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[Mu] SR measurement of the magnetic penetration depth in high-TC superconductors Riseman, Tanya Maria 1989

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fiSK M E A S U R E M E N T OF T H E M A G N E T I C P E N E T R A T I O N D E P T H IN H I G H - T C S U P E R C O N D U C T O R S By Tanya Maria Riseman B.S. Stanford University, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1989 © Tanya Maria Riseman, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 6224 Agriculture Road Vancouver, Canada Date: \ Abstract The effect of the magnetic field inhomogeneity in the mixed state of isotropic and aniso-tropic type II superconductors on the muon polarization in Muon Spin Rotation (/iSR) experiments is explored in this thesis. Calculations of the penetration depth from the second and third moments of the muon precession frequency distribution (completely analogous to the NMR Redfield lineshape) are discussed critically for oriented and un-oriented powder samples. In addition, a method of fitting the frequency distribution directly with models which take into account the variability of the penetration depth and the applied field as well as displacements of the flux tubes from their regular posi-tions in the flux lattice is presented. This method allows the penetration depth to be fitted directly from the frequency distribution. The method is demonstrated with //SR data on an unoriented sample of YBa 2 Cu 307, using a model of a highly anisotropic su-perconductor, and on an oriented sample of YBa2Cu307, using the isotropic Abrikosov theory, giving a hard-axis penetration depth in YBa2Cu3C"7 of approximately 1000 ± 400 A. 11 Contents Abstract ii List of Tables vi List of Figures vii Acknowledgements ix 1 Introduction 1 2 Basics of Muon Spin Rotation 3 2.1 Overview 3 2.2 Production of Low Momentum fj,+ 4 2.3 Experimental Apparatus 7 2.4 //SRData 11 3 Isotropic London Theory 16 3.1 Derivation of the Isotropic London Theory 16 3.2 London Equation in the Type II Phase 19 3.3 Choice of Bravais Lattice 22 3.4 The Redfield Lineshape 23 3.5 The Second Moment 25 i n 4 Anisotropic London Theory 28 4.1 Introduction 28 4.2 Derivation 29 4.3 Choice of Reciprocal Lattice 33 4.3.1 The Triangular Lattice 33 4.3.2 The Rectangular Lattice 35 4.4 The Anisotropic Redfield Lineshape 36 4.5 Discussion 36 5 A Simple Model 44 5.1 Introduction 44 5.2 Derivation 45 5.3 The Frequency Lineshape 46 5.4 The Second Moment 50 6 Data Analysis 52 6.1 Introduction 52 6.2 Experimental Frequency Distributions 54 6.3 Comparing Data with Theoretical Frequency Distribution 56 6.4 Estimation of the Penetration Depth from the Frequency Distribution's Moments 59 7 Results and Discussion 62 7.1 Justification for Smearing the Theoretical Lineshape 62 7.2 Data and Analysis 66 7.3 The Third Moment 66 7.4 Discussion of Results 69 i v 7.5 Possible Future Developments 74 A Moments of Theoretical Lineshapes 87 A.l Moments of the Isotropic Lineshape 87 A. 2 The Second Moment of the Kossler Lineshape 90 B Computer Programs 92 B. l The Redfield Lineshape 92 B.l . l SampleInputfileIAC_14KG_1000A_R0.DAT 92 B.1.2 Program ABRIK0_T.FOR 92 B.2 The Kogan Model 95 B.2.1 SampleInputfileI_25KG_3435A_R100.dat 95 B.2.2 Program K0GAN5. FOR 95 B.3 The Kossler Model 100 B.3.1 InputfileIC3_15kg_1000a_R0.DAT 100 B.3.2 Program JACK.TRIANGLE3.FOR 100 B.4 Limitations of Computer Simulations 104 B.5 Fitting Subroutines 105 B.5.1 Subroutine INIT-TH 105 B.5.2 Subroutine TRANS-TH 107 B.5.3 Function YOFX 109 B.6 Calculation of Constants Co and Cooo 110 B.6.1 Program CALC_SUM_M0M2. FOR 110 v List of Tables 7.1 Summary of fit results 67 7.2 Summary of penetration depth from second and third moments 69 vi L i s t o f F i g u r e s 2.1 Radial plot of positron emission probability 6 2.2 High transverse field (TF) geometry 9 2.3 Example of experimental apparatus 10 2.4 Single experimental time spectrum 14 2.5 Experimental two-spectrum asymmetry vs. time histogram 15 3.1 Isotropic Redfield lineshapes for square and triangular lattices 24 4.1 Coordinate and crystal axes 39 4.2 Anisotropic superconductor's local field distribution 40 4.3 Anisotropic London model, various angles 41 4.4 Anisotropic London model, various penetration depth ratios 42 4.5 Anisotropic London model, various applied fields 43 5.1 Kossler model, various applied fields . 48 5.2 Kossler model, various penetraton depths 49 5.3 Kossler model's lineshape is independent of applied field 51 6.1 Frequency fits of data 55 6.2 Fast Fourier transform vs. fit 61 7.1 Frequency fits of data 64 vii 7.2 Change of Redfield lineshape with inhomogenous penetration depths . . 65 7.3 Neutron scattering measurement of sample alignment 76 7.4 Penetration depth fit to unoriented data 77 7.5 Penetration depth fit to aligned data 78 7.6 Oriented YBa2Cu307 time spectrum 79 7.7 Lai.gSro^CuC^ sample with background signal 80 vm A c k n o w l e d g e m e n t s I'd like to thank Jack Kossler of College of William and Mary for his fruitful idea of how to neglect the penetration depth due to fields applied along the soft crystal direction and Moreno Celio, formally of TRIUMF and now of Dipartimento dell'Ambiente SEPA in Switzerland, for his inital computer code and guidance. I appreciate that Rob Kiefl of University of British Columbia acted as in loco advisor for Jess Brewer, when I could not benefit from his enthusiasm and support while he was out loco. I appreciate Ames Laboratory's Dr. V. G. Kogan's patient answers to my questions about his derivation, especially as I have never met him. I appreciate that Graeme Luke of Columbia University has allowed me the use of Figures 2.1, 2.3, 2.4, and 2.5 and to Jess Brewer for Figure 4.1. Thanks to Jess Brewer and Syd Kreitzman of TRIUMF for modifications to exsisting //SR analysis programs. Since I did not work in an experimental vacuum, I'd like to thank the countless members of the TRIUMF //SR group and outside collaborators. Excellent samples were provided by, among others, University of British Columbia's Department of Physics and AT&T Bell Laboratories. Special thanks to Tomo Uemura of Columbia University for the procurement of many high quality samples from distinguished laboratories through-out the world, including the oriented YBa2Cu307 sample from General Electric Cor-poration. Thanks to Ben Sternlieb of Columbia University for the neutron-scattering data on the alignment of the oriented YBa2Cu307 sample. ix C h a p t e r 1 I n t r o d u c t i o n The Muon Spin Rotation (/iSR) technique was the first method used to directly measure the penetration depth of the copper-oxide based perovskite type II superconductors (see Ref. [1] for Lax 85Sr0.i5CuO4 and Ref. [2] for YBa2Cu307). Due to the fact that this class of superconductors are usually produced in the form of sintered powder ceramics, the connections between the superconductor's grains tend to quite small relative to the size of their surfaces, leading to Josephson junction effects in the connections.[3] Since the effective penetration depth in a Josephson junction is much larger than in the interiors of superconductor's grains, some traditional methods of measuring the penetration depth, such as determining the lower critical field {Hcl) and flux expulsion below Hci, have proven to be unreliable. In contrast, above Hcl the magnetic field passes through a type II superconductor in the form of flux vortices which produce an inhomogeneous field distribution characterized by the penetration depth. Techniques such as NMR and //SR can directly measure the field inhomogeneity by observing the distribution of precession frequencies of a host spin (NMR [4]) or of an implanted probe (/JSR or Hydrogen [5]). Since the flux vortices are distributed evenly throughout the bulk of the material, the penetration depth derived from experiments (such as /J,SK) above Hcl should not suffer from the difficulties posed by the ceramic nature of the 1 CHAPTER 1. INTRODUCTION 2 perovskite superconductors. The copper-oxide perovskite superconductors are highly anisotropic [6] and their flux vortices only approximately form a regular triangular lattice. [7,8] These two effects complicate a careful study of the field inhomogeneity arising from the flux vortices. This thesis strives to go well beyond the simple calculation of the penetration depth from the second moment of an isotropic superconductor's field inhomogeneity; it explores more correct ways of deriving the penetration depth by including consideration of the anisotropy, the imperfection of the flux lattice, and local inhomogeneities of the applied field and penetration depth. Chapter 2 contains a introduction to /tSR with particular emphasis on the trans-verse field (TF) technique used to measure the local field inhomogeneity of our type II superconducting samples. Chapter 3 reviews the derivation of an isotropic type II superconductor's magnetic field inhomogeneity and includes explanations of terms such as "type II" and "penetration depth." Chapter 4 extends the derivation for anisotropic superconductors using tensor notation. Chapter 5 presents a simpler model for highly anisotropic type II superconductors and explains how to calculate the field inhogeneity for that model as well. Chapter 6 discusses the various approaches to analyzing /fSR data (and, by analogy, NMR data) using the models discussed in chapters 3 and 5. In particular, it discusses how to extract a penetration depth by fitting in frequency space or to calculate the penetration depth directly using the second or third moments of the data's frequency distribution. Chapter 7 discusses chapter 6's methods critically in general and in light of the particular properties of the high temperature copper ox-ide superconductors. Appendix A contains derivations of the second and higher order moments of the local field inhomogeneity for isotropic superconductors. Appendix B contains the computer code used in the simulations. C h a p t e r 2 B a s i c s o f M u o n S p i n R o t a t i o n 2.1 O v e r v i e w The use of muons as probes of solids, liquids and gases is collectively referred to as /LZSR: Muon Spin Rotation/Relaxation/Resonance, where the choice of word beginning with the letter "R" depends on the specifics of the application. (For details relating to this chapter, see Alex Schenck's book "Muon Spin Rotation Spectroscopy" [9] and references therein.) Muons are spin 1/2 leptons with a mean life of 2.197037 fis and a mass of 105.65839 MeVc - 2 = 206.76826 me = 0.11260965 m p , where me is the mass of an electron and mp is the mass of a proton. The positively charged muon can be treated as a light isotope of positively ionized hydrogen, while negative muons can be thought of as very heavy electrons. When a negative muon is implanted in a solid, it is quickly captured by one of the atoms and cascades down into the lowest muonic orbital. Since the lowest muonic orbital is comparable in extent with the radii of heavy nuclei, the muon usually undergoes nuclear capture, fj,~p —> nv^. As the negative muon has complicated behavior such as shortened lifetimes and rapid 3 CHAPTER 2. BASICS OF MUON SPIN ROTATION 4 muon spin depolarization, this thesis describes only the use of the positive muon to probe superconductors. Positive muons implanted in a solid will precess at a frequency = 7A,.Hr/0C in the presence of a local magnetic field H\oc (due to the applied field and/or local magnetic moments), where the muon gyromagnetic ratio is 7^ = 27r x 0.01355342 MHzG - 1 . If there are dipolar or hyperfine couplings with the host spins, there will be a continuous loss of muon spin polarization, termed "relaxation." In the absence of spin-spin cou-pling, any apparent relaxation of the muon polarization will be due to the dephasing effect of the local magnetic field inhomogeneities. Local field inhomogeneities can be caused by such things as an imperfect experimental magnet, some spin disorder in a ferromagnetic or antiferromagnetic sample, a "spin glass" sample, or the mixed state of a type II superconductor. This thesis is concerned with the dephasing ("relaxation") effects of the inhomo-geneous magnetic field distribution arising from type II superconductors, in particular the high temperature copper oxide superconductors. Since there has been no evidence for spin-spin coupling playing an important role in the copper-oxide based perovskite superconductors, only the dynamics of spin precession and the associated dephasing are discussed. The "R" word in /LtSR here refers to "Rotation" and "Relaxation." 2.2 P r o d u c t i o n o f L o w M o m e n t u m fi+ When protons are accelerated to an energy of 500-800 MeV by an accelerator such as the TRIUMF cyclotron and then hit a "production target", pions (TT) are produced via the reactions of the projectile proton (p) with the protons and neutrons (n) of the target's nuclei: p + p — • 7 r + + p + n CHAPTER 2. BASICS OF MUON SPIN ROTATION 5 p + n —> 7 r + + n + n —> 7T~ + p + p. The produced pions decay with a lifetime of 26.030 ns into muons (//) and muon neu-trinos ( t ^ ) according to 7 T + —• / i + + V fi As this is a two body decay, in the rest frame of the pion, the muon and the neutrino are emitted in opposite directions in order to conserve momentum. The muon carries a kinetic energy of 4.11913 MeV which implies a momentum of 29.7894 MeV/c. Because the pion has spin zero, the net spin of the muon and the neutrino must be zero. Since neutrinos are exclusively left handed, that is, their spins are antiparallel to their momentum, the muon from the pion decay must also be left handed. Therefore, the decay of positive pions produces 100% longitudinally polarized positive muons with total relativistic energy of 109.7775 MeV. The decay muons are emitted isotropically in the pion's rest frame. The muon itself decays with a mean lifetime of 2.197037 fis into an electron or positron and a neutrino-antineutrino pair: fj,+ —> e + + ue + Ufj, \i~ —> e" + ue + Up, where the combination of muon and electron neutrinos and antineutrinos is such that both lepton numbers are conserved. Due to the facts that this is a three body decay of a particle with spin 1/2 and that only left-handed neutrinos exsist, the positron/electron will be emitted in a direction which is correlated with the muon's spin. After integrating CHAPTER 2. BASICS OF MUON SPIN ROTATION 6 Figure 2.1: Radial plot with respect to fi+ spin of positron emission probability for e = 1.0 and e = e. over all possible neutrino momenta, the probability W per unit time that a positron will be emitted at angle 6 with respect to the [i+ spin is given by dW(e,0) = —[l + a(e)cos(0)}n(e)dedcos(6) (2.1) where a(e) = (2e — l)/(3 — 2e) and n(e) = 2e2(3 — 2e). The reduced positron energy is defined by e = E/Emax where the maximum positron energy E m a x = 52.8304 MeV is approximately equal to half the muon rest mass energy. The probability as a function of polar angle 9 is shown in Fig. 2.1 for the values of e = 1.0 and e — e = 0.682, the latter of which corresponds to the average positron energy. The energy average of a(e) is a = 1/3, while the low energy limit is a(e —> 0) = —1/3 and the high energy limit is a(e -> 1) = 1. CHAPTER 2. BASICS OF MUON SPIN ROTATION 7 The muon spin polarization will be very nearly 100% for muons produced by those pions which decay on the surface of the production target.[10,11,12] These muons are commonly referred to as "surface" muons and tend to have much lower momentum than those from pions which decay in flight. Various equipment (momentum slits, Wien filters) along the beamline allow one to select different muon momentum ranges and separate muons from beam positrons of the same momentum by virtue of their different velocities. The surface muons have a total range of about 140 mg/cm2 and a range straggling about 20 mg/cm2 in water so denser samples as thin as 100 /j,m can be studied. 2 . 