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Magnetic feedback and quantum oscillations in metals Van Schyndel, André John 1980

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el MAGNETIC FEEDBACK AND QUANTUM OSCILLATIONS IN METALS B . S c , McMaster Un i ve r s i t y , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES Department of Physics We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA by ANDRE JOHN/VAN SCHYNDEL MASTER OF SCIENCE in October 1980 0 Andre John Van Schyndel , 1980 In presenting th i s thes is in p a r t i a l ' fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date O c t . ) Q ) q % 0 ) i i A B S T R A C T A feedback technique is presented fo r the reduction of the Shoenberg magnetic in terac t ion in metals. The method allows the spin s p l i t t i n g parameter g c for extremal o rb i t s on the Fermi surface to be obtained from de Haas-van Alphen measurements, now e s s e n t i a l l y f ree from the o f t - t imes severe d i s t o r t i on s re su l t i ng from magnetic i n te rac t i on . The feedback technique a l so o f f e r s several advantageous s ide e f f e c t s , the most important one being a simple and r e l i a b l e method for determining absolute amplitudes of de Haas-van Alphen o s c i l l a t i o n s . E x p l i c i t formulae are derived showing the dependence of several key observable quant i t ie s on the amount of magnetic feedback, and these formulae are found to be in good agreement with experiment. The technique is appl ied to the determination of g c for the [110] y o s c i l l a t i o n s in Pb. TABLE OF CONTENTS page Abstract i i Tab 1 e of Contents . . . . i i i L i s t of Tables v L i s t of Figures v i Acknowledgements v i i i CHAPTER I - INTRODUCTION 1 CHAPTER II - SPIN SPLITTING OF LANDAU LEVELS 5 CHAPTER III - THE SHOENBERG MAGNETIC INTERACTION EFFECT 13 CHAPTER IV - REDUCTION OF MAGNETIC INTERACTION USING A FEEDBACK TECHNIQUE 20 CHAPTER V - EXPERIMENTAL DETAILS 27 5.1 Sample Preparation 27 5.2 Detection Apparatus 31 5.3 Modulation Co i l and Superconducting Magnet . . . . 36 5.4 Cryogenic Apparatus 37 5.5 Signal Processing 39 CHAPTER VI - EXPERIMENTAL TEST OF THE FEEDBACK TECHNIQUE 50 6.1 Prel iminary Considerations 50 i v page 6 . 2 Minimization of Sidebands 52 6.3 The Mass Plots 62 6 . 4 The Beat Pattern 69 6 . 5 Phase Information 8 6 6 . 6 The L i n e a r i t y of A J / A J vs. ( A ] / A 2 ) 2 91 6 . 7 Conclusions •••• 91 CHAPTER VI I - EXTRACTION OF THE g'c FACTOR FROM A j /A^vs» (A ^  / A^) •••• •••• 9 ^  CHAPTER VIII - A SEARCH FOR THE 4 MG OSCILLATIONS.. 9 9 8.1 Prel iminary Remarks 99 8 : 2 Review of the Standard Weak Modulation So lut ion 100 8 . 3 Large Modulation 103 8 . 4 Mod i f i ca t ion to the Apparatus and Analys i s for the F ^ 4MG Search 112 APPENDIX A: F l e x i b l e Gear Rotator 114 APPENDIX B: The D i screte Four ier Transform 119 APPENDIX C: Computer Programs 122 BIBLIOGRAPHY 134 V LIST OF TABLES Page I Four ier Coe f f i c i en t s p, q 16 II E f f e c t i v e Mass for Observed O s c i l l a t i o n s in Pb:H||/[110l 64 I I I Ranking the Terms 109 vi LIST OF FIGURES page FIGURE 1. The S p i n - s p l i t Magnetization at Absolute Zero 9 2. The dHvA Magnetization at F i n i t e Temperature . . . . 10 3 . Crysta l Diameter as a Function of Melt Heater Voltage 28 *t. Detection Arrangement . . . . 3^ 5. General Schematic Cryogenic Assembly 38 6. Fine Tuning C i r c u i t . . . . kO 7. Equivalent C i r c u i t for Tuning Arm kO 8 . Block Diagram of Apparatus . . . . kk 9 . C i r c u i t Diagram of Integrator and Adder kS 10 . Synchronization of the Time Window A8 1 1 . dHvA Osci 1 lations;! i.nLead Along [ 1 1 0 ] 51 12. Four ier Transforms Around 160 MG 53 13 . Fourier Amplitudes as a Function of Feedback Gain 5k \k. Mass Plots with No Feedback 65 1 5 . Mass Plots with Near-Optimal Feedback 6 6 16. Mass Plots at Optimum Feedback 70 17 . Beat Envelopes with No Feedback 73 18. Beat Envelope with Near Optimum Feedback Ik 19 . Ca l cu la t ion of Apparent Beat Periods 79 vi i page 20 . Ca lcu lated Solut ions for 80 2 1 . The Beat Envelope with Optimum Feedback . . . 84 22 . Ideal L.K. Beat Envelope 85 2 3 . Individual O s c i l l a t i o n s at the "Magic F i e l d " 88 2k. Individual O s c i l l a t i o n s at l /H- of Figure 22 89 2 5 . Phase D i f ference and Amplitude of A2 at a "Magic F i e l d " 90 26 . A . /A , vs. ( A . / A j with and without Feed-back^ 1 92 27 . Fundamental y Beat Envelope 97 28. Sample Rotator Assembly . . . . 115 29 . Sample Rotator Assembly with Dr iv ing Gear and Coi l Former . . . . . .1 117 ACKNOWLEDGEMENTS It is a s incere pleasure to thank Dr. A.V. Gold for h is support and d i r e c t i o n in the supervis ion of th i s work, and his c lose personal in terest in provid ing advice and encouragement. I am gratefu l to the National Sciences and Engineering Research Council for the i r f i n anc i a l support in the form of a Postgraduate Scholarsh ip. 1 CHAPTER ONE INTRODUCTION In 1930, de Haas and van Alphen not iced that the magnetization of bismuth o s c i l l a t e d as a funct ion of an ex terna l l y appl ied magnetic f i e l d at low temperatures. This remained a laboratory c u r i o s i t y for almost 20 years un t i l i t was rea l i zed that th is de Haas-van Alphen (dHvA) e f f e c t could be used as a powerful tool in the study of the Fermi surface of metals. Valuable information on the deta i l ed shape of the Fermi surface can be obtained from the frequency o f : the magnetizatton o scM la t i ons i . i n the -inverse f i e l d domain, and i t was soon found that the o s c i l l a t i o n s are exhib i ted by most metals in the per iod i c tab le . There is a l so a wealth of information contained in the harmonic content of the o s c i l l a t i o n s , in p a r t i c u l a r about the spin propert ies of conduction e lec t rons . Amplitude measurements of the fundamental frequency component are usual ly stra ightforward (although absolute determinations 2 of the amplitude require great care ) . However, various d i f f i c u l t i e s are encountered when studying the higher harmonics and these d i f f i c u l t i e s often make i t we l l -n igh impossible to obtain meaningful in terpretat ions of the data. The most serious of these d i f f i c u l t i e s is the s i g n i f i c a n t harmonic d i s t o r t i o n caused by the Shoenberg magnetic in terac t ion e f f e c t . In th i s thes i s , we present an o r i g i na l technique for the minimization of the Shoenberg e f f e c t , thereby al lowing the spin parameters to be determined r e l i a b l y , and without the use of cor rec t ion f a c to r s . These spin parameters are Lande spin s p l i t t i n g factors g c appropriate to cyc lot ron o rb i t s in the metal, and the re l a t i on of the factors g c to the harmonic amplitudes in the dHvA e f f e c t is reviewed in Chapter I I. In Chapter I I I we discuss the magnetic i n t e r a c t i o n , how i t a r i s e s , and how i t has been dealt with (to a very l imited extent) by ex t r ao rd ina r i l y tedious deconvolution of the experimental data. In the past, several attempts have been made to reduce the magnetic in teract ion experimental ly, but these have met with only modest success. A f ter summarizing these experimental approaches to the problem, we present in Chapter IV the p r i n c i p l e s of the feedback technique which is centra l to th i s thes i s. 3 Putting the idea of feedback to work in the laboratory is the subject of Chapter VI. The dependences of various key observable quant i t ies on the amount of feedback are ca l cu l a ted , and compared with experiment. The experimental apparatus used for the feedback measurements is described in de ta i l in Chapter V. This sect ion includes the c i r c u i t r y which the concepts developed in Chapter IV d i c t a t e , along with spec ia l design cons iderat ions to make the technique simple, p r a c t i c a l , and r e l i a b l e . Having developed the procedure for obta in ing data which are e s s e n t i a l l y f ree of magnetic i n te rac t i on , we present in Chapter VII the f i r s t app l i ca t i on of the feedback technique to the determination of g c for the y o s c i l l a t i o n s in lead along [110]. Recent observation of o s c i l l a t i o n s o f ' v e r y long period in lead using sound attenuation and the magneto-resistance (Shubnikov-de Haas e f fec t ) prompted a search for s im i l a r o s c i l l a t i o n s in the dHvA e f f e c t . The detect ion of such long period o s c i l l a t i o n s benef i t s g rea t l y by the use of very large modulation f i e l d s , of an amplitude larger than can be treated a n a l y t i c a l l y by ex i s t i ng formulat ions. In Chapter VIII we der ive an exact so lu t ion for the response of dHvA o s c i l l a t i o n s to a modulation f i e l d of a rb i t r a r y amplitude. This is fol lowed by the d e t a i l s of an experiment in which a concerted but un-successful attempt was made to find" the long-period o s c i l l a t i o n s . h Had they been found, the feedback technique would have shown them e i ther to be genuine dHvA o s c i l l a t i o n s , or o s c i l l a t i o n s generated by magnetic i n te rac t i on . The best we could do was place an upper l imi t on the i r amplitude in the three major symmetry d i rec t i ons [100], [110], and [ i l l ] . We conclude with some suggestions for fur ther work, both in the technique i t s e l f , and i t s app l i c a t i on . CHAPTER TWO SPIN SPLITTING OF LANDAU LEVELS IN METALS In the same year as de Haas and van Alphen's d iscovery Landau (1930) independently remarked that the magnetization of a metal would be expected to show o s c i l l a t i o n s because of the quant izat ion of the h e l i c a l o rb i t s of the conduction e lec t rons . Onsager (1952) predicted on the basis of general semi - c l a s s i ca l arguments that the p e r i o d i c i t y was simply re lated to extremal areas of the Fermi surface normal to the magnetic f i e l d . Short ly thereaf ter L i f s h i t z and Kosevich (1955) confirmed Onsager's p red ic t i on and pro-ceeded to work out express ions for the amplitudes of the o s c i l l a t i o n s . The resu l t of th i s rather beaut i fu l work, with some modif icat ions by Dingle, (1952) is equation [I], general form of the de Haas-van Alphen magnetization: 6 [la] M = I I A r s in [2irr(f- - Y ) ± 'n/k] orbits r [lb] A r= r " 3 / 2 D(B)(rX/sinh rX) exp (-rXTp/T) cos (rirS) [ lc] X = 2TT2 m* k DT/efiB C D [Id] S = g c m*/2m The symbol M refers to the o s c i l l a t i n g magnetization, and does not include the steady magnetization a r i s i n g from o r b i t a l quant izat ion and sp in . The o s c i l l a t i o n s are per iod ic in 1/B, and each o rb i t has i t s own c h a r a c t e r i s t i c frequency F. The phase factor TT/A is po s i t i ve for minimal crossect ional areas and negative for maxima. D(B) is a funct ion of the magnetic induct ion, and a l so of Fermi surface parameters. rX/sinh rX is a measure of the thermal broadening of the quantized o r b i t s . The imperfections of the c ry s ta l resu l t in a s im i l a r broadening, and are character ized by the Dingle temperature T^ of the c r y s t a l . The factor cos (TTTS) = cos (rirg m"/2m) is the one of c c major concern in th i s thes is s ince from i t g c i t s e l f is determi ned. The e lect ron spin w i l l interact with the appl ied magnetic f i e l d symmetrical ly s p l i t t i n g the Landau leve l s by 7 the amount g e 1i B/2m, where m is the mass of the f ree e l ec t ron , g is a s p l i t t i n g fac tor which may d i f f e r from i t s f ree e lec t ron value (2.0023) because of s p i n - o r b i t - c o u p l i n g . Each Landau level is thus s p l i t , re su l t ing in two sets of levels each separated by e fi B/m", but sh i f ted in phase by the amount 2TT (g m"/2m). Each set w i l l g ive s im i l a r o s c i l l a t i o n s in the magnetization with the same fundamental frequency F, and ha l f the amplitude of that in the absence of spin s p l i t t i n g . In 1/B, the 2 sets w i l l be d isp laced from one another by an amount gm"/2mF, so the resu l t ing magnetization w i l l become ]r [M(-+ gm"/4mF) + M( - - gm* AmF) ] . Z Q :B •The cosine spin< factor " i n the amplitude expression [lb] ' ' i ^ fol lows immediately when th i s average is appl ied to a. wave1- ' form of the kind given in [la] . To give a c l ea re r p ic ture of the e f f e c t of the spin s p l i t t i n g , let us examine equations [ 1 a ] - [ 1d] at absolute zero (T=0) and in a per fect c ry s ta l (T n=0). In th is case, [lb] reduces to A r = D(B) r " 3 / 2 cos nrS which are the Four ier c o e f f i c i e n t s of a cusp l ike waveform for S = 0. iA discontinuous change in the magnetization occurs when the uppermost Landau level becomes depleted as i t crosses 8 the Fermi energy. The e f f e c t of cos TTTS as we have jus t seen is to sum the contr ibut ions from the two sets of Landau levels spin s p l i t about the value at S = 0. The two sawtooth wave-forms, along with the i r sum is shown in Figure 1. With a waveform such as that shown in Figure 1, i t would be a simple matter to determine the phase s h i f t between the two sawtooth waves, and thereby determine S. At temperatures ava i l ab le to us in the laboratory, the thermal damping fac tors p r e f e r e n t i a l l y reduce the higher harmonics, resu l t ing in a waveform more l i ke that shown in Figure 2, which is an experimental recording taken, with typ ica l experimental parameters. The trace is not purely s i nu so ida l , but i t ;is qu i te evident that S cannot be measured d i r e c t l y from the waveshape. To exp lo i t the cos nrS dependence in the hopes of ext rac t ing the g c f ac tor from dHvA amplitude data, i t would be convenient to obtain a method in which the other amplitude factors played l i t t l e or no ro l e . Gold and Schmor (1976) showed that with some mani-pu lat ion of [lb] an algorithm could be obtained using the f i r s t three harmonic amplitudes to determine the value S. Forming the dimens ion 1 ess quantity a = A^/AjA^, [lb] g i ves [2a] a a [1 + 1/3 tanh 2 X] o Ilk a [1 + 1/3 tanh 2 X] where [2b] a CO (/3/2)(l - t an 2 ^ S ) 2 / ( l - 3 t a n 2 ^S) Figure 1. The Spin S p l i t Magnetization at Absolute Zero a, b : contr ibut ions from each of the 2 spin d i rec t i on s c: resu l tant magnetization Figure 2. The dHvA Magnetization at F i n i t e Temperature 11 and the subscr ipts refer to the l im i t i ng cases X -»• 0 and X °°. Using the harmonic content i t s e l f as an imp l i c i t gauge of the bath temperature, the hyperbol ic funct ions of X can be el Imki a ted between [lb] for r = 1, 2 and [2a] to give the simple re l a t i on [3a] A ] / A 3 = a . [ ( A j / A ^ 2 - 1/4 ( A ^ ) * ] where [3b] ( V V f j = 2 / 2 " exp (XTp/T) cos TTS/COS 2TTS is independent of the temperature T. The value a M can therefore be obtained as the slope of a s t ra i ght l i ne p lot of A^/A^ vs. 2 (A j /A£) as the temperature is var ied and the f i e l d held constant. From [2b] i t is c lear that the so lut ion for S w i l l be 2 obtained from a quadrat ic equation in tan TTS and the so lut ion-is^ therefore mult iva lued. The phys i ca l l y meaningful so lu t ion can be se lected with the aid of r e l a t i v e phase measurements and a rough estimate of the Dingle temperature T^ which can be obtained from the f i e l d dependence of the fundamental amplitude. This mult ivalued nature ar i ses from the fact that we do not know the absolute sign of the harmonic amplitudes. A further m u l t i -p l i c i t y ar i ses from the per iod ic nature of tan irS in [2b]. Equivalent so lut ions are ± S ± p where p is an integer. This d i f f i c u l t y is inherent in the use of quantum o s c i l l a t i o n s to determine g c and a r i ses from the p e r i o d i c i t y of the cosine in [ l b ] . One can use a band s t ructure ca l cu l a t i on to hopeful ly resolve the ambiguity. The three harmonic method o f fe r s the major advantage of focusing on the S dependence of the amplitude. Other methods require fur ther information about Fermi surface parameters or sca t ter ing rates s ince complete cance l l a t i on of the other amplitude factors is not accomplished. 