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Theoretical calculation of muon site in YBa₂Cu₃O Li, Qiang 1990

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T H E O R E T I C A L C A L C U L A T I O N O F M U O N S I T E IN YBa 2 Cu 3 Ojr By Qiang Li B. Sc. (Nuclear Physics)Peking University A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1990 © Qiang Li, 1990 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. Physics The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5 Date: / / Abstract The muon is a useful probe of magnetic fields in superconductors, but knowing the field seen by the muon is often of limited value until we know where the muon is in the crystal lattice. In this thesis I employ two independent theoretical methods to search for candidate muon sites: the potential energy field method, which seeks the minimum of the electrostatic potential of the and the magnetic dipolar field method, which compares the calculated magnetic field (due to the host electronic, atomic or nuclear dipolar fields) with the observed local fields at the muon. ii Table of Contents Abstract ii List of Tables v List of Figures vi acknowledgement viii 1 Introduction 1 1.1 History of Muon Site Determination 1 1.2 Space Group 3 1.3 Muon Spin Resonance/Rotation/Relaxation (uSTZ) 4 2 Methods of Muon Site Determination 7 2.1 Electrostatic Potentials 7 2.2 Dipolar Magnetic Fields 13 2.3 Conclusion 15 3 Explicit Calculations of Muon Sites In Y B a 2 C u 3 0 6 16 3.1 Potential Maps of A F M 16 3.2 Dipolar Magnetic Field Calculations 36 3.3 Comparison With Experimental Data 39 4 Calculations on Y B a 2 C u 3 0 7 43 4.1 Muon Site Decision Experiment 43 iii 4.2 Potential Map of Superconductor Y B a 2 C u 3 0 7 45 4.3 Dipolar Magnetic Field 46 4.4 Conclusion 47 Bibliography 50 A Programs 53 iv List of Tables 1.1 Transformations pmmm of Y B a 2 C u 3 0 7 4 1.2 Transformations pl23 of Y B a 2 C u 3 0 6 • 5 3.3 Properties and coordinates of ions 16 3.4 The magnetic field at z=0.03c 38 3.5 The magnetic field at z = 0.19c 38 3.6 Muon candidate site 39 4.7 The positions and properties of Y B a 2 C u 3 0 7 45 4.8 The nuclear magnetic moments of ions in YBa 2 Cu 3 0a; 47 v List of Figures 2.1 Potential Vi(r) vs R 10 3.2 Two dimensional potential map at Z=0.02c 19 3.3 Two dimensional potential map at Z=0.03c 20 3.4 Two dimensional potential map at Z=0.05c 21 3.5 Two dimensional potential map at Z=0.07c 22 3.6 Two dimensional potential map at Z=0.10c 23 3.7 Two dimensional potential map at Z=0.15c 24 3.8 Two dimensional potential map at Z=0.17c 25 3.9 Two dimensional potential map at Z=0.19c 26 3.10 Two dimensional potential map at Z=0.20c 27 3.11 Two dimensional potential map at Z=0.26c 28 3.12 Two dimensional potential map at Z=0.30c 29 3.13 Two dimensional potential map at Z=0.35c . 30 3.14 Two dimensional potential map at Z=0.38c 31 3.15 Two dimensional potential map at Z=0.40c 32 3.16 Two dimensional potential map at Z=0.42c 33 3.17 Two dimensional potential map at Z=0.45c 34 3.18 The contour potential map at Z=0.03c 35 3.19 The contour potential map of YBa 2 Cu 3 0 6 at Z=0.19c 37 3.20 Contour plot of magnetic field at 0.03c 40 3.21 Contour plot of magnetic field at 0.19c 41 vi 3.22 The crystal structure of antiferromagnetic YBa 2 Cu 3 06 42 4.23 The contour plot of the potential map at Z=0.19c 48 4.24 The crystal structure of superconducting YBa 2 Cu 3 07 49 vii acknowledgement It has been a pleasure to work in this group from which I have benefited and enjoyed so much. I would like to express my thanks to Rob Kiefl who has been not only helpful to my thesis work but also a good teacher for a broad area in /iSR. I would like to say that I have very much enjoyed working with Tanya Riseman, a helpful colleague and an interesting person to talk with. All I have achieved from my work should be credited to my supervisor Jess Brewer. I am grateful for what he has done, although I have a little trouble with his American accent. The data employed in this thesis were provided by several people; I very much appreciate those involved. viii Chapter 1 Introduction 1.1 History of Muon Site Determination The first /x+ site determination was performed in a single crystal of gypsum (CaS0 4 • 2H 20), in which the magnetic field due to the adjacent protons at fixed locations is well known. If the fi+ were mobile in the crystal, the dipolar fields of the protons would simply cause relaxation of the muon spin. However, if the muon replaces protons at fixed lattice sites, the Muon Spin Resonance (pSR) results should show exactly the same behavior as the proton NMR. The experiment showed the split-frequency muon precession expected for muons occupying proton sites, and thus gave the first positive identification of the /x+ site in a crystal. [1] During the following years the fiSH technique has been applied to metals, oxides, magnetic materials and superconductors to explore the internal magnetic and electronic structure. But without knowing the exact locations of the fx+ in the samples the information often has limited value. The local magnetic field at a stopped fi+ site was measured in a single crystal of ferromagnetic cobalt as a function of temperature between 4 and HOOK by Graf. [2] The data are consistent with a relatively smooth temperature dependence of the hyperfine field only if it is assumed that the fi+ is at an octahedral interstitial site; the calculated local field at the tetrahedral site would imply a discontinuous hyperfine field. These facts lead to the conclusion that octahedral interstitial sites are preferred 1 Chapter 1. Introduction 2 by the p+. The longlived metastable states of the u+ in the magnetic oxide a—Fe203 have also been studied. [3] Three separate frequencies were seen. This indicates that the muons are localized at three different local energy minima in the unit cell. Other muon site determinations have been performed on Cu, [5] Fe, [6] [7] and Si. [8] Lattice distortion plays a key role in determining the muon site in Alkali fluorides and semiconductors. One of the most interesting magnetic phenomena in a superconductor is the well known Meissner effect. When a type I superconductor is cooled in a magnetic field below its transition temperature, the magnetic flux is abruptly expelled, except for a thin surface layer where the field decays roughly exponentially on the scale of the London penetration depth Aj,. In a type II superconductor the magnetic flux penetrates into the interior of the sample in an array of "vortices" each carrying the flux quantum —• —* ip0 when the magnetic field is between Hc\ and Hc2. As a result, the internal magnetic field in the superconductor is inhomogeneous. The pSTZ technique can measure the mean magnetic field at the muon site (through the p+ precession frequency) and its dispersion (through the p+ relaxation rate). Experiments on superconducting GdBa2Cu307 were performed in the temperature range from 30mK to 130K by A. Golnik. [9] In zero external field two different frequen-cies were distinguished, which correspond to internal magnetic fields of 33 and 52 mT, respectively. This shows that muons find two magnetically inequivalent stopping sites in the sample. Two types of muon stopping sites are in principle possible: interstitial sites and oxygen vacancy sites. So far there is no solid evidence to show the p+ site in YBa 2Cu 307. Our own TRIUMF [46] group predicts that p+ stops primarily in a site close to the Cu0 2 plane. The PSI [48] and Tokyo [47] groups claim that the p+ stopping site is close to an oxygen Chapter 1. Introduction 3 on the CuO chain. The only point of agreement is that the muon site is l A away from some oxygen ion. So far no rigorous theoretical prediction has been made, due to the difficulties involved in evaluating electrostatic potentials and magnetic fields inside the supercon-ductors. The current theoretical method is to apply the potential equation which best generates the fi+ sites in YBa2Cu306 to the case of YBa 2Cu 307. During the round-table discussion at the Vth International Conference on pSTl in Oxford (1990), it became clear that one reason for the differences of opinion about the muon site in Y B a 2 C u 3 0 6 is the uncertainty regarding the orientation of Cu magnetic moments which should lie in the Cu-0 plane oriented along an a or b axis. The initial approach taken to locating the muon site focussed on the antiferromag-netic (AFM) material YBa2Cu306, which has a magnetic structure consisting of strong nearest-neighbor correlated spins in adjacent copper planes which are aligned antifer-romagnetically. This special case offers some hope of calculating the internal magnetic field at potential muon sites theoretically. The simulated results can be verified by comparing them with the experimental data. The first study of local fields at muon sites in antiferromagnetic YBa2Cu306+r was done by Nishida. [14] The experimental data show that there are at least two independent fj,+ sites. 1.2 Space Group A perfect crystal is constructed by an infinite regular repetition in space of identical structural units or building blocks. A space group, which consists of a Bravis lattice and a basis, displays all the symmetries of a crystal. Through the symmetry of the space group, one point generates several equivalent points at symmetric positions in a Chapter 1. Introduction 4 X Y Z -X -Y -Z -X -Y Z X Y -Z X -Y -Z -X Y Z -X Y -Z X -Y Z. Table 1.1: Transformations pmmm of YBa2Cu30*7 unit cell. Therefore, a space group fully represents a crystal structure. [10] [36] [13] Neutron-diffraction experiments [12] have shown that YBa 2 Cu 3 07 has a pmmm structure and YBa 2 Cu 3 06 has similar structure but with 4 transformations instead of 8 for YBa 2 Cu 3 07. i.e., every input ion site will generate 8 equivalent ion sites for YBa2Cu3C>7 and 4 equivalent ion sites for YBa2Cu30*6, due to its antiferromagnetic structure. In summary, the transformations of YBa 2 Cu 3 07 are pmmm and orthorhombic. Its startpoint, the origin of the cluster, is (0.0,0.0,0.0) which can be specified by users. The transformations of Y B a 2 C u 3 0 6 are pl23 which is termed for convenience and tetrago-nal. Its startpoint is (0.0,0.0,0.0). Through such transformations, (1,1,1) represents a set of 4 sites at (1,1,1), (-1, -1,1), (1, -1 , -1) and (-1,1, -1) for Y B a 2 C u 3 0 6 and a set o f 8 s i t e s a t ( l , l , l ) , ( - l , - l , - l ) , ( - l , - l , l ) , ( l , l , - l ) , ( l , - l , - l ) , ( - l , l , l ) , ( - l , l , - l ) and (1, -1,1) for Y B a 2 C u 3 0 7 . 1.3 Muon Spin Resonance/Rotation/Relaxation (pSTt) The muon possesses a magnetic moment and a spin of ~; hence it exhibits Larmor precession in a tranverse magnetic field. After stopping inside the target, the implanted Chapter 1. Introduction 5 X Y Z -X -Y Z X -Y -Z -X Y -Z Table 1.2: Transformations pl23 of Y B a 2 C u 3 0 6 highly spin polarized positive muons, precess in the local field and decay (fi+ —* e+ -+-ve + with a mean lifetime of 2.198/is. The decay positrons are emitted in directions correlated with the direction of the muon spin. Inhomogenenities of the local field will cause loss of muon spin ensemble polarization, termed depolarization, by dephasing. In this case information about the internal or external field distribution can be derived from the experimental distribution of precession frequencies which are propor-tional to the local field. In general, the precession frequency u> and depolarization rate a are given by w = f^H (1.1) where M 2 = ((H — (H))2}, u = {H) and 7M is the gyromagnetic ration of the muon. Specific equations have to be derived for specific cases. In this thesis two experimental techniques are referred to zero applied field (ZF) fiSH and Level Crossing Resonance (LCR). In ZF-fiS7l the fi+ undergoes Larmor —* precession around the local magnetic field. For a local magnetic field given by H = (Hx,Hy,Hz) the muon spin initialially polarized along the z direction evolves as P,(i) = ^ + Hx2^Hy2cos(yflHt) (1.2) = cos29 + sin^Bcos^nHt) Chapter 1. Introduction 6 where 6 is the angle between H and the z direction. The internal field components are assumed to have a gaussian distribution where A 2 = 72(if,-2). The average of the polarization in the z direction is Pz(t) = J J J piH^p^piH^cos^ + ain^cos^H^H Evaluation of this equation leads to the well-known Kubo-Toyabe formula for the evo-lution of a static spin in zero field: P*(t) = | + f (1 " 7 „ 2 A V ) « p ( - ^ 2 A 2 * 2 ) The application of LCR was first suggested by Abragam [15]; it quickly became one of the most valuable pSTZ techniques. The level-crossing occurs when the field strength is adjusted to match the energy splitting of the muon's Zeeman levels to those of the magnetic levels of the neighboring host nuclei (in this study, 1 7 0), which are determined mainly by their quadrupolar interaction energies. Details of the LCR technique can be found in references [16] [17] [18] [19]. Chapter 2 Methods of Muon Site Determination In this chapter two theoretical approaches are introduced to search for candidate muon sites: electrostatic potential mapping and dipolar magnetic field calculation. 