3 E x p e r i m e n t a l A p p a r a t u s The muon beam is produced with the muons' spins antiparallel to their momentum, as explained above. Just before the experimental area, there is a Wien filter which has mutually perpendicular electric and magnetic fields which are both perpendicular to the beam. The relative strength of the electric and magnetic fields is adjusted to discard the beam's positron contamination while transmitting the muons undeflected, which is easily accomplished using modest fields. Because the muon's spin will pre-cess in the magnetic field during its time of flight through the Wien filter, a strong enough magnetic field will rotate the muon spin as much as 90 degrees from the beam momentum axis. Rotating the spin until it is perpendicular to the momentum has the advantage that the anisotropic distributions of decay positrons, which we wish to de-tect, will be in the plane perpendicular to the last remaining contamination positrons from the beam, thus lowering the background. More importantly, spin rotation is abso-lutely essential for high transverse field (TF) /j,SK experiments in which the muon spin must be perpendicular to the experimental magnetic field but the muon momentum CHAPTER 2. BASICS OF MUON SPIN ROTATION 8 must be parallel to the field so as to avoid bending the beam out of alignment via the Lorentz law F = e\(v x B)\ = mv2/R. The Lorentz law implies that a charged particle (charge of q) with momentum p = mv perpendicular to an applied field B will have a radius of curvature R given by mv2 p qvB qB' which gives a radius of 9.64 cm at a field of 1 T for surface muons with momentum of 28.9 MeV. The muon beam leaves the beam pipe though a thin vacuum window and enters the /iSK apparatus (simple cryostat, dilution refrigerator, oven, etc.) containing the sample. The apparatus is inside of a magnet, which is usually placed such that the applied field is parallel to the muon momentum so that the beam is not deflected by the Lorentz force (see above.) If the magnetic field is applied parallel to the muon spin, the configuration is referred to as a longitudinal field (LF) geometry and the technique as Longitudinal Field Muon Spin Relaxation (LF-/aSR). If the magnetic field is applied perpendicular to the muon spin, the configuration is referred to as a transverse field (TF) geometry (Fig. 2.2) and the technique as Transverse Field Muon Spin Rotation (TF-//SR). If no external magnetic field is applied, the technique is known as Zero Field Muon Spin Relaxation (ZF-/iSR). The incoming muons and the decay positrons are detected with counters made of scintillating plastic connected by light guides to photomultipliers. Scintillator emits light when charged particles (such as positrons and muons) pass through it; the light is collected and guided to the photomultipliers by a Lucite light guide. The photomulti-pliers convert the light into an electrical signal, which is transmitted via coaxial cable to a counting room where fast electronics are used to intepret and store the information. CHAPTER 2. BASICS OF MUON SPIN ROTATION 9 F C o u n t e r JJ, B e a m B C o u n t e r L C o u n t e r Figure 2.2: High transverse field (TF) geometry, in which the UP and DOWN counters are not shown for the sake of clarity. The BACKWARD and FORWARD counters are for ZF and LF experiments and are not generally used in this geometry. Between the end of the beam pipe and the sample is thin piece of scintillator used to detect incoming muons. This muon counter must be thin, on the order of 250 /j.m, to minimize multiple scatterings in the scintillator, which cause muons to miss the sample and thus increase the background signal. It will also detect the beam contamination positrons and the decay positrons from previously undetected muons, but with far lower efficiency than for the muons. Most false events due to e+ will be rejected by the electronics. Between the magnet and the sample are the decay positron counters (Fig. 2.3). For a LF geometry, one (BACKWARD) will be between the beam pipe and the sample (with a hole cut out for the beam to pass through) and another (FORWARD) on CHAPTER 2. BASICS OF MUON SPIN ROTATION 10 Vertical Field Coil 30 cm Figure 2.3: Example of experimental apparatus the opposite side of the sample. For a TF geometry, the counters are placed surrounding the beam rather than intersecting it. One (UP) will be in the direction of the rotated muon spin and another (DOWN) 180 degrees opposite. Sometimes two more are used, LEFT and RIGHT (Fig. 2.2). If all six counters are used, it is possible to subtend all angles (save the hole for the beam entrance) and not waste any of the decay positrons. In time differential //SR, the idea is to map the time evolution of the muons' spin polarization. First, the beam rate is reduced until less than one muon on average enters the sample during the time period over which one wishes to examine the muons' behavior (typically 10 /is.) The signal from the muon counter (indicating an incoming muon) starts a clock, subject to a pile-up gate. When a decay positron is detected in one of the positron counters (BACKWARD, FORWARD, UP, DOWN, LEFT or CHAPTER 2. BASICS OF MUON SPIN ROTATION 11 RIGHT) the clock is stopped and the event is added to the appropriate time bin in the histogram corresponding to that positron counter. The accumulation of many single events allows us to make an ensemble average of the behavior of a single muon. There is a tendency to discuss the muons' interaction with the sample and the applied magnetic field in terms of a single muon. This is justified as long as the density of muons in the sample is low enough that muon-muon interactions are neglible, which is certainly satisfied. To study superconductors with critical temperatures of 40-120 K in the type II state, we used helium cryostats and a superconducting magnet that can produce fields as high as 30 kG. Since all measurements were taken above 1 kG, the external field was always applied parallel to the beam. The muon spin polarization was rotated with the Wien filter until perpendicular to the muon momentum. 2 . 4 LISK D a t a The most obvious feature of a positron counter's histogram is the exponential decay of the number of histogrammed events, reflecting to the muon's lifetime of 2.197 /is (Fig. 2.4). The error for each time bin is just the square root of the number of events in that bin. For two matched histograms, e.g., UP and DOWN, the histograms would be exactly the same if the decaying muons were completely depolarized, assuming that the two counter geometries and the photomultiplier efficiencies were exactly the same or their differences properly taken into account. If the muon polarization is predominantly in the direction of one of the counters, its histogram will have more events than the other. Since the muon's lifetime is very well known for positive muons, it is more informative to combine the pairs of histograms. The relative asymmetry A12(t) of two CHAPTER 2. BASICS OF MUON SPIN ROTATION 12 paired histograms H~i(t) and H2(t) is denned by Hi(t) - H2(t)  A l 2 { t ) ~ Hl(t) + H2(ty where the respective backgrounds have been subtracted from the individual histograms (Fig. 2.5). The only reminder of the muon's exponential decay will be that the error bars increase exponentially in size for the bins corresponding to longer times. To within a normalization and the exponential decay behavior of the error bars, this asymme-try Ai2(t) is completely analogous to the NMR free induction decay (FID) along the corresponding x, y, or z direction. —* For a TF geometry with the applied field B = Bz perpendicular to the x axis of the "1" and "2" positron counters, in the absence of relaxation the signal A12(t) = A0P„x(i) = A0 cos(lfiBt + <f>), where A0 is the asymmetry, y^B is the muon precession frequency, and (j) is the rela-tive phase. The maximum measurable asymmetry A0 will be ~ 25%, due to counter geometry, positron absorption and the kinematics of muon decay. If the local field (h = hxx + hyy + hzz) is not exactly equal to the applied field B — Bz, the muon's polarization in the x direction P^x(t) can be expressed as [13] If the local field h varies spatially, we need to average this result over all space. If the magnitude of the local field changes, the cosine term will appear to relax due to "dephasing": v ; , cos(7M/ii + <?!>)—> Gxx(t) cosiyjit + <j>), where Gxx(t) is some relaxation function. In liquid and gas phase muonium chemistry experiments, a Lorentzian relaxation function is almost always assumed, as the sample's CHAPTER 2. BASICS OF MUON SPIN ROTATION 13 molecules are usually rotating fast enough that the "motional narrowing" limit is in effect. In solid state JUSR, unless the muons are diffusing quickly or the magnetic structure of the solid is fluctuating rapidly, a Lorentzian is not appropriate. A Gaussian relaxation function Gxx(t) = exp(~"^t2) is often used as an Ansatz relaxation function until an appropriate theoretical description of the muon depolarization is formulated. In a high precision experiment, this Ansatz can lead to qualitative and quantitative misinterpretations. The purpose of this thesis is to accurately describe the muon's depolarization in high temperature superconductors in the type II phase. CHAPTER 2. BASICS OF MUON SPIN ROTATION 14 2 . 2 5 E 4 i i 1 1 1 1 1 2 . E 4 aft -"a 1.75E4 — -a 1.5E4 — m ™ -D J 1 1 1 O 1 . 2 5 E 4 m cn _ <l> a in a . m „ JSh 1.E4 — Hb 111 m ID -a m DJ • i m 7 5 0 0 W n J * -5 0 0 0 ft 2 5 0 0 -0 1 1 1 1 I^^T 0 0 ' 8 1 '^^ 0 1 2 3 4 5 6 7 8 TIME (Microsec) Figure 2.4: Single experimental time spectrum showing the exponential decay of the muons and muon precession. CHAPTER 2. BASICS OF MUON SPIN ROTATION 15 TIME (m ic rosec ) Figure 2.5: Experimental two-spectrum asymmetry vs. time histogram showing muon precession. C h a p t e r 3 I s o t r o p i c L o n d o n T h e o r y 3 .1 D e r i v a t i o n o f t h e I s o t r o p i c L o n d o n T h e o r y The following is a simple derivation of London's equation which assumes no electrical resistance ([14] pp. 7-9, [15] p. 5). Consider Ampere's law with a local magnetic field h and the superconducting cur-rent fs treated as the total current: - 4TT V x h = —fs. (3.1) c If ns is the number density of superconducting carriers (where two carriers of appro-priate momenta make up a Cooper pair) and q is the charge of the carrier, the current is h(r) = nsqv(f). The kinetic energy of the superconducting current is Ekin = Jns^mv2(f)df = JJ2(r)dr. The justification for naming 16 CHAPTER 3. ISOTROPIC LONDON THEORY 17 the "London penetration depth" will be clear by the end of this section. (In the majority of copper-oxide superconductors, the charge carrier is a hole with charge q = |e|. Notice that the penetration depth is independent of the sign of the carriers.) The kinetic energy can be rewritten in terms of the penetration depth as Ekin = ^ - j \ 2 \ V x h(r)\2df. — — 2 Since the potential energy density due to the magnetic field h is just ^ | / i ( r ) | , the total energy is 2 + A 2|V x h{f)\2)dr. (3.3) To derive the familiar London equations, one minimizes the energy e with respect to the magnetic field h: Se = 0 = j(2h-Sh + 2A 2(V x h) • (V x Sh)) df. (3.4) Rewriting the second term in the integral gives J (V x h) • (V x Sh) df = J ^ V -(ShxV xh) df+ J ^Sh • (V x V x h) df Using the divergence theorem, this becomes J (V xh)-(V x6h)dr = Js(6hxV x h) • dh + J' Sh • (V x V xh)df. We assume that Sh is parallel to dn on the surface so that the surface integral is zero. (In general, Sh not parallel to dh on the surface causes the phenomena of nucleation of superconductivity at the surface. Since the surface will contribute relatively little to the bulk properties that we are concerned with, this simplifying assumption is reasonable.) Using the product rule and one of Maxwell's equations, ^ ( V x h) • (V x Sh) df = Jjh • (V x V xh)df = I Sh-{V(V -h)-V2h)df J \f = J' vSh-(-V2h)df CHAPTER 3. ISOTROPIC LONDON THEORY 18 It is now obvious that Eq. 3.4 can be rewritten as —* _ —* l df 8e = 0 = J[2h-Sh + 2X26h • ( - V 2 K ) = 28h- J (h- X2V2h) df, which must be true for arbitray 8h, giving the London equation 0 = h-X2W2h. (3.5) The above certainly motivates the London equation from the standpoint of Maxwell's equations, but seems to avoid the quantum mechanical essence of superconductivity. Consider the canonical momentum ([15] pp. 44-45) - qA p = mv H . c The conduction carriers must be fermions, such as electrons or electron holes. BCS superconductivity arises when the conduction carriers with momenta p and — p are correlated over a distance £ with opposite spins, effectively making a boson out of two fermions. The two carriers within £ of each other and with opposite and equal momenta are called Cooper pairs. In the absence of an applied electric or magnetic field, one expects that the average momentum is zero, giving the ground state an average local velocity of /-\ {Vs) = . mc If, in an applied magnetic field, the wavefunction of the superconducting electrons remains the same as the zero field ground state, the superconducting current is - nsq2A cA Js = nsq{vs) = = - — - . (3.6) mc 47rAi! (This can be qualitively justified on the grounds that a superconductor has a gap in the energy spectrum between the ground state and higher energy states, which leads CHAPTER 3. ISOTROPIC LONDON THEORY 19 to a large fraction of the conduction electrons remaining in the ground state at finite temperatures and applied magnetic fields. The excited states are uncorrelated, and give rise to a portion of the current which is normal — i.e., has resistance.) Taking the curl of both sides of Eq. 3.6 and combining with Ampere's law (Eq. 3.1) gives V x V x h = —V x A = -C 4 7 T A 2 which gives us London's equation again. For a semi-infinite slab of type I superconductor, with its boundary at x = 0, in an —* —* external magnetic field hQ = h0z, the London equation gives the magnetic field h = hz exponentially decaying into the superconductor: h(x) = haexp if a; > 0, hence the nomenclature, "penetration depth." 3 . 2 L o n d o n E q u a t i o n i n t h e T y p e I I P h a s e If we allow magnetic field singularities at the points fj within the superconductor, the London equation can be modified in the following way: U A 2 ( V x V x ^ ) = ^ ^ ^ - f J ) , (3.7) j where the sum over j is restricted to an area of one square centimeter, to ensure normalization in this choice of units, and the singularities are assumed to form a regular triangular lattice as that will minimize the energy. [16] The constant <f>0 is the flux quantum for a carrier charge q = e ^ = H(|__l)o.207 xlO- 1 0 Gcm 2 , 2q CHAPTER 3. ISOTROPIC LONDON THEORY 20 where % is Planck's constant. One may think of the singularities as holes that skewers might punch into the superconductor. The superconductor can be considered to be a multi-holed torus. Magnetic field can leak in from any part of the surface; the skewer holes are external surfaces just like the original outer surface. Of course, there are no physical skewer holes in the material, but energy considerations allow the magnetic field to pass through the interior of the superconductor via the magnetic field singularities. The magnetic field singularities are commonly referred to as flux lines or flux vortices. Whether the superconductor is type I (no singularities) or type II (with singularities) depends upon its lowest energy state — more specifically, The symbol £ stands for the coherence length, which is the typical length scale over which the superconducting Cooper pairs of electrons are correlated. In the high tem-perature superconductors, A >^ £, so we are well within the type II regime. The London equation was derived assuming that only superconducting conduction electrons are present, as we ignored resistive losses. In reality, the flux line is not a dimensionless singularity: there is a region of radius £ around each fj which is not superconducting. Hence, the term "magnetic flux tube" is more descriptive of the actual physical pattern of the magnetic flux while the equally common terms "flux line" and "flux vortex" are more descriptive of the the magnetic flux in the London model. The London description is reasonably good in superconductors in which the coherence length £ is much, much smaller than the penetration depth A, since the model neglects the cores. Ultimately, we would like determine the distribution of the magnetic field in type II superconductors (isotropic and anisotropic) from the spectral distribution of the muon precession frequencies and thus determine the magnetic penetration depth A. With Type I Type II. (3.8) CHAPTER 3. ISOTROPIC LONDON THEORY 21 the goal of calculating the field everywhere in the superconductor (neglecting edge effects), we Fourier transform the London equation for a superconductor in the type II state (Eq. 3.7). (The Fourier transform is continuous, but the result will be a discrete summation due to the lattice of singularities on the right hand side of Eq. 3.7.) For simplicity, let us assume that the applied field is in the z direction and that the local —* field is constant in z and varies in x and y - i.e., h(x, y, z) = h(x, y) z = h(f) z. Using an integral representation of the delta function and the Fourier transform of the field h(r) - ^ Jh^exp(-ik • f)dkz dky, the London equation (Eq. 3.7) becomes °^(27r)2 J^exp(-ik-(r-fj))dkxdky = [y h%exp (ik • f)dkx dky + A2 V x V x Jexp(ik • f) dkx dky = J^y IJ hk e x P (}k • r) dkx dky - A2 j I — + — J exp (ik • r) dkx dky = J(h^ +X2k2h^)exp(ik • f)dkxdky. I have used k2 = k2 + ky2. At this point, we would like to set exp (ik • fj) = 1. For a single flux tube at the origin, the argument of the exponential is zero. For many flux tubes regularly spaced in the x-y plane, the exponential is one only when k = J?, where K is a vector in the reciprocal lattice of the of flux tube lattice. Assuming the latter case, we consider only the integrands after cancelling ex.p(iK • f) from both sides: nf4>0 = <j>0 £exp(i/? • fj) = hR(l + K2\2). 3 I have defined to be the number of flux tubes per unit area in the x-y plane so that nj<f>0 has units of gauss. Finally, the total local magnetic field is m = E h* exp (—iK • f)z = nJ(f>0 £ ^ t ^ - f l z . (3.9) it it CHAPTER 3. ISOTROPIC LONDON THEORY 22 If we were concerned with a single isolated flux tube, we would replace the discrete sum over recipocal lattice vectors with an integral over all k space. 3 . 