Radius (inches) \-100 245 245 260 250 255 $( degrees) Figure 44. Midplane focus as calculated from the measured magnetic field of shim set one in the narrow gap of Tank XBX-2. 114 collected from runs made in Tank XBX - 2. CHAPTER VI EXPERIMENTAL TESTING OF THE SEPARATOR AND CONCLUSIONS Separator testing. This chapter will give a brief review of some actual 1sotope separations performed in the 255º 1sotope separator to - . 2 - verify the basic accuracy of the theory and magnetic field measurements. Figure 45 shows the first experimental ion source at the bottom and the receiver at the top, mounted on the vacuum tank front, which is mounted on a transporter used in installing this equipment. Ions are formed in the source at the bottom of Figure 45 are accelerated to 35 key and travel in a 20-inch radius circle to the receiver shown at the top. Figure 46 shows the midplane ion beam outline and several º plane beam cross-sections. Figure 47 shows a close up of the most up-to-date ion beam receiver which has straight entrance slits and pockets. This straight image was obtained by using a curved ion exit elit at the source. Thus, unlike the calutron, the 255° separator receiver may be completely constructed without resorting to the machining of complex curves. The receiver operates within one-fourth inch of the theoretical position and at the theoretical focal angle. The angle of the focal plane has been determined from computer trajectories to be 42° from the incoming ion beam. Ions with - Am focus earlier than er, and those with + o m focus later than fr' These trajectories also show a slight curvature of the midplane section of the focal plane. Ions with om * 1/10 focus approximately one-tenth inch later in ø from the flat . . . . . focal plane. . . - 115 . . . L 116 w 11 " CEN 1 . 2 EX 11 Ihn . y 1. . ! " . . A 2 . . . TV IN * 11 " . Win SU IA 74 P M . . 124 mi . . l . . . V IN W . W M RE UY ' " F . 17 . + . - . . - - - - Figure 45. First experimental ion source and receiver for the 255° separator mounted on the vacuum tank front. --, ....SO ' . • Plane beam cross sections = 255° Midplane beam outline $12000! Receiver $=1270 ***** Doi $=640 Source $ = 0° Straight object Curved object Not to scale Figure 46. 255° isotope separator midplane ion beam outline and several 0-plane beam cross-sections. 118 Nr 1- 122 . be W Riz-.- --- . 4 . ". Live WO '1': ! . 2. ! -. i SMP . . . . S 71 W OT ** in . . gui, . .. ! 1 !1 ail In 1 11! " ad. " " . W 11' ) 2 1 . . ." W .. 3 * *7 ." i 11 2. . IV. 262 7. " . . - - . I . 27 I WAY i " I ' , ! 1 . 7 N 1.!!! . . , . MA 2 . , . - 11. 7 ni . ws - indist 11., S 7, . w H WA 17. . * A H. ol 1. . . LMU 5 " 1 WYX ', WI ' . . i : " yurt TI W " 16 IN ... f 1 1 4 1 . .. 5 7 * - ' It S . .. .- - VI - IT an - LIN - NE? 2 1 = . IM" 7 . W . 5 1. O hey 1 Y 11 ber 1217 W " " . DR. VW . . . VI. - .. - ' . LINO A. . ! 2 - ER Figure 47. 2559 separator straight focal pattern receiver. , 19 119 Figure 48 shows a graph comparing the best isotopic enrichments of the 255° separator to the best results of the 24-inch beta and 48-inch alpha calutrons. The 255° separator has provided beam currents equal to those of the alpha and beta calutron separators and has made possible the simultaneous collection of all of the isotopes of those elements where this was not previously possible due to the lower dispersion of the beta calutron. The higher resolution has been especially valuable in the isotopic separation of those elements above mass 100, where geometrical focusing plays a more important role in obtaining high 1sotopic purities. Figure 49 shows an electrical beam scan made during a mercury accelerating voltage was varied, and the varying current signul across a resistor from the collector pocket to ground drove the X-Y plotter ordinate. A voltage was generated for the abscissa which was proportional to the accelerating voltage variations. Figure 50 shows a receiver graphite faceplate after a thirty-six hour ion beam collection which used a straight 