13 CHAPTER THREE THE SHOENBERG MAGNET IC-INTERACTI ON EFFECT The d iscuss ion in the previous chapter assumed that the o s c i l l a t i o n s were measured as a funct ion of the magnetic i n -duction B. In p r a c t i c e , the o s c i l l a t i o n s are measured as a funct ion of the appl ied f i e l d H, re lated to B by [k] B = H + 4TT (1-6) M for a second degree surface with H p a r a l l e l ' t o a pr inc ipa l , ax i s . <5 is the demagnetizing f a c t o r . In the normal laboratory s i t u a t i o n , 4TT(1 -6)M/H i 10 ^, however s ince B is in the argument of a rap id ly o s c i l l a t i n g s inusoid (see [ l a ] ) , the co r rec t i on term often const i tutes a large part of one c y c l e . The necess i ty of d i s t ingu i sh ing between B and H was f i r s t pointed out by Shoenberg (1962). The subs t i tu t ion of [k] into [la] resu l t s in an i m p l i c i t equation for M as a funct ion of H',, convolving the harmonics into often hopeless contor t ions , thereby severely modifying the ideal amplitudes A and the phases of the harmonics. Recovery of the ideal amplitudes and phases is the centra l theme of th is thes i s. There are varying degrees of the sever i ty of th is magnetic in terac t ion (M.I.). In the l im i t of small M corrections can be made, but often the d i s t o r t i on s are so severe that i t is impossible to extract the ideal amplitudes and phases from the data. To see the e f fec t s of the term 4-rr(1 -6)M, le t us re-wr i te [1] by se t t ing x = 2TT (jf "" Y ) Z = KM Since | ku (1 - 6 ) M | « H , [ la] becomes [5] z = I C R s in [r(x-z) + -n/h] r Thi:s impl i c i t ' e q u a t i o n for z can be solved by a ser ies of success ive approximations in a scheme developed by P h i l l i p s and Gold (1969) where the n ^ approximation is given by z ( n ) = I C s in [ r ( x - z ( n _ r ) ) + T T A ] r=l r and z ( 0 ) = 0 . 8TT 2F / . . \ K = — (1-6) H 2 C = KA r r 15 While the gathering of terms can become quite tedious (n) a f te r a few steps, the procedure is convenient in that z is exact to 0(n) in the amplitude f ac to r s . This scheme has been ca r r ied out by P h i l l i p s and Gold, and the resu l t s are most conveniently d isp layed as a table of Four ier c o e f f i c i e n t s p r > q r in Table 1, where / \ n [6] M W = I [p r s in (rx + TT/4) + q cos (rx + TT/4) ] r=l r The i 'terationc" scheme-has been ca r r i ed out to many more orders by Perz and Shoenberg (1976) with the a id of a computer program designed to perform the a l gebra ic manipulat ions. To obtain the amplitudes of the re su l t ing harmonics in the in terac t ing theory, we merely f i nd the magnitude of the t h r term in the complex Four ier expansion i . e . , A' = ( P 2 + q 2 ) , / 2 r K r M r For the f i r s t 3 harmonics, the resu l t is [7a] [7b] [7c] A, = A 1 + 0(3) A„ = A, A 3 = A 3 1 1 / f 3 /2 1/2 + 0(4) 8/2 + 0(5) Term 0(1) 0(2) 0(3) 0(h) P l A l icA | ic^ A | 2/2 8 Q 1 KA A + Z 2/2 P 2 K A 2 A l-2 2/2 icA«A— K A« 1 3 + 1 - K 2 A , 2 A 2 /2 6/2 1 2 Q 2 K A 2 + 2/2 K A . A K 3A 1} ± — — + -/2 6/2 P3 3KA.A. A 3 " - p ^ 2/2 q3 3KA,A 3<2A3 ± + -2/2 8 K 3A 1} 2KJA.A. K A 2 A. + L Ul _ _ i . ^ 3/2 /2 % K A 2 ^ A 1 } 2 K A . A . ± 2 ± ' ± ' 3 + 2 < 2 A 2 A 0 /2 3/2 /2 1 2 Table 1 Four ier Coe f f i c i en t s p,q. From P h i l l i p s and Gold (1969). We recover the amplitudes A^ , A^, A^ of the ideal theory i f these amplitudes are s u f f i c i e n t l y smal l . If, as i t often happens, the ideal amplitudes are swamped by the terms generated by M.I., we obtain Shoenberg's " s t rong -fundamental" r e su l t s . These are found in [7b] and [7c] by taking the l im i t as and A^ approach zero, the resu l t is [8a] A] = A 1 + 0(3) [8b] A^ = - j K A 2 + 0(4) [8c] A 3 = | k A i + 0 ( 5 ) There are other not iceable e f fec t s of M.I., besides the d i s t o r t i o n of the harmonic content. If two or more fundamental dHvA frequencies are present, M.I. acts l i ke a mixer, and generates sidebands and combination tones. The simplest of these should be sum and d i f f e rence frequencies of fundamental o s c i l l a t i o n s from d i f f e r e n t o rb i t s on the Fermi sur face. If we consider ju s t the fundamentals from two extremal sect ions , we have (assuming a long rod 6 =0). 18 where the subscr ipts refer to the two sect ions . Since —• [4ir(M + M,)]<<1, we can wr i te [9] as H 2 a b [10] M = A s in (x - K m) + A, s in (x, - K , M ) a a a b b b F a b where x , = 2TT (—r* v L) + a ,b H a ,b 8TT2F . 1 a ,b and K . = — — a,b ,2 From [7a] we see that for one frequency alone the amplitude of the fundamental remains unchanged to second order, so that rep lac ing M on the r ight s ide of [10] by the ideal L i ' f s h i t z -Kosevich magnetization should) be a reasonable approximation for c a l c u l a t i n g the lowest/order combination terms. Vie then obtain [11] M = A s in [x - k A s in (x ) - K A, s in (x, )] a a a a a a b b + A, s in [x, - K , A s in (x ) - K . A s in (x, )] • b b b a a b a b Assuming the quant i t ies KA to be smal l , we keep only the l inear terms in such quant i t ies g iv ing K A 2 K A 2 [12] M V " s i n < 2 x J - - V ^ s i n (2x.) 2 a 2 b A a A b - — ^ — [(K + K , ) s in (x + x, )-(K - K , ) s i n ( x - x , ) ] l a b a b a b a b 19 Thus, to lowest order, the amplitudes of the sum and d i f fe rence frequencies are given by [13a] A s A A, a b 2 ( i e a + K b ) and [13b] A A, a b 2 The resu l t s obtained so far apply when the absolute amplitude of the dHvA o s c i l l a t i o n s is much smaller than the f i e l d spacing, AH - — or in the reduced notat ion, C r<<l. There are many cases where th i s is no longer t rue. The dHvA magnetization can approach or even exceed the f i e l d spacing. In such cases the magnetization formal ly becomes mult iva lued, and the resu l t ing magnetization is the one with the greatest thermodynamic s t a b i l i t y . When C r>l the magnetization is no longer uniform ins ide the sample, and Condon domains are formed (see Condon 1966, Condon and Walstedt The M.I. resu l t s discussed in th is chapter c l e a r l y a l t e r the temperature dependence from that given by the ideal L i f s h i t z -Kosevich (L.K.) amplitudes [ l b ] . For example, in the case of combination tones generated by M.I., the temperature dependence of the amplitudes of the sum and d i f f e rence frequencies from 1968). X X 20 CHAPTER FOUR REDUCTION OF MAGNETIC INTERACTION USING A FEEDBACK TECHNIQUE From the d iscuss ions of the previous chapter, i t is evident that magnetic in terac t ion must play only a very small ro le i f any information from the harmonic content is to be obta i ned. It is c lear that the absolute amplitude of the magneti-zat ion determines the r e l a t i v e s izes of the M.I. generated harmonics. One might consider reducing these troublesome M.I. e f f ec t s by exp lo i t i ng the temperature dependence. At a high enough temperature, the absolute amplitude can be made a r b i t r a r i l y smal l , thereby reducing the M.I. harmonics. Un-for tunate ly the L.K. second and higher harmonic amplitudes drop o f f f a s te r than the i r M.I. counterparts with increasing temperature thereby increas ing, not decreasing the waveform d i s tor t ion. The dependence on the demagnetizing factor 6 in [k] has been used with some success to minimize or control the e f fec t s of M.I. Everett and Grenier (1-978) have cut c r y s t a l s into 21 e l l i p s o i d s of varying aspect rat ios to study the dependence of the harmonic s t ructure on three d i f f e r e n t values of 6 . ln previous g c f ac tor measurements, Gold and Schmor (1969) have cut very thin ( 0 .5 m m ) disks with 6 % 0.9 to reduce M.I. Un-fo r tuna te ly , th i s th in disk method is tedious, and has some undesirable s ide e f f e c t s . The method about to be described avoids most of the undesirable features of the "d i sk method" and o f f e r s some advantages as w e l l . Experimental ly, one usua l ly modulates the q u a s i - s t a t i c background f i e l d H with a small perturbat ion h(t) produced by a modulation c o i l so that [15] B = H + 4TT ( 1 - 6 ) M + h(t) . The fact that the modulation f i e l d h(t) enters into the equation for B in the same way as M is the seed for the feedback idea. If we separate h(t) into two components [16] h(t) = h m + h f and we let h^ = - 3R, where 3 is an experimental ly adjustable feedback ga in, then the equation for B can be made independent of M, i . e . , i f [17] BM-= 4TT ( 1 - 6 ) M 22 the M. i . w i l l be e f f e c t i v e l y suppressed. It is therefore necessary to obtain a s ignal proport ional to M, adjust the gain accord ing ly , and apply th i s s ignal as a f i e l d to the sample. Suppression of M.I. by means of magnetic feedback was f i r s t achieved by Testard i and Condon (1970) in the course of the i r sound v e l o c i t y measurements in bery l l i um. In t h e i r work, a c o i l was wrapped t i g h t l y around a cubic sample, and a current in the c o i l approximated the equiva lent surface currents in the sample and thus could be made to cancel the dHvA magnetization. The appropriate current was found by Imposing a nu l l detect ion c r i t e r i o n on an external pickup c o i l . In the Testardi-Condon arrangement, the sample could not be rotated, and for a . cubic sample , the magnetization is inherent ly non-uniform. We have developed a d i f f e r e n t type of technique which allows the sample to be rotated, and in which the dHvA e f f e c t i t s e l f is used to e s t ab l i sh the cor rect amount of feedback. There are several such c r i t e r i a , and the ones which are eas ies t in p rac t i ce w i l l be discussed in turn. 1. Minimization of M.I. Combination Terms As we have seen, M.I. acts as a mixer in the sense that i f two genuine frequencies F 1 and F 2 are present, M.I. generates nFj ± mF2 where n and m are integers . These terms are not present in the ideal theory, of course, and the c r i t e r i o n becomes the minimization ( i d e a l l y the zeroing) of these combination f requencies . 2. Mass P lots The temperature dependence of the L.K. harmonics is given by A (T) a . r* „ % X e " r X r smh rX where X = Xm* T/H and X = 2ir2 k D /e f i . C D The M.I. terms have a d i f f e r e n t temperature dependence for each harmonic, so that optimum feedback is found when p lots of In A/T vs. T/H fo r the f i r s t , second, and th i rd harmonic are s t ra i gh t l ines with the slopes p rec i se l y in the r a t i o 1:2:3. 3- The Beat Envelope When 2 s igna ls are c lose in frequency, a l l the harmonics beat. The M.I. terms genera l ly beat at the d i f f e rence frequency of the f i r s t harmonic because i t is the strongest in amplitude. The feedback can be adjusted to make the harmonies beat at t h e i r proper f requenc ies . 4. Phase Information The r e l a t i v e phases of the harmonics of the ideal L.K. terms are e a s i l y c a l c u l a t e d , and s ince the M.I. terms add in a d i f f e r e n t phase, one need simply adjust 8 un t i l the L.K. phase re la t ionsh ips are e s tab l i shed . 2 5. L i nea r i t y of Aj/A^ vs. ( A j / A ^ As discussed in Chapter Two, (see [3a]), a p lot of A^/A^ 2 vs. (A^/A^) should y i e l d a s t ra i gh t l i ne i f the harmonic amplitudes fol low the ideal L.K. form. Curved p lots are obtained in the presence of M.I. As presented, the feedback technique seems to accomplish the same des i r ab le object ives which have prev ious ly been at ta ined by exp l o i t i n g the demagnetizing f i e l d . Perhaps the most tedious feature in the disk-method is the actual preparat ion of the sample. In each sample, only one d i r e c t i o n can be s tud ied, the one perpendicular to the plane of the sample. The demagnetizing f i e l d is very s en s i t i ve to o r i e n t a t i o n , so that spec ia l care must be taken to ensure that the external f i e l d H is p rec i se l y perpendicular to the plane 4 of the d i sk . Disk-shaped samples are f r a g i l e , and when using so f t mater ia l s such as lead, i t is d i f f i c u l t to keep the sample f ree of s t r a i n . The s e n s i t i v i t y of the detect ion apparatus is a l so dependent on ( 1 - 6 ) , so that reducing M.I. reduces the s e n s i t i v i t y by the same f a c t o r . The above drawbacks are a l l re lated to the requirement that the sample be a th in d i sk . In the feedback method there is no such cons t ra in t , and any e l l i p s o i d can be used. The e l l i p s o i d a l shape is necessary only to achieve a uniform induction f i e l d , and i t is f e l t that fo r some app l i ca t ions of the feedback technique, the sample need not even be e l l i p s o i d a l (d i scr iminat ion between genuine dHvA terms and M.I. terms; see below). In the case of a spher ica l sample a l l d i rec t i on s in the c ry s ta l can be studied in the same experiment, and only one sample need be prepared. The spher ica l sample is ev ident ly not subject to the prec i se o r i en ta t i on requirement of the d i sk , and a sphere is the optimum shape fo r mechanical s t a b i l i t y . F i n a l l y , the sphere has a large f i l l i n g fac tor fo r so leno ida l p ick-up c o i l s , and i t can be shaped qu i te p r e c i s e l y . There is a d i s t i n c t advantage to the feedback system when one is faced with the problem of deciding whether or not an observed frequency is genuine, or generated by M.I. Increasing the feedback gain from zero causes the M.I. generated terms to f a l l in amplitude, while the genuine dHvA frequencies stay the same or r i s e in amplitude. This can be very helpful in . cases where M.I. generates tens of sidebands,many of which may be larger than the genuine frequency ( c f . van Weeren and Anderson, 1 9 7 3 ) . To acquire the same information unambiguously from the disk method, one would have to make at least two disks with d i f f e r e n t aspect r a t i o s . A valuable side e f f e c t of using feedback is that the absolute amplitude of the dHvA o s c i l l a t i o n can be very e a s i l y measured. In the past, careful measurement of sample volume and geometrical coupling constants between the sample and the detection c o i l were required, as well as the net gain of the a m p l i f i c a t i o n system, with a l l i t s f i l t e r s . ( c f . Knecht 1 9 7 5 ) Using feedback, the only c a l i b r a t i o n constant which is required is the Gauss to amp r a t i o y of the modulation c o i l . When optimum feedback is attained, one simply measures the amplitude of the feedback current 1^ in the modulation c o i l , and the dHvA magnetization M is given absolutely by Y I F 3 Y I F  [ , 8 ] M = M l - f i ) = ~ 8 t T ( F O R A S P H E R E ) where 6 is the demagnetizing factor.' The feedback p r i n c i p l e could be used with advantage in measurements of the quantum o s c i l l a t i o n s in other e l e c t r o n i c propert ies of metals eg. Shubnikov-de Haas e f f e c t , u l t r a son i c attenuat ion e t c . One would s t i l l need to measure the dHvA magnetization o s c i l l a t i o n s in order to obtain the required feedback s i g n a l . For example, feedback could be used to determine whether a set of quantum o s c i l l a t i o n s i n , say, the u l t ra son ic attenuat ion are genuine ones or generated by M.I. As we sha l l see, an integrator is present in the feedback loop, br ing ing inev i tab le d r i f t s in the zero level of the feedback s i g n a l . When th i s occurs , there is a D.C. current added to the modulation f i e l d which appears as a D.C. s h i f t in the external f i e l d H. In our experiment, th i s d r i f t could e a s i l y be kept below 5 Gauss during the course of the measurements,which was i n s u f f i c i e n t to mate r i a l l y a f f e c t the amplitude data. However, these d r i f t s make the phase information less r e l i a b l e . A s l i gh t imbalance in the pickup c o i l does not turn out to be a problem s ince th i s would simply increase or decrease the modulation range. The c o i l s could e a s i l y be balanced to 3 re ject the homogeneous modulation f i e l d to 1 part in 10 , so that a modulation range of 1 kG might have been a l te red by at most ± 1 Gauss. 