3 C h o i c e o f B r a v a i s L a t t i c e The time has come to decide upon our Bravais lattice and hence our reciprocal lattice, as we cannot program a computer simulation without specifying them. If r-i, r2 and f 3 are a set of primitive vectors for a Bravais lattice, then the reciprocal lattice vectors A ,^ k2 and k3 are generated by [17] k = eijk 27T , r ' * r \ v (3.10) ri • (rj xrk) W. Kleiner, L.M. Roth and S.H. Aulter [16] found that for an isotropic superconductor, a triangular lattice of flux tubes has a lower energy than other possible configurations, such as a square lattice. Therefore, in the ideal lattice with flux lines spaced L apart, the primitive vectors are ri = Lx and r2 = L i — a: -f- j/ J . If we assume that r 3 = z so that applying Eq. 3.10 is easy, we find * • = j(V ^ k3 - 2irz. It is obvious that my choice of the z components is extraneous. An arbitrary vector K in the reciprocal lattice is K = nk\ + mk2, CHAPTER 3. ISOTROPIC LONDON THEORY 23 where n and m are integers. If we assume that the average field nj(j)0 = H0 in the superconductor is equal to the applied B, then the number of flux tubes per unit area is n = — = — L~2 and the distance L between flux tubes is 3 . 4 T h e R e d f i e l d L i n e s h a p e A nucleus with spin or a muon (spin 1/2) in a superconductor will precess at an angular frequency u> equal to its gryomagnetic moment 7 times the local field h. The x component of the observable polarization P* for a nucleus or muon in a local field h = (hx, hy, hy) is P;{t) = ^  + ^ ^ c o s ^ h t ) , (3.11) assuming an initial polarization of 100% in the x-direction. In particular, if one samples the local magnetic field due to the array of flux tubes using probes randomly sprinkled throughout the superconducting sample, one can accumulate a histogram of local fields called the Redfield lineshape. (See Fig. 3.1 for the isotropic Redfield lineshape. See Fig. 4.2 for an example of the spatial distribution for an anisotropic superconductor with a square flux lattice.) In the case of an isotropic superconductor, the local field is parallel to the applied field and the polarization simplifies to p;it) = cos(7ht). CHAPTER 3. ISOTROPIC LONDON THEORY 24 Frequency (MHz) Figure 3.1: Isotropic Redfield lineshapes for square lattice (A = 1365 A and B = 4 kG) and triangular lattice (A = 1000 A and B = 14.35 kG). If one looks at the spatial dependence of the magnetic field in a regular array of flux tubes, Eq. 3.9 has the field diverging at each of the flux line, while in reality it is has a maximum value of HC2. Consider a triangular-shaped unit cell formed by three flux tubes. There is a saddle point in the field midway between two flux tubes. Because the field distribution is the flattest at that saddle point, there is more area within 8h of the saddle point field hs than for any other point in the unit cell. This leads to a cusp in the Redfield lineshape. Near one of the flux tubes, the field is increasing very CHAPTER 3. ISOTROPIC LONDON THEORY 25 quickly; as there is relatively little area within Sh of that point, the Redfield lineshape will be relatively small for that field value. The lowest field is produced at the center of the equilaterial triangle formed by three flux tubes. There the field is almost as flat as at the saddle point, so the height of the Redfield lineshape at the minimum field is almost as high as for the cusp field. The cusp field is very close to the lowest field and it significantly lower than the average field, which is equal to the applied field in the absence of macroscopic demagnetization effects. The difference between the cusp and the low field cutoff depends on the applied field: at fields near HcX, they are almost the same field, while near Hc2, they are clearly distinguishable.[18] Also, they are more easily distinguishable for a square lattice of flux tubes (Fig. 3.1). Qualitatively, the lineshape looks like the right hand side of a Lorentzian distribution. 3 . 5 T h e S e c o n d M o m e n t It has been common for people involved in NMR and //SR to calculate second moment of the Redfield lineshape using BM2=6IP = (h2) - (h)2 with Eq. 3.9 and then converting the summations to integrals. (The superscript B on the M2 means that it is the second moment of the field distribution. * M2 refers to the second moment of the muon precessional frequency distribution.) Then some continue by fitting NMR or //SR data with a Gaussian relaxation function (see Sect. 2.4 and Chap. 6) and calculating the penetration depth by assuming that it is reasonable to substitute in that formula the Gaussian fit's second moment in place of the Redfield lineshape's second moment. This approach has two serious flaws. First of all, as the Redfield lineshape has a long high field tail (due to the divergence at the flux tube core), CHAPTER 3. ISOTROPIC LONDON THEORY 26 the second moment calculated will give a higher relaxation rate than a gaussian fit to the main part of the lineshape. This will lead to an overestimation of the penetration depth. (This problem is reflected in the fact that the Redfield lineshape has a strong third moment while a Gaussian has no odd moments at all.) It will also underestimate the local average field as the average field is larger than the cusp field. Secondly, a numerical calculation gives the second moment of the Redfield lineshape [18] for high magnetic fields as i2 BM2 = 8H2 £ 0.00371 (3.12) A rather than the integral result for low magnetic fields i2 ±2 BM2 = 8H2 / ° = 0.002016^7 (3.13) 167r3A4 A4 v ' in the classic paper by P. Pincus et al. [19] The numerical calculation leads to a pene-tration depth that is 16 percent larger. For a more complete description of the second moment, see Appendix A, where I have derived a series expression for an abitrary moment of the Redfield lineshape. This point is also discussed further in the last two chapters. Moreover, converting the summations to integrals is only justifiable when A Ak <C 1. As we will see in the next section, Ak is proportional to 2TT/L and L oc H~x. Thus, the conversion is only valid where (A/L) <C 1, which is considered to be a low field regime for type II superconductivity. Yet we would prefer to take data at high fields (A/L) > 1, since then the effects of phenomena such as pinning at defects will be minimized (so that our idealized model is qualitatively valid) and the second moment will be field independent (making fitting of data to theory easier). The high field formula for the second moment of the Redfield lineshape for an CHAPTER 3. ISOTROPIC LONDON THEORY 27 isotropic superconductor as derived in Appendix A is oo BM2 = B2Q4 Y, (~l)m(™ + l)<92mCm, m = 0 where Q is the dimensionless quantity Q=2^X ("2^") 5 Q allows the reciprocal lattice vectors Kj to be expressed in terms of dimensionless vectors Q as The constant Cm is the following sum: 1 Cm — E /•2(m+2) ' C#0 S In the high field limit, we can neglect the terms with m > 0, which leaves us with a field independent result lim BM2 = 0.00379o B— oo A 4 C h a p t e r 4 A n i s o t r o p i c L o n d o n T h e o r y 4 . 1 I n t r o d u c t i o n The effective mass tensor approach to anisotropic superconductors is completely anal-ogous to the isotropic case, only more tedious. In the following, I present a derivation published by V.G. Kogan [20] for a uniaxial anisotropic superconductor, with a few of the omitted steps filled in. If whatever mechanism which causes the anisotropy, such as the layered chemical structure of the new high Tc superconductors, is on a smaller scale than the coherence length £ of Cooper pairs of charge carriers, then one can ignore the particulars of the anisotropy and use a tensor to describe the carriers' relative ease or difficulty of moving in different directions. More precisely, the use of a mass tensor is valid in a Josephson-coupled layered superconductor, which has an interlayer spacing of s, when the coherence length £ exceeds ,s/\/2-[21] The assumption that the superconducting perovskites are three dimensional super-conductors is perhaps just barely valid. For instance, in YBa2Cus06.95, the chemical unit cell height is 11.7 A and the distance between copper-oxide planes on either side of the yttrium layer is 3.9 A . [22] The published values for £c have been drifting in 28 CHAPTER 4. ANISOTROPIC LONDON THEORY 29 time from as high as 9 A [23] to as low as about 2 A . [24] Since the coherence length £C(T = 0) is estimated by extrapolating HC2 from a very small region below TC (where it is measurable) to zero temperature, the value derived is very sensitive to how the data is analysed as well as to the assumption that HC2(T) behaves as theory predicts in the region where it is unmeasurable. A recent experiment [25] using a scanning electron microscope at low temperatures measured the coherence length £0f, directly and obtained a value of 77 A , which is a factor of two different from the extrapolated value of 32 A . [6] It is unfortunate that a direct measurement of the coherence length £c has not been made yet. One wonders whether at low temperatures we might be in a regime where there is just barely three dimensional superconductivity yet an effective mass tensor approach is not valid because the spacing between copper-oxide planes is greater than y/2£c. 4 . 2 D e r i v a t i o n To derive the anisotropic London equations, one minimizes the Ansatz energy 8TTE = J h2 + X2mIJ(V x /i)t(V x h)3 df, (4.1) where A is the average London penetration depth A ^ Mc2 4irnse2' M is the average mass of the charge carrier, with its anisotropy the normalized effective mass tensor of the carriers, m^: Mij = Mrriij = M TTlxx f^xy Wlxz TflXy Yflyy Yflyz YYlxz f^lyz nT*z where Tr(m2j) = 1 (4.2) CHAPTER 4. ANISOTROPIC LONDON THEORY 30 and Mij is the carrier mass tensor itself. If the external magnetic field is parallel to one of the crystal axes, then m mi 0 0 0 m2 0 0 0 m3 (4.3) We'll use as our model a layered superconductor in which the a and b crystal directions are equivalent (mi = m2) and the c axis is the hard direction (m3 mi). Let us assume that the applied field H0 is along the z axis, which makes an angle 9 with the c axis (see Fig. 4.1). Since we are considering an uniaxial superconductor, we can take the y axis as fixed relative to one of the crystal axes. Therefore, the rotation matrix is cos 6 0 sin 6 R{9)= 0 1 0 — sin 9 0 cos 9 The rotated mass tensor is mtj(0) = R-1 mij(0) R. or, more explicitly, mi3(9) = m-icos2 9 + m3sin2 9 0 (mi — m3)sin#cos# 0 (4.4) 0 mi (mx — m3) sin# cos 9 0 m3cos29 + m1sin29 Just as in the isotropic derivation (Eqs. 3.7, 3.5), minimizing the energy gives the London equations d2h • hi — A mkl^lsi^ktj-^ ^ jklst OxsOxt (4.5) where the penetration depth is defined by A? = A2m,j. CHAPTER 4. ANISOTROPIC LONDON THEORY 31 Once again, I've included singularities so that we can investigate the type II phase. (Note that the isotropic limit is = Ski and all the following results revert back to the isotropic case.) Assuming that the flux lines are along the z direction, hx = A 2 hy = A 2 ' d2hx d2K m. dy2 dxdy ro. 'd2K mi m " \d^dy)+mzz 'd2K ,dy2j d2hx' dx2 dxdy 'd2K d2hx + d2K ar y V. / \ dy2 oxoy Using V • h = 0 and assuming that dhz/dz — 0, this can be rewritten as (d2hS + (j)oJ2S(f-fu). (4.6) K = A mzzAhx ro, / i y = A 2 xdy2 ) ( &K \ +mzzAhymxz [Q^Q^J hz = A 2 'd2K mi 'd2hz + mxzAh (4.7) dx2 J " V ^ y2 where A = d2/dx2 + d2/dx2. Let us suppose that there is a regular flux line lattice f{ in the x-y plane. Just as in the isotropic case, in the course of taking the Fourier transform we remove the term —* —* —* —» exp(z'A: • rv) by only allowing k G {K}, where {K} is the set of reciprocal lattice vectors. The anisotropic treatment produces the following Fourier components of the local field, Kit Kit = nf(f)0(\2mxzK2)/d -nf<f)0(\2mxzKxKy)/d nf<f>0(l + \2mzzK2)/d (4.8) where d = (1 + X myyK + X2mxxK2)(l + X2mzzKl) - {X2mxzKKy) CHAPTER 4. ANISOTROPIC LONDON THEORY 32 and K> = I<l + Kl. Here the average field H0 equals rif<f>0, which is equal to the applied field in the absence of any flux exclusion. The total local field at a point f= [x,y] is —* — — ^ —* —* h(x, y) = Re hg exp(—iK • f). K Note that if there were only a single, isolated flux line at the origin (r0 = 0), there would be no restriction on k since exp(ik • f0) = exp(0) = 1. In this case, all the summations would be replaced with integrals. Notice that in the isotropic case the local field is parallel to the applied field, while in the anisotropic case there are transverse components of the local field which can be quite substantial. Physically this is because the screening currents, discouraged by the poor carrier coupling along the c axis, tend to flow in the a-b planes rather than perpendicular to the applied field. A measure of the size of this effect is found in the ratio of the magnetic portions of the line energies for the transverse and axial components:[20] £tr m2 — = — , (4.9) eax rnzz(mzz + m3)' which also gives the ratio of the mean squares of the tranverse and axial fields in a single vortex. For an angle of 45 degrees, mxz = 1/2 (mi — m3), mzz = 1/2 (mi + rn3), and the ratio is etr _ (mi - m3)2 tax (mi + m3)(mi + 3m3) For a ratio of m3/mx = 50, which corresponds to a ratio of A3/Ai = 7.07, etr/eax — 0.3117, which is quite substantial. In the limit that m3/mi goes to infinity, the ratio €tr/tax goes to one third. CHAPTER 4. ANISOTROPIC LONDON THEORY 33 Preliminary LF-^ uSR experiments at TRIUMF have detected small perpendicular field components in unaligned sintered powder samples, but certainly not of this mag-nitude. See the last section of this chapter for further discussion. 4 . 3 C h o i c e o f R e c i p r o c a l L a t t i c e 4.3.1 The Triangular Lattice It is reasonable to assume that the flux tube lattice will be a non-equilateral triangular array. The two lowest energy solutions have the following possible sets of anisotropic primitive vectors: [26,27,28] fi — Lxx Case I, f 2 = L x \ x + Ly*fy or h = ^ n TT Case 11. r2 = Ly \ y + Lx ^ x L. Dobrosavljevic and H. Raffy [26] found that the ratio of the x and y components of f 2 has two equivalent values: ^ = V 3 j — Case I r2x V myy or r2x ~ Case II. m In the first case, = Lxx and r 1/r 2 a ; = 2. Likewise, in the second case, f1 = Lyy and r i / r 2 y = 2. Just as in the isotropic case, the number of flux tubes per unit area must be H0 2 (j>0 V3LxLy CHAPTER 4. ANISOTROPIC LONDON THEORY 34 for an average magnetic field in the sample of H0, which equals the applied field B in the absence of flux exclusion. The distances L x and L y are determined by 2 <j>0 L x L y ~ V3H0 and by giving Lx L r / mi T \ ' _2_<f>o_ \ rai y/3H0V mxx " ~ L/5ffeV mi 1/2 1/2 Using Eq. 3.10, we conclude that the reciprocal lattice's primitive vectors (for Case I) are —* An arbitrary reciprocal lattice vector K is K = nk\ + mA;2, where n and m are integers. Having two different lattices which are equally probable effectively doubles the com-putation required for simulation. At the time I did my computer simulations, I did not understand L. Dobrosavljevic's and H. Raffy's notation, so I wrote my computer sim-ulation assuming a rectangular lattice. This reduces the number of different lattice cases from two to one. Qualitatively, using a rectangular lattice shouldn't matter too much, as the difference in the spectral distribution of frequencies between the two cases CHAPTER 4. ANISOTROPIC LONDON THEORY 35 is slight. The figure on page 841 of Superconductivity edited by Parks [29] shows the difference for the isotropic case near Hc2, where the difference is the most extreme. Recently, it was shown that in the high field limit, to lowest order in the energy, all orientations of the lattice's unit vectors relative to the crystal are equivalent.[28] I be-lieve that if one were to average over all possible orientations, the lineshape would still remain much the same because the ratio of the unit vectors is kept constant. 4.3.2 The Rectangular Lattice For a rectangular flux lattice (Fig. 4.2) in the x-y plane, with spacing L x and L y between flux tubes, the reciprocal lattice's primitive vectors are x and k2 = — y. An arbitrary reciprocal lattice vector K is K = nki + mk2 where n and m are integers. The spacings L x and L y are found from LxLy — (j)0/H0 and giving CHAPTER 4. ANISOTROPIC LONDON THEORY 36 It is interesting to note that, using L x and L y for the sides of a rectangular flux tube lattice's unit cell, the long axis of the magnetic flux's elliptical vortex is along the short side of the unit cell and the short axis is along the long side (see Fig. 4.2.) I find this mildly counterintuitive. 4 . 4 T h e A n i s o t r o p i c R e d f i e l d L i n e s h a p e Figure 4.3 shows the anisotropic lineshape for various angles relative to the hard axis. Figure 4.4 shows the anisotropic powder average lineshape as a function of various degrees of anisotropy and figure 4.5 shows the powder average lineshape as a function of applied field. Although my calculations are rough, it is evident that in order to simulate the shape of the unoriented powder data (which looks a bit like a capital italic letter A, see Fig. 6.1), it is necessary to go to ratios of A3/A1 ~ 50. Note that there is a tremendous amount of the distribution in the long tails. This will give an extremely rapid relaxation of the muon's polarization. 4 . 