1/8 x 4 inch ion exit slit. It shows that the beam fits the slot quite nicely. Figure 51 shows a normalized isotopic contamination plot for the 108ca isotope. The mass abundance assays used for this graph are from the best individual runs from the two preceding 48 alpha calutron collections and from the best run of this first 255° separator collection. This graph shows that a significant reduction has been obtained in the contamination of the 108ca isotope pocket by isotopes with lerge om Enrichment factor 600 800 99 Ru 108 Cd III Cd 113 Cd 1175n 119 Sn Isotope 138 La ONL beta, and 255° separators. Figure 48. Isotopic enrichments which compare the best single sample results of the alpha, 180 TO M081 M281 Alpha calutron Beto calutron 255° Separator 183 W 184w 186 W 1 Ν 206 Pb 208Pb - 6,660 26 Mg AW L UB WW 4,160 - 120 202 200 199 Beam current (microamps ) 121 198 204 196 Isotope Figure 49. Mercury ion beain scan using a 0.046 inch by 4 inch receiver beam defining slot in the 255° separator. -- - - .. - 2 3 iNCHES * .01 14 April PL W . dd PM "NA" w . - 06 /14 10 R . N 7 . * 2.0 L . 122 Figure 50. 2559 separator receiver beam defining faceplate (using a straight object) after a long experimental collection. EN LNB CE . I . S 1 4 . . 4 : LA 1 po . . a ON LU W 1 . A2 2 le C * sy Bu 4 Lit u 7 ir aq'-***-*...** - *.27 : mm. - het nie *was*** 1 .. --- . ***•=..* 100 o 255° separator , 1963, 88.6% • Alpha colutron, 1962, 84.0% A Alpha colutron, 1960, 82.4% Contamination factor 123 TTTT 0.014 Ver 106 108 110 112 114 116 Μα88 Figure 51. Isotopic contamination factors for the best 108ca isotope sample from each of three best different ORNL isotope separations. 124 from the 108ca. This result may be partly due to the fact that the 255° separator ton path length 18 0.59 times that of the 48 inch alpha calutron while keeping 83.5 per cent of its mass dispersion. The shorter path length reduces residual gas scattering. The 255º focal pattern 18 much smaller than the 0.400 inch alpha calutron focal width at B = 1 10°. This contamination reduction holds true for the 111ca and 118 ca isotopes, which were also extensively assayed. Conclusions. The approach used in developing the 255° separator has proved very rewarding and would not be changed significantly for solving another problem of this type. The n = 1/2 inhomogeneous magnetic field was first investigated theoretically and was shown to have very excellent beam focusing properties. This field configuration was shown to be the best for adaptation into the beta calutron tanks mainly due to its double directional focusing property and to its high dispersion. Relaxation calculations were then used to solve the magnetostatic problem of determining accurately a shim contour to provide an accurately shaped n = 1/2 inhomogeneous field. It was recognized from the start that the classical equipotential contours would introduce large field errors due to the large gap width to radius ratio used and to the abrupt truncation of the contours for mechanical considerations. The initial set of shims performed very satisfactorily and provided isotopic samples during the first experimental separations which were better than those obtained either from the 24 inch beta or 18 inch alpha calutron separators. . 125 The magnetic field measuring program provided + 0.01 per cent accurate midplane magnetic field shape measurements which were used to compute ion beam trajectories for determining the focal properties of the shims. This data was used in conjunction with the relaxation program to compute shim contour corrections which resulted in a factor .. :. , , of 10 reduction in the midplane magnetic field errors with the second . - set of shims. The effective field shaping error of the second set of shime is within 1 0.02 per cent of that shape which would produce a point image in the midplane. The focal width was reduced to 0.025 inch from 0.110 inch with the second set of shims for B = + 10°. A program to reduce the azimuthal field