27 CHAPTER FIVE EXPERIMENTAL DETAILS 5.1 Sample Preparation The experiments were performed on s ing le c ry s t a l s of lead. Previous use of the Czochralski method for c ry s ta l growth ( c f . P h i l l i p s and Gold 1969) showed i t s great success in producing s ing le c r y s t a l s of extremely low Dingle temperatures. The apparatus centers around a melt of zone ref ined lead (6NT grade) from Cominco L t d . , e l e c t r i c a l l y heated in a vacuum of 10 ^ to 10 Torr . A s ing le c ry s ta l seed is dipped into the melt, and the heat conduction through the seed is enough to keep a l l but the submerged port ion s o l i d . In the process of reaching equ i l i b r ium, the meniscus turns upward, and the seed is slowly pul led from the melt. Typica l growth rates are 0.5-1.0 cm/hr., 0.5 being the slowest poss ib le . The diameter of the resu l t ing s i n g l e - c r y s t a l cy l inder was found to be rather in sens i t i ve to pu l l i n g speed, but c r i t i c a l l y dependent on melt temperature. Figure 3 shows the dependence of c ry s ta l diameter on heater vo l tage. The c ry s ta l diameter is very sens i t i ve to heater vo l tage, and th i s d i c ta te s the need for a rather high" degree of long term s t a b i l i t y and measurement accuracy. The more d i r ec t measurement of temperature with the use of a thermocouple had inherent thermal lag and i r rep roduc ib i1 i t y drawbacks, requir ing the operator ' s constant a t tent ion during the growth process. In the voltage measurement, one need only set the voltage to obtain a c ry s ta l of any pre-determined diameter. Four f i gure accuracy was required in the absolute (A.C.) voltage measurement and a Sola 5008 constant voltage transformer provided the required s t a b i l i t y . To adjust the vo l tage, a var iac was used in conjunction with two 3 rheostats. The f i ne contro l had a range of ± 1 part in 10 of the absolute vo l tage. Crys ta l s ranging in diameter from 1 to 10 mm were pu l led reproducibly using th i s method. The experiment required a spher ica l c ry s ta l of roughly 7 mm diameter, so that c r y s t a l s pu l led for the feedback experiment had a diameter s l i g h t l y larger than t h i s . A f te r pu l l i n g a c ry s ta l roughly 5 cm long, i t was separated from the melt by r a i s i ng the voltage on the heater. Once the cy l i nder was removed from the growing apparatus, i t was c a r e f u l l y mounted in a rotat ing chuck. A hollow copper c i r c u l a r cy l i nder was used as a spark cut t ing t o o l . The wall was kept below 0.010 i n , and the tool was rotated during the cut t ing procedure s ince the tool erodes as well as the sample. As the rotat ing tool was lowered with i t s axis perpendicular to the ro ta t ing c ry s ta l c y l i n d e r , a spher ica l sample resu l ted i f the axes in te r sec ted . 30 The in ter sec t ion was c loser than 0.002. in , as th i s could be c a r e f u l l y adjusted during the f i n a l cut t ing stages by making sure that the tool wascutt ing on a l l of the c i r c l e inscr ibed in the c r y s t a l . When the sphere was near completion, there remained two points or " ea r s " on which the tool was cu t t i n g . One held the sphere to the unused part of the crys ta l c y l i n d e r , the other holds the endpiece to the sphere. It was des i rab le for the l a t t e r to cut through f i r s t , so the tool axis was pos i t ioned 5° away from being perpendicular to the c rys ta l ax i s , keeping both axes coplanar. Using th i s cut t ing procedure, c ry s ta l s forming better than l%spheres were usual ly obtained. Since lead is a strong absorber of X-rays, the surface of the c ry s ta l must be very good in order to obtain adequate Laue back - re f l ec t i on photographs. A su i tab le etching procedure was needed to remove the p i t ted surface layer generated by the spark erosion process. This procedure consisted of a 45 minute etch in strong etchant (250 cc g l a c i a l a ce t i c a c i d , 187.5 cc d i s t i l l e d h^O, and 62.5 cc 30% immediately followed by a wash in ethanol . A f te r c a r e f u l l y mounting the c rys ta l on a goniometer, 5 minute Polaro id X-ray photographs could be taken for o r i e n t a t i o n . The X-ray process included rotat ing to major symmetry d i rec t i ons to ensure both a s ing le c r y s t a l , and an .unambiguous f i n a l o r i e n t a t i o n . 31 5.2 Detection Apparatus In the induct ive method for measuring magnetic s u s c e p t i b i l i t y the sample is placed in a balanced pickup c o i l and a l so a separate modulation c o i l . The l a t t e r provides a time dependent (often s inusoida l ) dev iat ion in the steady background f i e l d , and the pickup c o i l is balanced to be i n sens i t i ve to th i s change. Any net magnetization ins ide the balanced pickup c o i l induces a dM voltage in i t proport ional to . The balanced pickup c o i l cons i s t s of two c o i l s , one for the detect ion of the tota l induction (the pickup c o i l ) and the other to buck out the cont r ibut ion from the modulation f i e l d (the bucking c o i l ) . Since large modulation was envisaged, mechanical r i g i d i t y was of primary concern s ince in th is regime v ibra t ions are the major source of noise. To th is end, the modulation c o i l took the form of a long so leno id , mechanical ly f ixed ins ide the bore of the main magnet providing the steady background f i e l d . The pickup c o i l and the counter-wound bucking c o i l , again to achieve maximum mechanical r i g i d i t y , were wound as two con-c e n t r i c solenoids d i r e c t l y on top of one another. The centre tap was made ava i l ab le to f i n e tune the balance when the c o i l s were cooled. It is true that some s e n s i t i v i t y is lost in th i s arrangement because the f lux due to the magnetization of the sample threads both the pickup and the bucking c o i l s . For the worst case of a long rod sample, th is loss for our c o i l s is less than a fac tor of 2, but the gain in s i gna l - to -no i se is well worth i t . 32 Some thought was given to the s i ze of the wire which should be used. A simple c a l c u l a t i o n taking into account the Johnson noise given by the Nyquist equation and the to ta l i n -duced s ignal S gives the s ignal to noise r a t i o S/N as a funct ion of the radius of the wire r. [19] S/N £ [ 4 T r M ] ( £ ) ( 4 k B T A f ) " 1 / 2 & (1) where M is the magnetization of the sample h is the height of the c o i l w is the width of the c o i l T is the temperature Af is the frequency bandwidth and p is the r e s i s t i v i t y of the wire. [19] has no maximum as a funct ion of r, so that r should be made as small as pos s ib le , with only the mechanical strength of the wire to be cons idered. The r e s i s t i v i t y dependence suggests pure copper wire or superconducting wire. When th i s so leno id-on-so lenoid arrangement is used, one must c a r e f u l l y c a l cu l a te the r a t i o of turns in the pickup to that in the bucking c o i l , as the combined width of the c o i l s is l imi ted by the ava i l ab l e space. One must a l so be carefu l to overwind the bucking c o i l so that in the process of ba lanc ing, turns need be removed, not added. In the f i n a l model, the pickup-bucking c o i l pa i r had an ins ide diameter of 0.300" and an outs ide diameter of 0.500". 33 The pickup c o i l had 9500 turns, and the bucking c o i l had 60163* turns of #46 copper wire. (0.0017 inches in diameter, insu la t ion included). The balancing was done by construct ing a modulation c o i l s im i l a r to that in the c ryos ta t , and p lac ing the sample-bucking c o i l arrangement in s ide . Turns were removed from the bucking c o i l un t i l zero pickup resu l ted. This could be done to an accuracy of h turn (^ l in lO^ at room temperature but worsening to ^ 1 in 10 upon cool ing to 4.2K) . For best noise immunity, the connection to the outermost windings was put at ground p o t e n t i a l . The spher ica l sample (radius a) and balanced pickup c o i l form" a very convenient detect ion arrangement. The spher i c i t y of the sample ensures a uniform magnetizing f i e l d ins ide, and concurrent ly forms a s p a t i a l l y inhomogeneous d ipo le magnetizing f i e l d outs ide the sphere as shown in Figure k. The c o i l former for the sample c o i l (a f u l l desc r ip t ion is given in Appendix A) formed the housing of an i n t r i c a t e rotat ion system designed to rotate the sample about an axis 90° away from the only d i r e c t i o n of access. Pippard and Sadler (1969) descr ibe a system which uses very l i t t l e space in the sample region by employing a Mylar gear wheel. Several modi f icat ions to th i s design were made to accommodate our spher ica l sample, and compactness requirements. The d e t a i l s of the construct ion are presented in Appendix A. 34 H_ + h m - / 3 M s< N \ \ l / / / , balanced pick - up * coil i ' ' (~l in I0 4) B (r<a): add uniform field j 4 ? M B (r>a): add M ^ (-f) 3 {2? cos0 + 0 sintf} (dipole field) Figure 4. Detection Arrangement Although [19] gives a 1/p dependence to the s ignal to noise r a t i o , we were working in the regime where the tota l noise was predominantly determined by the input noise of the f ront end d i f f e r e n t i a l a m p l i f i e r . For f r e -quency response cons iderat ions , however, (see sect ion 5-5) a superconducting sample c o i l was wound. Unfortunate ly, i t turned out that the large modulation employed in our measurement techniques rendered the c o i l normal in parts d 2 h of the modulation cyc le where —=- was large. The second dt de r i va t i ve is important s ince i t , along with the s e l f capacitance of the c o i l determines the internal induced currents . A c a l cu l a t i on performed a f te r the c o i l was wound and used showed that these induced currents exceeded the c r i t i c a l current for the wire at the f i e l d s in which we were working. 36 5.3 Modulation Coi l and Superconducting Magnet Both the use of the feedback technique, and the detect ion of long period dHvA o s c i l l a t i o n s benef i t g reat ly from the use of large modulation. Homogeneous modulation of >1 kG peak to peak amplitude could be achieved with the f i n a l apparatus. Wrapped by Richard C h r i s t i e , the c o i l took the form of an 0.0602m long solenoid with an O.OO986 m inner radius. Four layers of 316 turns each were wound using a niobium h$% t i tanium a l l oy superconducting wire (0.0065 inches in diameter, insu la t ion i n -c luded). The f i n i t e solenoid equation gives a Gauss-to-amp r a t i o ' of y=251 for th i s geometry. In order to modulate with amplitudes of £ 1 kG, several amps are required to power the c o i l . Since the transmission of th i s current would be a major heat leak to the helium bath, a simple c a l c u l a t i o n was done to optimize the diameter of the leads to the modulation c o i l . Taking the re-s i s t i v e heating and conduction into account, copper and brass wire gave roughly the same heat leak for t yp i ca l currents (the optimum diameters were, of course, very d i f f e r e n t ) . Brass was chosen because of i t s smaller temperature c o e f f i c i e n t of res i s tance. For typ ica l currents , the heat leak was estimated to be 0.1 Watt for the 1.4 mm optimized diameter brass wire. Superconducting wire was continuously soldered to the brass up to the maximum level of the helium bath. In add i t ion to screw mounts, grease was used as a low temperature glue to ensure the r i g i d i t y of the mount. The main magnet was b u i l t by American Magnetics (A.M.I. #10066) q and was rated at 80 kG with a homogeneity of 1 part in 10 over a 1 cm diameter sphere at i t s centre. Vapour cooled current leads were used to minimizethe heat loss ; the current at peak f i e l d was 65 Amperes. The flow rate of helium through these leads could be constant ly monitored whi le running. The Gauss to amp r a t i o of the main magnet was 1229. The assembly included a pers i s tent current switch, which allowed the magnet to run without an external power supply once i t was energized. 5.4 The Cryogenic Apparatus Housing the main magnet is an Oxford l i q u i d helium dewar with the usual l i qu id nitrogen jacket . The vacuum spaces contain super insu la t ion fo r maximum thermal i s o l a t i o n . Inside th i s outer dewar is an inner dewar with a t a i l extending into the main magnet core, ins ide the modulation c o i l . B u i l t by Peter Haas in the Physics machine shop at U.B.C., the inner dewar featured an ex terna l l y cont ro l l ed vacuum t i ght valve which when opened allowed a t rans fer of l i q u i d helium from the outer dewar to th i s inner dewar. Not only does th i s make the t rans fer process more convenient, but the helium trans ferred to the inner dewar (and sample) could be f i l t e r e d to remove s o l i d a i r p a r t i c l e s . . The vacuum jacket of the inner dewar allowed the pumping of the helium ins ide to a t t a in temperatures of about 1.2K. For best noise immunity, the inner dewar was i so lated mechanically from the main magnet, and modulation c o i l . This 38 TRANSFER VALVE INNER DEWARj HELIUM FILTER LEVEL DETECTORS Inner Dewar Outer Dewar LIQUID HELIUM (T-4-.2 K) LIQUID HELIUM (l.2K<TM.2 K) INNER DEWAR HELIUM RESERVOIR OF OUTER DEWAR SUPERCONDUCTING MODULATION COIL MAIN MAGNET COIL Figure 5- General Schematic Cryogenic Assembly 39 i s o l a t i on could be checked ex te rna l l y with a contact res i s tance t e s t . Both the outer and inner dewars contain l i qu id helium level detectors manufactured by American Magnetics. Figure 5 shows a composite drawing of the cryogenic apparatus. 5.5 Signal Processing As previous ly discussed (Section 5.2), the pickup c o i l consisted of 2 counter-wound solenoids balanced at room temperature to be i n sens i t i ve to uniform f i e l d s . A lead from the centre tap of these c o i l s was made ava i l ab l e at the top of the cryostat to f i ne tune the balance as the c o i l s were cooled. The major cons iderat ion in the design of the detect ion c i r c u i t r y was the e l iminat ion of any frequency dependent com-ponents in the feedback network. Since the pickup c o i l s , and modulation c o i l s would have cons iderable inductance, the fo l lowing procedures were used to e l iminate poss ib le phase s h i f t s caused by the i r reactance. Care was taken in the design of the f i ne tuning c i r c u i t to ensure that the c o i l s were not loaded to the point where phase s h i f t s might be important at the maximum frequency. The pickup c o i l ' s inductance was ca lcu la ted to be 18 mH as an upper l i m i t , and 500 Hz was chosen as the maximum frequency to be handled by the feedback network. The balancing arrangement is shown in Figure 6. The equ i -valent c i r c u i t for the balancing c i r c u i t is shown in Figure 7. ho i ± LOW NOISE DIFFERENTIAL AMPLIFIER P.A.R. 113 iqure 6. F i ne Tun i ng C i rcu i t • W r R 2 R* X 1^ Rj INPUT IMPEDANCE OF PAR 113 Figure 1. Equivalent C i r c u i t for Tuning Arm 41 The external load on the centre tap (C.T.) is at most R^  If R is the pa ra l l e l combination of R„ + R_ and R., 2 3 i R (R + R ) R = — R. + R„ + R, i 2 3 V.R Then V = 1 [21] o Rg+jiol_2+Rj+R V.R (R +R +R - M 2 ) (Rg+R^R) 2 + ( o j L 2 ) 2 The phase s h i f t is [22] <|> = tan R^ +R^ +R The D.C. res is tance of the counterwound c o i l was R^  = h2ti at 4.2K hence, for a phase s h i f t of <1° at 500 Hz, Rj + R > 450 ti As i t turns out, the s e l f capacitance of the c o i l was the determining f ac to r . The s e l f capacitance of a c o i l can to a good approximation, be represented by a p a r a l l e l capac i tor shown as C<. in Figure 6. The fact that th i s capacitance was the dominating inf luence on the frequency c h a r a c t e r i s t i c s was ascerta ined by unbalancing the f i n e tuning s l i g h t l y , and observing the frequency response of the pickup c o i l s to the modulation f i e l d . When performing th i s measurement, with R j » 450 fi, the frequency response of the pickup c o i l r o l l e d o f f to -3db at 90 Hz. When the s e l f capacitance is taken into account, the voltage in Figure 6 is given by [23] V = - -JL o coC J c s ( j ( a , L - J _ ) + R s ) V i where L = Lj + l Hence [24] ^o = l-a)2LC - jRcoC V i ( l - U 2 L C ) 2 + (wRC)2 If we assume L is neg l i g i b l y smal l , v 2 o V. i 1 2 1 + U R S C S ) From the frequency c h a r a c t e r i s t i c s , was determined to be 25.8 yF. Using the f u l l equation [24] with L.