5 D i s c u s s i o n Due to the surprisingly small experimental values for £c, I have not been able to decide whether or not the high temperature superconductors are in a regime in which this theory is strictly valid. Furthermore, there is another problem with it which I believe is more serious. For ratios of m^/m-i = 50 and 2500, the ratios of the lengths of the unit cell are Ly/Lx = 7 and 50 respectively for the applied field at an angle of 90 degrees to the hard axis of the crystal. At an angle of 45 degrees and m3/rai = 2500, Ly/Lx = 5.05 and etr/eax = 0.3329, which is very close to the limit of 1/3 as m3 goes to infinity. For CHAPTER 4. ANISOTROPIC LONDON THEORY 37 such a large ratio and 6 not too close to zero, For an applied field of 25 kG, L x — 128 A and L y = 646 A at an angle of 45 degrees and L x = L y = 287 A at an angle of zero degrees. The twinning planes in YBa2Cu307 are spaced approximately 150 A to 1500 A [30,31] apart. In the plane at 45 degrees to the hard axis, they are spaced from 212 A to 2120 A . Therefore the unit vector L y is comparable to the distance between twinning planes. It is obvious that one needs very high fields to make the zero angle unit vector Ly(0) much smaller than the twinning boundary spacing and that, for large anistropy, the non-zero angle unit vector Ly{0) eventually catches up with the twin boundary spacing. What is disturbing is that Bitter experiments [7,8] in the a — b plane have shown that in low field the flux tubes tend to pin along the twinning boundaries. This sug-gests that ratios of unit vectors may deviate from the theoretical angular dependence. (Certainly twinning planes as pinning sites will have a much different effect from ran-dom pinning sites.) I believe that any attempt to "realistically" simulate the effects of the twinning boundaries will radically alter the powder average anisotropic Redfield lineshape calculated. Also, this would introduce yet another fitting parameter to the list consisting of the muon's initial polarization, the phase, the frequency, A3 and Ax — not to mention any other empirical parameters included to account for such phenom-ena as pinning gradients, etc. This theory is in principle quite elegant but there is a practical concern that modifications to account for imperfections in the flux lattice may introduce too many parameters for fitting //SR or NMR data on aligned and unaligned powder samples. A theoretical concern is whether or not the predicted large local field components perpendicular to the applied field [20] are in fact observed. Due to these transverse CHAPTER 4. ANISOTROPIC LONDON THEORY 38 local field components (discussed previously), we expect that the muon polarization in a TF-/iSR geometry, U2 U2 , u2 would show a loss of oscillating asymmetry {h2 + h2z)/h2 compared to above the crit-ical temperature where the local field is parallel to the applied field. Certainly, flux expulsion from individual grains will also contribute to the presence of transverse com-ponents. If one fits with an incorrect model which neglects a portion of the lineshape, then one runs the risk of underestimating the asymmetry, so our fits of TF data with a Gaussian relaxation function which show a loss of asymmetry are consistent with but not necessarily indicative of this effect. Similarly, a LF-^SR experiement would also show a loss of asymmetry. Both TF-//SR and preliminary LF-//SR experiments did find a clear loss of asymmetry compared to above Tc, but the asymmetry decrease was much smaller than the dramatic predictions from the ratio of the line energies (Eq. 4.9). Figure 4.1: Coordinate and crystal axes; 8 is the rotation of the crystal's hard axis c with respect to the coordinate axis z. The angle <j> is a rotation within the x — y plane. CHAPTER 4. ANISOTROPIC LONDON THEORY 40 Figure 4.2: Contour plot of the local field distribution of an anisotropic superconductor with a square flux array and its crystal axes tilted at 45° with respect to the applied field of 4 kG. CHAPTER 4. ANISOTROPIC LONDON THEORY 41 X = 1 3 6 5 A av Ratio of X's - 70:1 B = 4kG (54.2 MHz) 1.0 0.8 0.6 0.4 0.2 0.0 r_J 1 1 1 1.0- 1 1 r 1 1 — 1 — 1 1  i b 1 c - 0.8- i 1 30 Degrees 1 45 Degrees - 0.6- j - j 0.4-- 0.2- 1 ! 0.0- i 20 30 40 50 60 70 80 20 30 4 0 5 0 60 7 0 80 1.0 0.6 0.2 0.0 d i 60 Degrees \— 20 30 +0 50 60 70 80 1.0- i i i e 0.8- -90 Degrees 0.6- -0.4- -0.2-0.0 - i i i U — , — 2 0 3 0 40 5 0 r equency (MHz 60 7 0 8 0 Figure 4.3: The anisotropic London model shown for various angles for an average penetration depth of 1365 A , a penetration depth ratio of 70 to 1, and an applied field of 4kG. The dashed line at 54.2 MHz corresponds to the applied field. CHAPTER 4. ANISOTROPIC LONDON THEORY 42 1.0 0.8 0.6 0.4 0.2 0.0 1 1 1 — 1.0- I 1 1 a b - 0.8-X = 1365 A X1 = 7 9 8 A X's ra t io = 1:1 0.6- X3 = 3991 A - 0.4- X's ra t io = 5:1 \ - 0.2- A i i i 0.0-—,—,—A-V—,— 20 30 40 50 60 70 BO 20 30 40 50 60 70 80 1.0 0.8 0.6 0.4 0.2 0.0^  1.0 0.8 0.6 0.4 0.2 0.0 1 1 1 1.0- 1 1 1 i i c d - 0.8- -X1 = 6 3 4 A X1 = 371 A " X 3 = 6 3 3 6 A 0.6- X3 = 18526 A -X's ra t io = 10:1 0.4- X's rat io = 50:1 i -0.2-o.o-r i i i 20 30 40 50 60 70 80 20 30 40 50 1 1 e 1 1 1 X, = 331 A ' X 3 = 23185 A X's rat io = -"""M*!! 1 70:1 1 20 30 40 50 1.0 0.8 0.6 0.4 0.2 0.0 60 60 70 80 20 30 40 50 60 requency (MHz 70 70 80 1 1 1 f I 1 X1 = 2 9 4 A " X3 = 2 9 4 0 8 A _ \^ 1 1 1 80 Figure 4.4: The anisotropic London model's powder average shown with an applied field of 4kG, an average penetration depth of 1365 A , and various penetration depth ratios. The dashed.line at 54.2MHz corresponds to the applied field. CHAPTER 4. ANISOTROPIC LONDON THEORY 43 (x1D+1) ( X I C T ) Frequency (MHz) Figure 4.5: The anisotropic London model's powder average shown with an average penetration depth of 1365 A , a penetration depth ratio of 70 to 1, and various applied fields. The dashed lines represent the muon precession frequency corresponding to the applied fields. C h a p t e r 5 A S i m p l e M o d e l 5 .1 I n t r o d u c t i o n In this chapter, I describe a simple model of the flux penetration in layered type II superconductors, which I call the Kossler model because it was first suggested by Jack Kossler of the College of William and Mary. It was subsequently developed by Moreno Celio while he was associated with TRIUMF in 1987 and later by myself. [32] Since the newly discovered copper-oxide based perovskite superconductors are highly anisotropic, a logical simplification is to treat them as a stack of uncoupled, layered two-dimensional superconductors. Triangular flux tube arrays are formed on the lay-ers, but using only the component of the field which is perpendicular to the layers. As any conduction out of the plane of the layers will be through some sort of Josephson junctions [33,21,34,35] inherent in the crystal structure, any flux tubes formed from magnetic field components which aren't parallel to the c direction will have a very long penetration depth. In the limit of very closely spaced layers and infinite penetration depth for tubes parallel to the layers, we could imagine that flux tubes moving from one layer to an adjacent layer are completely aligned; i.e., they don't have a chance to "kink" in between layers. 44 CHAPTER 5. A SIMPLE MODEL 45 With this in mind, the idea is to decompose the applied magnetic field into com-ponents perpendicular and parallel to the hard axis of the crystal, where the hard axis is perpendicular to the layers. The perpendicular component h± passes through unchanged. The parallel component h\\ forms a vortex lattice. 5 .2 D e r i v a t i o n Let the applied field B be in the z direction, which makes an angle 9 with the hard axis (see Fig. 4.1). (I explicitly assume no flux exclusion, that is, B is equal to the average field H0 in the superconductor.) The component of the field perpendicular to the hard axis, h± = -Bsm9, (5.1) passes through undeflected. The component parallel to the hard axis, /i|| = 5cos0, (5.2) forms an isotropic 2-D vortex lattice F(r, Bcos9) with penetration depth A. Eq. 3.9 defines F(f, B cos 9) as F(r, B cos 9) = B cos * £ ^ f^ P, (5-3) Re 6 where the summation is over the reciprocal lattice vectors of the flux tube lattice. The vector1 h(r) =-Bsin6x± +F(r, Bcos8)X]] (5.4) gives the overall internal field. Assuming that the layers (referred to as the a-b plane) are isotropic two-dimensional superconductors, the lowest energy configuration of the flux tubes is a regular triangular CHAPTER 5. A SIMPLE MODEL 46 array. [16] The Bravais lattice unit vectors of the flux tube lattice are = LgX, at he = Lg(l/2xa + V3/2xb) and the reciprocal lattice unit vectors are (5.5) JbeVo where, assuming no flux expulsion from the macroscopic sample, the distance Lg be-tween flux tubes is Vv^-Bcosf? and (j>0 is the magnetic flux quantum using a charge of 2 e . A reciprocal lattice vector Kg is Since the calculation of the muon's polarization is complicated by the choice of reference frame, I present the calculation here. For the purposes of this calculation, I assume that the muon's momentum is in the z direction, parallel to the applied field, and its polarization is in the x direction. I assume that the sample is an unoriented powder of minute single crystals. I ignore any of the powder grains' surface effects (e.g., the Meissner effect, flux expulsion, and shape or size effects) and I assume that every grain is in the same magnetic field. (5.6) where n and m are integers. 5 . 3 T h e F r e q u e n c y L i n e s h a p e CHAPTER 5. A SIMPLE MODEL 47 First we will consider the precession of a muon in a single crystal. Since our mag-netic field coordinates are relative to the hard axis c and our probe, the muon, has coordinates relative to the z axis, we must project (h\\,h±) onto the z axis and the plane perpendicular to z: hz = F(r, B cos 6) cos 6 + (B sin9) sin 9 (5.7) hp = F(f; B cos 9) sin6 - (B sin 9) cos 9. (5.8) The hp can be oriented arbitrarily in the x - y plane: hx = hp cos <j) hy = /ip sin (/>. The local field /\ = (hx,hy,hz) has a magnitude squared of h2 = h2 + h2 + h2 = hp2 + h2 = F2(r, B cos 5) + 52sin2 9. In a single crystal, the muon's polarization is given by P/(t) = ^  + ^ y ^ c o s ( 7 ^ ) -Since we have assumed that the layers are isotropic 2-D superconductors, the flux tube lattice will be oriented randomly relative to the angle 4> (Fig. 4.1). The single crystal muon polarization averaged over <j> becomes 0.5/u2 , (0.5^2 + fez2) , . . . , , Q , P» (*) = h2 + 1~2 cos(7M/\r). (5.9) In order to calculate the frequency distribution (the analogue of the Redfield line-shape), we must calculate the local field h for a uniform distribution of points within the unit cell of the flux lattice and for 9 varying from 0 to 90 degrees, and histogram the results weighted by (0.5/ij.2 + h2)/h2. CHAPTER 5. A SIMPLE MODEL 48 0.2-1.0 0.4 0.2 0.0 1 1 1 1 1 1 b 1 ^^\ 1 Appl ied field 15kG i 139 200 201 202 203 204 205 206 1.0 0.8 0.6 0.4 0.2 269 270 271 272 273 274 0.0 ! d I Applied field 25kG 275 337 338 339 340 341 342 343 404 405 406 407 408 409 410 F r e q u e n c y (MHz Figure 5.1: The Kossler model with a penetration depth of 1000 A for fields applied perpendicular to the copper-oxide planes in various applied fields of 10, 15, 20, 25, and 30 kG. The dashed line shows the muon precession frequency corresponding to the applied field. CHAPTER 5. A SIMPLE MODEL 49 Figure 5.2: The Kossler model in an applied field of 15 kG with various in-plane pen-etration depths of 354, 500, 707, 1000, 1414 and 2000 A . The dashed line shows the muon precession frequency corresponding to the applied field. CHAPTER 5. A SIMPLE MODEL 50 The resulting lineshape can be seen as a function of applied field in Fig. 5.1 and as a function of A in Fig. 5.2. Note that A is the penetration depth for the applied field normal to the planes. The penetration depth for the field parallel to the planes is infinite in this model. 5 .4 T h e S e c o n d M o m e n t Appendix A contains an approximate calculation of the second moment. Because the Kossler model gives the field's magnitude in terms of h2 — F2(r, B cos 6) + £ 2sin 2 9 and because the second moment can be evaluated by BM2 = {(h-B)2) = (h2)-B2, the second moment's expression is simple. Since the higher moments can only be evaluated by BMn — {{h — B)n), their evaluation is rather more complicated, and I did not attempt to evaluate them. Nonetheless, in the high field approximation, the second moment is field independent and is proportional to the inverse square of the penetration depth, just as in the isotropic case. This can be seen in Fig. 5.1, which shows that the Kossler model's lineshape is translatable along the frequency axis, and in Figs. 5.2 and 5.3, which show that the Kossler lineshape's width scales inversely with the square of the penetration depth. CHAPTER 5. A SIMPLE MODEL 51 2000A 414A OOOA 07A 500A 354A Frequency (MHz Figure 5.3: Same theory lines as in Fig. 5.2, but with the deviation of the frequency from the applied field scaled by a factor of A2 to show that the lineshapes are nearly the same. The lineshapes with 1414 and 2000 A look slightly different because they were made with 60 rather than 40 angular increments. C h a p t e r 6 D a t a A n a l y s i s 6 .1 I n t r o d u c t i o n Given typical experimental //SR data consisting of "time spectra" (histograms of //-e decay time intervals detected in various counters) and a suitable Fourier transform method, one has the option of fitting the data to theory with either time or frequency as the independent variable. Simple analytic functions, such as Gaussians [Gxx(t) — exp(—a2t2/2)] or Lorenzians [Gxx(t] = exp(—At)], are often used for fitting in time space as long as they give good fits (x2 per degree of freedom « 1). When simple analytic functions are inadequate, more complicated functions can be utilized using numerical interpolation tables of Gxx(t). A theoretical frequency distribution A(f), such as that furnished by the Kossler model or the isotropic Redfield lineshape, can be used to fit the data in either the time domain [by generating Gxx(t) from A(f)] or the frequency domain [by comparing A(f) to the experimental frequency spectrum]. (If one wishes to test consistency, one can convert the experimental frequency distribution into a relaxation function by the same method and "fit the data to itself" in the time domain.) If it is necessary to modify the theoretical frequency distribution in some manner, such as convoluting one 52 CHAPTER 6. DATA ANALYSIS 53 or more of the distribution's parameters with its own distribution, then it is easier to fit with frequency as the independent variable and to do the actual convoluting within the fitting routine. If more than one theory has the same parameter transformation properties, the same fitting routine will be appropriate for all of them. Less attractive alternatives are (a) to convolute the frequency distribution and then Fourier transform it into the time domain throughout the fitting routine (very slow), or (b) to fit using a large interpolation table with time as the independent variable (fast, but very tedious for the table maker and aesthetically unpleasing). This chapter covers • some details of proper discrete Fourier transforms, • procedures for generating the data's frequency distribution by fitting, • how the Kossler model's lineshape and the isotropic Redfield lineshape depend upon the penetration depth and the applied field, • how to fit them with frequency as the independent variable, and • how to fit with Gaussian convolutions of applied field and penetration depth. The last section explains how to estimate the isotropic London magnetic penetration depth A using the second and third moments of the frequency distribution and how to estimate the penetration depth due to applied fields parallel to the hard axis of an anisotropic superconductor using the second moment of the Kossler model. CHAPTER 6. DATA ANALYSIS 54 6 . 2 E x p e r i m e n t a l F r e q u e n c y D i s t r i b u t i o n s Experimental time spectra were converted into frequency distributions in two ways. The first was to fit with 27 non-relaxing cosines of variable amplitudes but fixed fre-quencies and a common (variable) relative phase. The frequencies were regularly spaced about the peak frequency, with the difference between frequencies (A/) given by where T is the time range of the fit in microseconds. Less optimal fits can use smaller A/'s, but as we were limited to 27 frequencies by the pre-existing fitting program, I chose the optimal case with a time range T chosen such that the signal had clearly completely relaxed well before the end of the time range. The span of the 27 frequencies was chosen to ensure that it was much larger than the square root of the second moment of the data's frequencies — i.e., to make certain that all potentially nonzero parts of the frequency distribution were within the range of fitting frequencies. A related relation is that the maximum frequency measurable (i*max) is where At is the data's time bin size. With time bin sizes of typically 0.625 ns, i^ rnax = 800 MHz; data taken in a transverse field of 15 kG has a muon precession rate of about 200 MHz, which is well below Fmax- Typical fits are shown in Fig. 6.1. The second method of converting time spectra into frequency distributions was to use a standard fast Fourier transform program with zeroth order phase correction. Originally, our fast Fourier transform (FFT) program was deficient in that it only phase corrected to within a few degrees. The resulting admixture of some of the imaginary frequency spectrum in the real spectrum was seen as superfluous wings. This problem led Jess Brewer to develop the first method (fitting with 27 non-oscillating CHAPTER 6. DATA ANALYSIS 55 0.013 0.011 0.009 0.007 0.005 • 0.003 • 0.001--0.001 Q Unor iented 200 201 202 203 204 205 206 207 192 193 194 195 196 197 198 199 0.013 0.011 0.009 0.007 -\ 0.005 0.003 0.001 -0.001 C Oriented c _ L B 192 197 198 199 F r e q u e n c y (MHz Figure 6.1: The results of fitting data with 27 non-relaxing cosines for (a) an unoriented sintered sample of YBa2Cu307, (b) an oriented sintered sample of YBa2Cus07 with the muon spin perpendicular and the applied field parallel to the crystal's hard axis (c) and (c) the same oriented sample of YBa2Cu3C*7 with the muon spin parallel and the applied field perpendicular to the crystal's hard axis (c). CHAPTER 6. DATA ANALYSIS 56 cosines) as a way to get the correct frequency distribution. However, with the phase correction properly calculated, the real part of the Fourier transform spectrum (Fig. 6.2) is virtually identical to the results of fitting (Fig. 6.1). This agreement gives me confidence that the frequency distributions that I generate either way are correct. 