variations does not appear to be worthwhile due to the large effort involved and to the small perturbations which they probably add to the three dimensional focal pattern. The Hall probe measuring system used has provided very accurate field shape - si determinations and seems to be the most expedient field measuring method for mapping or monitoring magnetic fields. Accurate absolute measurements are obtainable with the use of nuclear magnetic resonance equipment. Two of these 255° isotope separators have been in constant use since May 28, 1963 and have performed isotope separations on 10 different elements at competitive outputs with an average increase of two in the enrichment factors over those obtained with other ORNL separators. ." - + S BIBLIOGRAPHY 1. 2. Svartholm, Nils. "Velocity and Two-Directional Focusing of Charged Particles in Crossed Electric and Magnetic Fields," Arkiv for Fysik, 2:195-207, No. 20, 1949. Artsimovich, L. A., et al. "A High Resolving Power Electro- magnetic Isotope Separator for Heavy Elements," Atomnaya Energiya, 3:483-91, No. 12, 1957. (Translated by the ORNL Library Staff March 4, 1958.) 3. Bretscher, M. M. "Focusing Properties of Inhomogeneous Magnetic Sector-Fields," Oak Ridge National Laboratory Report ORNL-2884. Oak Ridge: 1960, pp. 4, 8. 4. Goldstein, Herbert. Classical Mechanics. Reading, Massachusetts: Addison-Wesley Publishing Co., Inc., 1959, pp. 291, 228, 223, 234. 5. Tasman, H. A., and A. J. H. Boerboom. "Calculation of the Ion Optical Properties of Inhomogeneous Magnetic Sector Fields, including the Second Order Aberrations in the Median Plane," z. Naturforschg, 149:121-129, 1959. 6. Beiduk, F. M. , and F. J. Konopinski. "Focusing Field for a 180° Type Spectrograph," The Review of Scientific Instruments, 19:594-98, No. 9, 1948. 7. Huster, Von E., G. Lehr, and W. Walcher. "Ein B-Spektrometer mit Weitwinkel-Doppelfokussierung," 2. Naturforschg, 102:83-84, 1955; eingeg. am 10. Oktober 1954. 8. Verster, N. F. "Spherical Aberration of a Double Focusing Beta Ray Spectrometer," Physica, 16:815-16, No. 10, 1950. Weber, Ernst. Mapping of Fields. Vol. l of Electromagnetic Fields. 2 Vols. New York: John Wiley and Sons, Inc., 1950, pp. 45, 47. 10. Leitzke, M. H., "A Generalized Least Squares Program for the IBM 7090 Computer," Oak Ridge National Laboratory Report ORNL-3259. Oak Ridge: 1962. 11. Whitehead, T. W. "A Study of Magnetic Fields for Use in the ORNL Isotope Separations Program." Unpublished Master's thesis, The University of Tennessee, Knoxville, 1963. 127 - ! -* APPENDIXES We are the = 7 EP 23 APPENDIX A DERIVATI ON OF THE VECTOR POTENTIAL FORM The vector potential components A, and A, may be expressed with 7 a power series as V 2 land 8 WiWi імя імя 4679.99 - E _ V Z . (A-1) (A-1) ISO MFO no The magnetic induction field B is calculated from g=x À. (A-2) For the cylindrically symmetrical magnetic field being investigated, B. = 0, thereby imposing the restriction on the vector potential coefficients that (A-3) Substituting the expressions in Equations (A-1) into Equation (A-3) gives 8 filom n = 0 بے . (A-4) mo n=o For Equation (A-4) to be true requires that 129 - - - . 130 in , which imposes the restriction that can and B. be zero for 1 and n greater than zero. We may define À such that Ñ À = 0 with no effect on B, since this amounts to an adjustment of À by addition of the gradient of an arbitrary scalar function whose curl 18 zero. Using a power series definition o + --- ..... 1.mn 2 r .. . and the divergence expressed in cylindrical coordinates as } (A-5) produces 1.mn ro 2 IH 3 [a" 1.** MFO n=0 mA cannon 1=0 MFO no 00 00 00 NB. nzn- Imn (A-6) 1=0 mFO n=o • IN By the previous restriction that B. = 0, Equations (A-6) reduces to - E - 9 . 131 МА PO lo MPO neo leading to E forme 200, (A-7) mpo which requires that all terms in Anmn containing m higher than o must vanish and that come = 0 for all m. Thus An 18 not a function of º. omo Examining the r component of the curl of A, JA, DA B. - Jooz, produces Mo IM: Sie mBummałom-2n & Wi8W! Mo IM8 na ro0 zn-1 (A-8) Forro which by previous restrictions reduces to a IM8 ) mBom-1 naorrten-i. (A-9) Omo Polo CE ten . 