2.1 Electrostatic Potentials The principal interactions of the u+ in ionic lattices are the static long-range Coulomb force and the overlap force, both of which are envisioned mainly as simple two-body forces; electronic polarization effects ( in which an ion may be polarized by the resultant electric field due to other ions in the crystal) are considered to be less important. The u+ interaction potential is the sum of all two-body interactions between the muon and other ions. The concept of an additive two-body interaction is only an approximation to reality; the wave mechanical calculations made by Lowdin [22] showed that an appre-ciable fraction of the lattice cohesive energy cannot be represented in terms of two-body interactions. Moreover, the effect of electronic polarization cannot be represented by a two-body interaction. However, the polarization effects are largely suppressed by the high symmetry of the structure so that the approximation is adequate for most purposes. The Coulomb potential can be employed to describe the potentials of ions at dis-tances much larger than their ionic radii; in insulating YBa2Cu30x (as in most ionic oxides), this is a fair description of all the positive ions. In metallic YBa2Cu307, atten-tion should be paid to the screening effect of conduction electrons on the field due to 7 Chapter 2. Methods of Muon Site Determination 8 ions fax away from the muon sites. For nearby ions, in this instance O 2 - , the calculation is more difficult. I resort to a familiar approximation, namely the Morse-like potential, which is frequently used for quantum mechanical calculations in solid state physics to calculate the potentials between / i + and 0 2 - : V(r) = i + B e x p ( - ^ ) - L # , (2.3) where A = Ze and r is the /x + —O 2 - distance. The first term arises from the Coulomb interaction between fi+ and 0 2~. The second term is the Morse-like potential, which represents the major part of the 0 2 - potential. The semi-empirical constants B and p are determined by experiment, and r 0 is the equilibrium fi+—02" bond lengh. The third term arises from the interaction of /x+ with the induced electric dipole moments by the muon on the O 2 - ions, p. The induced moment was determined from the following procedure. A unit dipole po was placed on a test O 2 - ion giving a dipole moment of pi = poi on this ion, where i is a unit vector along x. The electric field at the test ion produced by these other dipole moments is given by Ej = (Eix, Eiy, Elz). This procedure was repeated with unit dipole moments along the y and z axes of the test ion, p 2 = poj and p 3 = p0k; the corresponding electric fields calculated at the test ion are given by E 2 = ( £ 2 1 , E2y, E2z) and E 3 = (E3x,Ezy,Ezz). The net dipole moment at the O 2 - ion can be written This moment is induced by the electric field due to the point charges of all the other ions Eo plus the electric field due to induced dipole moments on all the other O ions. Thus, p = a 0 [E 0 + (l/poXPxE! + p v E 2 + p,E 3)], where c*o is the polarizability of 0 2" in YBa 2 Cu 3 Cv The last two equations can be Chapter 2. Methods of Muon Site Determination 9 written as a set of three simultaneous equations: px = CiolE0x + (l/p0)(pxElx + pyE2x-rpzE3x)], (2.4) pv = a0[E0x + (l/po)(pxElv + pvE2y+pzE3y)], (2.5) P x = ao[E0x-r(l/p0)(pxEl2 + pvE2i + pzE3i)}. (2.6) If E 0 and (l/po)(pxEi + pvE2 + pzE3) could be derived from a lattice-sum calculation, then these equations would be solved for px, p„, and pz. In order to fix the parameters in Eq.(2.2), I simplified the model. First, no con-sideration was given to any screening effects, which would introduce a screening factor to the first term. However, one reference is given here for those interested in solving screening effects. Given the electron density, the charge distribution in the p+—02~ could be estimated as for OH", [20] and the screening factor could be fixed. Second, the dipolar part is suppressed by the high symmetry of the structure, so it was neglected. Recent ZF-pSTZ and wTF experiments on 170-doped YBa 2Cu 307 [21] place the p+ approximately 1 A away from some oxygen ion; LCK-pS1Z measurements suggest specifically the 0(2, 3) oxygen in the Cu0 2 planes. Previous ZF-pSTZ experiments [23] [24] on A F M Y B a 2 C u 3 0 6 provide some constraints on the local field. Based on this foreknowledge, and assuming that the muon's site preference will not be much affected by oxygen deficiency in YBa 2Cu 306, I narrowed my search down to a small ensemble of candidate sites within one unit cell. This allowed me to fix certain necessary parameters in the semi-empirical equations with which I calculated the crystal potential in superconducting YBa 2Cu 307 to better locate the muon sites, V;.( r) = 2 + !2l (19.4-25r[A]) r r 0 where Q is the valence of the ion and r 0 is the ion's hard core radius. The first term Chapter 2. Methods of Muon Site Determination 10 Figure 2.1: Potential Vj-(r) vs R is the Coulomb potential and the second term is the repulsive overlap potential. This choice of parameters ensures that the muon-oxygen bond length is 1 A. The total potential at the /x+ is in the form v;ot = £ v ; ( r ) t=i The summation has been carried out in two regions defined by a spherical boundary with finite radius. Upon the asssumption of zero net charge in the unit cell, the coulomb contribution of ions outside this sphere goes to zero as long as the distance from that unit cell to the fi+ site is long enough compared with its dimensions, because this unit cell can be considered as a point with zero net charge. The contributions from included ions are simply summed. The exponential term is convergent. Therefore the Chapter 2. Methods of Muon Site Determination 11 convergence of this sum is ensured by assuming that the net charge of the unit cell is zero. Another general method for calculating the interionic potentials, based on the den-sity functional approximation to the energy of an electron gas, is introduced here but not used in this thesis because of the difficulties of finding the electro densities in YBajCuaCv Gordon and Kim [26] assumed that the interatomic interactions between all charges must be evaluated from the additive atomic densities. For a pair of ions, AB, the total density PAB{R) 1S assumed to be the sum of the separated ion densities pA(r) and PB^)', the total energy