6 . 3 C o m p a r i n g D a t a w i t h T h e o r e t i c a l F r e q u e n c y D i s t r i b u t i o n In this section, I discuss how to fit with the isotropic London theory and with the Kossler model presented in Chapter 5. As I was not satisfied with my predictions using the anisotropic London theory, I did not bother to attempt fitting the data with it. The point is moot anyway; I will show in the next chapter that the Kossler model with a convolution over applied fields gives a satisfactory fit, which implies that the additional parameter necessary to use the Kogan model will overspecify the unoriented powder /J.SR data. How does one take a theoretical frequency lineshape (or the the results of the Fourier transform to check for consistency) and fit an experimental time spectrum? One needs to know how the lineshape changes with the average muon frequency / = (,ylt/2ir)B, where B is the average local field, and with any parameter which determines its line width. With the Redfield lineshape A(f) due to an isotropic Abrikosov flux lattice, the second moment is independent of field to lowest order in the applied field, and is inversely proportional to the penetration depth (A) to the fourth power (see Appendix A). [Please note that my discussion of the moments is in terms of the local magnetic field h(f), whereas my simulations are with respect to f(r) — (•yfJ,/2ir)h(r), so as to CHAPTER 6. DATA ANALYSIS 57 be more easily compared with the data.] More generally, the n^n moment (Mn) scales with the penetration depth as Mn^ oc 1 to lowest order in the applied field B. If Ax is changed to A2 with B constant, the original frequency distribution A\(f) is transformed to A2(f) according to Ar(/) = A 2 ( / o + ( / - / 0 ) ^ , (6.1) where / = (7 /^27r)/i(r) is the local muon frequency and f0 = (j^/2ir)B. For the isotropic superconductor, the deviation of the microscopic local field distri-bution h(r) from the average field B itself scales as A - 2 : —* , , „ „ x-^  exp(—iK • r) 1 in the high field limit. Therefore, if the applied field B is changed and the penetration depth A remains constant, the same lineshape is simply translated along the frequency axis. I have verified this qualitively for the Kossler model by doing simulations at various fields (see Fig. 5.1). One could try to estimate the error of this approximation to lowest order in B by considering the general second moment Eqs. A.2 and A.6 M2(B) = B2Q\C0 - 2CXQ2 + 3C 2Q 4 + • • •) Isotropic M2{B) w B2Q\C0 - 3C2Q4 + •••) Kossler, where the constants Cm are defined by Eq. A.3 and Q = '2~^\(^rf-)~1^2- As the constants Cm are calculated by a sum over reciprocal lattice vectors, one could estimate the error in cutting off the calculation of Cm's at a finite number of reciprocal lattice vectors by examining its convergence criterion. Likewise, as we chose B such that Q <C 1 so that CHAPTER 6. DATA ANALYSIS 58 the series for Mn is convergent, we can calculate the error due to cutting off M2(B) after a finite number of terms. The zeroth order approximation is reasonable to use as a transformation rule in fitting programs as long as Q2 oc 1/(\2B) is small and the average field and penetration depth of the fitting program's theory lineshape are within an order of magnitude of the sample's. Fitting in frequency space entails fitting the data's frequency lineshape (from either the fast Fourier transform or the fit to 27 non-relaxing cosines) to the theoretical lineshape from the isotropic or Kossler models. The theory lineshape is modified by (a) translation to adjust for the applied field, (b) scaling the linewidth using Eq. 6.1, and (c) a multiplicative normalization factor. I decided to include a Gaussian distribution of applied fields in my simulation. Section B.5 describes the initialization and transformation subroutines used in a program called XYFIT, which uses MINUIT subroutines to fit data. By a simple extension, I could have also included a Gaussian distribution of pene-tration depths, but did not bother once I saw that the Gaussian distribution of applied fields was sufficient to fit my data. If one wished to include the variation of the penetration depth as a function of angle for the oriented powder (the hard axis was oriented within a Gaussian distribution with an angular variance of about 5.45 degrees), then one could use the A^  and Ay from the diagonal approximation for the rotated mass tensor (Eq. 4.4), A, 2 = Ax2 cos2(0)-f-A32sin2((9) CHAPTER 6. DATA ANALYSIS 59 \ z 2 = Ax2 sin2(0) + A 3 2 cos2(0) (6.2) for data taken with the applied field parallel to the hard axis. Because Ai, the pen-etration depth when the field is parallel to the hard axis, is much smaller than A3 in high temperature copper-oxide superconductors, if one neglects the fact that the oriented powder is not perfectly aligned by assuming that A^  = Ai, then Ai will be overestimated. If one believes and uses the ratio of the penetration depths R = \\/\^ as determined from other experiments as a fixed parameter in the fit, this approach will not add any additional parameters to the fit and may yield a better estimate for the two penetration depths. However, I do not believe that the data contain enough independent information to permit a useful determinination of the ratio i i as a free parameter in the fits. 6 . 4 E s t i m a t i o n o f t h e P e n e t r a t i o n D e p t h f r o m t h e F r e q u e n c y D i s t r i b u t i o n ' s M o m e n t s From Eqs. A.2 and A.6, to zeroth order in the applied field, the second moment of both the isotropic Redfield lineshape (rigorously) and the Kossler lineshape (approximately) is For Co = 4.33306 from the summation over 172 reciprocal lattice vectors, this works out to give BM2 = 0.00370^. A4 Using fM2 = B M 2 ( 7 / i / 2 7 r ) 2 = BM2(0.013554MHz)2, CHAPTER 6. DATA ANALYSIS 60 A can be expressed in terms of the second moment *M2 by / C \ 1 / 4 A M 2 = ^8.2063013 x 105 (jjf) = 1306.8(/M2)"1/4. The isotropic Redfield lineshape has a third moment of BM3 = B3QeC00o. Using Cooo = 4140.028 from the summation over 172 reciprocal lattice vectors, the third moment estimates the penetration depth as ,n N i /6 A M 3 = V8.2063013 x 105 (jj^) = 3630.0 (fM3)-^6. CHAPTER 6. DATA ANALYSIS 61 0.013 0.011 -0.001 198 199 200 201 202 203 204 205 206 207 208 F r e q u e n c y (MHz Figure 6.2: The results of fitting unoriented YBa2Cu307 data with 27 non-relaxing cosines (dashed line between points) are compared with the real frequency spectrum of a Fast Fourier Transform of the same sample (solid line). C h a p t e r 7 R e s u l t s a n d D i s c u s s i o n 7.1 J u s t i f i c a t i o n f o r S m e a r i n g t h e T h e o r e t i c a l L i n e -s h a p e It is very noticeable in a quick comparison of the data (Fig. 7.1) with the theoretical lineshapes (Figs. 3.1, 4.4, and 5.1) that the latter are consistently too sharply cusped. There are three obvious ways to "smear" lineshapes which are justifiable from a physical point of view: • Distribution of Penetration Depths: The penetration depth will vary due to inho-mogeneities in the concentration of oxygen or other dopants. It could vary in the area around twinning boundaries and other defects. The simplest assumption is a Gaussian distribution of penetration depths. • Distribution of Applied Fields: The applied field that each superconducting grain experiences may vary slightly due to partial anisotropic flux expulsion from neigh-boring grains. Also the magnetic field will prefer slightly to be in the voids be-tween grains. As the distribution and size of the voids are not uniform, there will be an uneven effect throughout the sample. Macroscopic field gradients due 62 CHAPTER 7. RESULTS AND DISCUSSION 63 to flux pinning will be seen as a variation in the local average field. Again, in the absence of detailed knowledge, a Gaussian distribution of applied fields is a reasonable choice. • Distribution of Flux Tube Positions About Their Ideal Flux Lattice Positions: Due to weak pinning forces or a free energy whose minimum is shallow, the flux tubes could be displaced from their ideal flux lattice positions. In fact, this is clearly seen in Bitter decoration experiments.[7,8] According to Brandt,[36] moderate displacements due to either weak pinning or random disorder have approximately the same effect as a Gaussian distribution of applied fields. This means that we cannot determine the cause of the Gaussian-like smearing of the lineshape on the basis of the lineshape alone! Since Bitter decoration shows obvious disorder (enough to duplicate the largest effect Brandt shows in his simulation [36]), it swamps any distribution of applied fields. Note that trying to fit using smearing due both to variation of the penetration depth and to inhomogenous fields or an imperfect flux lattice assumes that the two effects are uncorrelated, which is a dangerous assumption to make. This is an incentive to decide which is the dominant effect and only simulate that one. (Another incentive is to avoid adding more parameters to the theory than the data can determine.) Since averaging over the applied field makes the lineshape become Gaussian shaped (Fig. 7.5) while averaging over penetration depths makes the lineshape trumpet shaped (Fig. 7.2), I chose to fit by varying only the applied field because the data (Fig. 7.1) looks mostly Gaussian. CHAPTER 7. RESULTS AND DISCUSSION 64 -0.001 200 201 202 203 204 205 206 207 -0.001 192 0.013 0.011 0.009 0.007 0.005 0.003' 0.001--0.001 192 b Oriented c||B 193 194 195 196 197 198 193 194 195 196 197 198 F r e q u e n c y (MHz 199 c Oriented c_ |_B 199 Figure 7.1: The results of fitting data with 27 non-relaxing cosines for (a) an unori-ented YBa2Cu307 sample, (b) an oriented YBa2Cu307 sample with the muon spin perpendicular and the applied field parallel to the crystal's hard axis (c) and (c) the same oriented sample of YBa2Cu307 with the muon spin parallel and the applied field perpendicular to the crystal's hard axis (c). CHAPTER 7. RESULTS AND DISCUSSION 65 200 200 1 (li c; : J 1 1000A +- 75A • I ^ T ~ r 190 192 194 196 198 200 190 192 194 196 . 198 200 196 198 200 200 F r equen c y (MHz Figure 7.2: The isotropic Redfield lineshape is shown convoluted with Gaussian distri-butions of penetration depth with variances of 25, 50, 75, 100, 150, and 200 A . The dashed vertical line indicates the frequency corresponding to the applied field of 14.35 kG. CHAPTER 7. RESULTS AND DISCUSSION 66 7 .2 D a t a a n d A n a l y s i s The c-axis alignment of the YBa2Cu307 sample was measured with neutron scattering (Fig. 7.3.) A fit assuming a gaussian distribution of angles about the mean c-axis alignment gives a standard deviation of 5.45 ±0.11 degrees. Figure 7.1 shows the frequency data for the unoriented sample and for two orien-tations of the aligned sample for easy comparison. Figure 7.4 shows asymmetry versus frequency data (from fitting to 27 non-relaxing cosines) for a typical unoriented sin-tered powder sample of YBa2Cu307 using a fit to the Kossler model with a Gaussian convolution of applied fields with variance aj (see Table 7.1 for results of fit). Figure 7.5 shows the same for the aligned sintered powder sample of YBa2Cu307 (See Table 7.1). The fitted curves shown are theoretical lineshapes convoluted with a Gaussian distribution of applied fields with variance cr/, as discussed in the previous section. The fitting was done using the routines in section B.5. Figure 7.6 shows the time spectrum for the aligned sample, with the theory line shown being the fit to 27 non-relaxing cosines, which demonstrates consistency. 7 . 3 T h e T h i r d M o m e n t It is rather disturbing that even the oriented powder's lineshape is quite Gaussian. In general, the second moment of a theory lineshape will not be equal to the second moment of that lineshape convoluted with a Gaussian distribution of fields. Because the flux lattice is imperfect in high temperature superconductors, fitting theory lineshapes convoluted with a Gaussian distribution of fields is very reasonable, as I have argued before. Hence fitting data with a Gaussian and deducing a penetration depth using Eq. 3.12 with the second moment of the Gaussian fit is questionable if not completely CHAPTER 7. RESULTS AND DISCUSSION 67 554: GE oriented powder, fit to isotropic lineshape Normalization 0.011335 ±0.00013 Penetration Depth Ax 1785.9 ±2.1 A Average Frequency 194.533 ±0.072 MHz try 0.4795 ±0.0076 MHz 14.35 kG ±5.3 G 35.38 G ±0.56 G 408: UBC unoriented powder, fit to Kossler lineshape Normalization 0.011398 ±0.00032 Penetration Depth Aa 945.13 ±0.72 A Average Frequency 202.552 ±0.017 MHz Gj 0.250 ±0.016 MHz 14.94 kG ±1.3 G 18.45 kG ±1.2 G Table 7.1: Summary of fit results incorrect. The one thing a Gaussian lacks is any odd moments, so it cannot fit the high fre-quency tail well. This suggests to me that an alternative way of deducing a penetration depth may lie in the calculation of the third moment, if the third moment can be ex-tracted accurately from the data. A strong benefit is that if the local average field is a Gaussian distribution about the applied field, the measured third moment will equal the true third moment in the limit when the variation in field is small compared to the average field. The third moment of the magnetic field distribution is given by BM3 = J [B(f)-B]3df. If the calculated average field Best is off slightly from the true average field B by B = Best + AB CHAPTER 7. RESULTS AND DISCUSSION 68 then the true third moment is B M 3 t r u e = J [B(f) - (Best + AB)}3 df = J [B(f) -Best}3 df- 3AB J [B(f) - Best}2 df +3(AB)2 J [B(f) - Best] df- (AB)3 = B M 3 e s t - 3AB(BM2est) + 3(AB)2(AB) - (AB)3 ~ B M 3 e s t - 3 A B ( B M 2 e s t ) , to lowest order in AB. Assuming that there is no variation of the penetration depth, we can now average over a Gaussian (or any symmetric) distribution of B and use only one value for Best, namely the calculated average field of the data, to get (BM3true) = (J [B(f) - Best}3 df) - (ZAB j \B(f) - Best}2 df) = (BM3est) + (3AB)(BM2est) = (BM3est). Note that (AB) — 0 is equivalent to (B) = Best. Also I have used the fact that, in high fields, the second moment is approximately independent of field. Thus I have shown that, to lowest order in the variation of the applied field, the measured third moment is equal to the true third moment. It is independent of the smearing effects due to a variation in the local field or due to random flux lattice imperfections. Using Eqs. A.2, A.4 and A.3, the penetration depth in an isotropic superconductor can be expressed in terms of the second and third moments by (see Table 7.2) and A 2 _ l2<t>o (Cooo\1/3 _ 8 2 f ) 6 o 0 1 3 (£ooo\1/3 X 2 ( 7 2 ) A(/M3> ~ 2*>y/3 \ f M 3 ) - 8 2 0 6 3 0 - 1 3 \JM3) A ' { 7 - 2 ) CHAPTER 7. RESULTS AND DISCUSSION 69 Xfu Xtheory fM2 A M 2 fM3 XMs 554: Oriented 1786 A 0.3178 1635* A 1740+ A 0.0162 2794* A Redfield Lineshape IOOO A 2.271 1064+ A 7.691 2583+ A 408: Unoriented 945 A 0.3445 1135* A 1706+ A 0.0029 2468* A Kossler Model IOOO A 0.572 1502+ A 0.659 n.a. Table 7.2: Summary of penetration depths calculated from second and third moments for an oriented powder YBa 2Cu 30 7 sample, the isotropic Redfield lineshape, an unori-ented powder YBa2Cu307 sample, and the Kossler model. Note that * M2 is in units of (MHz)2 and fM3 is in (MHz)3. Calculations marked by * were made using Eq. 7.3. Calculations marked by + were made using Eqs. 7.1 and 7.2. Alternatively, if one has a theoretical lineshape, such as the Kossler model, whose parameters (applied field, pentration depth) transform the lineshape the same way as for the isotropic theory, one can simply scale the moments: f M n t h = B M n t h = f\exp\2n This second method (see Table 7.2) avoids the question of whether or not C00...0 N A S been calculated to adequate accuracy. 