= 18 m.H did not change the r o l l o f f frequency, proving our assumption that L is n e g l i g i b l e . The frequency c h a r a c t e r i s t i c of the c o i l is the l im i t i n g factor in the feedback loop, and in fact sets the upper bound to the usable feedback band. The modulation c o i l was energized with a Crown M600 D.C. a m p l i f i e r . The c o i l was placed in a se r ie s combination with a 5 fi non- induct ive monitor r e s i s t o r . The inductance of the c o i l h3 produced a large phase s h i f t even for frequencies as low as 50 Hz. This s i t ua t i on is e a s i l y remedied by using current feedback. A tap was taken between the 5 r e s i s t o r , and the modulation c o i l (see Figure 9), and used to provide a feedback voltage for the operat ional amp l i f i e r feeding the Crown a m p l i f i e r . The Crown amp l i f i e r gain is then immaterial so long as i t is large enough and the current gain of the op-amp-Crown conf i gurat ion is determined by only 2 r e s i s t o r s . Using th is idea, the phase s h i f t s were el iminated up to 1 kHz, above which the gains needed would set the system into o s c i l l a t i o n s . There is a safety precaution which must be emphasized at th i s po int . Since the Crown amp l i f i e r can d e l i v e r enough current to damage the apparatus, i t s ga in, whi le s t i l l being s u f f i c i e n t to assure l i n e a r i t y , must be kept below the point where acc identa l d isconnect ion of the feedback re s i s t o r resu l t s in catastrophe. The Crown gain se t t ing can be used as a l i m i t , without harming the overa l l open-loop ga in, s ince the op-amp provides most of the gain and merely saturates i f the Crown cannot de l i ve r the current needed (see Figure 9)• The Crown gain was set to de l i ve r the maximum al lowable output current for a 15V input s i gna l . The voltage appearing on the wel l -balanced pickup c o i l s is dM proport ional to -pp M can be retr ieved ea s i l y by analogue in tegra t ion . The s ignal proport ional to M was then added to the modulation s ignal and fed to the modulation c o i l as a current . A block diagram of the apparatus appears in Figure 8. The c i r c u i t diagram appears in Figure 9. kk balanced pick up coil h V W 1 LA. balance trim + Differential amplifier Voltage Gain ° ~ 5 0 0 d M dt Superconducting Modulation Coil h m « 1 kG P-P - / 3 M ~ 1G P-P Integrator Spectrum Analyzer X-Y Recorder oscillator h m - A / W -Adder ' h m-/3M Power amplifier Figure 8. Block Diagram of Apparatus INTEGRATOR FEEDBACK GAIN ADJUST > FROM PAR 113 10 K - W A r 10 K INVERTER ADDER AND AMPLIFIER CLOSED FOR NO FEEDBACK Figure 9- C i r c u i t Diagram of Integrator and Adder A simple arrangement was used fo r tes t ing the frequency response of the en t i r e c i r c u i t r y . The procedure involved un-balancing the pickup c o i l s l i g h t l y , and observing the s ignal a f te r integrat ion on one channel of a dua l - t race o s c i l l o s c o p e . The feedback path was broken at the adder, so that only the modulation s ignal was fed to the modulation c o i l . The modulation s ignal from the o s c i l l a t o r was monitored on the second trace of the 'scope. The two traces were c a r e f u l l y superimposed at some low frequency, and then the frequency of the modulation was increased un t i l a separat ion between the 2 traces became not iceab le . The usable feedback band determined in th i s way was 0.2 to 79 Hz. It must be pointed out that th i s l imi ted bandwidth is not very r e s t r i c t i v e . The low frequency l im i t was imposed by a 10 sec time constant on the integrator which could be increased in the fu ture . In the procedure used here, the lowest frequency is that of the f i e l d modulation and dHvA terms can always be made to appear at higher time frequencies. I n i t i a l l y the lead sphere was or iented c lose to the [110] d i r e c t i o n . So o r i en ted , lead exh ib i t s a pa i r of strong y o s c i l l a t i o n s with F^ 'v 17.9, and a s ing le a o s c i l l a t i o n with F^d& O M G . We wene most fortunate to bb.ta i n^the use of a d i g i t a l^ spectrum analyzer (Hp3582A). wh-ich is essent ia 11 y' an on-1 i ne, Four ier t r an s -former. • When appropr ia te ly set up, th i s computer could resolve the harmonics of the y doublet, as well as the sidebands appearing at F ± nF due to M.I. This " appropr ia te " setup wiell now be cx y descr ibed. 47 A t r i angu la r wave was used to provide a ramp in the modulation f i e l d , and i t s amplitude was chosen to sweep through enough dHvA o s c i l l a t i o n s for the reso lut ion of the analyzer to exceed ^ = 18 MG. When a Hanning window is used, (see Appendix B) h o s c i l l a t i o n s of the fundamental were enough to e a s i l y resolve a l l the harmonics. In p rac t i ce i t was not d i f f i c u l t to sweep through about 5 fundamental y o s c i l l a t i o n s at 60 kG; th i s number of cyc les could be increased by working at lower f i e l d s 2 H 2 because of the H dependence of the f i e l d spacing AH % y . At 60 kG, the f i e l d at which most of the work was done, AH^ * 200 G, and a peak to peak ramp of 1 kG sweeps through 5 y osci11 at ions. The spectrum analyzer obtained i t s input d i r e c t l y from the front-end d i f f e r e n t i a l amp l i f i e r (see Figure 8). The modulation frequency was chosen to be 1 Hz, hence the time window on the analyzer should be a l i t t l e less than 0.5 sec, t r iggered at the beginning of the r i s i n g ramp of the t r i angu la r modulation. Figure 10 shows the synchronizat ion of the t r i gger and the time window. If a l imi ted number of cyc les is cons idered, the o s c i l l a -t ions are e s s e n t i a l l y per iod i c in H, and a l inear f i e l d ramp w i l l transform the dHvA o s c i l l a t i o n s to the time domain where the i r time frequency is given by h F f , [25] f = , m ° d H I 48 TIME WINDOW O P E N 1 C L O S E D \- O P E N L-O 0.5 1.5 TIME (SEC) Figure 10. Synchronization of the Time Window 49 where H is the steady background f i e l d h is the P-P modulation ramp amplitude ^mod ' s t ' i e m ° d u l a t i o n frequency F is the dHvA frequency and D is the duty cyc le of the modulation waveform. For a t r i angu lar wave, the duty cyc le is 1/2 so that 2h F f , ^ _ mod H 2 For f m o d = 1 Hz, H = 60 kG, h = 1 kG, the y o s c i l l a t i o n s appear at a frequency of ^ 10 Hz, the a ' s at ^ 89 Hz. 50 CHAPTER SIX EXPERIMENTAL TEST OF THE FEEDBACK TECHNIQUE 6.1 Prel iminary Considerations I n i t i a l experiments were performed on a s ing le c ry s ta l lead sphere or iented with the to ta l appl ied f i e l d along [110]. In th i s d irect ion:, lead exh ib i t s strong y osci 11 at i ons con-s i s t i n g of two frequencies separated by 0.42 MG at F^ ^ 18 MG. These 2 frequencies y 3 , have approximately equal amplitudes of A a : % A b £ 1,. Gauss at 1.2K in a f i e l d of 60 kG. There Y Y are a lso somewhat weaker a o s c i l l a t i o n s at a frequency of 160 MG, and under the above cond i t ions , A^ £ 0.03 Gauss. These o s c i l l a t i o n s are shown in Figure 11 (H ^ 60 kG, T ^ 1.2 k) It turned out that the expansion parameter KA^ was about 0.4 in the condit ions under which we usual ly worked, so we expected the harmonic d i s t o r t i o n to be quite high. In th i s chapter, we interpret the dependence of several observable quant i t ies on feedback gain. The technique appears to agree well with the expected resu l ts for no M.I. at the optimum feedback gain using the c r i t e r i a presented in Chapter IV. Much of what fo l lows, therefore is the ana lys i s of the quant i t ie s under non-optimal feedback cond i t ions , to ensure an understanding 51 Pb M || [110] y oscillations or , y plus M.I. generated c r i m y Figure 11. de Haas-van Alphen O s c i l l a t i o n s in Pb:H|J [110] . . Above: y o s c i l l a t i o n s , below: ^ 4 y o s c i l l a t i o n s which were experimental ly suppressed to br ing out the a o s c i l l a t i o n s and M.I. sidebands 52 of the mechanisms involved, and at the same time, i t c l e a r l v shows the very tedious mathematics one avoids by usinq the feedback technique. While i t is important to check th is once, one must remember that at the optimum feedback p o s i t i o n , the cor rec t ion factors in any of the resu l t s are no longer needed. For each quantity d iscussed, we s h a l l , in turn, look at the cases of no feedback, non-optimum feedback and optimum feedback. In some cases, a va r ie ty of feedback sett ings were used. 6.2 Minimization of Sidebands In order to obtain an idea where the optimum feedback set t ing was, the minimizat ion-of-s idebands c r i t e r i o n was used f i r s t . It is the simplest and perhaps a l so the most dramatic. Using the t r i angu la r modulation, about 5 fundamental y o s c i l l a t i o n s were swept out, and analyzed by the spectrum analyzer. About seven sidebands at F i nF v were e a s i l y resolved. Using the minimization of sidebands c r i t e r i o n , the object was to adjust the feedback gain un t i l these sidebands reached a minimum. Changing the gain from zero in e i ther d i r e c t i o n produced the expected re su l t s , the sidebands decreased for negative feedback, and increased for po s i t i ve feedback. Examples of the Fourier amplitude spectrum for several feedback sett ings appear in Figure 12. Figure 13 is a quant i ta t i ve presentat ion of the dependence of the a , a + y , and a-y amplitudes on the feedback gain. 53 positive _ without negative feedback feedback ^ feedback ar or a: J ~ optimum negative feedback Figure 12. Four ier Transforms Around 160 MG As A Function Of Feedback Gain. (The transforms do not s tar t at F=0) H = 61.79 kG T = 1-.2K 5k .(M.I.) * if f * \ A j t y - A f f J x ( f a ^ A r ) ± A r J 1 ( f / £ A a ) 6 = 1 -4 T T ( 1 - S ) H 2 0.20r —> Sign of Feedback 0 0.2 0.4 0.6 0.8 I TT (1 -8 ) - i - £ 1.2 1.4 Figure 13- Fourier Amplitudes As A Function Of Feedback Gain 55 To compare the observed amplitude with theory, let us use the convenient notat ion: 2TT F f = A" = 1m (1-6) M e =• 1 - 7 — 7 ^ — r - M = 4TT(1-6)M 4IT(1-6) Y,<* Y,« We expect the strongest M.I. cont r ibut ion from the e f f e c t of the Y term on a. The fundamental dHvA magnetization fo r a is given by F [26] M = A" s in [2ir(-=^-> 1/2) - TT/4] a a D+n where h is a small change in H. The phase factors need not concern us for the present, and F h we can expand the denominator for << 1 to obtain a f i r s t H 2 order approximation M = A cos [f (h+eA cos f h)] a a Y Y assuming that the dominant contr ibut ion to the tota l magnetization in B is A'^  cos f h. This can immediately be decomposed into harmon i cs 56 [27] M ( l ) = A J n ( f e A " ) cos f h a 0 a y a - A* J , ( f e A * ) [ s i n ( f + f ) h + s i n ( f - f )h] a I a Y a y a y - A* J 0 ( f e A * ) [ s i n ( f + 2f ) h + cos(f - 2 f ) h ] a 2 a y a y a y + A* J , ( f e A * ) [ s in ( f + 3f ) h + s i n ( f - 3f )hj a 3 a Y a Y a y where J (x) is the Bessel funct ion of the f i r s t kind of order v v and argument x. This f i r s t . s t e p generates sidebands around f , and the next step in the ca l cu l a t i on is to allow these sidebands and f i t s e l f to interact with the y o s c i l l a t i o n s . The equations a are ident i ca l to those in the f i r s t step with the subscr ipts r e fe r r i ng to the corresponding reversed ro les . One only needs carefu l bookkeeping. This step is ca r r ied out only to the cont r ibut ions . For example, when f interacts with y, we qet terms in f and f '. The resu l t is as fo l lows: a+Y a-y 57 [28] M ( 2 ) = A* J Y + A" J Y A" J Y + A" J Y - A'"' J Y + A x J Y + A" J Y + A" J Y + A" J Y + A" J Y f eA* )cos[ f - ( f + f •)•] Y a+Y Y a Y f eA" ) [-cos{f +(f - f ')}] Y a-y y a y f eA* ) s in (f + f ) Y a y a f eA* ) s in (f - f ) Y a+zy Y a+zy f eA*) s in (f - f ) Y a y a f eA* . ) s in (f + f , ) Y a-zy Y a-2Y f eA*^ ) [-cos(f + f ^ )] Y a+Y Y c +Y f eA* ) [ - co s ( f - f )] Y a+3y Y a+3y f eA* )cos(f. ~ f ) Y a-y Y OL-y f eA* . )cos( f + f . ) Y a~3Y Y a ~ 3 Y terms at f re -quency f terms at f re -quency f a + y terms at f re -quency f a-Y terms at f re -quency f a+2y terms at f r e -q U e n C y f a - 2 Y (2) In th i s second step M , the amplitudes A are to be taken from those generated in the f i r s t step M ^ . This expansion should be a reasonable approximation to the gain dependence in our p lots of the sideband amplitude (Figure 13) The expansion reduces, to the simpler form, given e a r l i e r ([13a], [13b]) i f we consider only the f i r s t 2 combination f r e -quencies, and allow only y to interact with a. We then obtain 58 A" t = A J . ( f eA ) + A" J . ( f E A ) a+y a 1 a Y y \ y a and using the small argument approximation J i ( x ) - TT , x<< 1, th i s reduces to e A" A" A* « — f J t ( f + f ) a+Y 2 a Y S i m i l a r l y , , , eA wA' c A" ~- - f J t ( f . f ) oi-y 2 a y A'^  remains unchanged in th i s approximation. This is the s i m p l i f i e d resu l t obtained e a r l i e r with e = l for no feedback (see [ 13a] , [ 13b]) . We immediately see that the simple approximation is i n s u f f i c i e n t to descr ibe the large e f f e c t M.I. has on our data, s ince A* does 3 ' • a indeed change, and the sidebands do not r i s e l i n e a r l y as th i s pred i c t s . In order to use the more de ta i l ed formulae [28], we must f ind the absolute magnetization of at least one of the fundamental terms. This can be done in a few ways. The most s a t i s f y i n g way is to measure A'^  by observing the feedback f i e l d at the optimum 3 . The r e l a t i v e magnitudes of A5^ and A'^  canbe read d i r e c t l y o f f the Four ier transform. An a l ternate method is to f i t the gain dependence of the a o s c i l l a t i o n s to J (f eA" ) . When there is no feedback, o a Y e = 1, and the magnetization at th i s point is M , = A" J (f A" ) • ' a,e = l a o a Y 59 At optimum feedback, e = 0, and M n = A , hence r a,e=0 a T T ^ = J (f A ) M „ o a y e=0 which y i e l d s A". Using the feedback technique and [18], A' was measure to be 6.2 Gauss in a f i e l d of 61.82 kG. (Keep in mind that 4TT(1-<$) is included in th i s notat ion) . An independent f i t to the J curve o y ie lded 6.0 Gauss. This is good agreement e s p e c i a l l y s ince in these ear ly stages, the feedback was not set p a r t i c u l a r l y care -f u l l y . The amplitude of the a o s c i l l a t i o n s was then obtained by the i r r e l a t i v e strength in the Four ier transform. This gave A* to 0.189 Gauss. Using the derived formulae for A" , A" , , and A" , we obtain • a' a+Y a - Y the feedback dependence: A*(e) = A*(0) J (f eA* ) a a o a y A* - (e) = A*(0) J . ( f eA" ) ± A * J . ( f eA" ) ± A *J , ( f e A* (e a±y a l a y y 1 y a y 1 a a+2y = A*(0) J . ( f eA") ± A * J . [ f A * J n ( f eA" )] a l a y y 1 Y a 0 a y + A" J . [ f A" J_( f eA" )] . y 1 y a 2 a y The resu l t s of th i s c a l cu l a t i on are shown as s o l i d curves along with the measured data in F igure 13. It should be mentioned that there are no adjustable parameters in the sense that a l l the quant i t ie s [28] [29] 60 needed for the calculations were measured by Independent means. The agreement is exceptionally good for A q , and the general trends for the sidebands seem correct. An independent check on the A^(e) dependence is conveniently afforded by Aoki and Ogawa (1978) who used rod shaped samples. In their data the a osci l lat ions have a very small amplitude, and only the sidebands appear in the transform. We expect A* a ^ Q ^ a ^ ' We measure A^ = 6.0 Gauss, for a sphere, (6=1/3), so that for a rod (6=0), A ^ w i l l have increased by the ratio (].j/3) = 3/2. The argument of the Bessel function becomes f A * = 2* 0 60 MG) ( 3 / 2 ) { 6 Q ) ^ 2 > 3 6 >  a Y (61.82 kG) 2 This is very close to the f i r s t zero of J q , which is 2.405. Extrapolating our data, which was possible at a later date from an amplitude vs. f i e ld measurement, (see Figure 27) we can obtain a more rea l i s t i c estimate at the f ie ld used by Aoki and Ogawa. At their f i e l d , 50.5 kG, we measure A* = 4.10. For their f i e l d , f A* = 2 7 7 ( 1 6 0 " G ) (3/2) (4.10)'= 2.42 a Y (50.5) 2 which again is in good agreement with their observation of small fundamental a amplitude. We notice from Figures 12 and 13 that the sidebands never com-pletely disappear. There are many reasons why this might occur, the most obvious being a frequency dependence of the feedback network. There is a clue to the origin of this imperfection in the 61 amplitudes of the sidebands f , and f r e l a t i v e to each other. r a+Y a-y The data show a change in t h e i r r e l a t i v e amplitude, so the mechan i'sm must expla in th i s change. The frequency response of the feedback network might be such a mechanism i f the time frequency of the y term is c o r r e c t l y fed back, but that of the higher a term is not; however, th i s is not the case, as was shown by a simple experiment. Decreasing the modulation frequency by a fac tor of 10 decreases a l l the feedback components by the same amount. There is no doubt that the system's response is very f l a t below 10 Hz, yet even when a was made to appear at only 7 Hz, the sidebands at optimum feedback s t i l l appeared with the same amplitude. It was thought that a s p a t i a l l y inhomogeneous feedback f i e l d might a l so cause the res idual sidebands. While the feed-back f i e l d i t s e l f is only of the order of 1 Gauss, the super-imposed t r i angu la r modulation was about 500 Gauss at i t s peak value. This inhomogeneity would not however, cause the r e l a t i v e sideband amplitudes to change. Even i f the f i e l d is s p a t i a l l y inhomogeneous, there would be no dependence of the feedback f i e l d on o r b i t i . e . F and F would both be fed back the same. This Y a inhomogeneity may impose a l im i t on the minimum amplitude of the sidebands, but would not change the i r r e l a t i v e amplitudes. Eddy currents induced in the sample would cause a s im i l a r inhomogeneity. The magneto-resistance of lead gives i t a skin depth of about 1 meter fo r our t yp i ca l f i e l d s and f requencies . The skin depth, then should not be a problem, but i f eddy current inhomogeneities 62 were present, they would not account for the non-vanishing s idebands. The most l i k e l y explanation is phase smearing in the sample. This sample had been thermally cyc led between room temperature and 4.2K three times before these data were taken. If d i s l oca t ions and s t r a i n had b u i l t up, the change in e lect ron density would change the frequency of the o rb i t s in the immediate area, making the optimum feedback a funct ion of the o s c i l l a t i o n s under cons iderat ion. 6.3 The Mass Plots The temperature dependence of one dHvA harmonic amplitude is given by (see [lb]) \ ( T ) a 7 f n F 7 x X = 2 * 2 < k B T / ( e * B ) . 