132 SA All B terms must vanish except Bo to keep B. finite at the origin and B18 arbitrarily set equal to zero since it will have no influence on the field shape. Thus the vector potential as previously restricted has only an A, component expressed genrally as (A-10) n=o Examining Equation (A-9) and using the symmetry condition that B (,2) = - B(r, - 2), reveals that the Ann coefficients in Equation (A-10) must vanish for all odd n's. W APPENDIX B DERIVATION OF THE THREE DIMENSIONAL MAGNETIC FIELD AND VECTOR POTENTIAL FIELD EXPANSIONS The following appendix will present the derivation of the three dimensional megnetic field expansion and of the expression for the vector potential coefficients in terms of the magnetic field coefficients. Three dimensional magnetic field expansion. The field component equations for a magnetic field which has cylindrical symmetry and which 18 mirror symmetrical with respect to the magnetic midplane 0 = 0 are B. = Bo {(1 + a20p + 220p@ + 2300* + 2.400* + asop5 + ...) + (202 + 2120 + a2202 + 43203 + ...)02 +(80*85* ...} , and B = Bo (bol + b120 + b2202 + b320 + 64204 + ...), + (bos + b23P + b23p2)o® + 60505 + .. (B-1) Maxwell's equations expressed in normalized coordinates for the field shape used give dB dB = 0 s = 0 (B-2) 133 ! -.- 3 134 from the curl of B and het en of TJ Bu (B-3) from the divergence of Substituting the expressions for the field components in Equation (B-1) into the curl condition in Equation (B-2) produces (bo1 + b110 + b2102 + b3103 + 64204 + ...) + 3(bos + b132 + b23p2 + ...)02 + 5050* + ... - (220 + 2a20p + 3a3002 + 424002 + 58500* + ...) , - (212 + 2a220 + 3a3202 + ...)02 - 21404 + ... = 0. ......... Adding and equating the sum of coefficients of equal powered terms equal to zero produces ..... boi = ajo , ........ 021 = 2820, b21 = 3aso, . b31 = 4240 , b42 = 5850 , 135 b23 = 232 , and bos = 5 814 · (B-4) Substituting the field component expressions in Equations (B-1) into the divergence condition in Equation (B-3) produces {(012 + 2b210 + 363182 + 464208 + ...)o + (b13 + 2b230 + ...), + b220 + b2202 + bains - boje - 61202 - b2103 + bospa + 62203 - bolo + ...)0 + (bo3 + b238 - boso + bo3p2 + ...)03 22120 + 222202 + 2a3202 + ...), + (4204 + 489.40 + .../09}+ ... -0. + ... = 0 . (B-5) Adding and setting the sum of coefficients of equal powered terms equal to zero produces 8o2 = - 820 NIP 802 = - 220 - 2 210 , 212 = -3230 - 220 + { 420 , - - (12840 + 3a30 - 2820 + aro) , 229 - - ] (20850 + 4640 - Bago + 2220 - 820) , a23 a 50 + lado - 3azo + 2a20 - 136 ao4 = a + សម , and 204 = 240 + į 230 - Že 420 + žt 420 , and 234 = 5850 + 20.40 - 290 + 220 - 230 · (B-6) Expressing the off-midplane b terms as a function of the midplane B. coefficients produces bos -- 230 - 220 + Ž 220 -- 4240 - 230 + $ 220 - 240 113 = - 4240 } b23 = 832 , and bos = 250 bos = 250 +240 - zo 290 + žo a20 - 210 · Unino (B-7) induct: calculated f expresses the field Three dimensional vector potential expansion. The magnetic induction Bºmay be calculated from '= 3 x A, which expresses the field components of the cylindrically symmetrical field studied in terms of the vector potential as components of vlindrically ymmetrical field studied in terms of (B-8) Bo - Folo+ Š {co + 1)ąs} . (B-9) Using the magnetic field component expressions of Equation (B-1) and the form of the vector potential as derived in Appendix A and substituting them into Equations (B-8,9) leads to the expression of the vector potential coefficients in terms of the midplane field expansion if . A . . NA M 1 WTO 1 . . ! r. " 4 LA 1.11 M 1. . 1 So UNCLASSIFIED . . V . V ORNL 5 - - Pixy Ti. 1, . . . . . " .. 744 40F4 . . 