is therefore given by EAB = E[pAB] = E[pA + pB] so that the interaction energy can be written as EINT = E[pA + PB] - E[pA] - E[pB) This is the essence of the approximation to the interionic potential. The energy of an isolated, closed-shell ion is given by E[p) = C„ J[p(r)f3dr + C, J Mr)]*' 3* - Z f p(r)/r<ir \j j y r ^ * ' +1*[*•)]*•)* in which Z is the nuclear charge, p is the ion density and Ck = (3/10)(3TT2)2/3 and Ce = -(d/tyS/v)1'3. The first four terms represent the kinetic, exchange and coulomb energies respectively, while the final term is an estimate of the electron pair correlation energy. The correla-tion energy density ec[p(r)] is simply an interpolation between the high and low density limits. Chapter 2. Methods of Muon Site Determination 12 The Coulombic interaction of this pair is given by V. = ZAZB/R - ZB I flHh - ZA I ^ d r 2 + / / '" ( r i )'" ( r'Wr ; J TIB J riA J J r 1 2 (2.7) The electron-gas contribution to the interatomic interaction is Vg = fUpA(r) + pB(r)]Eo\pA(r) + pB{r)] -PA(r)EG[pA(r)] - PB{r)EG[pB{r))}dr The final form for Eint is thus, EUR) = ZAZB/R - ZB I Hdnljr, _ ZA I SSiDldr, + / / " ^ " B ^ d r , d r 2 J rlB J r1A J J r12 + J {[PA{T) + PB(r)]EG[pA(r) + pB(r)] -pA(r)EG[pA(r)] - PB(r)EG[pB(r))}dr in which EGW)] = Ck[p(r)}2/3 + Ce[p(r)}1/3 + *Mr)} and r , r \ A , r \ B and ri2 are functions of the internuclear separation, R. Here only nearest-neighbour interactions axe included in the lattice summations. The next-nearest-neighbour interactions lead to an overestimate of the cohensive en-ergy. Rae [27] pointed out that the exchange energy in the equation includes a self-energy contribution which, though negligible for an infinite electron gas, is significant for a small, finite number of electrons and leads to an overestimate of the exchange energy in the interaction of two light atoms or ions. A correction has been made by replacing the exchange term by a modified contribution, Ve(GKR) = Ve(GK)[l - 8/36 + 2S2 + 1/3S4} Chapter 2. Methods of Muon Site Determination 13 in which 6 is a solution of (4iV) - 1 = 63(1 - 9/86 + 1/463) and N is the number of electrons. It has been concluded that for solids such as the alkali and alkaline-earth halides and the alkaline-earth oxides, which are largely ionic, the modified electron-gas approx-imation is a reliable non-empirical method for the calculation of interionic potentials. [28] [29] [30] One different potential assumption is also used by W.K. Dawson et al. [32] They assume that the muon-oxygen bonding potential has the form v = ( t + 1 4 4 e y where r is the /x + —O 2 - distance in A and a, b, and c are semiempirical constants (a = +0.359,6 = 0.0531, c = +8.32). The calculations are carried out within a finite sphere of radius on the order of 50A. Convergence of the lattice sum is ensured by assuming that the net charge of the unit cell is zero. 2.2 Dipolar Magnetic Fields In the zero field (ZF) case, [31] the local magnetic field at the / i + is usually split up into two contributions: H = Hkf + Hdip where the dipolar field contribution is from the O 2 - interaction. The hyperfine field H/i/ results from the contact interaction between the fi+ and any polarized electrons density at the muon site. When covalent bond effects occur as in muon-oxygen bond formation, Hhj is mostly due to these effects. Generally HH/ = 3f[n+(r*/i) — n~(rM)], Chapter 2. Methods of Muon Site Determination 14 where n +(rM) is the density of spin-up electrons at the / i + and n~(rM) the corresponding spin-down density. In the case of a—Fe20*3 the hyperfine effect is due to electron spin transfer into unoccupied metal 3d-orbitals, which causes spin polarization of the oxygen 2p-orbitals. These 2p-orbitals overlap with the muon s-orbitals, resulting in a nonzero spin density at the muon site. In order to estimate H/ , / for a muon participating in a "muoxyl bridge", Sawatzky [33] pictures a simple arrangment of one oxygen ion surrounded by metal ions; only those metal ions which form a direct link with the muoxyl bridge were taken into account. For this simple structure H^/ is assumed to be the sum of the contributions of the linkages: H^/ = C • - A21t)cos2di + A2„]S,-where S, is the unit vector of the magnetic moment of the metal ion, C is a constant, 9 is the angle between the metal-oxygen and muon-oxygen directions and A2„ and A2„ are the magnitudes of the spin polarization of oxygen orbitals with er and 7r symmetry. The spin density at the muon is proportional to (Aa2 — Air2)cos26i + AT2. Here I neglect Hhf in the case of antiferromagnetic YBa2Cu306, because of too small electron spin density at muon site, and assume that the field at the muon arises mainly from the dipolar magnetic fields of surrounding atoms. The dipolar field is H , = (^j [3(/i, • r\)f, - /«,•] where H ; is the field generated by the i t h magnetic moment, r,- is the vector distance from that magnetic moment to the muon and fa is a unit vector in the direction of the magnetic moment. The resultant magnetic field at any point is the sum of such contributions over the lattice. It is calculated by summing explicitly over the contributions from lattice points within a certain radius R (the "Lorentz sphere") and replacing the summation over points beyond R by an integral. The accuracy of this Chapter 2. Methods of Muon Site Determination 15 method obviously increases with the radius R; an estimate of the error may be obtained by recording the variation of the calculated result with the value used for R. M. Bonn [34] has proved that a simple procedure for evaluating a sum over a simple lattice is to obtain by direct summation the contributions by the lattice points X(l) within a certain radius R and to replace the summation over the points beyond this radius by an integral. 