63 In the high temperature l im i t (X ~ 3), the hyperbo l ic s ine funct ion 1 X can be replaced with i t s exponential approximation: s inh X £ 2 Then, l e t t i n g \ = 21r2m'kB/(e'R ) = 146.9 ^ , we have [31] A r (T ) a T e ' r X y T / H y = ^ 1 m hence, a p lot of £n ^ vs. T/H should y i e l d a s t ra i gh t l i n e of s lope -rAy. This approximation usua l ly ho lds , but in the case of the fundamental y osci11 at ions at [110], the value of X at 1°K and a f i e l d of 60 kG is about 1.4, and the er ror in the exponential approximation becomes about 6%. A simple co r rec t i on term can be used, and i t fol lows from an i t e r a t i v e scheme to determine y from data which arenot s a t i s f a c t o r i l y f a r into the high thermal smearing l im i t (X ^ 3)• To develop th i s scheme, we s t a r t with the bas ic hyperbo l ic form , CTA = rX = 2 r X e " r X r sinh rX . -2rX 1-e where C is a temperature independent constant. Then at constant f i e l d we have A 2 r v £n ( y ) + £n ( l - e A ) + constant = -rX or ) A ( l - e " 2 r X y T / H ) Jin < A r U 6 _ ' ? = - rXy T/H If n is the order of i t e r a t i o n , the value y is obtained from [32] 4 h ^ ( l - B - 2 r X w ( n ' l ) T / M \ } - r X y ( n ) T / H , (0) _ , . . , (n-1) (n) where y ->- °°. Convergence is achieved when y = y to the des ired degree of accuracy. When the y o s c i1 l a t i on s are observed without feedback, a p lot of Jin j vs. T/H resu l t s in curves for a l l but the f i r s t harmonic. This is shown in Figure 14. App l i ca t i on of near-optimal feedback promptly stra ightens out these curves as shown in Figure 15. Table I I shows the e f f e c t i v e masses measured with near optimal feedback in the [110] d i r e c t i o n , along with the i r counterparts derived from data given by P h i l l i p s and Gold (1969). Table II E f f e c t i v e Mass for Observed O s c i l l a t i o n s in Pb:HJj [110] Osc i11 at ion y(Measured) y(Derived) a 1.08 1.1010.01 a+y , a-y 1.74 1.66±0.02 y fundamental 0.565 0.56±0.01 Y 2nd Harmonic 1.00 1 .12±0.02 Y 3rd Harmonic 1.35 1.68±0.03 [Values of y (measured) are derived from the slopes of Figure 15 for non-optimal feedback] Figure 14. Mass Plots With No Feedback 66 Near-Optimal Feedback Figure 15- Mass Plots With Near-Optimal Feedback 67 The agreement for the fundamental o s c i l l a t i o n s is good, which is to be expected because they are e s s e n t i a l l y unaffected by M.I. The combination terms ot+y, ' a -y, are generated by M.I., and in the simple approximation A A A - «JC (f ± f ) a± y 2 a y The temperature dependence is -A(y +y )T/H [ 3 3 ] A a ± y ( T ) = T e Our measured value comes from the s lope of a p lot of A In '"TY VS. T/H, and the derived value is the sum of y and y . . T 2 Y In the ideal theory, the r a t i o y ^ r y ^ y ^ is 1 : 2 : 3 . The derived values in Table II are y^, 2y^ and 3yj> s ince y^ can be measured with minimal in ter ference from M.I. Upon measurement of these va lues, we found them not in accord with th i s r a t i o suggesting the presence of res idual M.I. In order to take account of the res idual M.I., present because the feedback gain was not set to optimum, the conventional theory of M.I. is app l i c ab le . In the conventional theory, i f A represents the L.K. amplitude, and A 1 is the in terac t ing amplitude, [ 3 4 ] A^ = A 2 ] l 68 To find the amount of M . I . s t i l l present, we calculate A^ from [34] adjusting e unti l the temperature dependence plot yields the Ideal effective mass. -X . . _ .. -2X Writing A C, X e and A 2 = ^ 2 X e to extract the temperature dependence, we obtain [35] where and [36] A 2 = K2 X e" 2 X {l-?X + 1/2 £ 2 X 2 } 1 / 2 In X[1-SX+1/2S 2X 2] 1 / 2 Plotting the left side of this equation against T/H should yield an ideal slope of -2Xy^ - 165. A numerical calculation gave the value 5 = 0.62 as the best f i t line with the desired slope. With this value of ?, the correction factor [1-?X + V 2 t ; 2 X 2 ] 1 / 2 attains the values 1.47 to 0.71 between 3.4K and 1.2K, respectively which is appreciable, but not drastic enough to warrant further terms in the expansion. The same treatment can be applied to the third harmonic. The conventional interacting theory in the presence of feedback gives (see [7c]). [37] 3K E AjA 2 + 1/2 3< e AjA 2 ft A3 3 < 2 e 2 A ; V " / 2 8A„ 69 - 3X As before = ^ X e , and introducing n = F , the equation becomes [38] A^ = A 3 { l -3?nX + 1/2 (3?nX ) 2 [ l - l/2 a] + ( 3 / 4 ? 2 n X 2 ) 2 } 1 / 2 The cur ly brackets give the cor rec t ion f a c t o r , so that p l o t t i n g A 3 SLn r - / 9 vs. T/H should give a slope of ~3Xy for the appropriate x{ y u Y choice of r\. Using the value of t, ca l cu la ted before, a value of n = 1.1 y i e ld s the cor rect slope for the th i rd harmonic temperature. dependence. The agreement of the c a l cu l a t i on with the expected resu l t s suggests we c o r r e c t l y understand the mechanics of the non-optimal feedback s e t t i n g , and hence we can use the slopes of the mass p lots as a c r i t e r i o n for r e a l i z i n g optimum feedback. The mass p lots made at optimum feedback are shown in Figure 16. The largest source of sca t ter was the temperature measurement; th i s is indicated in Figure 16 by a systematic s h i f t in the points of each l i n e which were taken at the same temperature. The r a t i o of the slopes are now 1:2:3 with in h%. The y^ data were known to be i n s u f f i c i e n t l y far into the high thermal smearing l i m i t , and the cor rec t ion term developed e a r l i e r was employed in the i r ana l y s i s . 6.4 The Beat Pattern The dominant contr ibut ion to magnetic in terac t ion usua l ly comes from the fundamental dHvA frequency s ince i t is usua l ly the Figure 16. Mass Plots at Optimum Feedback .020 .025 .030 .035 .040 .045 T/H ( K / k G ) strongest. If there are two neighbouring frequencies present, and beating occurs, the terms from M.I. at the second harmonic frequency w i l l beat with a frequency equal to that of the funda-mental beats rather than at twice the fundamental beat frequency as would normally be expected. The same is true fo r a l l the M.I. harmonics. As a r e su l t , the beat frequency w i l l only be pro-port iona l to the harmonic index (r) i f there is no M. i . present. In lead along [ 1 1 0 ] , we have such a s i tua t i on with the y frequencies (see Figure 1 1 ) . The fundamental is about a fac tor of 10 stronger than the second harmonic, and the fundamental beat frequency is roughly 0 .42 MG. In the ideal theory we can represent the magnetization due to the two y frequencies (•Ya,yb) by M a = I A a s in [27rr(£ip - y) -r M b = I A* s in [ 2 T r r ( ^ p - - y) + TT/4] r . -.total ..a , ..b and M = M + M where 6 F = F a - Fb and F ^ F 1 3 The sign reversal of the phase fac tor IT/4 is required s ince the area of one of the corresponding Fermi surface areas is a maximum, whi le the other is a minimum.(see Ogawa and Aoki ( 1 9 7 8 ) ) . A a - A b If we let n = 1 , then elementary tr igonometr ic A a + A b manipulations lead to 72 [39] M t 0 t a 1 = I (A;|+Abr)Vf {(l+n 2) + ( l - n 2 ) s i n(2T : r 6 F / B ) } 1 / 2 r x s in [2irr - v) + if^l where = tan '{n tan ( i r r D S F - ir A ) } r u B and F = r — Unfortunately, the phase of the beats is very s en s i t i ve to o r i e n t a t i o n . This is c l e a r l y shown by the large spread in f i e l d values of the minima reported in the l i t e r a t u r e . This s e n s i t i v i t y resu l t s from the r e l a t i v e l y low symmetry in the y o s c i l l a t i o n s along [110]. Beats in o s c i l l a t i o n s corresponding to o rb i t s of higher symmetry such as 3 at [100] do not appear to be so sen s i t i ve . The beat envelopes fo r t h e f i r s t three y harmonics without feedback are shown in Figure 17. The envelope of the f i r s t harmonic appears as we expect, however that of the second harmonic c l e a r l y has the p e r i o d i c i t y of the f i r s t . The th i rd harmonic beat envelope has a part which is beating at t h r i c e the fundamental beat frequency however, the pos i t ion at the f i e l d corresponding to a f i r s t harmonic maximum has higher amplitude, ind ica t ing that at least some of the amplitude is due to terms generated from M.I. With near-optimal feedback, shown in Figure 18, the second harmonic appears to be approaching a beating pattern at twice the fundamental frequency. The th i rd harmonic is s t i l l a f fected by M.I. e f f e c t s . 73 Pb y oscillation envelopes H|| C MO] 0.015 0.016 0.017 0.018 '/H (kG" 1 ) —*> Figure 17. Beat Envelopes Without Feedback (A., A_, A in Gauss) Ik I L I L_ .015 .016 .017 .018 l/H (kG"1) Figure 18. Beat Envelopes With Near-Optimum Feedback (A,, A „ , A_ in Gauss) 75 Concentrating on- the second harmonic, we observe a sequence of alternating large and small beat maxima. The apparent beat period also alternates. To explain these results we can ca l l upon a result derived ear l ier (see [7b]) for the second harmonic amplitude in the presence of M.I., and feedback, namely, , 1 i c e A 2 K E A? 2 1/2 . 8TT2F where K = — = — H If we take the limit as A 2 approaches zero, we obtain Shoenberg's "strong fundamental" result [Al] A 2 = ± A 2 K e Upon substitution of the beating amplitude of. the fundamental Aj into [4l], we can find the contribution of Aj at the second harmonic. We see from Figure 17 that i t is a good approximation to take the amplitudes of the individual y osci l lat ions to be equal, A ( a ^ = A ( ^ . The magnetization due to the fundamentals is then f - 6F M ( l ) = A^ {sin £2TT( H 2 - y)- TT/4] p" - — , + sin [2TT( h 2 - Y) + TT/4]} or [42] M^) = 2A1{sin[2^(^- - y) ] COS[2TT(JI) - TT/4]} 76 The envelope is given by .. 2A, cos [ ^ - IT A ] so that the magnetization amplitude appearing at the second harmonic due to the f i r s t is ~M(2) = " J L L { / ( A 2 C O S2 [ 2 ^ ^ ) _ or [43] M ( 2 ) = - K A 2 e{l + cos [2u(~) - TT/2]} The genui.ne second harmonic gives M = A 2 ( s i n [2^(2I+iI - 2y) - TTA] + s in [2ii - 2y) + if A ] } or [44] M = 2A 2 s in [ 2 7 T ( ^ f - 2y)] cos - irA)' Adding the 2 contr ibut ions gives the to ta l magnetization amplitude at the second harmonic [45] M = 2A 2 cos - * A ) - KB A 2 [1 + c o s ( ^ F - - ir/2) ] The cont r ibut ion from the f i r s t harmonic can be v e r i f i e d by using an a l te rna te approach. From [12] for two f requenc ies , the cont r ibut ion at the sum frequency is 77 A 2 O [ 4 6 ] ASUM = s i n ( 2 x a } " -V- S i n ( 2 V e A a A b - — f — [ ( K A + K B ) s in (x a + x b ) ] Subst i tut ion of the two y fundamental frequencies g ives: ^2 A S U M = { S l n [ 2 T T ( 2 F + 6 F - 2 Y) - TT/4] + s in [ 2 T T ( 2 F " 6 F - 2 Y) + } n A 2 £ - 4" " 2 k s i n f 2 ' 7 7 ( 2 7 f - 2 Y ) 1 -Elementary tr igonometr ic manipulations lead to: [47] M = - K E A 2 [cos ( ^ L - Tr/2) + 1Isinl2Tr which reproduces the amplitude in [43]. The in terac t ing resu,lt [45] should f i t the near-optimal feedback data. In order to obtain values fo r Aj and to f i t the curve, one can f ind the zeros , and match the period r a t i o , that i s , i n s i s t that every other beat period be shorter by the observed amount. The zeros of the ca lcu la ted second harmonic amplitude are determined from 2A 0 cos (2TT~ F _ - TT/4) - K e A? { l + c o s C 2 ^ - TT/2) } = 0 z n I n or 2A [48] -J- cos - Tr/4) = 1 .+ cos - TT/2) K A* e H H 78 148] Is a transcendental equation which can be solved numerically, 2A2 and iteration leads to a value of — - r for which the ratio of the tcAjC beat periods agree with the observed response. The details are simply c ler ica l and wi l l not be included, however, a plot of the ratio of the two apparent second harmonic beat periods as a function K A 2 E of — appears in Figure 19. At e = 0 which is equivalent to saying there is no M.I., the corresponding ordinate is 1, indicating the equivalence of a l l beat periods. The value of 0.737 on the ordinate corresponds to that observed in the near-optimal feedback setting of Figure 18. This ordinate corres-ponds to 2A — | =4.91 KA J e A similar calculation was done assuming A2<0 which yields the result: 2A„ — | = - 3.41. KAJE Using these solutions, the calculated interacting second harmonic was plotted along with the calculated f i r s t harmonic beat envelope in Figure 20. The biggest difference between the positive and 2A2 negative solutions for — j - ' s t n e relative phase of the second KAJ e harmonic minima with respect to the f i r s t harmonic minima. Upon comparison with the measured data (see Figure 18), i t becomes obvious that A 2 is indeed negative. With the negative solution Figure 19. Ca lcu la t ion of the Ratio of the Apparent Beat Periods of vs. KA 2 E ( 0 . 2 0 4 , 0 . 7 3 7 ) K A f € 2 A o BEAT ENVELOPE AMPLITUDE (Arbitrary units) 08 the agreement is r e a l l y qu i te good, e s p e c i a l l y in the r e l a t i v e phase of the f i r s t harmonic minimum with respect to the second harmonic minimum. The ca l cu l a t i on pred ic t s the 2 minima of the second harmonic to be displaced from a fundamental minimum by A ( f^ H L L) = 0.748 rad and 4.33 rad . The observed resu l t from Figure 16 is A (-^fp) = 0.75 rad and 4.5 rad. The c a l c u l a t i o n of the in terac t ing resu l t for the th i rd harmonic fol lows s im i l a r c a l c u l a t i o n s , but the complexity is much greater, e spec i a l l y in the transcendental equations determining • Since the procedure worked well with the second harmonic, the th i rd harmonic equations were solved by computer. An option in the program enabled us to take the Four ier transform at each f i e l d se t t ing requested which is more in keeping with the way the data was obtained exper imental ly. This program is i n -cluded in Appendix C. One could correct the temperature dependence of the near-optimal feedback data by ext rac t ing th i s dependence from the ca l cu la ted in terac t ing r e su l t . This was done at a f i e l d corresponding to a maximum in the ideal second harmonic beat envelope. At th i s f ixed f i e l d H q , the arguments of the cosines in the in terac t ing resu l t s are constant, and the temperature dependence is extracted from A.j and A^. H q is independent of feedback gain s ince i t is the f i e l d at a maximum of the ideal beat envelope. The temperature dependence i s : 82 A 2 ( j ) a 2A„ 2 K A j£ COS {~T^~~. ~ T T A ) + { 1 + C 0 S ( 2 T ^ - TT/2)} 2 K A £ a 2A 2 {cosC2^ - TT/4) + -^A- []+cos(^- - TT/2)]} o 2 o but K A^E K £ 2 e X 2 K 5 -|E i 2A, 2? 2X 2 E 2 The quantity X = Ay T/H contains a l l the temperature dependence, so, l e t t i n g X be the value of X where • o 2 2 K A , £ K A j £ 2A, ( -^—) Q ( = ~3.4l in our case) then K A 2e 2A„ X X '2A 2 > K A 2 £ j l e t t i n g a = cos (^rr— ~ TT/4) H O and b = {1 + cos - TT/2)} f2A, 2 KA^ e we obtain a temperature dependence of A 2 (T ) = 2A 2 [a + bXj hence, p l o t t i n g 83 In vs. T/H should y i e l d a s lope of -2Xy. In our data, a = 0.642, b = 0.352, and we obtain, a slope of 167 kG/K which corresponds to y = 0.568. This r e su l t , ca l cu la ted from the second harmonic near-optimum feedback data is in exce l lent agreement with that derived from the fundamental amplitude (see Table l l ) . The above agreement again demonstrates the understanding of the deta i l ed mechanisms involved in near-optimal feedback, and we now move on to the optimum feedback p o s i t i o n , the resu l t s appearing in Figure 21. With optimum feedback, the beat frequencies of the f i r s t three harmonics are in the r a t i o 1:2:3. From the ideal non- in teract ing beat envelopes which are now ava i l ab le to us thanks to the use of optimum feedback, i t appears that there is a favoured f i e l d se t t ing with in each fundamental beat cyc le where the three beat envelopes are simultaneously c lose to their maximum va lues, and the slopes are not very large. This occurs at roughly 1/3 of the way into. the beat envelope p lot ted against 1/H (shown as 1/H^ in Figure 22). These pos i t i on s , a f f e c t i o n a t e l y ca l l ed "magic f i e l d s " are the optimum f i e l d s to perform a three harmonic measurement. The amplitudes of the harmonics are c lose to but not at t h e i r beat maxima, so that a f i e l d dependence measurement is needed to determine the actual r e l a t i v e amplitudes from those measured at the "magic f i e l d " . Simulation of such a p lot appears in Figure 22, wi th the correct ion = .for the fundamental shown 84 0.015 0.016 0.017 0.018 ' / H (kG- 1 ) — Figure 21. The Beat Envelope With Optimum Feedback ENVELOPE AMPLITUDE 58 86 The corresponding empir ica l r e su l t for the fundamental appears in Figure 27. 