137 coefficients of B • Performing these operations in Equation (B-8) leads 2+ A12p + A2202 + A3 zp® + A42D 4 + ...) doo ro - . -. *. - + 40P(A04 + A140 + A2 402 + ...) + 605 (B-10) where only odd powered o terms are nonzero due to the symmetry condition on B. Using the expression for B. in Equation (B-1) and collecting and equating the coefficients of equal powered terms equal to zero produces Ao2 = -roBoboi , A12 - roBobil, A2 = A32 Bob31 A4, = 041 , A04 = - roBob21, Age. - - roBobsa , A42 = - roBobaz , robobos, roBobis , roBobas , and Ž robobos · A14 = A24 = Aos = (B-11) 158 Performing the indica'. d substitutions and operations in Equation (B-9) produces + Abo) + p (2A20 + A10 - 100) + 02 (3Ayo + Azo - A10 + Aco) + pº(4A40 + Azo - Azo + A20 - 400) + 04(5A50 + A40 - A30 + A20 - A10 + 10o) +05(6A80 + Aso - A40 + Ago - A20 + A10 - A (A12 + Aoz) + 0(2A22 + A12 - Aoz) + 02 (3A3 2 + A22 - A12 + App) 4 03 (4A42 + A32 - Aza + A12 - A02)} + 0*{(A14 + A04) + P(2424 + A14 - A4) 4 + 124 + A14 - AC (B-12) Using the expression for Bfrom Equation (B-1), and adding and setting the coefficients of equal powered terms equal to zero produces A00 = 3 roBo (See Equation (2-5)), A10 = - 3 roBo , A2 = { roBoano , A3o = { robo(azn - 430) , A80 = roBo (230 - 120 + { 270) , t . - -... ... 139 - ap2 - Aso – 5 roBo(ado - $ 850 + 220 - 3 010) , 100 = { robo(250 - 300 + } 230 - 220 + 110) , Az2 = YoBo (802 + { bol), z ro Bo (212 - 892 - Boa), As2 = robo(#22 -2,2 + 202 + { boa) , roBo (232 - 222 + 412 - zace - 2boa) , A14 = roBo (804 + 003) , and A24 = } ro Bo (214 - 204 - į b03) · ao + a22 + 2 - 2a02 - 2 i • .. . . . - - .-' A24 . - a04 - (B-13) . - .4 . : Substituting expressions for the b's in terms of the a's yields . ..gi . . the vector potential coefficients in terms of the midplane B. coefficients - through terms of the sixth order as - ixli. - 9 :-. Boaio , * -- *-- ADD = roBo , A20 – roBo A20 = z ro Boazo , POBO (20 – į 2.0) , Ako = robo(830 - 3 120 + ] 2:0) , roBo(aso - ] 230 + {a 220 - 240) : *- '.•. - • - I 220 + Win 140 Aco = robo(aso • 24 + 230 - ] 220 + 420) , A2 = - roBoazo , roBoao , A12 = - ro Boazo , Az2 = - Ž PO Boaso , Boa3o , Ag2 = - croBoa 40 , 2 Boa 50 g A42 = -roBoaso , A04 - 1a robo (3290 + tipo - 210) , Ź robo (62.40 250 - 420 + {230) Az: - f ro Bo(10850 + 2240 - Ž 180 + 820 - 410) , and 106 = ZO Yo Bo (10850 + 4a 40 - 230 - 220 - 210. (B-14) A complete list of the three dimensional field coefficients in terms of the midplane B. coefficients through terms of the fifth order is as follows: apo = 1 , a10 = a10 , azo = a 20 , a3o = 830 , 840 = 840 , a50 * a50 402 - a20 - NIN o , Y u - 212 = - Zago - - 220 + 410 , (12240 + 3ago - 2220 + a10) , (20850 + 46.40 - 3230 + 2220 - 210), 204 = 840 + { 230 - ] 220 + 210 , + 2840 - 490 + ] 220 - į 810 , - ... - - .. - . .. . . .. boi = 210 , .. . .. . bin = 2820 , b2i = 3a30 , b31 = 4840, 641 = 5850 , - 230 - 220 + 230 , 019 = - 4880 - 230 + ſ 220 - 240 , b23 = 232 , and . -. -. - * * * * * * * * * ਟੇt bos = 850 + £ 84 – ਫੈਠ ਕ30 + ਫੈਚ 20 · ਫਿਰ 830 · (B-15) -- : ' - -- I ', · · - . " '{." ਤੇ " -- . - - - - .: . . APPENDIX C MEASURING EQUIPMENT LIST AND PERTINENT SPECIFICATIONS (1) 4590/ER3401 Kintel differential amplifier Gain: -2,-6, -20,-200 with 1 per cent vernier control. The -20 position was used for the field measurements. e n *,** Gain Stability: $ 0.02 per cent for 8 hours with constant . *:-1- h temperatures and with power line at 115 + 3 volts, im -,.. and + 0.05 per cent for 40 hours. dc Linearity: + 0.01 per cent of full scale output with load impedance of 1000 ohms and for either signal polarity: Frequency Response: 0 - 1.4 cps. Common Mode Rejection: 180 db at dc, 1.30 db at 60 cps with input guard returned to common mode source and -.. . , ' . *.* ..*.!. ---- with an unbalance of up to 1000 ohms in series with either input lead. Floating input and output with 1013 ohm isolation -- - - - - . . - . . between the input leads and the chassis. . . . . . . . . Drift: $ 0.7 microvolts on a gain of -20 for 40 hours after a - - - . - - - 30 minute warmup. Noise: Less than 1 microvolt peak to peak in a 0-3 cps band. - . . 