2.3 Conclusion From the potential maps I was able to narrow down the areas of candidate fi+sites in the sample and then apply the dipolar field calculations on these areas. The candidate muon sites are then those points which fall into the potential minima and have the correct magnetic field. Chapter 3 Explicit Calculations of Muon Sites In Y B a 2 C u 3 0 6 3.1 Potential Maps of A F M Fig. 3.22 depicts a unit cell of the YBa 2Cu 306 crystal. The positions and properties of the ions of YBa 2Cu 306 are shown in Table 3.3. The ion positions are taken from T. Siegrist. [35] Hard core radii are from Handbook of Chemistry and Physics. [36] The valences of barium and yttrium can only be 2+ and 3-f, but Cu can be 1+ or 2-h Copper ions on the Cu-0 chain are thought to have no magnetic moments; they are therefore 1+ ions, and those on the Cu-0 2 plane are 2+ ions. [37] [38] [39] Cu2 has a slightly smaller radius than Cul. ion X-axis (a) Y-axis (b) Z-axis (c) valence haxd core radius Cul 0.5 0.5 0.5 1+ 0.74A Cu2 0.5 0.5 0.143 2+ 0.7A Y 0.0 0.0 0.0 3+ 0.893A Ba 0.0 0.0 0.314 2+ 1.34A 01 0.5 0.5 0.347 2- 1.2A 02 0.0 0.5 0.12 2- 1.2A 03 0.5 0.0 0.12 2- 1.2A Table 3.3: Properties and coordinates of ions in YBa 2 Cu 3 0 6 . Here a = b = 3.856A amd c = 11.666A. Origin is at Y(0,0,0). The calculations are performed over one unit cell at the center of a finite sphere of more than 400 ions generated by a software package, ORTEP. Twenty planes in-tersecting the z axis at even intervals are chosen in the unit cell, and the potential is calculated on each plane. The deepest minima of the potential maps are the expected 16 Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu3C*6 17 muon sites. The exact positions can be decided by finely adjusting z values. The following are the potential maps at constant distance along the z direction. The two dimensional potential maps on each plane are represented graphically by single charac-ters which stand for the value of the potential at that point, using the table below to decode the potential map. Generally * represents the highest potential, capital letters are the next highest and numbers are the lowest. The length scales are in units of the lattice constants a = b = 3.856A and c = 11.666A. Coppers are at the four corners of the unit cell, oxygens are at the centers of edges and Yitrium is at (0,0,0). The graphs that follow indicate that there is a very large increase in potential energy close to the ions and that the potential has (naturally) the symmetry of the crystal. Surprisingly, a large region close to the Cu-0 chain in YBa 2Cu 306, which might make an attractive home for a muon, doesn't have a potential minimum. Instead, the minima occur in two planes at 0.03c and 0.19c. Chapter 3. Explicit Calculations of Muon Sites In YBa2CusOs 18 Value of Potential (F) Character Printed F > 3 6 * 35 < F < 36 Z 10<F<11 A 9 < F < 10 9 1< F < 2 1 0 < F < 1 0 - 1 < F < 0 a -26 < F < -25 z F< -26 j Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu306 Z = 0.20000E-01 YrrQ^S ?5 Y=0 Y=0.5 55555 "15555555 .J5555555 555555555 5.555555"" 5555555 j555566666666666666666666( >555666666666677777777766( .6 66 66666555 "6666655 666666" 6666666. X=-.5 555566666666777889AAA98877] "5566666666778ACGJMNMJGCA81 A . . 556666666789DKV*******VKD9876666< 66666678AGU***********UGA876f" 66666?8AH***************HA8/t 66666679FZ***************ZF9766 6666678CQ*****************QC8?c 6666679H*******************H97( "66677BN*******************NB7J 66678CU*******************V 6666/8DZ*******************l__ 666678DZ*******************ZD876 666678CU*******************UC87c 666677BN*******************NB?7t 6666679H*******************H976( 666667oCQ*****************qC87f 6666679FZ***************ZF97( 6666678AH***************HA87( 66666678AGU***********UGA876( 6666666789pKy*******VKD98766( 66666666778ACGJMNMJGCA877666t, __,566666666777889AAA98877766666l 5555566666666667777777776666666666555555555£ _5555556666666666666666666666666665555555555£ 555555556666666666666666666666666§5555§5555555 "5555555556666666666666666666665? X=0.0 51 55J 55555555555555555555555555555555555555555555555555555555555 X=0.5 Figure 3.2: Two dimensional potential map at Z=0.02c Chapter 3. Explicit Calculations of Muon Sites In YBaiCuzOe 20 t30000E-01 555566666667 .555566666677 5555:A::::~~" ~ 55566666. "6 6 6666Z79CJY*********YJC977 881*************188? 9FX**+**********TF9 BK***************KB CP***************PC 66678DS***************SD8 "6678DS***************§D8 -6678CP*************** 66678BK*************** 666??9FT*************T 6666678BI************* [BJ 666??§BHP*******Pfi ..6666778ACGKN0NKGCA877 666666677789ABBBA987776.. "666666667777888777766666 666666666667777766666666 566666666666666666666666 556666666666666666666666 5555566666666666666666665 666666 X«-.5 X=0.0 X=0.5 Figure 3.3: Two dimensional potential map at Z=0.03c Chapter 3. Explicit Calculations of Muon Sites In YBaaCuzO^ 5544445555555566666779EQ***********qE97766666555555554444555 X=0.0 5544445555555566666779DN***********ND97766666555555554444555 Figure 3.4: Two dimensional potential map at Z=0.05c Chapter 3. Explicit Calculations of Muon Sites In YBavCusOe 22 X=-.5 S * * * * * * * ***** ***** ***** x=o.o X=0.5 Figure 3.5: Two dimensional potential map at Z=0.07c Chapter 3. Exphcit Calculations of Muon Sites In YBa^CusOs • 0.10000 . -0.5 ******YF97 ******]tr ****** { *****Q|r Y-0 ******************* ******************* ¥*****************U *****************! Y-0.5 ******* ******* ******* 9EQ****** ****QFA7665555555556AQ***************QA6555555555667AFQ**** K H E * : ****Q X=-.5 X=0.0 A F Q * * * * * *****QE9766555555557I*****************I7555555556679EQ****** ******KB76655555556AU*****************UA65555555667BK******* ******pn86655555556E*******************E65 ******YF9765555555 ^^^citc^^^G^Y655555557I****^*****:**^*!****,*CI?o555555679G******** X=0.5 Figure 3.6: Two dimensional potential map at Z=0.10c Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu30e Z - 0.15000 Y=-0.5 * * * * * * * * * * H 8 **********£7 * * * * * * * * * V B 7 *********Kgg •*******YC7 ******** *******H *****XF9 ***S^^7 §§66515. Y=0 J * * * * * * * * * * * * * * * * * ?***************** w***************y 51* ************* * j i ~ n************* iN***********N Y=0.5 v *********** X s *********** v********** 69K********** 7CV********* ********* H******** 9FX****** ••SIB7,,. ****UFg6 ******G85 * * * * * * Y E -*******0 ******** ******** ******** ******** ******** ********n *******09 ******YE^ ******Gr ****U **SIB ECA86554 ?65555"" " 155 -.5 ***SIB8766 *****XF976 *******g ******** • * * * * * * * V C 7 * * * * * * * * * K g *********VB76 •*****»***E76 **********H86 **********X86 5556A0*************0A6 "5581***************I8 BV***************VB6 7F*****************?? j*****************j K*****************R S*** ***** 58G******* 7EY******* 90******** fu ********* H********* 8K********* 8K********* X=0.0 7H********* 456D********* 44590******** 457EY******* t458G******* 4568EU***** IS*** 8ACEI 55 . J D G J K BIS**** X****** H******** ********* a 7CV***+***** g K * * * * * * * * * * SV********** *********** !