6.5 Phase Information The value of the argument of the s inusoid descr ib ing the dHvA e f f e c t is qu i te large (^10^). Absolute phase measurements thus require great p rec i s i on in f i e l d and o r i en ta t i on i f they are to be considered r e l i a b l e . Fortunate ly , the phase re la t ionsh ips between harmonics can be measured r e l i a b l y . However, s ince we are cons ider ing d i f f e r e n t frequencies ( i . e . , F^, F^=2F^ , F^=3F^), the r e l a t i v e phase must be defined with some care. The standard d e f i n i t i o n of the phase s h i f t between a va r i a t i on of the form s in (u)t+ij>j) and i t s r**1 harmonic sin(roit+ij^) involves the construct ion of a reference s inusoid with frequency u> cross ing zero with a po s i t i ve s lope at some a r b i t r a r y t = 0 (any convenient t = 2mir/cj where m = 1, 2, 3 . . . would a l so do). Associated with th i s fundamental reference is another at a frequency no cross ing zero with a po s i t i ve s lope at the same t = 0. If the phase d i f f e rence between the fundamental s ignal and the reference is <f» j and the corresponding quant ity for the r t h harmonic is , then the quantity r ^ - ^ is a constant and serves as the d e f i n i t i o n of the phase d i f f e r e n c e . When deal ing with a s ignal which is the sum of two c lose f requenc ies , the short-range modulation may not be enough to resolve the ind iv idua l frequencies in the Four ier transform. In th i s case, one must c a l cu l a te the resu l tant phase in order to compare with the experiment. 87 We have a lready presented the re su l t s of a numerical sub s t i tu t i on in the envelope equation fo r a pa i r of beating o s c i l l a t i o n s (see [33] plotted in Figure 2 2 ) . The two inverse f i e l d s 1/Hj and l/h^ correspond to Figures 23 and 2k where the ind iv idua l o s c i l l a t i o n s are p lot ted out to determine the r e l a t i v e phase. With reference to Figure 2 2 , both the second and th i r d harmonics have undergone one zero cross ing between 1/Hjand 1/H 2 but the f i r s t has not. At l/h^ we therefore expect the second and th i rd harmonics to have the opposite phase re l a t i on sh ip to the f i r s t harmonic when compared to that at 1/H^. This is indeed shown in Figures 23 and 2k. 1/Hj corresponds to one of the "Magic f i e l d s " (see sect ion 6 . 4 ) . Figure 23 shows that at the f i r s t of these f i e l d s i . e . the one with the lowest value of 1/H, the phase d i f f e rence between the f i r s t three harmonics is zero . Genera l iz ing [6] to al low for the phases i|>r in the presence of beats, and feedback, we obtain 150] A 2 = A 2 s in (2x+^ 2 ) - 1/2 KC A 2 s i n ( 2 x + 2 ^ 1 ) . In add i t i on , from [39] Tor small n we see that and i|>2 can a t t a i n only 2 values,^-0 and TT. This makes the two terms in [50] e i ther in phase or TT out of phase depending on the s ign of A 2 , and the value of T(>2. If the two terms compete in the presence of M.I., we expect a phase reversal of the measured phase of A 2 i f the magnitude of the second term in [50] exceeds that of the f i r s t . In any case the measured phase d i f f e rence should fo r small n (narrow beat waists) be 0 or TT. The measured phase d i f f e rence as a funct ion of feedback gain F igure 23. Ind iv idua l Osci11 at ions Near the "Mag i c F i e l d " l/H, of F igure 22 Figure 2k. Individual O sc i l l a t i on s Near 1/H 2 of Figure 22 Figure 25. Measured Phase D i f ference and Amplitude of y at a Magic F ie ld (61.739 kG) 91 at a magic f i e l d appears in Figure 25. Included in th is f i gu re is the second harmonic amplitude dependence. From 150] we see that should be a l i near funct ion of e for small n in [39]. Phase measurements at other parts of the beat cyc le were not r e l i a b l e s ince the amplitudes of the o s c i l l a t i o n s were changing qu i ck l y , and the large modulation smears the phase. 6.6 The L i nea r i t y of A j / A ^ vs. ( A j / A ^ In Chapter I I, we found that information leading to the g c 2 factor comes from the s t ra i gh t l i n e p lot of A^/A^ vs. (h^/A^) . This l i ne is s t ra i gh t only i f the ideal L.K. behaviour is r e a l i s e d . Figure 26 shows plots of th is kind for data without feedback and with optimum feedback. The l i n e a r i t y and low sca t ter in the graph with feedback is su rp r i s i ng l y good to one who has made these p lots using other techniques to deal with M.I. This f i gu re shows very dramat ica l ly the d r a s t i c way in which M.I. i n te r fe res with amplitude information, and how th i s inter ference has been succes s fu l l y removed by the feedback technique. 6.7 Conclusions Before using the feedback technique to measure quant i t i e s such as the g c f a c t o r , we must be conf ident that the technique is working proper ly , and know the l im i t s with in which we can work. This chapter demonstrates the consistency and the e f fect iveness which feedback has in reducing M.I. I50r-Pb H|| CIIOJ y -o sc i l l a t ions 92 XlO' 100 500 Figure 26. 1000 1500 2000 Aj/A^ vs (A,/A 2)' With and Without Feedback. H is constant at 61.13k kG, and the bath temperature T is va r ied . 93 The ca l cu la t i ons involv ing non-optimum and near-optimum feedback give cons istent agreement with the experimental r e su l t s , demonstrating the understanding of the ro le of feedback in the experiment. Only for the a + y sideband amplitude (Figure 13) do we f ind a systematic dev iat ion from theoret i ca l expectat ion. This is presumably re lated to the non-vanishing of the a + y and a -y sidebands at optimum feedback, and is not yet f u l l y understood. In every other case of optimum feedback, the data conform to the resu l t s expected for ideal L.K. behaviour, and i t must be stressed that in each sect ion of th is chapter, the phrase "optimum feedback" refers to the same feedback gain i . e . , the optimum set t ing fo r one experiment is the same for a l l the others. This consistency gives one confidence that the same optimum feedback se t t ing w i l l g ive r e l i a b l e g c f ac tor measurements. While most of the chapter deals with non-optimum feedback, i t c l e a r l y demonstrates that at the optimum feedback gain no correct ions for M.I. need be app l ied . 94 CHAPTER SEVEN EXTRACTION OF g FACTOR FROM A^/A^ vs. {A}/A2)2 PLOTS To apply the a lgor ithm presented in Chapter II le t us r e c a l l a few r e s u l t s : 2 2 [3a] A , / A 3 = am [ ( A ^ r - 1/4 ( A , / A 2 ) ] [3b] ( A 1 / A 2 ^ 0 = 2 ^ 6 X P ( X T D / T ^ C O S i r S / c o s 2 i t S A? [2b] a - (/3/2)( l-tan nS) /(1-3 tan »S) - Um ( — ) OO X » ( A ^ / A 2 ) Q is independent of the temperature T X + » A j A 3 From [3a] we see that the slope of the graph of |A^/A 3| vs. 2 (Aj/A,,) (holding the f i e l d H constant, and varying the temperature T) is . [2b] can e a s i l y be inverted to give [51] t a n 2 rrS = 1 - /3 <v V ±^~-W 95 The value |a } from a least squares f i t to the points in Figure 2 6 is '|'a ] = 0.392. The square root in [ 5 1 ] y i e l d s an imaginary resu l t for a = + 0.392 so we must conclude that a =-0.392, This implies A ^ / A ^ < 0. The real so lut ions of [51] are S = 0.330 or 0 . 1 9 7 . These values are modulo 1 because of the 2 p e r i o d i c i t y of the funct ion tan TTS. T O decide between these two p r i n c i p a l values for S we measure the absc issa intercept in Figure 26 to obtain, 1 /4 (A / A ) . as can be seen from [3a]. 1 2 0 [3b] can ea s i l y be inverted to give Dingle temperatures corresponding to the two so lut ions for S . u (A./An)n cos 2TTS [ 52 ] T = - f - £n [ ( 1 2 ° ] D X » 2 / 2 cos TTS From the least squares f i t to the points in Figure 26, 2 1 /4 ( A J / A 2 ) Q = 9 1 . 0 . Using [ 5 2 ] and the experimental parameters used in the experiment, along with the e f f e c t i v e mass p found in Chapter VI, our previous so lut ions for S correspond to the fo l lowing Dingle temperatures S = 0.330, T D = 1 . 4 2 K , A T /A < 0 S = 0.197, T D = 0 . 7 4 9 K , A. T/A 2 > 0 The Dingle temperature can a l so be obtained from the f i e l d dependence of the fundamental amplitude and [ l b ] ; only a rough estimate is necessary. In the approximation X * 3, the complete f i e l d and temperature dependence is given by 96 Aj cx(T//B) exp {- Xy ( T / B ) ( l + Tp/T)} so that [ 5 3 ] Jln(A 1/B) = -Xy (T/B) (1 + Tp/T) From [ 5 3 ] we see that a p lot of In ( A J / B ) vs. 1/B can give the Dingle temperature. The fact that A^ is beating does not change t h i s , as long as the points used on the graph are at the same pos i t i on in the beat c y c l e . The obvious choice is to use the f i e l d and the amplitude at the maxima of the beat pat tern. Figure 2 7 shows a p lot of A^ vs. 1 / H , and a least squares f i t to the maxima for T = 1 . 2 5 K gives -Xy ( T / B ) ( 1 + T D / T ) = - 1 6 4 . 5 or JQ = 0 . 7 5 0 K It is quite evident that S = 0.197 is the proper p r i nc i pa l va lue. The phase measurements give A ^ / A 2 > 0 cons i s tent with th i s choice. We are thus l e f t with only the tr igonometr ic m u l t i p l i c i t y according to which poss ib le so lut ions are S = ± 0 . 1 9 7 ± P, P e 1 each of which gives ident i ca l experimental r e su l t s . For simple metals such as lead, where the band s t ructure can be derived from a weak pseudopotential together with the sp in -o rb i t i n t e r a c t i o n , a physical argument given by Pippard (1969) r e s t r i c t s the range of poss ib le values of S according to the 98 inequal i ty S 1 (rn* /m) + (s/2) where, s is the number of Bragg re f l e c t i on s undergone by an e lectron in one cyc lot ron o r b i t . This value is 3 for the t, o rb i t normal to [110] which gives r i s e to the y o s c i l l a t i o n , so that with rrT/m = 0.560, we have 0 < S < 2.06 Of the 4 values of S which f a l l into th i s interva l (0.197, 1.197, 0.803, 1.803) two give A]/A2 < 0 which is incons is tent with the phase and Dingle temperature c r i t e r i a . We are thus l e f t with the two poss ib le so lu t ions . S = g m"/2m = 0.197 and 1.803 c c corresponding to g c = 0.704 and 6.44 r e spec t i ve l y . The ult imate choice between these 2 r e l i e s on a band ca l cu l a t i on which includes the sp i n - o rb i t i n te rac t i on . CHAPTER EIGHT A SEARCH FOR THE 4MG OSCILLATIONS 8.1 Prel iminary Remarks Quantum o s c i l l a t i o n s of unusually long period 4MG) have been observed recent ly in lead using the Shubnikov-de Haas e f f e c t (Tobin e t . a l . , 1969) and sound attenuation (ivowi and Mackinnon, 1976). It has been suggested that these long o s c i l l a t i o n s might a r i se from small pockets of e lectrons in the 4th B r i l l o u i n zone. While pockets of th i s kind appear in the empty l a t t i c e band s t ruc ture , a l l r e a l i s t i c band ca l cu l a t i on s f i t t e d to the Fermi surface data show the 4th zone to be empty, ( c f . Anderson and Gold, 1965). We are thus led to wonder whether the long o s c i l l a t i o n s might be an a r t i f a c t generated by M.I. A concerted e f f o r t was made.to detect s im i l a r o s c i l l a -t ions in the dHvA e f f e c t with the hope that they could then be studied with the feedback technique. Unfortunately, no evidence for these long o s c i l l a t i o n s could be found, so that on an upper l im i t on the i r amplitude resu l ted. In the process of 100 the search, some useful ideas were developed inc luding the exact so lu t ion to the problem of large modulation. 8.2 Review of the Standard Weak-Modulation Solut ion F i e l d modulation, followed by phase-sens i t ive de tec t i on , is the most widely used technique for observation of the de Haas van Alphen e f f e c t . The problem of c a l cu l a t i n g the e.m.f. induced in a pick-up c o i l surrounding the sample has been solved in de ta i l for weak modulation f i e l d s . There are, however, circumstances which warrant rather large modulation f i e l d s , large enough so that some of the approximations made in the weak-modulation treatment may no longer be v a l i d . One such circumstance is the detect ion of long-period o s c i l l a t i o n s such as those reported by Tob i ri e t . a 1 . (rl 969) havfing frequencies F ^ li MG. It is- then des i rab le to modulate with an amplitude ~ 1 kG which is a s i zeab le f r a c t i o n of the q u a s i - s t a t i c back-ground f i e l d . We f i r s t review the standard formulation for weak modulation, and then develop an exact, e x p l i c i t so lu t ion for a rb i t r a r y strength of modulation f i e l d . In i t s present widely-used form, the modulation f i e l d h is s i nu so ida l , small with respect to the large background f i e l d H, and is p a r a l l e l to i t . Thus, the sample experiences a net f i e l d H + h s in ait. The large background f i e l d is made to sweep s lowly, so that in t reat ing the modulation f i e l d , we can regard the background f i e l d as e s s e n t i a l l y constant. The 101 c r i t e r i o n for th is assumption is (dh.) , < < dJi dt R M S dt The treatment for weak modulation is well known, and w i l l ju s t be out l ined here. Without loss of genera l i t y , we can ignore various constant phase f ac to r s , and wr i te the o s c i l l a t o r y part of the magnetization simply as: M • r 2 T r F i M = S m [H + h ( t ) ] where h(t) = h s in cot and u> is the modulation angular f r e -quency. The equation is for a .reduced magnetization, with the amplitude factors incorporated into i t . We might a l so add that we must work at a low enough frequency to so that we have no f i e l d inhomogeneity due to eddy currents . In the conventional approximation, the denominator is ex-panded, and only the l inear term in 77 is re ta ined, so that rl M ^ s i n [ * f (1 - ^ ) ] Since the approximation is made in the argument of a rap id ly o s c i l l a t i n g s ine funct ion , (—rr~'Xj 10^ t y p i c a l l y ) one must be n carefu l to s tate the j u s t i f i c a t i o n c o r r e c t l y . 102 The c r i t e r i o n to be s a t i s f i e d must assure that the argument is at most f i r s t order in •p- . This is true i f and only i f the second order term is very much less than 2TT. The second order term i s : 2TTF , h j t h 2 H V H ; So the j u s t i f i c a t i o n is H or simply Fh 2 . — « 1. If th i s inequa l i ty is not s a t i s f i e d , the argument of the s ine must be taken to be at least quadrat ic in h. This "weak modulation" c r i t e r i o n is usua l ly met in the normal laboratory s i t u a t i o n , and M can be developed in a Four ier ser ies as fol lows ~ . r 2 TTF h(t ) \ 1 M .= s i n [ — (1 — ) ] 2TTF /, . v 2TTF 2TTF,, . ^ . 2TTF (hsinwt) .--rr— : —r - (hs inwt) i 1 i H2-^'=» - i H - i H 2 H -r-r Ve e -e e 2i 103 Us^ng the ident i ty 00 - i n y - i u s in y I J>> " e we obtain 2TTF e H -intot - i H -e The c o i l surrounding the sample gives a voltage pro-dM port ional to . Taking th i s d e r i v a t i v e , • J - % - I no) J (—r—) cos (-rj n u t ) dt n ,,Z H n=-°° H We now separate the t and H dependences; , and f ind a f t e r a l i t t l e man ipulat ion This is the conventional resu l t • for weak modulation. We note in passing that the same resu l t is obtained i f the sign of the two phase factors ' s reversed. 8 . 