7 .4 D i s c u s s i o n o f R e s u l t s From Table 7.2, one can see that the underestimation of the isotropic superconductor's second moment due to the arbitrary cutoff of the computer simulation at a frequency CHAPTER 7. RESULTS AND DISCUSSION 70 of 204.11 MHz (compared with average frequency of 194.4425 MHz) causes the pen-etration depth to be overestimated by 6% (1064 vs. 1000 A ) . Since the amplitude of the high frequency tail is so small, experimental noise will ensure a larger overestimate of the penetration depth, all other things being equal. The 50% overestimate of the penetration depth using the second moment of the Kossler model shows that the as-sumptions used to derive the second moment formula, in particular the assumption that the local field is approximately equal to its component in the direction of the applied field (h ~ hz), are not valid. For the isotropic Redfield lineshape, the inconsistency of the penetration depth as calculated from the third moment (2583 A ) with the true penetration depth (1000 A) indicates that the calculation of the constant C0oo has not been carried through until the convergence criterion was satisfied. (17 x 17 reciprocal lattice vectors were sufficient to calculate Co but not Cooo- I did not properly determine what the convergence criterion was.) Nonetheless, we can use the ratio of the measured and theoretical lineshapes (Eq. 7.3) to estimate the true penetration depth of both aligned and unoriented samples. It is unfortunate that both the second and third moment methods of calculating the penetration depth have serious difficulties. The calculation of the second moment is fairly stable against changes of the frequency range over which the calculation is performed. However, the distribution of average local fields due to the displacements of flux tubes from their ideal flux lattice positions means that the measured second moment is a function of the mean size of the displacements as well as of the penetra-tion depth. Assuming that none of the signal is lost in the experimental noise, the second moment will always underestimate the penetration depth if there are flux lat-tice imperfections. Alternatively, the second moment will overestimate the penetration depth if there is loss of the high frequency tail in the noise, assuming that the flux lat-tice is perfect. Since these two effects compete, the second moment is not necessarily CHAPTER 7. RESULTS AND DISCUSSION 71 a reliable measure of the penetration depth for both unoriented powder samples and single crystals. Experimentally, the distribution of local average fields compensates for the loss of the Redfield and Kossler models' lineshapes' high frequency tail in the noise as far as the second moment goes; for example, the oriented sample's second moment gives a penetration depth of 1635 A , which is comparable to the fitted value of 1786 A , while the third moment gives a penetration depth of 2794 A . In calculating the penetration depth with the third moment, the flux tube displace-ments will not partially compensate for any loss of the high frequency tail because they will not strongly affect the third moment, as I have argued in the previous section. As can be seen in Table 7.2, the third moment calculated for the aligned sample gives a penetration depth that is inconsistent with both the fitted value and the value as calculated from the second moment. Because the third moment is higher order than the second moment, it will be even more sensitive to any loss of the high frequency tail in the noise. Being an odd moment, the experimental third moment is highly sensitive both to the choice of frequency range, and, more importantly, to the presence of a background signal. While every care is taken to minimize the background signal, it is always present to some degree (see Fig. 7.7.) Even a flux exclusion of a few gauss out of 15 kG will ensure that the average frequency of the background signal will be higher than that of the "foreground" signal from the superconductor. (I have neglected all flux exclusion in my simulations, except in so far as the average field should be the applied field minus the amount of flux exclusion.) Therefore, the presence of any back-ground signal will have drastic consequences on the calculation of the third moment. An authoritative discussion on the proper way to calculate the third moment of u^SR data is beyond the scope of this thesis. The theoretical lineshapes with Gaussian distributions of applied field were able to produce good fits to the frequency distributions. The Gaussian distribution's variance CHAPTER 7. RESULTS AND DISCUSSION 72 0 7 , 35.4 G for the aligned sample and 18.5 G for the unoriented sample, is difficult to interpret without a detailed understanding of how displacements of the flux lattice produce the apparent variance. (Brandt [36] states only that the effect of moderate displacements of the flux lattice can be approximated by a Gaussian distribution of applied fields.) Also, it is not clear why the variance for the two samples differ by a factor of two, unless the size of the variance depends on the orientation of the powder's crystal grains with respect to the applied field, which is not implausible. From the last section of Chapter 6, the diagonal approximation for the rotated mass tensor for an anisotropic superconductor gives the penetration depths in the planes of A2 = A2 cos2(0) + A2sin2(0) and K = A2-Integrating over the distribution of hard crystal axis orientations, as given by neutron scattering, results in e2 = |(A12cos2(^) + A32sin2(^))exp sm(6) d0 ~ 0.33 A x 2 + 0.65 A 3 2 (7.4) for ag = 5.45 degrees. If the penetration depths have an anisotropy of R — \3j\\, then (A*) ~ A^O.33 + 0.65R2. (7.5) Averaging the x and y penetration depths ((A )^ and Ax) gives roughly (Vanes) = + Ai) = y (l + V0.33 + 0.65 R2) (7.6) CHAPTER 7. RESULTS AND DISCUSSION 73 which, for a ratio R = 5 and an apparent mean planar penetration depth (Xpianes) = 1786 A as measured in the aligned sample, implies a penetration depth in the planes of Ai = 704 A . This result indicates that the penetration depth derived from aligned powder samples is highly sensitive to the degree of alignment (ag) and to the ratio of anisotropy (R). Since I have not developed a lineshape appropriate for fitting the aligned sample with the hard axis (c) perpendicular to the applied field (run #567), I cannot estimate the ratio of the penetration depths via comparison with the data of the same sample with the hard axis (c) parallel to the applied field (run #554.) In fact, both the Kossler model and the anisotropic mass tensor theory may not be appropriate when the hard axis (c) of the copper-oxide superconductor's crystal is perpendicular to the applied field, because of the very small coherence length in that direction and because of the flux pinning at twin boundaries. In summary, I have derived the penetration depth for an oriented sample and an unoriented sample of YBa2Cu307 sintered powder by fitting their frequency distribu-tions with a Gaussian convolution of the Redfield lineshape and the Kossler model respectively. Fitting the unoriented sample to the Kossler model yielded 945 A . Fit-ting the aligned sample to the isotropic Redfield lineshape yielded 1786 A , neglecting the variance of the crystal alignment, and 700 A , including the alignment variance and assuming an anisotropy of 5 in the penetration depths. I have estimated the pene-tration depths of the samples using the second and third moments of the frequency distribution and explained the difficulties of this method. Without a reason to prefer one model over another, analysis of more data which would clarify any differences in dopant concentrations, or a reliable estimate of the ratio of the pentration depths, I cannot estimate the penetration depth in the YBa2Cu30"7 superconductor's planes any better than 1000 ± 400 A . CHAPTER 7. RESULTS AND DISCUSSION 74 7 .5 P o s s i b l e F u t u r e D e v e l o p m e n t s It would be desirable to develop a phenomenological model more reflective of the details of flux pinning, flux entanglement and so on than the simple Gaussian convolution I assumed, if it does not overspecify the information in the //SR data and NMR data. There has been increasing evidence that the flux lattice in high temperature copper ox-ide superconductors is mobile,[37,38] perhaps acting mOre like a liquid than a solid,[39] in different temperature regimes near the critical temperature for different supercon-ducting compounds. As long as the flux tubes' movement is much slower than the lifetime of the muon, we need not worry about motional narrowing effects causing us to overestimate the penetration depth.[40,41] However, as the flux tube mobility is particularly large in the Bismuth-based copper oxides, we should be concerned that this effect may cause us to misinterpret the temperature dependence of the penetration depth near Tc. Without the development of single crystals of unvarying chemical composition or more highly aligned sintered powder samples, I believe that the extraction of the ab-solute penetration depth from high temperature copper-oxide superconductors using //SR can not proceed much further from what I have presented within this thesis. NMR does have the advantage over //SR of being capable of measuring small single crystals, whereas /fSR must have a sample of at least 1 cm in diameter (which may, of course, be composed of a "mosaic" of small single crystals with a thickness of at least 200 /mi. [42]) Since I have not seen any NMR penetration depth estimates for the copper-oxide superconductors in the literature that did not use a second moment calculation,[4,5] I hazard to say that NMR will not proceed much further either without a great deal of improvement in measurement sensitivity. CHAPTER 7. RESULTS AND DISCUSSION 75 On the other hand, for a series of samples with identical morphology (both micro-scopic and macroscopic), identical degrees of inhomogeneity in the penetration depth and identical degrees of flux lattice disorder, any of the standard methods of estimating the penetration depth from TF-//SR data should miscalculate it by a common factor. Thus TF-//SR results should be quite reliable as a relative measurement of the pene-tration depth. CHAPTER 7. RESULTS AND DISCUSSION 76 Figure 7.3: A Gaussian fit to the GE YBa2Cu307 oriented powder's neutron-scattering rocking curve, which shows the distribution of orientations in the sample. The fit gives a Gaussian standard deviation of 5.45 ±0.11 degrees. CHAPTER 7. RESULTS AND DISCUSSION 77 4 0 8 : P l a n a r p e n e t r a t i o n d e p t h wi th G a u s s i a n c o n v o l u t i o n 2 0 0 2 0 1 2 0 2 2 0 3 2 0 4 2 0 5 2 0 6 2 0 7 F r e q u e n c y ( M H z ) Figure 7.4: The results of fitting the unoriented YBa2Cu307 sample's asymmetry vs. frequency data to the Kossler lineshape convoluted with a Gaussian distribution of applied fields. CHAPTER 7. RESULTS AND DISCUSSION 78 Figure 7.5: The results of fitting aligned YBa 2Cu 30 7 sample's asymmetry vs. frequency data to an isotropic Redfield lineshape convoluted with a Gaussian distribution of applied fields. CHAPTER 7. RESULTS AND DISCUSSION 79 554: O r i e n t e d GE y b c D TF=13DDDdac T=4.33K , 08 I 1 1 1 1 1 1 1 1 r -0 ,2 . 4 . 6 . 8 1 1 . 2 1 . 4 1 , 6 1 . 8 2 TINE ( m l c r o s e c ) Figure 7.6: Oriented YBa2Cu307 with the muon spin perpendicular to the crystal's hard axis in a field of 15 kG. Note that the data is shown in a rotating reference frame very close to the applied field. The theory line shown is the fit to 27 non-relaxing cosines. CHAPTER 7. RESULTS AND DISCUSSION 80 8 5 1 : L a * 3 r a 2 C u 0 4 _ ) ( T F = 2 . 6 k G T = 5 . 0 0 ( 5 ) K .1 i 1 1 1 1 1 1 1 1 r 3 3 3 3 . 5 3 4 3 4 . 5 3 5 3 5 . 5 3 6 3 6 . 5 3 7 3 7 . 5 3 8 F r e q u e n c y ( M H z ) Figure 7.7: Sample of Lax.8Sr0.2CuO4 at 5.0 K in an applied field of 2.6 kG. The large background signal is due to muons stopping in the cryostat instead of the sample or due to the sample having a chemical phase impurity. B i b l i o g r a p h y [1] G. Aeppli, R. J. Cava, E. J. Ansaldo, J. H. Brewer, S. R. Kreitzman, G. M. Luke, D. R. Noakes, and R. F. Kiefl. Magnetic penetration depth and flux-pinning ef-fects in high-Tc superconductor Lai.gsSro.isCuCv Physical Review B, 35(13):7129, (1987). [2] D. R. Harshman, G. Aeppli, E. J. Ansaldo, B. Batlogg, J. H. Brewer, J. F. Car-olan, R. J. Cava, M. Celio, A. C. D. Chaklader, W. N. Hardy, S. R. Kreitzman, G. M. Luke, D. R. Noakes, and M. Senba. Temperature dependence of the mag-netic penetration depth in the high-Tc superconductor Ba2YCu30g_5: Evidence for conventional s-wave pairing. Physical Review B, 36(4):2386, (1987). [3] D. H. Wu, C. A. Shiftman, and S. Sridhar. Field variation of the penetration depth in ceramic YBa2Cu 30 ! /. Physical Review B, 38:9311, (1988). [4] J. T. Markert, T. W. Nob, S. E. Russek, and R. M. Cotts. NMR of 8 9 Y in nor-mal and superconducting YBa2Cu30g_5. Solid State Communications, 63(9):847, (1987). [5] N. Niki, T. Surzurki, S. Tomiyishi, H. Hentona, M. Omori, T. Kajitani, and R. Iegi. 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Schwall, and M. R. Beasley. Upper critical fields and reduced dimensionality of the superconducting layered compounds. Physical Review B, 21(7):2717, (1980). [35] K. K. Likharev. Superconducting weak links. Reviews of Modern Physics, 51(1):101, (1979). [36] E. H. Brandt. Magnetic field density of perfect and imperfect flux line in type-II superconductors. I. Application of periodic solutions. Journal of Low Temperature Physics, 75(5/6):355, (1988). [37] T. T. M. Palstra, B. Batlogg, L. F. Schneemeyer, and J. V. Waszczak. Thermally activated dissipation in Bi2.2Sr2Cao.gCu208+,$. Physical Review Letters, 61:1662, (1988). [38] J. Z. Sun, K. Char, M. R. Hahn, T. H. Geballe, and A. Kapitulnik. Magnetic flux flow and its influence on transport properties of the high T c oxide superconductors. Applied Physics Letters, 54:663, (1989). [39] P. L. Gammel, L. F. Schneemayer, J. V. Waszczak, and D. J. Bishop. 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A p p e n d i x A M o m e n t s o f T h e o r e t i c a l L i n e s h a p e s A . l M o m e n t s o f t h e I s o t r o p i c L i n e s h a p e The nth moment of the distribution of local fields B is given by B S{B-BfW{B)dB JW(B)dB where W(B) is the weighting factor for the field B, thus defining the magnetic field distribution. In the case of a vortex lattice, we need to take the spatial average over 7^ * (^x, y) B = f[B(x,y)-B]ndxdy  n $ dx dy ' where we only need to integrate over a single unit cell of the lattice. The field distri-bution of a type II superconductor is given by the following Fourier transform K —* — where K is the reciprocal lattice to the vortex lattice. Note that the average field B is equal to the term in the sum corresponding to K — 0. The nth moment of the magnetic field is now E p e x p ( t A ' - r ) dx dy f dx dy Only the exponential term exp(z(Jif1 + K2 H h Kn) • f) has an f = [x, y] dependence. When it is averaged over the unit cell, it will be be zero unless K\ + I<2 + I<3 + • • • + Kn = 0. 87 APPENDIX A. MOMENTS OF THEORETICAL LINESHAPES 88 This gives J = 1 A'j^O 3 K = — / - » \ >?• with the restriction that Y%=i Kj = 0. To simplify the notation, I define * 2TTA V 2 <j)0) 2TT\ so that the reciprocal lattice vectors can be expressed in dimensionless form where Q is a linear combination — • ^ A. Cj = ra(a + rb(a, ra and rj are integers and G = [x - (1/V3)y] and & = [2/x/3]y for a triangular vortex lattice. If A ^ > L(Q <C 1), the obvious next step is to neglect the one in the denominator of 1/(1 + \ 2 K 2 ) . However, we can expand the denominator and later cut off the series where we need to, according to the accuracy we desire. Expanding, 1 - (QIQ2 1 i + \*K] (Q/C)2 + i OO = (_l)mQ2(m+l)^-2(m+l)_ ^ ^ m=0 This expansion is fine as long as K2 ^ 0 and Q < 1. The condition J^ i + K2 = 0 simplifies the case n — 2: B M 2 " 0 ^ 1 B 2 E -, v_ + A2 A' 2) 2 = B2J2(Q/C)\l + (Q/02)-2 00 = 52x:(Q/c)4v2(-ir(m+i)(Q/o CO = B2Q4 ^ ( - l ) m ( m + l)Q2mCm (A.2) m=0 2m APPENDIX A. MOMENTS OF THEORETICAL LINESHAPES 89 where I have defined the constant C„ E £2(m+2) ' (A.3) Since Cm is on the order of one (with the accuracy depending on how many reciprocal lattice vectors you go out to) and Q2(m+1) goes to zero very quickly, I need only calculate the first few m terms to get a good approximation of the nth moment. The term m = 0 corresponds to the field-independent approximation of neglecting the one in the denominator. Since Cm is a constant, once it has been satisfactorily calculated, the value of Q (Q oc l/(\2B) ) is all that is required to calculate the second moment. By using the appopriate calculation of Cm, this expression for the moment can be used for a different kind of flux lattice (triangular, square, hexagonal). I have calculated the value Co = 4.33306 for a triangular lattice by summing over a 17 x 17 array of reciprocal lattice vectors. For arbitrary n, expanding the denominator gives B Mn = B N \ Y J £ ( - l ) m Q 2 ( m + 1 ) C " 2 ( m + 1 ) m=0 Collecting terms, I get the following: J M N = BnQ2n n E Q2m3{-l)m> £ j=l mj=0 2(l+mj) or Mn = BnQ2n £ ( Q 2 E - i m 0 ( - l E - i m j ) C m i , my=0 where C m u m 2 i . . . i m n is defined by B a ™1 , " l 2 r " > m n 1 \ EE---E >2(l+mi) / -2 ( l+m2) ^ 2 ( l + m „ ) <i C2 Cn SI S2 S" \T,i^=° - where O#0Vj In the case of n = 3, CO OO CO J M 3 = B 3 Q 6 E E E g 2 ( m i + m 2 + m 3 ) ( - i ) m i mi =0 m,2 =0 rri3 =0 + m 2 + m 3 ^ m i ,m2,ii (A.4) APPENDIX A. MOMENTS OF THEORETICAL LINESHAPES 90 where C m i , m 2 , m 3 is defined by C, mi,m2,m3 \ ' >2(l+m2) /.2(l+m3) S>2 S3 zlJ zlJ zl^ ..