1 dc Input Resistance: Greater than 30 megohms. Output Impedance: Less than 0.25 ohm, de to 509 cps. 143 144 Maximum Output Capability: + 2 volts dc. Sensitivity: 5 microvolts for gain -20 using the digital voltmeter with a 100 microvolts least digit sensitivity. Input Range: 0-100 mv (2) 456B Kintel 5-digit voltmeter Measurement Range: 0.0000 to £ 1.9999 "rolts. Range and Polarity: Constant for this application. Measurement Accuracy: 0.01 per cent of reading # ) digit. Input Impedance: 10 megohms at null. Input Filter: 60 db attenuation at 60 cps. Average Reading Time: 4 seconds. Output: 10 line decimal. Display: Visual readout, (3) 473 Kintel digital readout. (4) 453M/ER3401 Kintel mechanical stepping switch scanner. (5) ER3401 Kintel Programmer Buffer which performs the following functions: (a) Counts the characters per word punched. (b) Performs a 10 line decimal output to a 1-2-4-8 BCD serial output conversion to drive the paper tape punch. (c) Performs a 10 line decimal to parallel staircase voltage output conversion to drive the printer. (a) Controls the X-Y analog plotter. (6) 560A Hewlett-Packard paper tape printer which has an analog output. U .. . " . 145 (7) BRPE-2 Teletype paper tape punch which operates at 30 characters per second. (8) F-8 Varian nuclear magnetic resonance fluxmeter, 1-52 kilogauss. . . .. * . APPENDIX D 255° ISOTOPE SEPARATOR CONSTRUCTION AND OPERATING PARAMETERS Magnet Construction 1. The two shim plates are installed in a standard beta calutron vacuum tank which has two pieces of iron cut out of the diffusion pump manifold throat.. Vacuum tank inside dimensions are 16 inches wide, 51 inches deep, and 68 inches high. 2. Coil core dimensions -- 63 inches deep by 83 inches high. Gap width -- 12.1454 inches. 4. Equilibrium radius -- 20 inches. 5. Shim taper angle -- 8°38', 6. Focal Angle -- 254.56º. 7. Midplane field constants are - n = a10 = - Ž, and 240 = 256 , and aso = 522 146 147 8. Weight of two shims is 2,800 pounds. 9. Shim iron analysis is 0.13 per cent carbon and 0.40 per cent manganese. 10. B working range is 1-9 kilogauss. li. The effective midplane magnetic field deviation at B = 8.5 kilogauss is 0.02 per cent, 12. Field shape change as B. changes from 3.6 to 8.5 kilogauss is 0.06 per cent. 13. Minimum air gap is 10 inches. 14. Rough shim dimensions are 3 inches thick by 60 inches in diameter. 15. Field regulation is + 0.01 per cent over 10 minutes. Vacuum System 1. System volume is 2,450 liters. 2. Two 20 inch oil diffusion pumps with a net pumping speed of 10,000 liters/ second. 3. Vacuum is provided by a standard beta calutron system. 4. Operating pressures are < 2 x 10-5 Torr. Electrical Supplies 1. Accelerating voltage is 20 - 40 kilovolts with l ampere of current. Stability is 0.01 per cent. 2. Focusing lens potential is (-)5 to 40 kilovolts at 1 ampere of current. Stability is 0.1 per cent. . . . . . . . . . . . . . . . . . - *. on " "..." va come Win W .. " ! , 7 V '') LA 1. . VER WWW . , ' LUM . ... " ". TB2 148 3. Arc supply is 30-300 volts dc at 0-10 amperes. Filament current 18 0-500 amperes dc. 4. High voltage recycles automatically. Ion Source 1. Electron bombardment type with slit extraction. A typical slit size is x 5 inches. 2. Source efficiency is 5-10 per cent. i mi 3. Manually adjusted arc chamber and oven heaters. 4. Arc supplies are electronically regulated. 5. Maximum beam current to date is 135 ma of magnesium. . Receiver 1. Uses completely straight pockets, with straight entrance slits, and with a flat faceplate. 2. Pockets are made mostly from graphite and water cooled copper except where a special chemical reaction is desired for better retention. . - -. - .-. . - . - -4 . * * 2- DATE FILMED 3 / 23 /165 2K Share 1 S li LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accu- racy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the Use of any information, apparatus, mathod, or process disclosed in this report. As used in the above, "person acting on behalf of the Commission” includes any em- ployee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor, i END