3 Large Modulation By large modulation we mean that our i n i t i a l assumption about the l i n e a r i t y of the s ine argument breaks down. In p a r t i c u l a r , for the long o s c i l l a t i o n s having F ^  k MG in an dM „ v „ • /27TFhx . ,2TTF TI\ . , NTU — % - V 2nco J S i n (-rj- + n;d s i n (ntot + — ) at , n ,,c n L C n=l " H 104 appl ied f i e l d of 50 kG, for maximum response, we should modulate over something l i k e one cyc le of the waveform which makes h ^ 325 Gauss. Our c r i t e r i o n for neglect of the second, and high order terms was h « /P /F Any such long o s c i l l a t i o n in the dHvA e f f e c t would be swamped by the strong y o s c i l l a t i o n s with a frequency of 17 MG, making /P7F % 2.7 kG (H = 50 kG) For th is s i t u a t i o n , i t cannot be sa id that h is then very much less than / H 3 / F , and i t was f e l t that a deeper study into the e f f ec t s of the quadrat ic , and higher order terms, was warranted. A common prac t i ce is to exp lo i t the zeros of the Bessel functions to e l iminate the unwanted o s c i l l a t i o n s . Our ob jec t , in par t , is to determine poss ib le s h i f t s of these zeros when using large modulation f i e l d s . We now present an exact Four ier decomposition which is v a l i d fo r any strength of modulation. In our bas ic equat ion, M = s i n [rjrr r j H+hcoso3t we can use the cosine phase of the modulation without loss of genera l i t y , s ince the resu l t cannot be dependent on the o r i g i n 105 of time. This choice of phase makes M an even funct ion of time t which s i m p l i f i e s the planned Four ier expansion of the s ine argument. Because only the even, cosine terms can surv ive , we can wr i te the expansion as: 2TTF r u . L. T = Z a cos ncot H + hcoscot „ n n=0 where 2TT/CO a = 2co_ TT 2TTF H + hcoscot dt and CO a n = 7 2TT/CO 2TTF H + hcoscot cos ncot dt These integra l s may be reduced to standard form, and we read i l y obtain (Gradshyeyn and Ryzhik, 1965) ao = 2TTF ATTF h_ H - 2 v n < - ' n where P =" h/H H nr. 106 and 0 < p < 1 We now wri te for M 2 3 M = s i n [ 2 a 0 ( l / 2 - p cos uit + p cos 2wt - p cos 3^t + . . . ) ] 2 3 ( i a Q -2iagpcosa)t 2 ' a Q P cos2cot - 2 i a Q p cos3<*>t = Im, { e e e e . . . Making use of the i den t i t y e - i y c o s y = J ( . „ " J n ( v ) e - m y n=-co and i t s complex conjugate e . y co sy = l ( l ) n ^ e - n y n = - » We obtain the fo l lowing M = In, Je'a° I ( - i ) n J n(2a 0p) e ,mt J ( i ) n J n ( 2 a o p 2 ) a 2 i n u t ( n r n 2 . . . = -«>L k - i n k [ e i B t ^ - ) k k n k ] } Again, we need the time d e r i v a t i v e , which is e a s i l y obtained: 107 dM dt = Im r iwt | (-)H] X I 6 This is the exact so lu t ion for dM dt , given that M = sin [• 2TTF •I H + hcoswt h The solution is valid for arbitrary h, provided only that j | < l -In order to obtain a tractable and useful formula for dM , it is necessary to find suitable approximations for the inf in i te sums and products in the exact solution. This can be done to any desired accuracy. We are usually interested in the phase-sensitive detection at a particular harmonic of the modulation frequency OJ. For the nth time harmonic (nw), the required integers (±n) are related to the various integral indices occurring in the exact result by: where n^ can be any integer between -«° and °°. This equation determines a l l the sets {nk> for any desired time-harmonic nu>, each set giving one term in the solution. A procedure wi l l now be given for ranking these sets in order of importance. n 108 The par t of the s o l u t i o n which determines the r e l a t i v e magnitude of a p a r t i c u l a r t ime harmonic is f J (2a p K ) k=l n K 0 In most c a s e s , 2agp' < -«l f o r k i 2,. In a l l c a s e s , 2agP^<<l f o r a la rge enough va lue of k, s i n c e 0<p<l. For sma l l e r va lues of k, a l l J n (2a^p' <) are of order 1, and a l l must be cons ide red . k In the normal l abo ra to ry s i t u a t i o n , there is but one such term. k > When 2agP « 1 , which i s u s u a l l y the case f o r k—,2 , one may rank the se ts by t h e i r r e s u l t in the f o l l o w i n g order of magnitude e s t i m a t e . Given the set (n^} , the cor responding term i s approx imate ly : » (a.p ) If — — . k n k ! c where the product s t a r t s at a va lue k c > which i s the lowest i n -teger s a t i s f y i n g 2aQp^« l ( t y p i c a l l y k c = 2 ) . In p r a c t i c e , n ^ O on ly f o r smal l k(k~3), making t h i s a qu ick method of rank ing . One can see that the order depends somewhat on the va lues of ag and ,p . Table III i s an example of t h i s r ank ing , a long w i th the order of magnitude of the cor responding terms. It i s done f o r the second t ime harmonic, and t y p i c a l va lues were chosen f o r ag and p. 109 n = 2 , a = 5 x l 0 2 , p = TO TABLE III Ranking The Terms Order of Mag. nl n 2 n 3 nk 1 +2 0 0 0 1 0 ' 3 0 ±1 0 0 10" 3 ±k ±1 0 0 10 - 6 ±2 ±2 0 0 io"6 ;i 0 ±1 0 IO-9 ±4 ±3 0 0 10 " 9 ±1 ±1 ±1 0 10" 9 ;3 ±1 ±1 0 10" 9 +3 +i ±1 0 The next term is of order 10 The most important term is invariably n^  = ± n, n^ = 0 for k ^ 1. We shall now calculate it separately, and compare it to the result for weak modulation. 110 l m | e ' a ° ( - i ) n J n ( 2 a 0 p ) n J ^ a / ) e " ' n U t (-i na>) + e , a ° ( i ) " ( - ) " j n ( 2 a 6 p ) n J Q ( 2 a 0 p K ) e 1™ (ina, )j oo k ( ' a n l = 2nw s !n (nut ) J n-(2a Q P ) n JQ(2aQp ) lm.7i, I • ( - J ) n e = 2nu J n ( 2 a Q p ) n J 0 ( 2 a Q p ) sin(nuit) s i n ( a Q - y - ) To compare t h i s to the e a r l i e r convent iona l r e s u l t , which s t a r t e d w i th the s i n e phase of the modu la t ion , l e t us sub-s t i t u t e t -> t - ^ to get ^ £ -2no> J n ( 2 a Q p ) n < l 0(2a 0 p k ) s in(nwt - ^ ) s i n ( a 0 - !f) k\ . / ^ . nir \ . / . nir - -2naj J n ( 2 a Q p ) n J Q ( 2 a o p K ) s in(nwt + !f )s\n(aQ + >f ) By c o n t r a s t , the r e s u l t f o r weak modulat ion is dM _ . /2TrFhx . i ^ . niTv . /2TTF , = -2nu J n ( — — ) s i n (nut + —) s i n (-q— + —) In compar ison, there are two d i f f e r e n c e s . F i r s t l y and perhaps most impor tan t l y , the measured dHvA frequency is d i f f e r e n t . The convent iona l r e s u l t i s F, whereas the exact r e s u l t g ives This means that there i s a second order c o r r e c t i o n in the measured f requency. Under most c i r cums tances , t h i s s h i f t i s s m a l l , but g iven the high degree of accuracy which dHvA work boas t s , t h i s in some cases may be important . It i s important to note that a l though on ly the f i r s t term in the s o l u t i o n was taken , t h i s f requency c o r r e c t i o n i s e x a c t , that i s , none of the h igher terms change t h i s r e s u l t . The o ther d i f f e r e n c e is the ampl i tude c o r r e c t i o n \ J 0 ( 2 a ( A = i - k S ince 2aQp' % 1, 2a Q p « 1 f o r k > 1. One can show that t h i s i n f i n i t e product converges to S 1. In the f i r s t , o r d e r : te rm, we can say that the ampl i tude f o r s t rong modulat ion i s sma l l e r than the convent iona l r e s u l t , and a l s o that i t does not s h i f t the zeros of the Bessel f u n c t i o n . The obvious a l t e r n a t i v e to expansion in a Fou r i e r s e r i e s i s an expansion in a T a y l o r s e r i e s , namely Gather ing a l l the terms f o r any harmonic nco i s a fo rmidab le t ask , but i f we r e t a i n terms on ly to second order in (77), we c o s t o t + (TJ) c o s tot - (77) c o s tot + . note t h a t t h e dHvA f r e q u e n c y becomes w h i c h a r e the l e a d i n g terms i n a T a y l o r s e r i e s o f our e x a c t r e s u l t 8.4 M o d i f i c a t i o n s t o t h e A p p a r a t u s and A n a l y s i s f o r the F ^ 4 MG Se a r c h The major c o n s i d e r a t i o n i n the d e s i g n o f the d e t e c t i o n a p p a r a t u s was s e n s i t i v i t y and s i g n a l t o n o i s e . The s h i f t i n emphasis from t h e f r e q u e n c y r e s p o n s e d i c t a t e d s e v e r a l modi-f i c a t i o n s . W h i l e the s a m p l e - d e t e c t i o n c o i l arrangement remained t h e same, i t s o u t p u t now d r o v e t h e p r i m a r y o f a t r a n s f o r m e r (P.A.R. Model AM-l) t o t a k e advantage o f the low o u t p u t impedence lOOfi) o f t h e d e t e c t i o n c o i l s . The s i g n a l was d e t e c t e d on t h e second harmonic o f a 41 .7 Hz s i n u s o i d a l m o d u l a t i o n f i e l d w i t h a P.A.R. 124 p h a s e - s e n s i t i v e d e t e c t o r . I t s n o t c h f i l t e r (Q=50) was used t o b l o c k t h e fundamental and a K r o h n - H i t e (model 3322R) bandpass f i l t e r (Q=l) was c e n t e r e d on the second harmonic. 2 The a m p l i t u d e o f t h e m o d u l a t i o n was made t o v a r y as H whic h kept t h i s a m p l i t u d e s p a n n i n g the same number o f dHvA 113 c y c l e s at any f i e l d H. Using the zeros o f the Bessel f u n c t i o n response of the observed magnet iza t ion ori the modulat ion ampl i tude (see s e c t i o n 8 . 3 ) , the dominant o s c i l l a t i o n s in any d i r e c t i o n cou ld be at tenuated by about a f a c t o r of 5 0 , a l l o w i n g an inc rease in s e n s i t i v i t y of the same f a c t o r w i thout s a t u r a t i o n . The r e s u l t i n g s i g n a l was d i g i t i a l l y recorded w i th 20 b i t r e s o l u t i o n on magnetic tape. These data were sub-sequent ly F o u r i e r t ransformed w i th the use of the main U . B . C . computer (Amdahl 470 ) and a program o u t l i n e d in Appendix C. With t h i s arrangement, a l l of the o s c i l l a t i o n s in lead seen p r e v i o u s l y were e a s i l y i d e n t i f i e d , however, no s ign o f the k MG o s c i l l a t i o n s appeared. The search inc luded examinat ion of the Fou r i e r t ransforms at the second harmonic (^ 8 MG) to a l l ow fo r the p o s s i b i l i t y of a sp in s p l i t t i n g zero at the f i r s t harmonic. The r e s u l t of t h i s negat ive experiment p laces an upper bound on the ampl i tude of these long per iod o s c i l l a t i o n s in magne t i za t i on . In each of the three major symmetry d i r e c t i o n s [ 1 0 0 ] , [ 1 1 0 ] , [ i l l ] , t h e i r ampl i tude must be less than 1 par t in 10 of the magnet iza t ion of the dominant o s c i l l a t i o n s in each d i r e c t i o n . Th is l i m i t in abso lu te terms i s about \ M G 2 0 0 U G ' The o s c i l l a t i o n s of Ivowi and Mackinnon ( 1976) and Tobin e t . a 1 . ( 1 9 6 9 ) thus remain an enigma. It i s f e l t that t h i s area of study would bene f i t g r e a t l y by a c o l l a b o r a t i o n of the feedback technique w i th the Shubnikov-de Haas e f f e c t o r sound a t t e n u a t i o n , where these o s c i l l a t i o n s appear v i v i d l y . APPENDIX A FLEXIBLE GEAR ROTATOR An apparatus was b u i l t to ro ta te the sample about an a x i s which was 90° away from the a x i s of the magnet bore (the on ly d i r e c t i o n of access) based on an idea g iven by Pippard and Sad le r (1969). The m o d i f i c a t i o n s made to the o r i g i n a l des ign were ex tens i ve enough to warrant f u r t h e r d e s c r i p t i o n in t h i s appendix . Our compactness requirement r e s t r i c t e d the s i z e of the apparatus to a degree where the mechanisms would be s u b s t a n t i a l l y sma l l e r than any that had p r e v i o u s l y been b u i l t s u c c e s s f u l l y . The e n t i r e apparatus i s cons t ruc ted from nylon rod except fo r a Mylar gear . Th is c i r c u l a r Mylar gear was cut from a p iece of 0.003 inches t h i c k shee t . A s p e c i a l j i g was made to cut 32 t r i a n g u l a r teeth w i th a razor b lade in roughly c i r c u l a r s t a r t i n g m a t e r i a l . A square hole (s ide length 0.075 inch) was cut in the cen t re w i th a punch. Through the square h o l e , a r e t a i n e r fas tened a r i ng to the gear so that the a x i s o f the r i ng was pe rpend icu la r to the gear a x i s , and Figure 28. Sample Rotator Assembly i n t e r s e c t i n g i t ( s e e F igure 28). The r e t a i n e r , so p l a c e d , was welded to the r i ng w i th a s o l d e r i n g i r o n . The square ho le ensured the absence o f s l i p p i n g when the gear was tu rned . The s p h e r i c a l sample was glued to the r i ng w i th a smal l drop of G . E . v a r n i s h . Care was taken to apply a minimum amount of va rn i sh to the sample as d i f f e r e n t i a l c o n t r a c t i o n would cause s t r a i n upon c o o l i n g . A f t e r a l l ow ing 2k hours f o r the v a r n i s h to d ry , the assembly was p laced i n s i d e a c y l i n d r i c a l tube by bending the Mylar gear to conform to the shape of the tube. When in p l a c e , smal l a x l e p ins held the gear a x i s s t a t i o n a r y wh i l e s t i l l a l l o w i n g i t to r o t a t e . As the gear r o t a t e s , i t f l e x e s to r e t a i n i t s c y l i n d r i c a l shape, and ro ta tes the sample about the a x i s o f the gear . Only the teeth at the top of the gear prot rude from the c y l i n d e r , where they mesh w i th a d r i v i n g gear . The 16 tooth d r i v i n g gear was made by pushing a hot brass negat ive in to a c y l i n d r i c a l nylon b lank , and subsequent machining prov ided a coup l ing to the top of the c r y o s t a t . The body of the c o i l former held the d r i v i n g gear in the proper p lace to mesh w i th the Mylar gear . In order to keep the teeth of the d r i v i n g gear i d e n t i c a l to those of the Mylar gear , one f i n d s that the d r i v i n g gear must ro ta te about an a x i s which is o f f c e n t r e . The d r i v i n g gear w i th the r o t a t o r assembly and c o i l former i s shown in F igure 29. A be ry l l i um-coppe r sp r i ng was used to ensure in t imate contact of the gears when coo led 117 BeCu Spring Slots in both sides 2 - 5 6 Bolt to fit in slots »ff centre driving gear (16 teeth) to mote Mylar gear 0 . 5 0 0 " 0.281" c ^ i 1 1 i/32 hole for positioning Coil former 0 .853 Teeth of Mylar gear Axle pins Slot in base for positioning Threaded inside ^ 1 6 - 2 0 NF Spherical Sample in sample ring JIR-20 NF Figure 29. Sample Rota tor With D r i v i n g Rear , and Coi1 Former to l i q u i d hel ium temperatures. The c y l i n d e r ho ld ing the sample was i nse r ted in to the bottom of the c o i l former and held r i g i d l y w i th a 7/16-20 (NF) nylon b o l t . The l o c a t i o n of i t s proper r o t a t i o n a l p o s i t i o n was found by pushing a temporary w i re through a smal l ho le in the bottom of the c o i l former, and in to a s l o t m i l l e d in to the base of the sample c y l i n d e r . Th is w i re was removed a f t e r t i g h t e n i n g the nylon b o l t . The e n t i r e assembly was inse r ted in to the t a i l of the inner dewar shown in F igure 5- At the top of the c r y o s t a t , p r o v i s i o n was made to ro ta te the c r y s t a l e i t h e r by hand, or by e l e c t r i c motor. 119 APPENDIX B THE DISCRETE FOURIER TRANSFORM Since both the spectrum a n a l y z e r , and the computer programs use d i s c r e t e F o u r i e r t rans fo rms , the b a s i c de-f i n i t i o n s w i l l be presented in t h i s appendix . The d i s c r e t e Fou r i e r t ransform is de f ined by A(k) = V X. e " 2 i r I j k / N k = 0, 1, . . . , N-l j=0 J where X j , j = 0 , 1, . . . , N-l i s a set of complex numbers . The inverse t ransform i s B ( J ) = V A(k) e 2 l V i j k / N j = 0, . . . N- l k=0 where B( j ) = NX j . The f a s t Fou r i e r t ransform programs supp l i ed by l i b r a r i e s o f ten requ i re the input data to be e i t h e r symmetric or a n t i -symmetr ic . Any set of data can be separated in to i t s a n t i -symmetric and symmetric components. If the se t X. con ta ins 120 the o r i g i n a l data v a l u e s , then the ant isymmetr ic va lues are g iven by X N /2+ l - j = I ( X N /2 - j + l " X N /2+j+ l ) j = 2 ' • • • N / 2 " 1  x 3 = X N / 2 = °' g A s i n e t ransform can then be app l i ed to X . The symmetric S va lues X. are g iven by S 1 X N/2+1-j = 2 ( X N/2+j+l = X N / 2 - j + l ) j = 2 ' • • ' N / 2 " 1 S S XI = X T X N /2 = X N / 2 . s A cos ine t ransform can then be app l i ed to X . App ly ing a window to the o r i g i n a l data va lues u s u a l l y r e s u l t s in a t r adeo f f of r e s o l u t i o n and s i d e f e b e s . The la rge s i d e l o b e s encountered in the use of a square window can h ide f requenc ies of sma l l e r ampl i tude which are a c t u a l l y f a r away in the space of the v a r i a b l e k. The Hanning window is a good compromise, s i n c e not much r e s o l u t i o n is l o s t , but the s i d e l o b e ampl i tude decays very q u i c k l y in k space. If the o r i g i n a l data i s in t ime t , a p p l i c a t i o n of a Hanning window s imply i nvo lves m u l t i p l i c a t i o n 2 of the o r i g i n a l data by s i n [ T r ( t - t g ) / T ] where t i s the sma l l es t va lue of t and T is the du ra t i on of the reco rd . In e f f e c t , the Hanning window rounds o f f the sharp corners on the edges of the data where the window is opened and c l o s e d . 