^  Cl C~2 <3 ^ \Ci + C*2+C*3 =0 , where Cy # 0 Vj / I have calculated the value Co,o,o = 4140.208 for a triangular lattice by summing over a 17 x 17 array of recipricol lattice vectors. Thus, one can calculate the third moment using Q and the constants C m i , m 2 , m 3 , with the constants only calculated once for the particular symmetry of the flux tube array. A . 2 T h e S e c o n d M o m e n t o f t h e K o s s l e r L i n e s h a p e The calculation for the isotropic case assumed that the local field was exactly parallel to the applied field. If this is not the case, the muon's polarization is given by U2 U2 i 1,2 W = ^ + ^ « » ( 7 , « ) . If we neglect the non-oscillating term, the nth moment's integrand is modified by the weighting (h2 + h2z)/(h2). This makes for a truly horrific integral. If we neglect any deviation of hz from h, we can calculate the second moment easily. Doing so, we must use only the lowest order term. The Kossler model's local field is definitely not parallel to the applied field, so the assumption for convience that they are ensures that the following calculation of the second momnet is only approximate. The total field is which makes calculating the second moment is easy, but higher moments are difficult. The second moment can be written in two ways: B M _ f[B(x,y)-B)2dxdy f dx dy f[B2(x,y)-B2}dxdy f dx dy The second form is very convenient: if B± forms a flux lattice and B\\ goes though unchanged, then B \ x , y ) - B 2 = B\ + B \ - B \ - B \ = F2(f, B cos 6) - (B cos B)2 + (B sin 6)2 - (B sin 6)2 = F 2 (r,£cos0)-(5cos0) 2 . (A.5) APPENDIX A. MOMENTS OF THEORETICAL LINESHAPES 91 (Remember that the applied field is equal to the average local field: B — B.) Therefore, in the Kossler model, the second moment for a single crystal at a par-ticular angle 0 with respect to the applied field is given by fico = J [E^Bcosg(exp(«^ • r))/(l + A2I\V)]2 dxdy J dxdy ' where Kg is determined by B cos 0. Making the appropriate substitution and expanding the denominator (which is rather suspect for cos0 ~ 0), oo Q2(m+2) * ~ W M a = (5cos*)2 E ( - l ) m (™ + ^ ( e o s ^ ^ where Cm and Q are the same as in the isotropic case. Averaging over the angle 9, M T/2 B M2 = T BM2{0)d{-cos 0), Jo we find that we need to evaluate integrals of the form fir/2 rl T (cos 0)~md(-cos 0) = I x-mdx Jo Jo f l o g ( x ) | o if m = 1 -m+1 1 if m > 1. o Each of the terms blows up (log(0) and 0 - m + 1 ) , but f rtunately the signs alternate, so I assume that they cancel. It's nice that log(l) = 0, so that B2 dependence of the second moment is not present. The second moment is then BM2 = B2 \Q2CO + £ ( - i r + 1 ^ ± ^ g 2 ( m + 2 ) C m } . (A.6) I m=2  M ~ L J Discarding the higher terms, the second moment is approximately BM2 « B2Q2C0. This is the same second moment (to lowest order) as for an isotropic superconductor with the same penetration depth in the same applied field! Due to the rough approxi-mations, this result is independent of applied field and proportional to A - 4 . Unfortunately, in order to calculate the higher moments, one has to use (B2±(r,0) + Bf{(r,0)y/2 - BT dxdy I Bcos6M = j dx dy The difficulty of calculating the higher moments is not a practical concern, as they are not crucial to the transformation algorthm (discussed in Chapter 6) which is used to fit experimental data with this model. A p p e n d i x B C o m p u t e r P r o g r a m s B . l T h e R e d f i e l d L i n e s h a p e The following is the FORTRAN program ABRIKO.T.FOR and its sample input file IAC_ 14KG_1000A_R0 .DAT, referred to in the code as F0R004.DAT, which gives the penetration depth and the applied field. The output files F0R007.DAT = ATC_14KG_1000A_R0.LOG and F0R008.DAT = ATC_14KG_1000A_R0.DAT are the log file, containing essentially the same information as the input file, and the data file, containing the Redfield lineshape A(f). I have not included samples of the output files. Note that the input fields' format is generalized for several programs; since this program is for an isotropic superconduc-tor, some input variables such as NTHETA and RTHETA are not used. B . l . l Sample Input file IAC_14KG_1000A_R0.DAT RNUM(XY POINTS), STEP, START, FINISH (RNUM=N**2/2. N=1+(FINISH-START)/STEP) 20000.0, .0025, .00125, .49875 DELTA, SHIFT, IFR (MAX 400) ; (SHIFT +DELTA*IFQ)*TRUE_F = MAX FIELD RECORDED. 0. 00025, .95, 400 NMAX (RECIPROCAL LATTICE VECTORS), LAMBDA (A), TRUE_F (G), CONST, RATIO 45.0, 1000.0, 14350.0, 246.39, 0.0 NTHETA (100 MAX), RTHETA (=NTHETA), QC0NV (0 FOR NO, 1 FOR YES), C0NV (MHZ) 1, 1.0, 0, .0 VARIATION IN LAMBDA, NUMBER OF STEPS TAKEN IN LAMBDA, var in B, # of Std devs 0.0, 0, 0.0, 3 B.l.2 Program ABRJ.K0_T.F0R C Frequency line shape for an isotropic superconductor, using the c Abrikosov triangular lattice of flux tubes. C This version uses a triangular lattice of flux tubes, and hence c K-space is triangular. Space is scanned in a rectangular fashion. DIMENSION P(400) 92 APPENDIX B. COMPUTER PROGRAMS DIMENSION PP(400) REAL QQ REAL NMAX, Kl, K2, CX, CY REAL RNTHET, RNUM, LAMBDA REAL CONV, CONV.SQ, RTHETA, SLOPE INTEGER QCONV, LIMIT OPEN (4, STATUS = 'OLD', SHARED, READONLY) READ (4, *) READ (4, *) RNUM, STEP, START, FINISH READ (4, *) READ (4, *) DELTA, SHIFT, IFR READ (4, *) READ (4, *) NMAX, LAMBDA, TRUE.F, CONST, RATIO READ (4, *) READ (4, *) NTHETA, RNTHET, qCONV, CONV C For efficiency, we only need to sample 1/2 of the unit triangle C in real space. PI=ACOS(-1.0) SLOPE = .5*SQRT(3.0) C {UNITS GMU:MHz/g LAMBDA:Angstrons TRUE_F:Gauss FLUXq:Gauss-Angstron**2 > GMU=0.01355 C FLUXq = 0.207E10 C CONST- ((2.0*PI*LAMBDA)**2)*TRUE_F/FLUXq C COHL = 34.0 C CORE = COHL/SqRT(FLUXq/TRUE_F) C IBAD = 0 C MAXBAD = 20 C CORE = .118158 CONV.Sq = (C0NV/(TRUE_F*GMU*DELTA))**2 WRITE (7,*) 'Number of XY points sampled is ', RNUM WRITE (7,*) 'IFR = ',IFR, ' DELTA = ', DELTA, 'SHIFT of ', SHIFT WRITE (7,*) 'Gives range and min freq (MHz) of ', DELTA*IFR*GMU*TRUE_F, t SHIFT*TRUE_F*GMU WRITE (7,*) 'Linewidth and delta in MHz ', CONV, DELTA*TRUE_F*GMU WRITE (7,*) 'Number of angles(IGNORED), reciprocal vectors sampled = ', k NTHETA,NMAX WRITE (7,*) 'Lambda i s ' , LAMBDA, 'Ratio (IGNORED)'.RATIO WRITE (7,*) 'Applied f i e l d (G) is ', TRUE_F, '. CONST is ', CONST DO 66 K=1,IFR P(K)=0.0 PP(K) = 0.0 66 CONTINUE C Note: Do loops with real valued increments can scew up on the fi n a l APPENDIX B. COMPUTER PROGRAMS c value since #'s are not exact. The [+ start] is to take care of this. DO 100 Y = START, (FINISH + START*1.1)*SL0PE, STEP DO 100 X = START + STEP*JINT(Y/(SLOPE*STEP)), (FINISH + START*1.1), STEP C PRINT *, 'X, Y = ', X, Y C Warning! This wil l double count the line x = .5 i f the # of steps is odd QQ = 0.0 DO 60 Kl = -1.0*NMAX, (NMAX +.5), 1.0 DO 60 K2 = -1.0*NMAX, (NMAX +.5), 1.0 CX = Kl CY = (2.0*K2 - K1)/SQRT(3.0) FF=1.0/(1.0+CONST*(CX**2.0+CY**2.0)) qQ=QQ+FF*C0S(2.0*PI*(CX*X+CY*Y)) 60 CONTINUE M=INT((qq- SHIFT)/DELTA)+1 53 CONTINUE IF ((M.LT.IFR).AND.(M.GT.O)) THEN P(M)=P(M) + 1.0 ENDIF 100 CONTINUE C PRINT *, 'qCONV ', qCONV IF (qCONV.Eq.l) THEN DO 44 K = 1, IFR DO 44 J = K - LIMIT, K + LIMIT IF ((J.LT.1).OR.(J.GT.IFR)) GOTO 42 PP(K) = PP(K) + P(J)*CONV_Sq/(CONV_SQ +(K-J)**2.0) 42 CONTINUE 44 CONTINUE c Normalise the output. AMPMAX =0.0 DO 45 K = 1, IFR IF (AMPMAX.LT.PP(K)) THEN AMPMAX = PP(K) ENDIF 45 CONTINUE DO 46 K = 1, IFR P(K) = PP(K)/AMPMAX 46 CONTINUE PRINT *, 'Output was convoluted.' ELSE c Normalise the output. AMPMAX =0.0 DO 47 K = 1, IFR IF (AMPMAX.LT.P(K)) THEN AMPMAX = P(K) ENDIF 47 CONTINUE DO 49 K = 1, IFR P(K) = P(K)/AMPMAX APPENDIX B. COMPUTER PROGRAMS 95 49 CONTINUE PRINT *, 'Output was NOT convoluted.' END IF DO 50 K = 1, IFR FREQ= SHIFT + DELTA*(K-l) WRITE(8,*) FREQ*TRUE_F*GMU, P(K) 50 CONTINUE 110 CONTINUE STOP END B . 2 T h e K o g a n M o d e l The Kogan model is essentially a tensor extention to the isotropic Lodon theory. The Abrikosov lattice and hence the Redfield lineshape are generalized. However, in order to simplify the programming, I used a rectangular vortex lattice. As I felt that the results of the simulation were disappointing and that the behavior of the lattice's unit cell was incompatible with flux pinning at twinning planes and grain boundaries, I did not bother to make a triangular version. The following is the FORTRAN program KOGAN_5.FOR and its sample input file I_25KG_3435A_R100.DAT, referred to in the code as F0R004.DAT, which gives the pen-etration depth and the applied field. The output files F0R007.DAT = K5_25KG_3435A_ R100.L0G and F0R008.DAT = K5_25KG_3435A_R100.DAT are the log file, containing es-sentially the same information as the input file, and the data file, containing the aniso-tropic Redfield lineshape A(f). I have not bothered to include samples of the output files. B.2.1 Sample Input file I_25KG_3435A_R100.dat RNUM(XY POINTS), STEP, START, FINISH ("1/2 OF UNIT CELL) 1250.0, .01, .005, .495 DELTA, SHIFT, IFR (MAX 400) 0.0012, .885, 200 NMAX (RECIPROCAL LATTICE VECTORS), LAMBDA (A), TRUE_F (G), CONST, RATIO 10.0, 3434.7757, 24926.0, 5608.397081, 100.0 NTHETA (100 MAX), RTHETA (=NTHETA), qCONV (0 FOR NO, 1 FOR YES), CONV (MHZ) 50, 50.0, 0, .2 ! Lambda soft = 740A (= 1/2 isotropic calc.) Lambda hard = 7400A. B.2.2 Program K0GAN5 . FOR C Based on Kogan's treatment of an anisotropic layered superconductor C with lambda_l = lambda_2 » lambda_3. Only really valid near Hcl, where APPENDIX B. COMPUTER PROGRAMS C we can consider a single vortex. DIMENSION P(400) DIMENSION PP(400) C SIZE OF M (mass tensor*C0NST) ARRAYS IS NTHETA REAL MXX(IOO) REAL MXZ(IOO) REAL MZZ(IOO) REAL F(100) C SIZE OF QQ ARRAY IS 3 - X.Y.Z AXIIS REAL QQ(3) REAL NMAX, Ml, M3, Kl, K2 REAL RNTHET, RNUM, LAMBDA REAL CONV, C0NV_Sq, RTHETA, FIX INTEGER qCONV, LIMIT OPEN (4, STATUS = 'OLD', SHARED, READONLY) READ (4, *) READ (4, *) RNUM, STEP, START, FINISH READ (4, *) READ (4, *) DELTA, SHIFT, IFR READ (4, *) READ (4, *) NMAX, LAMBDA, TRUE_F, CONST, RATIO READ (4, *) READ (4, *) NTHETA, RNTHET, qCONV, CONV C For efficiency, we only need to sample l/8th of the unit square C in real space. For lOOOO points sampled in the unit square, we C a step size of .01 for x and y. We will only sample 1/4 * 10000 points. PI=AC0S(-1.0) C {UNITS GMU:MHz/g LAMBDA:Angstrons TRUE_F:Gauss FLUXq:Gauss-Angstron**2 > GMU=0.01355 C FLUXq = 0.207E10 C C0NST= ((2.0*PI*LAMBDA)**2)*TRUE_F/FLUXq C COHL = 34.0 C CORE = COHL/SqRT(FLUXq/TRUE_F) C CORE = .118158 C THE MASS TENSOR M1*M1*M3 = 1. WE HAVE LAMBDA**2 * Ml = LAMBDA_S0FT**2. C WE HAVE LAMBDA**2 * M3 = LAMBDA_HARD**2. I ASSUME THAT LAMBDA_S0FT C IS FIVE (OR RATIO) TIMES LARGER THAN LAMBDA-HARD. C -> Ml = 1/CUBER00T(RATI0**2). M3 = RATI0**2 * Ml. C I'm just throwing in some constants like lambda**2 and a (2*pi/L)**2, c which is from normalising the K's to integer multiples, for ease. Ml = (RATI0**(-2.0/3.0))*C0NST M3 = (RATI0**(4.0/3.0))*C0NST C0NV_Sq = (C0NV/(TRUE_F*GMU*DELTA))**2 APPENDIX B. COMPUTER PROGRAMS WRITE (7,*) 'Number of XY points sampled is ', RNUM WRITE (7,*) 'IFR = ',IFR, ' DELTA = ', DELTA, 'SHIFT of ', SHIFT WRITE (7,*) 'Gives range and min freq (MHz) of ', DELTA*IFR*GMU*TRUE_F, t SHIFT*TRUE_F*GMU WRITE (7,*) 'Linewidth and delta in MHz ', CONV, DELTA*TRUE_F*GMU WRITE (7,*) 'Number of angles, reciprocal vectors sampled = '.NTHETA,NMAX WRITE (7,*) 'Lambda average i s ' , LAMBDA, 'Ratio of '.RATIO WRITE (7,*) 'LI = ',LAMBDA*SQRT(M1/C0NST),'. L3 = ',LAMBDA*SQRT(M3/C0NST) WRITE (7,*) 'Applied f i e l d (G) is ', TRUE_F, '. CONST is ', CONST DO 4 ITHETA = 1, NTHETA THETA = ITHETA*PI/(2.0*RNTHET) MXX(ITHETA) = (M1*(C0S(THETA)**2) + M3*(SIN(THETA)**2)) MZZ(ITHETA) = (M3*(C0S(THETA)**2) + M1*(SIN(THETA)**2)) MXZ(ITHETA) = (Ml - M3)*SIN(THETA)*C0S(THETA) C To take into account that the unit c e l l turns into an rectangle c as theta changes from 0, I use f(theta). This also affects the c reciprocal lattice vectors K. NOTE: MYY = Ml. F(ITHETA) = (MXX(ITHETA)/M1)**0.25 4 CONTINUE DO 66 K=1,IFR P(K)=0.0 PP(K) = 0.0 66 CONTINUE C THETOT = 0 . 0 c sq_q = 0 . 0 C Note: Do loops with real valued increments can scew up on the fi n a l c value since #'s are not exact. The [+ start] is to take care of this. DO 100 RX = START, (FINISH + START), STEP CUTOFF = RX - 0.5+STEP DO 100 RY = START, (RX + START), STEP DO 1 ITHETA = 1, NTHETA C To advoid double counting the diagonal boundry. IF (RY.GT.CUTOFF) THEN FIX = 0.5 ELSE FIX = 1.0 END IF X = RX/F(ITHETA) Y = RY*F(ITHETA) C PRINT*, 'Xmax,Ymax,X,Y= ', 0.5*F(ITHETA), 0.5/(F(ITHETA)**2) , X, Y THETA = ITHETA*PI/(2.0*RNTHET) C Sum over reciprocal vectors. First, Kl = K2 = 0.0 QQ(1) = 0.0 q q(2) = o.o qq(3) = l . o c Kl = 0, K2 > 0 (use Fl) and Kl > 0, K2 = 0 (use F 2 ) APPENDIX B. COMPUTER PROGRAMS DO 62 K = 1.0, (NMAX +0.5), 1.0 C I = K*F(ITHETA) C2 = K/F(ITHETA) Fl =(1.0 + MXX(ITHETA)*(C2**2))*(1.0+MZZ(ITHETA)*(C2**2)) Fl = Fl - (MXZ(ITHETA)**2) * (C2**4) F2 =(1.0 + M1*(C1**2))*(1.0+MZZ(ITHETA)*(C1**2)) C Case Kl = 0, K2 > 0 qq(i) = qq(i) + (MXZ(ITHETA)*(C2**2))*2.O*COS(2.O*PI*(C2*Y))/FI c qq(2) = qq(2) - o qq(3) = qq(3) + (1.0 +MZZ(ITHETA)*(C2**2))*2.0*C0S(2.0*PI*(C2*Y))/F1 c Case Kl > 0, K2 = 0 c qq(i) = qq(i) + o c qq(2) = qq(2) - o qq(3) = qq(3) + ( 1 . 0 +MZZ(ITHETA)*(CI**2))*2.O*COS(2.O*PI*(CI*X))/F2 62 CONTINUE DO 60 Kl = 1.0, (NMAX + 0.5), 1.0 DO 60 K2 = 1.0, (NMAX + 0.5), 1.0 C I = K1*F(ITHETA) C2 = K2/F(ITHETA) C_Sq = Cl**2 + C2**2 FF =(1.0 + M1*(C1**2) + MXX(ITHETA)*(C2**2))*(1.0+MZZ(ITHETA)*C_Sq) FF = FF - (MXZ(ITHETA)**2) * C_Sq * (C2**2) C {FOR C I , C2 VECTORS EITHER BOTH POSITIVE OR NEGATIVE } qq(i) = qq(i) + (MXZ(ITHETA)*(C2**2))*2.O*COS(2.O*PI*(CI*X+C2*Y))/FF qq(2) = qq(2) - (MXZ(ITHETA)*CI*C2)*2.O*COS(2.O*PI*(CI*X+C2*Y))/FF qq(3) = qq(3) + (1.0 +MZZ(ITHETA)*C_Sq)*2.0*C0S(2.0*PI*(Cl*X+C2*Y))/FF C {FOR C I VECTOR POSITIVE AND C2 NEGATIVE, OR VICE VERSA > qq(i) = qq(D + (MXZ(ITHETA)*(C2**2))*2.O*COS(2.O*PI*(CI*X-C2*Y))/FF qq(2) = qq(2) - (MXZ(ITHETA)*CI*C2)*2.O*COS(2.O*PI*(CI*X-C2*Y))/FF qq(3) = qq(3) + (1.0 +MZZ(ITHETA)*C_Sq)*2.0*C0S(2.0*PI*(Cl*X-C2*Y))/FF c PRINT *, 'Ci, C2, qi-3= ', c i , C2, qq (D, qq(2), qq(3) 60 CONTINUE c sq_q = sq_q + (qq(i)**2 + qq(2)**2 + qq(3)**2 - 1.0) ABSI = sqRT(qq(i)**2 + qq(2)**2 + qq(3)**2) M=INT((ABS1 - SHIFT)/DELTA)+1 IF ((M.GT.IFR).OR.(M.LT.l)) GOTO 55 C Since the (fields/true_f) are such that qq(l) does not equal qq(2), c we need to do a complete shperical average, c However, integration over psi is t r i v i a l . C The muon will precess about the f i e l d true_f*ABSl ABS2 = qq(3)**2 +.5*(qq(i)**2) +.5*(qq(2)**2) AMPL = ABS2/(ABS1**2) IF (ITHETA.LT.NTHETA) GOTO 53 AMPL = AMPL*.5 53 CONTINUE P(M)=P(M) + AMPL*FIX*SIN(THETA) 55 CONTINUE 1 CONTINUE APPENDIX B. COMPUTER PROGRAMS 100 CONTINUE C WRITE (7,*) 'RMS DEV. FOR THETA = \THETA,' IS ',SQRT(SQ_Q/RNUM) IF (ITHETA.LT.NTHETA) THEN RMSF = RMSF + (SQRT(SQ_Q/RNUM)*SIN(THETA)) THETOT = THETOT + SIN(THETA) ELSE RMSF = RMSF + (SQRT(SQ_q/RNUM)*SIN(THETA))*.5 THETOT = THETOT + SIN(THETA)*.5 END IF C WRITE (7,*) 'Total calculated rms deviation is ', RMSF/THETOT C WRITE (7,*) 'total calculated rms Ireq (MHz) is ', RMSF*TRUE_F*GMU/THETOT PRINT *, 'QCONV ', qCONV IF (QCONV.EQ.1) THEN DO 44 K = 1, IFR DO 44 J = K - LIMIT, K + LIMIT IF ((J.LT.l).OR.(J.GT.IFR)) GOTO 42 PP(K) = PP(K) + P(J)*CONV_SQ/(CONV_SQ +(K-J)**2.0) 42 CONTINUE 44 CONTINUE c Normalise the output. AMPMAX =0.0 DO 45 K = 1, IFR IF (AMPMAX.LT.PP(K)) THEN AMPMAX = PP(K) ENDIF 45 CONTINUE DO 46 K = 1, IFR P(K) = PP(K)/AMPMAX 46 CONTINUE PRINT *, 'Output was convoluted.' ELSE c Normalise the output. AMPMAX =0.0 DO 47 K = 1, IFR IF (AMPMAX.LT.P(K)) THEN AMPMAX = P(K) ENDIF 47 CONTINUE DO 49 K = 1, IFR P(K) = P(K)/AMPMAX 49 CONTINUE PRINT *, 'Output was NOT convoluted.' ENDIF DO 50 K = 1, IFR FREQ= SHIFT + DELTA*(K-l) WRITE(8,*) FREQ*TRUE_F*GMU, P(K) 50 CONTINUE APPENDIX B. COMPUTER PROGRAMS 100 110 CONTINUE STOP END B . 3 The Kossler Model The following is the FORTRAN program JACK_TRIANGLE3.FOR and its sample input file IC3_5KG_1000A_R0.DAT, referred to in the code as F0R004.DAT, which gives the penetration depth and the applied field. The output files F0R007.DAT = J T C 3 _ 5 K G _ 1000A_R0.LOG and F0R008.DAT = JTC3_5KG_1000A_R0.DAT are the log file, containing essentially the same information as the input file plus some calculations of the second moment, and the data file, containing the anisotropic Redfield lineshape A(f). B.3.1 Input file IC3_15kg_1000a_R0.DAT RNUM(XY POINTS), STEP, START, FINISH (RNUM=N**2/2. N=1+(FINISH-START)/STEP) 1250.0, .01, .005, .495 DELTA, SHIFT, IFR (MAX 400) ; (SHIFT +DELTA*IFQ)*TRUE_F = MAX FIELD RECORDED. 0.0002, .99, 150 NMAX (RECIPROCAL LATTICE VECTORS), LAMBDA (A), TRUE_F (G), CONST, RATIO 14.0, 1000.0, 14925.5, 0.0, 0.0 NTHETA (100 MAX), RTHETA (=NTHETA), qCONV (0 FOR NO, 1 FOR YES), CONV (MHZ) 40, 40.0, 0, .0 variation in Lambda, # of steps for lambda, var. in B, # of Std. Devs 0., 0, 0, 3 ! START_THETA, END_THETA (DELTA THETA = 90/NTHETA),ROCK_THETA, THETA_0 ! 