*********** *********** X=0.5 Figure 3.7: Two dimensional potential map at Z=0.15c Chapter 3. Exphcit Calculations of Muon Sites In YBaaCu^O^ Z « 0.17000 : Y=-0 .5 Y=0 Y=0.5 #* * * * * * * *G8665555554557CN* * * * * * * * * * E 8 6 6 5 5 5 5 ^ * * * * * * * *QB76b55555544568DM** * * * * *MD^ * * * * * * * *H9665555555445569ELTZ*ZTLE9655445555555669H** * * * * * * * * * * * * * * p 7 6 6 5 5 5 5 5 5 5 4 4 4 ^ ^ ******nf i§6655555555544455 *****MC8766555555555444445556665554^ m X=- .5 jfrp***** x=o.o J8 JGDA876 * * * P G B 8 * * * * * M C 8 7 * * * * * *Qf )8 * * * * * * * N B 7 6 * * * * * * * * H 9 6 * * * * * * * * Q B 7 6 5 * * * * * * * * * E 8 6 6 * * * * * * * * * G 8 6 * * * * * * * * * H $ 6 j 7 A J Y * * * * * * * Y J A ] J 7 C N * * * * * * * * * N C 1 >8DP******** *PD{ 678ADGJK 8 B G P * * * * 8 C M * * * * * * 8 D Q * * * * * * * 7 {3N** * * * * * * 9 H * * * * * * * * * B Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * X=0.5 Figure 3.8: Two dimensional potential map at Z=0.17c Chapter 3. Explicit Calculations of Muon Sites In YBa^Cu^Oe 26 Figure 3.9: Two dimensional potential map at Z=0.19c Chapter 3. Explicit Calculations of Muon Sites In YBaaCu^Oe 27 Figure 3.10: Two dimensional potential map at Z=0.20c Chapter 3. ExpHcit Calculations of Muon Sites In YBaiCu^O^ 28 555555555555566666666789CIQZ***ZqiC9876666666655555555555555 Figure 3.11: Two dimensional potential map at Z=0.26c Chapter 3. Exphcit Calculations of Muon Sites In YBaaCuzOs L30000 AL****** X« 9HZ***** 7CN***** **** - . 5 ^^f 6 ^| | f If ppp 56l66677777776f66666 578BGP*******PGB r9DN***********N )ET************* Es***************sp * * * * * * * * * * * * * * * * * L ?******************* j * * * * * * * * * * * * * * * * * * * j * * * * * * * * * * * * * * * * * * * >666666667BR*******************RB766666666 " 366666667BR*******************RB26666666' )6666o667AQ*******************QA76666666 >666666679J*******************J976oo666o >666666678E*******************E876666666 X=0.0 555555! 7AL*****************LA76 78DS***************SD87c 6l9ET*************TE97f 6779DN***********ND9?7c 566666678BGP*******PGB8766e 566666666667777777666( 566666666666666666666( 566666666666666f 566666 mm .CRJD96! ***0E96! ****NC7! ****ZH9( *****}• *****] Ml 555555555555-, 1555555555555J )555555555555? 1555555555555! 59EO**** rcR***** )HZ***** IT.****** jff****** x* 0 .5 Figure 3.12: Two dimensional potential map at Z=0.30c Chapter 3. Exphcit Calculations of Muon Sites In YB&iCuzOe Z - 0.35000 Y=-0 .5 * * * * * * * * * * $ * * * * * * * * * T Q *********K? * * * * * * * * * C 6 * * * * * * * * K * * * * * * * Q A Y=0.5 * * * * * * * * * * * T * * * * * * * * * * K********** g £ * * * * * * * * * * K * * * * * * * * * ^ q * * * * * * * * X=- .5 * * * * * *PB6544555555555555555555555555555555555554456BP**** * * * X=0.0 55 ...3765 * * S I B 7 * * * • * ! * * * * * * p * * * * * * * * * * * * * * * K * * * * * * * * * 0 6 * * * * * * * * * £ ? * * * * * * * * * T 9 * * * * * * * * * * g **********£ B I S * * * * * * * * * p * * * * * * * * * * * * * * * K * * * * * * * * * * * * * * * * * * * K********** 7 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * X=0.5 Figure 3.13: Two dimensional potential map at Z=0.35c 0=X * * * * * * * * * * * * * * * * * * * * * * * * * * n ********T *******rt(] ******** -*****rt ***Aia89 . f r f r -"9 0=X ***A1Q89 * * * * * r t06 * * * * * * * £ *******rt< * * * * * * * * * * * * * * * * f l f l *********Q S'0=A ******** ******** n * * * * * * * 7******* rjrt****** o * * * * * * "DM**** " "A** sssgfl " I f vS9§aiA** rt**** ****** cm****** 1 * * * * * * * flfl******* Q******** 9-0-=• 0008S*0 Chapter 3. Explicit Calculations of Muon Sites In YBa^Cu^Oe 0.40000 0 " **ZMDB655455555555555555555555555555555555555555545568D8Z*** X=- .5 * * R I B ? E vqJD96S IFC975E >55§55 55! 55J X=0.0 **RIB75555^ Figure 3.15: Two dimensional potential map at Z=0.40c Chapter 3. Explicit Calculations of Muon Sites In YBavCu30e 33 Figure 3.16: Two dimensional potential map at Z=0.42c Chapter 3. Explicit Calculations of Muon Sites In YBaaCu^O^ 34 = 0.45000 : - - 0 . 5 Y=Q Y=0.5 ****MC8665555555555555555555 X=- .5 ***UIB866555555555555555555555555555555555555555 X=0.0 5 6 555555555555555555555! 15f 55 ] 6 6 § § 5 5 5 § 5 5 § 5 § ^ VQJEA7665555555555555^ Figure 3.17: Two dimensional potential map at Z=0.45c Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu306 35 -0.6 H 1 1 1 1 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Figure 3.18: The Contour Potential Map of YBa 2 Cu 3 0 6 at Z=0.03c. Coppers are at the four corners. Larger number lines represent lower potentials. Chapter 3. Explicit Calculations of Muon Sites In YB&iCu30s 36 The contour maps in Fig. 3.18 and Fig. 3.19 show the inverted muon potential energy — U(x, y) in the a — b plane (Cu ions at the four corners). Larger-numbered contour lines represent lower potential energy and indicate likely fi+ sites. The scales are in units of the lattice constants (a, 6, c). Yttrium is at the center. Fig. 3.18 describes the plane 0.03A away from Y plane and between the CuC>2 planes - and Fig.3.19 the plane 0.8lA below the Cu02 planes —i.e., toward the CuO chains. These contour maps confirm the strong fi+—O2- attraction that binds all muons closely to oxygen ions. For convenience, I define the z = 0.12c copper oxide plane as a reference plane; here yttrium is at (0,0,0). The first muon site (0.45a,0.06,0.03c) is 1.07A "down" away from 02 in the plane. The other site is (0.3a,0.06, 0.19c) which is 1.125A "above" away from 02 in the same plane. There are another six sites at symmetric positions on these planes. 3.2 Dipolar Magnetic Field Calculations For antiferromagnetic YBa 2Cu 306 the large Cu ion paramagnetic moments dominate the dipolar magnetic field, which is thus ~ 2000 times larger than that from the nuclear moments. The magnetic structure consists of strong nearest-neighbor correlated spins in adjacent copper planes which are aligned antiferromagnetically. The moments are constrained to lie in the tetragonal a — b plane; I assume that they point along either the a axis or the 6 axis. From neutron scattering measurements, the ordered moment on the Cu ions in the Cu02 plane is estimated to be 0.66 /XB[40]. J.W. Lynn[41] reports that at lower temperatures the Cu moments in the oxygen-deficient CuO chain layers also order antiferromagnetically, with a moment that can be quite substantial (0.5 / / B ) ; however, in my calculations I only consider the contributions from Cu moments in the Cu02 planes. Chapter 3. Explicit Calculations of Muon Sites In YBa^CuzOe 37 Figure 3.19: The contour potential map of Y B a 2 C u 3 0 6 at Z=0.19c Chapter 3. Explicit Calculations