122 APPENDIX C COMPUTER PROGRAMS The computer programs used to generate the r e s u l t s in the body of t h i s t h e s i s are l i s t e d in t h i s appendix . For the most p a r t , the programs are w r i t t e n to be s e l f exp lana to ry in regards to t h e i r use. The "Data Reading and A d j u s t i n g " program reads the data from the Stevenson i n t e r f a c e a f t e r i t has been converted to EBDIC from ASCI I . The convers ion was done by a standard t r a n s l a t i o n rou t ine ("TRANS) in the U . B . C . Computer l i b r a r y . A f t e r reading the d a t a , the For t ran program, by use of the f u n c t i o n sub-programs, a l lows the user to c rea te the proper x and y coord ina tes from the a v a i l a b l e d a t a . The program then conver ts the data to a format compat ib le w i th a l l of the remaining programs. The " S y n t h e t i c Data Genera to r " program a l lows the user to c rea te any data he p leases and puts i t in the proper format . Th is program was l a r g e l y used to t e s t the other programs, and B u i l t by A. Stevenson p resen t l y at TRIUMF. check the r e s o l u t i o n . The "Window" program was used to cut down the s i d e -lobes of the t rans fo rm. Instead of us ing the Hanning window, the data were m u l t i p l i e d by a s imple s i ne f u n c t i o n spanning 0—r r over the window. Th is g i ves more r e s o l u t i o n than the Hanning window, and the s ide lobes are s t i l l not too l a r g e . Th is s i ne window was used because the data at one end of the window (the h igh f i e l d end) were the most impor tant , and they were not cut o f f so d r a s t i c a l l y as w i th the Hanning window. The a n a l y s i s program takes the F o u r i e r t rans fo rm of the data prepared by the e a r l i e r r o u t i n e s . One can choose the r e s o l u t i o n and the window in k space f o r the t rans fo rm. The power spectrum fea tu re was most o f ten used. The " P l o t t i n g " program accepts data from a l l of the prev ious rou t ines so that rea l space and Fou r i e r space data can be p l o t t e d . P l o t t i n g can be done on the p r i n t e r , the g raph ics t e r m i n a l , or the hard copy Calcomp p l o t t e r . The 1/H ax i s i s l a b e l l e d in the p r i n t e r p l o t s . The M. I . S imu la t i on program uses the formulae developed fo r M. I . in Chapter I I , and c a l c u l a t e s the r e s u l t of our experiment desc r ibed in Chapter V I . Data Reading and Adjust ing }2h 1 DIMENSION 4t5O)7Dt50) 2 T4T4CH4R1/'*'/ 3 04T4 CHAR2/'0'/ 4 " P R I N T 3 U 5 314 F 0 R * A T (' M : J"9ER OF RECORDS? C*25aNJ M < * E R OF o»T4 PT P M ? 3 ) (14) •) b READ 315.NNN 7 3t5 F0R*4T(T4) 8 vjPPa2S*N'gM 9 - R l TE f 3. <91 )NPP 10 91 F3RMATCT5) 1 1 00 20 LTNFi l ,M*N 12 R E 4 0 ( 3 , l ) ( 4 ( T i » n ( I ) . I « l . S O ) 13 t F0R««4T(50(41,lx;F8,0n 14 M J U l , 5 0 , 2 15 n a m , 16 C IF(4(I),NE,CHAR1.0R.*(Tt1.NE,Crl4R2) PRJ.NT3 , L I*E 17 3 FORMATC A8MDR**I»ITY I s L.TNE',13) 18 o m a v F u > i ( 0 ( i n l D ( I ) > 19 OtIl)sXFUM(0 ( I n,0(in 20 2 c r i M T t M U E 21 w R T T E ( 2 . 2 i ) f 0 f I i . 0 ( T - l l . t s 2 , 5 ( i , 2 ) 22 21 F0R"4T(2F.14.7) 23 20 cnNTjvjJE 24 3T0P ?5 E^O 26 FUNCTION VFUM(X.Y) 27 vFUMsvolo'. 26 5 E T 'J R ^  29 ENO 30 FilMCTTON XF'JM(K.V) 31 XFUSal , / ( l . 22<»*y ) 32 PETJHS 33 END 125 Synthetic Data Generator 1 DIMENSION PTCtonOl 2 PTs3.1«15926 J PRINT 200 4 200 F0R*AT(» ENTER NU MBER 1? OAT* POINT P A I R S C i a ) ' ) 5 READ 201,NNN 6 201 FTRMAT(TU) 7 W9IT£(2,2051MN\) 8 205 FORMATCIS1 9 PRINT 202 10 202 FORMAT ( • ENTER H"IN,MM4X TN MS, (2F6,0)>) 11 READ 203,HMTN,MMAX t2 205 FORMATC2F6.0) 1J H T N C a ( H « A X - H M l N ) / F t . O A T ( N M N ) i a DO 100 1=1,1000 15 P T ( T ) s 0 ' . 16 100 CONTINUE 17 PRINT J IB 1 FORMATC'TO CREATE THE J U M OF T * E * P (-3*n «C13 t 2. « B T * A * T « 0 } . . , Ts 1 / M ' / 19 1' ENTER fl.A.3.'.'. C5F6.0l'.'.'.OR ZER3 TO STOP' ) 20 6 READ ?,B,A,0 21 2 FORMATC5F6.0) 22 IMA.EQ'.o'.) 30 TO 99 23 PRINT 7.3,A,3 ?U 7 F0R*AT(3E10.3) 25 MSHMAX 26 5 00 3 T s l . N N N 27 HsH-HTNC 28 T s l . / M 2R PT(I)sPT(I)+T*EXP(-B * T)*C0SC2.*PI*A» T-3) 30 5 CONTINUE 31 GO TO 6 32 99 M = h " A X 33 Tsl'./M 3a no 101 I « I , N V J N 35 HsH-HINC 36 Tsl'./M 37 *RlTEC2 . ' 4U,'TtT) 38 « F O R M A T (2E1«.7) 39 101 CfNTlMUE U 0 S T O P U l E NO 126 Wi ndow t DIMENSION X(100n),V(1000) 2 PT3i,tUt5"26 J READ(5,1)NNN U 1 F0RM»T(I51 5 W9ITE(2,6)NNN j, 6 FTRMATdSl 7 00 2 T»1,NNN B R E A D n , 5 i x m . v i r n <t ^ C O R M A T (2E10.7) 10 2 CONTIMU6 11 FsPI/<X(NNNl-XM 12 OO a T3l,\iNN 15 ya)sv(T)*STNfF*fX(T)-x(l ))1 ltt *RITE(2.5)X(I),Y(I) 15 5 F0RMAT(2El'a'.7) tfc 0 CONTINUE 17 STOP 18 E NO I' 1 2 1 Ana l y s i s Program 1 REAL T(1000),°Tf1000) 2 PI2»2'.*3, 1415^265 3 REA0(3,315)NNN a 315 P 0 R"ATO5) 5 XNMN«FLOAT(NN)N) b 00 "» t»t»NNN 7 QEAO(3»tO)TfI),BT(I) 8 10 FORMAT (?E1<»'.7) 9 9 CONTINUE 10 PRINT 1 11 1 F 0 R " A T ( ' 0 A N A I > S T S ? » ) 12 REAO ?,INAL 1 3 ? FORMAT(Il) 1 4 GO T0r3.4.5,b,7. (><>).mi IS 99 STOP tb 17 3 FORHIT!'OEOURIEP "EAL TRANSFORM'/' F M I N. F« A X , N'JMF ? ( 2F b . 0 . t 4 ) ' ) 70 18 READ 8» FMIN.FMAX.NIMF 1 9 A F 0 R « A T ( 2 F b , 0 , l 4 ) 20 *RITE(2.437)^UMF 21 437 F 0 R M A T ( I 5 ) ?2 FT^CsfFMAX-FMlMi/FLOATfM.IMF) 23 F s F ^ I N 2a 00 I t Ht.NUMF 25 9IJMS0'. 2b 00 12 J*l.NMN 27 S J M s S U ^ + P T ( J ) * C P S ( P I 2 * F * T ( J ) ) 28 12 CONTINUE 29 SlJMsSU M/XNNN 30 *RITE(2.13) F.3UM 31 13 F0RMAT(2Eia'.7) 32 FsF*FINC 33 11 CONTINUE 34 STOP 35 36 4 3tt PRINT 34 FORHATCOFOURIER IMAGINARY TRANSFORM/' FMIM,FHAx.NUMF?(2F6 . 0 ,14 ) 37 READ 8,FMIN,FM 4 1 < ,NSJMr 38 *RTT£(2.4371\IIHF 39 FnC»CFHAX-FMlsjl/FLOAT (MJMF) ao FsFMTN 41 00 31 Iat.NUMF U2 SJ M»0'. «3 00 32 Jsl.MMM au S.JM33!JH*PT(J)*CPS(PI2*F»T(J)) US 32 CONTINUE "6 S;J13SL)M/XNMN 47 WRITE(2 I13)F,3JM 48 F«F*PINC U9 31 CONTINUE 50 STOP 51 5 PRINT 54 52 54 FORMAT ( ' OPOWER SPECTRLJW,'/' FM T N, F* A X , NJMF ? (2F6 , 0 , I« ) ' ) 53 9EA0 R,FMTN,FMAX,NUMF 54 wRlTEf2.a37)NUMF 55 FINCB(FMAX-FHIN)/FLOAT(NJMF) 5b F»FMT.N 57 00 51 Ist.NUMF 128 i SB 3'J^loO, 59 SUM2aO, 60 DO 52 J«t,NNS 61 S J M l s S U ^ l * P T ( J ) * C 0 S ( 9 T 2 * r * T ( J ) ) 62 S U M 2 s S U " 2 * P T ( J ) « S I N ( P l 2 * M T U ) ) 65 52 CONTINUE 6a 3UM«(SU*1 /XNNN)**2*(3UM2/XNNN1**2 65 wRITE(2.15)F,3UM 66 FaF+FINC 67 5 1 CONTINUE 6B STOP 69 6 PRINT 7U 70 7a FORMAT ('OLAPlACC TRANSFORMI/I 3M IN,3MA X,NJ*3?(2F6,0,I a ) • ) 71 REAO 8,SMTN,9MAX,NUM3 72 «RITEC2.U37)\|JMS 75 S T N C « ( 3 M A X - 3 M I N I / F L O A T ( N j M S ) 7a SSS^IN 75 OD 71 Ixl.NUMS 76 3UMaO'. 77 00 72 J•1,NNN 7B SJM«3UM*PT(.n*EXP(-3*T(J)) 79 72 CONTINUE «0 SJ Ma3J M/XNNN SI W9ITE(2.13)3,SUM 82 Sa3*3INC «S 71 CONTINUE Bit 3T0P 85 7 PRINT 9tt 86 9a FORMAT('OLAPL*CF- B0*ER SPECTRUM'/' 8MIM,3MA X,NJMS?C?F6.0, 87 READ 8,SMtN,SMAX,NJMS 88 wRITEf2.U37)NUMS 89 PRINT 105 90 105 FOR^AT(> THE ANGULAR *RE3'IENC Y? ' ) 91 READ 106,A 92 1 06 FORMAT(F6'.01 93 STNC«f 3MAX-3MlN1/ri.0AT(NjM3) 9U SaS"IN 95 00 91 Ial.NUMS 96 SUMlaO, 97 SUM2oO, 98 00 92 Jsi.NNN 99 S ' J M l s S U M i + P T ( J ) * E X P ( - 3 * T ( T ) l * C 0 3 ( A * T ( J ) ) 100 S'JM2 = S U « 2 * P T ( J ) * E X P ( - S * T ( J ) ) « S I N ( A * T ( J ) 1 101 92 . CONTINUE 102 SUMa(3UMl/XNNN)*«2+(3JM2/XNNN)**2 105 wRITF(2,15)3,SJM l o a SsS*3TNC 105 91 CONTINUE 106 STOP 107 END 129 P lo t t i ng t DIMENSION xnoon),YUOo<n,CH4nm 2 L O G I C A l M QUE J REAL l<75) 4 OAT A CHAR/• 1,1*1/ 5 C REAO IN THE DAT* 6 READ(3.2)N 7 2 FORMAT(151 8 X M 4 X « . 1 '.E*50 9 YMAXaXMAX 10 XMIN«1,E*S0 t l YMJNsXMIN 12 OO 3 T 31» N is REAO(3.«)x(n.yi ,n 14 . a FORMAT(2El«'.71 15 IF(X(I)'.LT.XMlMlXMTNaXfT) 16 IF(X(t)^GT.XMAxiXMAXaX(I) 17 I F ( V ( T ) . L T . V M l N 1 Y M t N i Y ( t ) 1R IF (Y(t)'.GT.YMAXlYMAX»Yf I) 19 3 CONTINUE 20 C F J NO SCALING FACTORS 21 X3»8,/(XMAX-XMIN) 22 YSa8./(YMAX-YMIN) 23 XSRs50,*XS/8. 2tt Y S P « 5 0 , * Y S / 8 . 25 C PRINTER PLOT 26 PRINT 7 27 7 F 0 R M A T ( • 00 YO'J '*ISH A PRINTER P L 0 T ? , . , K I N 0 L Y ENTER Y OR N') f 28 REAO 9,.QUE 29 A F O R M A T U i l 30 IF (LCOMC (1 , 0 J E , »V« T.NE'.OlGO TO 9 31 PRINT 10 32 10 FORMAT ( i HOW MANY PRINTER PAGES wOULO YOU LIKE?'. .,(121 ' 1 33 REAO 11,NP 34 11 F0RMAT(I2) 35 TNC»N/(NP*60) 36 00 l a I•1.75 37 L(I)=CHAR(11 38 ta CONTINUE 39 IPNTst UO 00 12 1=1.N,INC a i L ( I P N T ) s C H A R ( t 1 42 I?NTaIFIX((Y(I)-YMIN ) *v3P*t1 U3 L ( T F N T ) a C H A R ( 2 l 44 w R I T E ( 2 . 1 5 1 x r l ) . Y ( I 1 , ( L ( J 1 , J a t , 7 5 1 US 15 FORMAT(I i , 2 ( E l U , 7 , I X ) , • (', 754 1 ) «6 12 CONTINUE U7 9 PRINT 16 48 16 F0RMAT ( * 00 YOj WISH 4 PFN PLOT?,,,(Y,N)•) 49 RE40 17.QUE 50 17 F0RM4T(4H 51 IF(LCOMC (t »QJE, ' Y i ) ' . NE'.0)STOP 52 C PLOT 53 CALL P L 0 T ( X S * ( X ( l ) - X M l N ) , Y S * ( Y ( 1 ) - Y M I N ) , 3 ) 5a 00 5 I82.N 55 CALL P L 0 T ( X S * ( x m - X M ! N l , Y S * ( Y C I ) » Y M T . N ) , 2 1 56 5 CONTINUE 57 C»LL PLOTNO 13Q M.I . S imu la t i on 1 DIMENSION X»l (5O0),XA2f500)> XA3C500) 2 DIMENSION XM(5l?),Xl(25«>),X2C256) 3 SEAL M,NliD,K4PP4,NCYC a INTEGER y . s w 5 COMMON XA1M»K,XA2MAX,XA3MAX,HTM4X.MIMIN 6 DATA Y/'Y'/ 7 PSI(RDUM)aATiNfETA*TAN(Pt*BDUM*DELf*MT-PI/«.)) ft A f R D U , A A D U ) » ( A A D U * A A D J * ( 1 . - E T A ) / ( l . * E T A n f t ( S r j P T ( 2 . ) / 2 . ) * 9 133RTm.*ETA*ETA ) t U.-ETA*ETA)»SIN(2,»Pi»KDU*DELF*HI)) 10 M(NU0.ALPHAn)«Ai*8IN(X*P3t(l.)J 11 . t+A2*CSINf2,*X*PSI ( 2 . n.0.5*NUD * S INC2.*X*2>PSI f I . ) ) ) 12 ?*A'3'*fSlNf 3 , » X » P 3 I ( 3 . ) ) . 1 . 5 * N U 0 » A L P H A ! 3 » 13 3 (SIN(3'.*X*P3I (1 . )*P3I (2.5 )-0.25*NUO*3lN(3.*X*3*PSI ( l . 1 ) ) ) i a A T ( A B . B f l , A L . 3 L ) s S 3 R T ( ( A 8 * S I N ( A L ) + 3 8 * S l M ( B L ) ) * * 2 15 1 •(AR *CO3(AL)*«9 *C0S rBL))**2) 16 PT»3 ,1«I59 17 PRINT 1 18 1 FORMAT (' PLEASE E N T E R l A M B L l T ' J O E S OF TM£ HI3HFST FREQUENCY'/ 19 1 'FIRST,3EC0ND, AND THll?0 HAR*TNTC (IN S A J 3 S ) , AND ETA...CUFU.0) ' ) 20 READ 2.AA1,AA2,AA3,ETA 21 2 PORMAT(UFa.O) 22 A A l s A A l / 1 0 0 0 . 23 AA23AA2/1OOn. 2U AAJ3AA3/1000. 25 PRINT J 26 3 FOR«AT<» PLEASE ENTER £P 8 T L ON . A N1 JEL T A , . ( ?Fa . 0 ) ' ) 27 READ a.EPSlL.OEl TA I 28 a F0RMATC2FU.0) 29 PRINT 5 30 5 FORMAT (' PLEASE ENTER T ME MEAN FREQUENCY F,'/ 31 1 ' AND THE DIFFERENCE IM FREQUENCY OELF (IN M G ) ( 2 F U . 0 )') 32 READ 6,F,DELF 33 6 F0R M A T ( 2 F « , 0 ) 34 PB1000,*F 35 DELFsi000'.*DELF 56 H K A P P A « 8 . * P T * P t « ( t ,-OEi_'TA)*F 37 PRINT 9 58 9 F 0 R M A T ( ' DO YOU "ISH A NON-SMEARED PL 3 T ?.'. . f Y , N 1 ' ) 39 READ 10.NSM UO 10 FORM AT(Al^ 01 IF(NSM,ME.Y)SO TO 200 U2 PRINT 7 U5 7 F 0 R M A T ( ' PLEASE ENTER ,^TN,HMAX < I n '< 31 , NUMH , ' , , ( 2Fa . 0 , I 3 ) ' ) ait READ 3,MMIN,HMAX,NUMH U5 8 F 0 R M A T C 2 F a ,0,131 U6 C CALCULATE THE NQN.sMEAREO A x P L l T J O E HIMINsl'./HMAX US HIMAX«l'./HMIS U9 H r i N C » ( H I M A X - H I M l N ) / F L O A T ( N U M M ) 50 HlaMIMIN 51 XA1MAXB0, 52 XA2MAX80. 53 XA3MAX30, SO DO 100 Tal.NjMH 55 HTaHI*HTINC 56 KAPPAaHKAPPA«MI*MI 57 A 1 a A ( l a ( A A l ) 131 58 X A t m a A l 50 I F C X A t m'.GT.XAT^AXmi'MiXsXAHI) 60 A2=A(2,,AA2) 61 P S I 2 a P S I ( 2 . ) 62 P 3 I l s P S l M . ) 63 XA2(I)»AT (Aa,-'.s*KAPP4*Al *A1 *EPSIL» B3Ta.2.*P311 1 6U IF(XA?CT)'.GT.XA?«AX)XA2H 4X = XA2(I) 65 A3»A(3 , .AA3) 66 PSI3aPSIC3.) 67 AlNTaATn'..-0.25*«APPA . A|*Al/A2*EPSIti psn*o 3i2,3,.pgTn 6S P S I I N T 3 A T A N C ( C 0 3 ( P 3 I l * o S i a ) » 0 ' , a 5 « i < A P P A * A l * A l / A a * E o 3 l L * C 0 S C 3 . * P S t l 69 1 ) ) / C S l M ( P 3 n * B S l 2 ) - 0 . 2 5 * K A P P A * A l * A l / A a * E P S I L * S l N ( 3 . * P 3 I l 70 2 ) ) ) 71 XA3(I1=A3*AT(l,.-1.5 * A?*A2/(Al*A3)«<APPA*Al*At / A a*EPSII.»AlNT,P3I3 72 l . P S I I N T ) 73 I F ( X A 3 ( ! ) ' . G T . X A 3 M A X ) X A 3 M A X s X A 3 ( I ) 7u too CONTINUE 75 C NORMALIZE 76 00 101 I s l , N J * n 77 X A t ( I ) s X A t ( I ) / X A l * A X * l 0 . 78 X A ? t I ) a X A 2 ( n / X « 2 M A X * l o , 79 X A 3 U ) » X A 3 ( I ) / X A 3 < » A X * l o . 80 101 CONTINUE 81 C PLOT 82 XSCALE»15'./FLOAT (NUM4) S3 C»LL °ISTR(ETA,FP9IL) 8U CALL PL0Tf2'.,XA1 ( 1 ) , 3 ) 85 00 102 T=2,NJM4 86 CALL P L 0 T ( X S C A L ' F » F L 0 A T ( I ) » 2 , , X A i m . 2 ) 87 102 CONTINUE 88 C*LL PL0T(2.,XA2 ( 1).3) 89 DO 103 T = 2 ,N J M H 90 CALL PLOT(XSCALF*FLOAT(T)+2,,XA2tI)•2) 91 103 CONTINUE 92 CALL 3 L 0 T ( 2 ' . , X A 3 ( t ) . 3 ) 93 no 10fl T a 2 # N J » M oii CALL P L 0 T f X S C A L F * F L 0 A T ( I ) + 2 . , X A 3 ( I ) . 2 ) 95 10a CONTINUE 96 C SMEARED AMPLITUDE 97 200 IFCNSM.EO'.YICALL PLOT ( ? 2 . .0,,-3) 9fl PRINT 11 99 11 FOR MAT(' 00 VOiJ «*ISH A SMEARED PL 0 T ? . . . ( Y , N 1 ' 1 100 PEAO 12.SM 101 12 FORMAT(Al) 102 IF(SM'.NE.Y)GO TO 999 103 PRINT 15 10K 15 FORMATC' ENTER THE P-P MODULATION IN <3 (Fd'.O)') 105 READ 16. PPMOD 106 16 FORMATCFo'.O) 107 PRINT 17 108 17 FORMAT( 1 PLEASE ENTER HMTN.HMAX (IN <G),AND NUMH . , ,1 , 109 1 1 (2F«'.0.13) • ) 110 READ I8.HMIN.HMAX.NUMH 111 18 FORMAT(2Fa,0,I 3) 112 HIMINB1),/HMAX 113 HTMAXal./HMTN 110 CALL PL9TR(ETA,FP3IL) 115 PM00sPPM0P/2. 116 H I I N C a ( H I M A X . H I H l N ) / F L O » T ( N U M H ) 117 HI aHIMIN I 13? U S KA l^AXaO. 119 XA2MAXX0. 120 XA3HAXaO. 121 OO 300 Ts1 ,NJMM 122 HlsHI*MTINC 123 HTMINC»(l'./f r./Hl-PHOOj-l *./(t'./Ht*PM00n/512. 12<l HAMF«PI/(HIMINC#512») 125 HTMS«J,/(1,/MI*PMOD) 126 MT^iHIMS 127 C GENERATE TME ST3NAI. FROM j)NE MOOULATTON 9*EE<> 128 DO 301 T=1.512 129 Hl*3HTM+HIMINC 130 KAPPAsHKAPPA*HT*Ml 131 A 1 aA ( 1 , « A A 1 ) 1 3 2 A 2 » A ( ? , , A A 2 ) 133 A3aA(3,.AA3) 13U Xs2.»ol * cF*HtM. ,.5) 135 C THF SINE IN THE NpxT LINE T3 THE MANNING *TNDO* 136 X M(J)sMfEP3rL*!<»P PA*AUAl/A2,A2*A2 /CA l*A3-n* 137 13INtHANF*tMIM-HIM3)) 138 301 CONTINUE 139 C SEPARATE XM INTO ITS ANTI3V" METflTC (Xt),AND SYMMETRIC fX2) mo c CO MPONENTS 1 a 1 00 302 Ja2,255 1U2 XI (257-J )3(XM ( 2 s 7 -J1-X-(257 + jn/2. U 3 X2(25 7-J)s(XM(25 7-J) + XM(257*jn/2. 1UU 302 CONTINUE t"5 XKllao'. 1«6 X1C256) = 0'. 1«7 X2C1 )sXM(i) ll«8 X2t2S6)sXM(256) 1U9 C TAKE THE FOURIER TRANSFORM 150 CALL S 5 t 2 f X 1 , X n 151 CALL C512(X2,X2i 152 C CALCULATE THE FIE|.n SPACING OF THE FTRST HAR"ONTC 153 FSPlal,/(HI*Ml»F) 150 C THE NUMBER OF CYCLES PER "«30 SWEEP,'.. 155 NCYCaPPM00/F3Pl 156 C THIS IS rfHERE THE FIRST HAR«ONIC IS IN THE TRANSFORM 157 C FORM WINDOWS 1 5 8 I*L'SNCYC / 2 ' * l ' . 159 T*iL2aMCYC*l , 5 * l ' 160 H L 3 = NCYC * 2 , 5*1, 161 lxR3a.NCYC*3.5*l . 162 C FIND THE MAXIMA OF THE A*9L!TU0E SPEC T RU *,'., 163 XMAXsO, 16U DO 3in JatWLlitwL? 165 A«PaS3RT(Xl(J)**2*X2(J1 * » 2 ) 166 IF (A-P'.GT.XMAXIXHAXSAMP 167 310 CONTINUE 168 XA1 (I)aXMAX 169 IF(XMAX'.GT,XA1MAX)XA1MAXSXMAX 170 XMAXaO, 171 DO 311 JsT^I. 2» rwLl 172 AMP«S3RT(X1(J1**2*X2(J^**2) 173 IFfAMP.GT*. XMAX)XMAX3AMp 17U 311 CONTINUF 175 XA2(I13XMAX 176 IF(XM4X'.GT,XA2MAX)XA2MAX«XMAX 177 XMAXaO, 133 1 78 00 31? Jat"(L3.tw»3 179 A«PsS3RT(Xl(.n**2*X2(!)*«2) 180 IF(AMP,GT'.XMAX)XMAXaAM8 181 312 CONTINUE 182 X 4 J(I1» X M 4 X 183 IF(XMAX'.GT,XA3MAX)XA3*AXaXMAX 18(1 300 CONTINUE IRS C °LOT'..'. lSfc C FIRST HARMONIC IN SQU4RE9, 187 C SECOND HARMONIC IN TRIANGLES. 188 C THIRO HARMONIC IN X'S 189 C 190 C N0RM4LIZE 191 00 320 Ial,NUMH t92 X41(I ) SXA1(I)/XA1MAX *10 . 193 x42(I)aX42(T)/X42MAX»lo, 19« X43msXA3(T)/XA3M4X*lo. 195 320 CONTINUE \ab C "LOT 197 XSCALEa15'./FL'14T(NUMH) 199 00 003 Ia1,NJMH 199 xRNTsX3C4LE*FL04T(I)*2'. 200 CALL STMBr)LfXPNT,XAl{t),. l«.0.0'.,-n 201 CALL 8YMBOL(XPNT,XA2(n,.lO,2,0.,-l) 202 CALL SYMROL(XPNT.XA3(h..lO,O.0.,-l) 203 003 CONTINUE 200 999 CALL PLOTNO 205 STOP 20(, E NO 207 SUBROUTINE PLSTR(ETA,E = STL ) |208 C O M M O N X A 1 M A X , X A 2 M A X , X A 3 - A X , M I M 4 X , M I M I N 209 CALL 4XrS(l'.,0..'FIRST HARMONIC AMPLITJ0E'« 20. 10.,90,, 210 10'. ,XA1MAX*100.) 211 CALL 4X13(1.5,0'. # 'SECOND H4RM0NIC 4MP LITJOE',25.1 0 , , 212 1 90. , O'. , XA2MAXMO0'. ) 213 CALL 4X13(2'., 0.."'THIRD HARMONIC 4MPLITJOE' ,20, 1 0 . , 210 190,,0'.,XA3MAX*100.) 215 CALL AXIS(2 ,.,0. t'l/H(<5*«-l)i.-ll,15.,0.0.HlMlN. 216 1(HI«4X-HIMIN)/JS.) 217 CALL PLOT(2 ,.,lo ' . ,3) 218 CALL °L0TM7..10,,2) 219 C4LL "LOT(17,,o'.,2) 220 CALL 9T«ROL(1 7.?,~10.,.?8,22,0'.,-n 221 CALL S V * 8 O L ( l 7 , u 8 , 1 0 . , , 2 » . , = , , 0 , , l ) 222 CALL NUMflFR(l7.76,10.,.29,ETA,0.,2) 223 CALL SYMBOLd?.?."., .2R.20 .0. .-1) 220 CALL SY-BOLd 7.UR.9. , . ?R, ts ' , 0 . , 1 ) ??5 CALL NUMRER( l7 .76 ,9.,.28,EP3lL,n . . 2 ) 2?6 RETURN 227 END 134 BIBLIOGRAPHY Anderson, J . R . and G o l d , A . V . Phys. Rev. 139, No. 5A, Al459 (1965). A o k i , H. and Ogawa, K. J . Low Temp. Phys. 32. "131 (1978). Condon, J . H . Phys. Rev. 145, 526 (1966). Condon, J . H . , and Wa ls ted t , R .E . Phys. Rev. L e t t . 2J_ 612 (1968). D i n g l e , R.B. P roc . Roy. Soc. (London) A211, 500 (1952). D i n g l e , R.B. P roc . Roy. Soc. (London) A211, 517 (1952). E v e r e t t , P.M. and G r e n i e r , C . G . Phys. Rev. B ]8_, 4477 (1978). Gradshyeyn, I .S . and Ryzh i k , I.M. Tab les o f In teg ra ls and P roduc t s , Academic P r e s s , N . Y . , London (1965). G o l d , A . V . and Schmor, P.W. Can. J . Phys . 54_» 2445 (1976). Ivowi , U.M.O. and Mackinnon, L. J . Phys. F'6_, 329 (1976). Knecht, B. J . Low Temp. Phys. 2]_, 619 (1975). L i f s b i ' t z , I.M. and K o s e v i c h , A .M . Sov ie t Phys. - JETP 2_, 636 (1956). Ogawa, K. and A o k i , H . , J . Phys. F. <8, 1169 (1978). Onsager, L. P h i l . Mag. 43, 1006 (1952). P e r z , J . M . and Shoenberg, D . J . Low Temp. Phys. 25, 275 (1976). P h i l l i p s , R.A. and G o l d , A . V . Phys. Rev. J_78, 932 (1969). P i p p a r d , A . B . Phys i cs of M e t a l s , V o l . 1 : E l e c t r o n s , ed . by J . M . Ziman, Cambridge U n i v e r s i t y P r e s s , London, P.113 (1969)• P i p p a r d , A . B . and S a d l e r , F .T. J . S c i . I ns t , s e r i e s 2 2_, 101 (1969). Shoenberg, D. P h i l . T rans . Roy. Soc. (London) A255, 85 (1962). T e s t a r d i , L .R. and Condon, J . H . Phys. Rev. B j_ , 3928 (1970) . T o b i n , P . J . , S e l l m y e r , D . J . and Averbach, B .L . P h y s . Le t t 28A, 723 (1969). van Weeren ,J .H . and Anderson, J . R . J . Phys. F 3_ p^2109 (1973). 

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