0.0, 16.0, 9.0, 0.0 B.3.2 Program JACK_TRIANGLE3.FOR c A treatment of an anisotropic layered superconductor, where there is no c coupling between layers. The screening currents can only move in c the planes. The component of the applied f i e l d perpendicular to the c layers wil l form a flux lattice, while the component parallel w i l l c pass through undeflected. I assume a square lattice for ease of c programmming. Idea by Jack Kossler, i n i t i a l programming by Moreno c Celio. C This version uses a triangular lattice of flux tubes, and hence c K-space is triangular. Space is scanned in a rectangular fashion. DIMENSION P(400) DIMENSION PP(400) REAL qq, H, HZ, HT REAL NMAX, Kl, K2, CX, CY REAL RNTHET, RNUM, LAMBDA APPENDIX B. COMPUTER PROGRAMS REAL CONV, C0NV_SQ, RTHETA, SLOPE REAL LA, TF, FLUXQ INTEGER qCONV, LIMIT DOUBLE PRECISION FREQ, Ml, M2, SUM, R, PI, q REAL FLAG1, FLAG2, REL INTEGER K OPEN (4, STATUS = 'OLD', SHARED, READONLY) READ (4, *) READ (4, *) RNUM, STEP, START, FINISH READ (4, *) READ (4, *) DELTA, SHIFT, IFR READ (4, *) READ (4, *) NMAX, LAMBDA, TRUE_F, CONST, RATIO READ (4, *) READ (4, *) NTHETA, RNTHET, QCONV, CONV C For efficiency, we only need to sample 1/2 of the unit triangle C in real space. PI=AC0S(-1.0) SLOPE = .5*SQRT(3.0) C {UNITS GMU:MHz/g LAMBDA:Angstrons LA:LAMBDA x E-3 c TRUE_F:Gauss TF:TRUE_F x E-3 FLUXQ:Gauss-Angstrom**2 X E-9> GMU = 0.01355 FLUXQ =2.07 TF = TRUE_F * .001 LA = LAMBDA * .001 C CONST= .50*SQRT(3.0) * ((2.0*PI*LAMBDA)**2)*TRUE_F/FLUXq Triangle CONST = 2.0*SQRT(3.0) *((PI*LA)**2) * TF/FLUXQ C CORE = COHL/SQRT(FLUXQ/TRUE_F) Not used CONV.SQ = (C0NV/(TRUE_F*GMU*DELTA))**2 WRITE (7,*) 'Number of XY points sampled is ', RNUM WRITE (7,*) 'IFR = ',IFR, ' DELTA = ', DELTA, 'SHIFT of ', SHIFT WRITE (7,*) 'Gives range and min freq (MHz) of ', fc DELTA*IFR*GMU*TRUE_F, SHIFT*TRUE_F*GMU WRITE (7,*) 'Linewidth and delta in MHz ', CONV, DELTA*TRUE_F*GMU WRITE (7,*) 'Number of angles, reciprocal vectors sampled = '.NTHETA,NMAX WRITE (7,*) 'Lambda soft i s ' , LAMBDA, 'Ratio of '.RATIO WRITE (7,*) 'Applied f i e l d (G) is ', TRUE_F, '. CALC. CONST is ', CONST DO 66 K=1,IFR P(K)=0.0 PP(K) = 0.0 66 CONTINUE C Note: Do loops with real valued increments can scew up on the fi n a l c value since #'s are not exact. The [+ start] is to take care of this. APPENDIX B. COMPUTER PROGRAMS DO 100 Y = START, (FINISH + START*1.1)*SL0PE, STEP DO 100 X = START + STEP*JINT(Y/(SLOPE*STEP)), (FINISH + START+1.1), STEP C PRINT *, 'X, Y = ', X, Y C Warning! This w i l l double count the line x = .5 i f the # of steps is odd DO 1 ITHETA = 1, NTHETA THETA = ITHETA*PI/(2.0*NTHETA) C PRINT *, 'X, Y, THETA = ', X, Y, THETA FIELD = COS(THETA) C Calc'ing the flux lattice due to the component perp. to the layers QQ = 0.0 DO 60 Kl = -1.0*NMAX, (NMAX +.5), 1.0 DO 60 K2 = -1.0*NMAX, (NMAX +.5), 1.0 CX = Kl CY = (2.0*K2 - K1)/SQRT(3.0) FF=FIELD/(1.0+C0NST*FIELD*(CX**2.0+CY**2.0)) qq=QQ+FF*C0S(2.0*PI*(CX*X+CY*Y)) 60 CONTINUE HZ = (qq*C0S(THETA) + SIN(THETA)**2) HT = (qq*SIN(THETA) - SIN(THETA)*C0S(THETA)) H = SqRT(HZ**2.0 + HT**2.0) AMPL = ((.5*HT)**2.0 + HZ**2.0)/(H**2.0) M=INT((H- SHIFT)/DELTA)+1 IF (ITHETA.LT.NTHETA) GOTO S3 AMPL = AMPL*.5 53 CONTINUE C PRINT *, ' HZ, HT, H =', HZ, HT, H C PRINT *, ' AMPL, M = ', AMPL, M IF ((M.LT.IFR).AND.(M.GT.O)) THEN P(M)=P(M) + SIN(THETA)*AMPL ENDIF 1 CONTINUE 100 CONTINUE SUM = O.ODO Ml = O.ODO M2 = O.ODO FLAG1 =0.0 FLAG2 =0.0 C PRINT *, 'qCONV ', qCONV IF (qCONV.Eq.l) THEN DO 44 K = 1, IFR DO 44 J = K - LIMIT, K + LIMIT IF ((J.LT.l).OR.(J.GT.IFR)) GOTO 42 PP(K) = PP(K) + P(J)*CONV_Sq/(CONV_Sq +(K-J)**2.0) 42 CONTINUE 44 CONTINUE c Normalise the output. AMPMAX =0.0 DO 45 K = 1, IFR APPENDIX B. COMPUTER PROGRAMS 103 IF (AMPMAX.LT.PP(K)) THEN AMPMAX = PP(K) ENDIF 45 CONTINUE DO 46 K = 1, IFR P(K) = PP(K)/AMPMAX 46 CONTINUE PRINT *, 'Output was convoluted.' ELSE c Normalise the output. AMPMAX =0.0 DO 47 K = 1, IFR IF (AMPMAX.LT.P(K)) THEN AMPMAX = P(K) ENDIF 47 CONTINUE DO 49 K = 1, IFR P(K) = P(K)/AMPMAX 49 CONTINUE PRINT *, 'Output was NOT convoluted.' ENDIF DO 50 K = 1, IFR FREQ= (SHIFT + DELTA*(K-i)) * TRUE_F*GMU WRITE(8,*) FREq, P(K) SUM = SUM + P(K) Ml = Ml + FREQ*P(K) M2 = M2 + (FREq**2)*P(K) IF (FLAG1 .LE. .005) THEN IF (P(K) .GE. .5D0) THEN FLAG1 = FREq ENDIF ELSEIF (FLAG2 .LE. .005) THEN IF (P(K) .LE. .5D0) THEN FLAG2 = FREq ENDIF ENDIF 50 CONTINUE Ml = MI/SUM M2 = M2/SUM q = M2 - (Ml**2) REL = 2.0D0 * PI* DSqRT(q / 2.0D0) WRITE (7, *) 'Av. freq.=>, Ml, ' FWHM=', FLAG2 - FLAG1 WRITE (7, *) 'Gaussian r e l . rate=', REL STOP END APPENDIX B. COMPUTER PROGRAMS 104 B . 4 L i m i t a t i o n s o f C o m p u t e r S i m u l a t i o n s Basically, I calculated the magnetic field at regular intervals in the unit cell of the flux tube array and from that calculated amplitude of the muon polarization function (0.5/i p 2 + h2)/h2 in the case of the Kogan and Kossler models. The amplitudes were histogrammed so that they may be compared with the muon frequency distribution in actual data. Given a relatively fine bin size for the histogram, the choice of various cutoffs and the like made to reduce computation time affected the line shape different ways. Since the strength of the lineshape is proportional to the percentage of area in the unit cell within 6f of the frequency / = _?7^/(27r), the area nearest the center of the tube has the steepest change in magnetic field so it produces the weak high frequency tail in the spectrum. The magnetic field blows up like log(l/r) at the flux vortex in the London approach. I've cut this part of the spectrum off arbitrarily at the frequency determined by 7^-* (SHIFT + IFR*DELTA) because it's so weak that we wouldn't be able to see it in the Fourier transforms of our actual data. Besides, as all three simulations use a London approach, they ignore the fact that there is a core area in the flux tube in which the field is reaches some maximum value. It is reasonable to cut off the summation of the reciprocal lattice vectors when 1/|7\| ~ £ 4 2 0 ] where £ is the coherence length. Practially, it is sufficient to choose a cutoff such that the errors in the lineshape are no longer obvious to the eye. As both the number of points sampled in the unit cell and the number of reciprocal lattice vectors used appear quadratically in the algorithms (which I tried to make as efficient as possible without making the code obscure), computation time easily gets out of hand. Cutting off the number of reciprocal lattice vectors used in the summation (Eqs. 3.9, 4.8 and 5.4) has the effect of introducing little peaks into the spectrum. The larger the number used, the greater the number of peaks and the smaller they are. The peaks seem to be strongest near the saddle point cusp of the distribution in the case of the Redfield lineshape, and near the applied field cusp in the case of the Kossler model. For the Kossler model, a cutoff of 7 leaves unacceptable peaks. With 14, they were tolerable, except for the larger values of A (1000A- 2000A), which give a wider distribution. With 21 the results are very nice, but the calculation is rather time consuming. With the spatial averaging in the Kogan model, the cutoff error peaks do not accumulate in any particular portions of the spectra. One thing I did neglect was the effect of making the cutoff circular in K space. The error due a square cutoff gets smaller as Kcutoff 1 S increased. Of course, the greater the number of points considered within the unit cell, the smoother the histogram becomes, especially the high frequency tail. A coarse his-togram binning (a few hundred bins for the spectrum) is necessary to avoid seeing the distracting "ringing" near the cusp, which is due to using too small a bin size for the APPENDIX B. COMPUTER PROGRAMS 105 number of points sampled per unit cell and perhaps due to the cutoff of the reciprocal lattice vector. To simulate the sintered powder samples, I did a numerical average over the angle 9. Obviously, taking too few angles makes the spectrum rough. The smaller the penetration depth, the larger the frequency distribution and the greater the number of angles needed to get a smooth distribution from a collection of histograms with sharp peaks at cusp frequency and long high frequency tails. With the Kossler model, relatively few angles (40) were needed, as the spectrum are compact compared to the Kogan model, where I found it necessary to sample 60 angles for small penetration depths and 90 angles for penetration depths above 1000A. I did not bother to optimize the Kogan model's input values for smoothness of the frequency distribution, since the qualitative trends were quite obvious as the anisotropy increased. B . 5 F i t t i n g S u b r o u t i n e s Jess Brewer has written a generalised fitting program called XYFIT which allows cus-tomization for specific applications by modifying the function Y0FX, a routine which returns the theoretical value of an independent variable Y for a given value of the in-dependent variable and the current values of the various fitted parameters. I modifed the code such that the theory lineshape was initialized once by INIT-TH and the entire lineshape transformed to the fitting parameter's values by TRANS_TH before extensive calls to Y0FX. Since YOFX is called by the routine XYFCN, not included here, which cal-culates chi squared (x2) for the fitting routine, it is essential from an efficiency point of view to transform the lineshape after each change of any one of the fitting parameters rather than transforming every time YOFX is called. The subroutine INIT-TH uses a theory lineshape such as contained in ATC_14KG_ 1000A_R0.DAT or JTC3_5KG_1000A_R0.DAT, referred in the code as F0R008.DAT. The theory's inititialization file, containing information such as penetration depth, applied field, and binning, is referred in the code as F0R004. DAT. Examples of theory initializa-tion files are IC_14KG_1000A_R0.DAT and ITC3_5KG_1000A_R0.DAT, which were shown in previous sections of this appendix. B.5.1 Subroutine INIT_TH c Modified April 1989, -Tanya Riseman c Extended to allow ini t i a l i z a t i o n of theory line c and transformation of theory to parameters outside of YOFX c function calls so that efficiency of chi squared calculation is c improved. Therefore, added lots of extra parameters for passing c between INIT.TH, TRANS.TH, and YOFX. TRANS_TH is specific to APPENDIX B. COMPUTER PROGRAMS c f i t t i n g isotropic superconductor freq. distributions with c Abrikosov Redfield lineshape or powder-average extremely c anisotropic SC's with Kossler model's lineshape. Obviously, c modifications can be made to do angular averging over restricted c angle ranges given a formula for the penetration depth c involving angle, etc. cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx SUBROUTINE INIT_TH(NSTD_DEV, LAMBDA, FREQ_0, IFR, SHIFT, DELTA) CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX C i n i t i a l i z e the theoretical lineshape function c IMPLICIT NONE Ccc Passed parameters REAL P.TRANS, P_ORIG, FREQ COMMON /P0L_HIST/ P_TRANS(400), P_0RIG(400), FREq(400) INTEGER NSTD.DEV, IFR REAL LAMBDA, FREQ_0, SHIFT, DELTA Ccc input variables (most throw away) REAL RNUM, STEP, START, FINISH INTEGER NTHETA, QCONV REAL NMAX, TRUE_F, CONST, RATIO, RNTHET REAL CONV, CONV_SQ, VARYJL, VARY_B INTEGER NVARY.L ccc subroutine variables INTEGER K REAL GMU GMU=0.01355 ! BEGINNING of i n i t i a l i z e theoretical lineshape Initialize the lineshape in histogram P_0RIG(K) OPEN (4, STATUS = 'OLD', SHARED, READONLY) READ (4, *) READ (4, *) RNUM, STEP, START, FINISH READ (4, *) READ (4, *) DELTA, SHIFT, IFR READ (4, *) READ (4, *) NMAX, LAMBDA, TRUE_F, CONST, RATIO READ (4, *) READ (4, *) NTHETA, RNTHET, qCONV, CONV READ (4, *) READ (4, *) VARY_L, NVARY_L, VARY_B, NSTD_DEV C only use NSDTJDEV, LAMBDA, and TRUE_F from this f i l e C {UNITS GMU:MHz/g LAMBDA: Angstrons TRUE_F} FREQ_0 = TRUE_F * GMU DO 120 K = 1, IFR APPENDIX B. COMPUTER PROGRAMS READ (8, *) FREQ(K), P_ORIG(K) 120 CONTINUE ! end of i n i t i a l i z e theoretical lineshape RETURN END B.5.2 Subroutine TRANS_TH cxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx SUBROUTINE TRANS_TH(NSTD_DEV, LAMBDA, FREQ_0, & IFR, SHIFT, DELTA, N0RM_D, LAMBDA_D, F_D, FSIGMA_D) CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX c transform the theory lineshape so that i t has the applied f i e l d c F_D and the penetration depth LAMBDA_D. C C THIS section of code calculates the THEORETICAL c LINESHAPE by taking a lineshape in f i e l F0R008.dat c and varies the applied f i e l d via C averaging the average freq F_d by +- FSIGMA_D c by assuming a gaussian distribution of (applied fields *GMU = F_D) c about F_D, with the square of the second moment of FSIGMA_D. c C PROBABILITY = EXP(-.5*((freq - F_D)/FSIGMA_D)**2)) C = EXP(-.5*((F_D*N*DELTA)/FSIGMA_D)**2)) C IMPLICIT NONE C Common and passed variables REAL P.TRANS, P_0RIG, FREQ COMMON /POL.HIST/ P_TRANS(400), P_0RIG(400), FREq(400) INTEGER NSTD_DEV, IFR REAL LAMBDA, FREQ_0, SHIFT, DELTA REAL N0RM_D, LAMBDA_D, F_D, FSIGMA_D c subroutine variables REAL L.FRAC, F, AMPMAX, PROBABILITY, STUFF INTEGER K, K_NEW, N, K.DELTA REAL P_TEMP DIMENSION P_TEMP(400) c GMU=0.01355 C Initialize the lineshape in histogram P_TRANS(K) CCCCCCC Adjust lineshape from 1000A to LAMBDA.D via L_FRAC CCCCCCCC L_FRAC = LAMBDA.D/LAMBDA IF (LAMBDA_D .GT. LAMBDA) THEN C case of lineshape narrows: DO 31 K = 1, IFR ! zero the array (not efficient) APPENDIX B. COMPUTER PROGRAMS 108 P_TEMP(K) =0.0 31 CONTINUE DO 32 K = 1, IFR F = FREQ_0 + (FREQ(K) - FREQ_0)/((L_FRAC)**2.0) C from end of program: FREQ =TRUE_F*GMU*(SHIFT + DELTA*(K-l)) K_NEW = NINT(1.0 + ((F/FREq_0) -SHIFT)/DELTA) P_TEMP(K_NEW) = P_0RIG(K) P_TRANS(K_NEW) = P_TEMP(K_NEW) 32 CONTINUE ELSEIF (LAMBDA.D .LT. LAMBDA) THEN C case of lineshapes widens: DO 33 K = 1, IFR ! zero the array (not efficient) P_TEMP(K) =0.0 33 CONTINUE DO 34 K = 1, IFR F = FREQ_0 + (FREQ(K) - FREq_0)*((L_FRAC)**2.0) K_NEW = NINT(1.0 + ((F/FREq_0) - SHIFT)/DELTA) P_TEMP(K) = P_0RIG(K_NEW)*((1.0-L_FRAC)**2.0) P_TRANS(K) = P_TEMP(K) 34 CONTINUE ELSE ! (LAMBDA_D .Eq. LAMBDA) DO 36 K = 1, IFR P_TEMP(K) = P_0RIG(K) P_TRANS(K) = P_TEMP(K) 36 CONTINUE ENDIF CCCCCCC Adjust lineshape from 1000A to LAMBDA_D CCCCCCCC CCCCCCC Shift lineshape over in freq space via DIFF_F CCCCCCCC ccccccc But, as we want to preserve FREq(K), just remember CCCC CCCCCCC DIFF_F = FREq_0 - F_D CCCCCC CCCCCC Smear lineshape by averaging over a gausian distribution ccccc cccccc of applied fields. ccccc IF (FSIGMA_D .GT. 0) THEN ! vary FREq_0 (APPLIED FIELD) C from end of program: FREq =TRUE_F*GMU*(SHIFT + DELTA*(K-l)) K.DELTA = NINT(FSIGMA_D/(F_D*DELTA)) STUFF = -.5*((F_D*DELTA)/FSIGMA_D)**2 DO 130 N = 1, NSTD_DEV*K_DELTA PROBABILITY = EXP(STUFF*(N**2)) C PROBABILITY = EXP(-.5*(N/K_DELTA)**2) DO 132 K = 1, IFR IF (((K+N).LE.IFR).AND.((K-N).GE.1)) THEN P_TRANS(K) = P_TRANS(K) + P_TEMP(K+N)*PROBABILITY P_TRANS(K) = P_TRANS(K) + P_TEMP(K-N)*PROBABILITY ENDIF 132 CONTINUE 130 CONTINUE APPENDIX B. COMPUTER PROGRAMS 109 ENDIF ! vary FREQ_0 (APPLIED FIELD) c Normalise the output. AMPMAX =0.0 DO 47 K = 1, IFR IF (AMPMAX.LT.P_TRANS(K)) THEN AMPMAX = P_TRANS(K) ENDIF 47 CONTINUE STUFF = N0RM_D/ AMPMAX DO 49 K = 1, IFR P_TRANS(K) = P_TRANS(K)*STUFF 49 CONTINUE RETURN END B.5.3 Function YOFX CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX c FUNCTION YOFX(FIELD, X, NPAR, FREQ_0, IFR, SHIFT, DELTA) CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX c c f i l e F0R008.dat contains the theory lineshape (frequency vs. asymmetry) c f i l e F0R004.dat contains the input f i l e that made the theory lineshape, c as i t has the applied f i e l d TRUE_F and the penetration depth LAMBDA c and the number of standard deviations NSTD_DEV to calculate c the sum over fields over (3 or 4). c f i l e F0R002.dat is the log f i l e of the f i t . c C User also supplies FUNCTION YOFX (T.X.N ) which returns the C theoretical value of VD at VI=T given the N MINUIT Parameters X. FUNCTION YOFX(FIELD, X, NPAR, FREQ_0, IFR, SHIFT, DELTA) C Common and passed variables REAL P, PP, FREQ COMMON /POL.HIST/ P_TRANS(400), P_0RIG(400), FREq(400) INTEGER NPAR, IFR REAL FREq_0, SHIFT, DELTA REAL X DIMENSION X(l) C function variables INTEGER K_NEW REAL DIFF_F, YOFX C DIFF_D = FREQ_0 - F_D = FREQ_0 - X(3) C K_NEW=NINT(1.0+(((FIELD+DIFF_F)/FREQ_0) -SHIFT)/DELTA) K_NEW=NINT(1.0+(((FIELD + FREQ_0 - X(3))/FREq_0) -SHIFT)/DELTA) APPENDIX B. COMPUTER PROGRAMS 110 IF ((K_NEW .GT. IFR) .OR. (K_NEW .LT. 1)) THEN YOFX =0.0 ELSE YOFX = P_TRANS(K_NEW) ENDIF RETURN END B . 6 C a l c u l a t i o n o f C o n s t a n t s Co a n d Cooo The program CALC_SUM_M0M2 .FOR shown below calculates the constants Co and Cooo, which are defined in appendix A. B.6.1 Program CALC_SUM_M0M2.FOR C Some sums are calculated for comparison of Abrikosov theory with c with approx's to results of simulations or with data IMPLICIT NONE INTEGER Ml, Nl, M2, N2, M3, N3 INTEGER K_SQ1, K_SQ2, K_SQ3 INTEGER NMAX,SQ_NMAX REAL C_0, C_000, E_0, E_000 NMAX = 17 Sq_NMAX = NMAX**2 WRITE (7,*) 'MAX # OF REC. VECTORS is ', NMAX C_000 =0.0 C_0 = 0.0 DO 110 Ml = -1*NMAX, NMAX, 1 WRITE (7, *) 'M1=',M1,'C_0 =', C_0,' ++C_000 =', C_000 DO 110 Nl = -1+NMAX, NMAX, 1 K_SQ1 = Ml**2 + Nl**2 - M1*N1 IF (K_SQ1 .GT. 0) THEN E_0 = 1.0/(K_Sqi**2.0) C_0 = C_0 + E_0 DO 120 M2 = -1+NMAX, NMAX, 1 DO 120 N2 = -1*NMAX, NMAX, 1 K_Sq2 = M2**2 + N2**2 - M2*N2 IF (K_Sq2 .GT. 0) THEN DO 130 M3 = -1*NMAX, NMAX, 1 DO 130 N3 = -1+NMAX, NMAX, 1 K_Sq3 = M3**2 + N3**2 - M3*N3 IF (K_Sq3 .GT. 0) THEN E_000 = 1.0/(K_Sqi*K_SQ2*K_Sq3) APPENDIX B. COMPUTER PROGRAMS 111 C_000 = C_000 + E.000 ENDIF 130 CONTINUE ENDIF 120 CONTINUE ENDIF 110 CONTINUE WRITE (7, *)'.BRANDT B_000 =', C_000, 'with remander ='. E_000 WRITE (7, *)',BRANDT B_0 =', C_0, 'with remander =', E_0 WRITE (7, * ) ' . C_000 =', (27.0/64.0)*C_000, & 'with remander =', (27.0/64.0)*E_000 WRITE (7, * ) ' , C_0 =', (9.0/16.0)*C_0, 'with remander =', (9.0/16.0)*E_0 STOP END 

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