of Muon Sites In YBa^CusOe 38 The calculated results are in Table 3.4 and Table 3.5. Here the magnetic field B = xjBx2 + Bv2 + B z \ X-axis Y-axis Z-axis Field(Gauss) 0.05 0.10 0.03 308.88 0.10 0.00 0.03 318.57 0.45 0.00 0.03 307.86 0.05 0.00 0.03 287.17 0.50 0.05 0.03 302.79 Table 3.4: The magnetic field at z=0.03c. X-axis Y-axis Z-axis Field(Gauss) 0.25 0.30 0.00 0.00 0.19 0.19 268.45 308.83 Table 3.5: The magnetic field at z = 0.19c. Only the points at which the local magnetic field has a magnitude of around 300G are listed here. Contour maps of the magnetic field strength on these two planes are shown in Fig. 3.20 and Fig. 3.21. Chapter 3. Explicit Calculations of Muon Sites In YBa^Cu^Oe 39 3.3 Comparison With Experimental Data Experiments [14] on A F M YBa2Cu306 below 100K obtain a dominant ZF-fiS1Z signal at about 4 MHz, which indicates an internal field of about 300 Gauss. Meanwhile, the LCR experiment shows that the muon sites in YBa2Cu307 are 1 A away from the 0 in the Cu0 2 plane. If we assume that the muon occupies the same site regardless of oxygen deficiency, the overlap between the regions with potential minima and those with the magnetic field within uncertainty will be the candidate muon sites. Table 3.6 lists the muon sites in YBa2Cu3C*6. X-axis a Y-axis 6 Z-axis c Distance A Field(Gauss) 0.45 0.00 0.03 1.070 307.86 0.30 0.00 0.19 1.125 308.83 Table 3.6: Muon candidate site in YBa 2 Cu 3 0 6 . Here a = b — 3.856A and c = 11.666 A . Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu306 40 eft? # $ 1- 2. 00E+02 too. 0 00. 0 2= 2. 50E+02 99. 6 99. 9 3- 3. 00E+02 96. 7 99. 1 4- 3. S0E*02 92. 1 97. 7 5« 4. 00E*02 87. 3 96. 0 6- 4. 50E+02 8 1 . 6 93. 6 7- 5. 00E+02 74. 9 90. 6 8* 5. SOE + 02 70. 3 88. 3 9- 6. 00E*02 66. 3 86. 1 H 0 - 6. 50E+02 62. 8 84. 0 11* 7. 00E + 02 59. 4 81. e 12» 7. 50E+02 56. 2 79. 6 13- 8. 00E+02 52. 7 77. 0 14- 8. 50E*02 49. 5 74. 5 15- 9. DOE+02 46. 7 72. 1 16* 9. 50E+02 44. 3 70. 0 17. 1. 00E+03 42. 0 67. 9 -18- 1. 05E+03 39. 9 65. 8 19- 1. 10E+03 37. 9 63. 7 20- 1. I5E+03 36. 0 61. 7 4 .5 Figure 3.20: The contour plot of magnetic field at Z = 0.03c. It shows only a quarter of the plane. Coordinates are in' the unit cell length. Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu306 Figure 3.21: Contour plot of magnetic field at 0.19c Chapter 3. Explicit Calculations of Muon Sites In YBa2Cu306 42 C u 1 Figure 3.22: The crystal structure of antiferromagnetic Y B a 2 C u 3 0 6 Chapter 4 Calculations on Y B a 2 C u 3 0 7 4.1 Muon Site Decision Experiment ZF— and WTF—fxSTZ measurements were made on normal and 1 7 0 enriched samples of YBa 2Cu 307 at a temperature of 103K. In the latter samples, 38 % of the naturally abundant and spinless 1 6 0 was replaced by 1 7 0 . At this temperature the correlation time for muon hopping is much longer than the muon lifetime so that the nuclear dipolar fields seen by the muon are essentially static. The ^SR time spectra for the 1 7 0 enriched sample are compared with corresponding spectra in the pure 1 6 0 control sample. The depolarization rate in the 1 7 0 substituted sample is enhanced in both ZF and wTF(24G), which suggests that the muon is close to an oxygen ion. The rms internal field was estimated by fitting the wTF relaxation function to a sum of two Gaussians. The fitted results yield an rms internal field (i?2)1^2 along the applied field direction equal to 1.31G in the unsubstituted YBa2Cu3C"7 and 2.65G in the 1 7 0 enriched sample. If the muon is much closer to a single oxygen (i.e. if it forms an hydroxyl-like bond) such that the dipolar fields from other oxygens can be neglected, one obtains a powder averaged (B2Z)X^2 due to a single 1 7 0 equal to L\0 =3.64G. The 1 7 0 nuclei then exert an effective static dipolar field on the muon given by Bdip = (^JN/R3)Sg[3(f -q)f—q] where Sq is the component of nuclear spin along the electric field gradient direction q and r is the unit vector between the muon and 1 7 0 nucleus separated by distance r. 43 Chapter 4. Calculations on YBa2Cu307 44 Averaging over all angles between Bdip and z yields A 0 = \{n~tN/r*)[S(S + l){3(f • qf + l}]1'2, from which r = 1.01 A can be estimated provided that the muon is bonded to a single oxygen and the bond axis is perpendicular to q. LCR was introduced in order to decide to which oxygen the muon is bonded. Prin-cipally, LCR is based upon the idea that when two systems are prepared so that an energy splitting in one system matches a splitting found in the second system, an ex-change of energy or polarization may occur between levels of the two systems. In this study, the system consists of a spin 1/2 muon in an external field interacting with a spin 5/2 nucleus 1 7 0 in an electric field gradient plus external field. Whenever the Zeeman energy of the muon matches the quadrupolar level splitting of 1 7 0 , there may be a resonance. The spin Hamiltonian for a p+ in the crystal is: H = Hdip + HQ + Hz (4.8) Hdip = ^ ^ [ s M . I - 3 ( s M . r ) ( I . r ) ] (4.9) H q = 4 7 ( 2 J - l ) [ 3 ( 7 - 2 " / 2 ) + ^ 2 " ^ 2 ) 3 ( 4 ' 1 0 ) Hz = (u„ + uo) • H e x , (4.11) where Hnp is the dipolar interaction between muon and 1 7 0 , HQ is the quadrupolar interaction of 1 7 0 , Hz is the combined Zeeman energy of the muon and the 1 7 0 in the external field, r is the separation between nucleus and p+, sM and I are the nuclear magnetic moments of /x+and 1 7 0 and 6 = (Vxx — Vyy)IVxz. Here VJj is the electric field gradient (EFG) tensor. Due to the two splittings between |l/2) and |3/2) and between |3/2) and |5/2), two resonances are expected at approximately Br = ^^QQ/iy^) ^ d 2B r, assuming 6 = 0. Chapter 4. Calculations on YBa 2Cu 307 45 From the data the best estimate of the average quadrupolar parameter is e2qQ/h =6.6MHz, which is very close to that measured by NMR for the 0(2,3) [42] planar sites in the absence of the muon[42]. These results suggest that the muon is bonded to an oxygen in the Cu0 2 plane at a distance of lA. 4.2 Potential Map of Superconductor Y B a 2 C u 3 0 7 ion X(a) Y(6) Z(c) Valence Hard core radius Y 0.0 0.0 0.0 3+ 0.893A Ba 0.0 0.0 0.347 2+ 1.34A Cul 0.5 0.5 0.5 2+ 0.74A Cu2 0.5 0.5 0.143 2.3+ 0.7A 01 0.5 0.5 0.312 2- 1.2 A 02 0.0 0.5 0.12 2- 1.2A 03 0.5 0.0 0.12 2- 1.2A 04 0.5 0.0 0.5 2- 1.2A Table 4.7: The positions and properties of YBa 2Cu 307. Here a = 3.856A, b = 3.87A and c = 11.666A. Eq. 2.1 is used to generate the potential maps of YBa 2Cu 307, which have a structure similar to those of YBa 2Cu 306 except that there is no potential minimum between two Cu—02 planes. Furthermore a minimum-potential area appears around the bridging oxygen 04 due to the two extra negative ions on the Cu chain, which decrease the potential around that region. Further studies of region will be conerned with the dipolar magnetic field. The plane at z = 0.19c has a potential map similar to those of YBa 2Cu 306. The contour plot of this plane shows that there is also a very strong interaction between the muon and the ions. Muons are evidently bonded to the oxygen ions in the copper-oxygen plane. Chapter 4. Calculations on YBa^CuzOj 46 4.3 Dipolar Magnetic Field For YBa2Cu307 the dipolar fields are due to weak nuclear moments, primarily from ^Cu and 6 5 C u nuclei but with minor contributions also from Ba and Y nuclei. For 170-doped samples the substantial contributions from 1 7 0 nuclear moments must also be included. The spin-3/2 copper nuclear moments will precess about the electric field gradient, which is along the c-axis for Cu ions in the Cu0 2 plane and along the chain for Cu ions in the CuO chains. This quadrupolar precession is assumed to average out all transverse field components; those remaining are all treated as static (i.e., any muon-nuclear "flip-flop" dynamics are neglected). Within these conditions, the nuclear moments are assumed to be oriented along the electric field gradient and the resultant local fields are characterized by a gaussian width A as usual. Chapter 4. Calculations on YBa2 CU3O7 47 Ion Nuclear Magnetic Moment pi, Abundance % ^Cu 2.226 69.01 6 5 C u 2.385 30.91 1 7 0 -1.8937 38 1 3 5 B a 0.8365 6.59 1 3 7 B a 0.9357 11.32 89y -0.13682 100 Table 4.8: The nuclear magnetic moments of ions in YBa2Cu30x. The calculations indicate that the muon relaxation rate of 1 7 0 doped sample should be larger than that in the unsubstituted sample, which is consistent with the exper-iments. The minimum potential region at z = 0.19c yields a theoretical value of A consistent with those measured experimentally. Other regions with potential minima don't have dipolar magnetic fields of the right magnitude. Note that no consideration has been given to any lattice relaxation or hopping of muons, which would significantly affect the relaxation. 4.4 Conclusion The combined results of potential maps and dipolar magnetic field calculations indicate that the most likely muon site is on the plane at z = 0.19c, which is between the Cu—02 plane and the Cu—O chain and 0.84A "down" from the Cu—02 plane, which is indicated on Fig.4.4. All the other regions either have minimum potentials but wrong dipolar magnetic fields or have the correct magnetic field but don't fall into the minimum potential areas. In conclusion, the most probable muon site is at (0.3a, 0.06,0.19c) and corresponding symmetric sites. Chapter 4. Calculations on YBa^Cu^ 48 0.6 0.4 H 0.2 0.0 - 0 . 2 H -0.4 2. 00E-01 4. 00E-0I 6. 00E-0I 8.00E-01 1.00E»00 1. 20E*00 l.40E«00 1.60E»00 i.80E»oo 2. OOE*00 2. 20E»00 2. 40E«00 2. 60E+D0 2. 8DE«00 3. OOE'OO 3. 2DE*00 3. 40E'00 3.60E*00 3. BOE'OD 4.O0E*00 94.5 100. 0 34. 1 — 33. 9 93. 6 93.4 93. 2 92.9 92. 7 92. 4 92.0 91.6 69. 9 89.2 85. S 81. 2 68.5 41.1 24.3 13.4 100.0 99. 9 99.9 93. 8 99. 7 99. 6 99.5 99. 4 99. 2 98. 9 98. 4 97.7 96. 3 93.9 89.9 77. t 48. I 29.4 16.6 -0.6 -| 1 1 1 1 r -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Figure 4.23: The contour plot of the potential map at Z=0.19c Chapter 4. Calculations on YBa 2 C u 3 0 7 49 Cu2 ^ s i t e Cu1 Figure 4.24: The crystal structure of superconducting YBa 2 Cu 3 07. Bibliography [1] A. Schenck and K.M. Crowe, Physics Review, 26:57, 1971. [2] H. Graf et al., Phys. Rev. Lett, 37:1644, 1976. [3] E. Holzshuh, A.B. Dension, W. Kundig, P.F. Meirer and B.D. Patterson Phys. Rev., B27.-5294, 1983. [4] C. Boekma and P.L. Lichti, Phys. Rev. B., 31 1233, 1985. [5] S.R. Kreitzman et al., Phys. Rev. Lett. 63:1865,1986. [6] B. Foy, N. Heiman and W.J. Koss\er,Phys. Rev. Lett, 30:1064, 1973. [7] Y.J . Uemura, W.J. Kissler, X.H. Yu, J.R. Kempton, H.E. Schnoe, D. Opie, C. E. Stronach, D.C. Johnston, M.S. Alvarez and D.P. Goshun, Phys. Rev. Lett., 59, 1045, 1987. [8] R. Kiefl et al., Phys. Rev. Lett, 60, 224, 1988. [9] G. Golnik et al., Phys. Lett. A, 125:71, 1987. [10] C. Kittel, Introduction to Solid State Physics, 1976. [11] J. Pearson, Handbook of Lattice Spacings and Structure of Metals, 1963. [12] Y. Le Page et al., Phys. Rev. Lett, 36 3617, 1987. [13] J. M. Tranquada, D.E. Cox, W. Kunnmann, G. Shirane, M. Suenage, P.Zolliker, D. Vaknin, S.K. Sinha, M.S, Alvarez, A. J. Jacobson and D.C. Johnston,Phys. Rev. Lett, 60 156, 1988. [14] N. Nishida et al., J. Appl. Phys. Pt.2, 26:L1856, 1987. [15] A. Abragam, CR. Acad. Sci. Ser. 2, 299 95, 1984. [16] S.R. Kreitzman, Hyperfine Interaction, 31:13, 1986. [17] R.F. Kiefl, Hyperfine Interaction, 49 233, 1988. [18] S. Kreitzman, J.H. Brewer, P.R. Harshman, R. Keitel, D.LI. Williams, K .M. Crowe and E.J. Ansaldo, Phys. Rev. Lett, 56 181, 1988. 50 Bibliography 51 [19] R.F. Kiefl, Eyperfine Interaction, 32 707, 1986. [20] J.B. Bates, J.C. Wang and K.A. Perkins, Physical Review B, 19:4130, 1979. [21] Proceedings of the 5th International Conference on Muon Spin Rotation, 1990. [22] P. Lowdin, A Theoretical Investigation into Some Properties of Ionic Crystals, 1954. [23] J.H. Brewer et al., Phys. Rev. Lett, 60:1073, 1988. [24] R.F. Kiefl et al., Physica C, 162-167:161, 1989. [25] B.G. Dick and A.W. Overhauser, Phys. Rev., 112 90, 1958. [26] R.G. Gordon and Y.S. Kim, / . Chem. Phys., 56:3122, 1972. [27] R.I.M. Rae, Chem. Phys. Ze«,18:574-7, 1972-73. [28] W.C. Mackrodt and R. F. Stewart, / . Phys., C12431, 1979. [29] C.R.A. Catlow, K.M. Diller and M.J. Norgett, J. Phys. C, 10 1395, 1977. [30] W.C. Mackrodt Superconductivity, 343, 1988. [31] P.F. Meier, Hyperfine Interaction, 8 591, 1981. [32] W.K. Dawson et al., J. Appl. Phys. 64:5809, 1988. [33] G.A. Sawatzky and Van der Woude, J. Phys., C647, 1974. [34] M. Bonn and K. Huang, Dynamics Theory of Crystal Lattice, 1956. [35] T. Siegrist, S. Sunshine, D.W. Murphy, P.J. Cava and S.M. Zahurak, Phys. Rev. B 35:7137, 1987. [36] The Chemical Rubber Co., Handbook of Chemistry and Physics, 1974. [37] Y. LePage, W.R. Mckinnon, J.M. Tarascon, L.H. Greene, G.W. Hull and D.M. Hwang, Phys. Review. B, 35:7245, 1987. [38] J.M. Tranquada et al., J. Appl. Phys., 64:6074, 1988. [39] F . Beech, S. Miraglia, A. Santoro and R.S. Ruth, Phys. Rev. B, 35 8778, 1987. [40] J.M. Tranquada et al., J. Appl. Phys., 64:6074, 1988. [41] J. W. Lynn et al., J. Appl. Phys., 64:6065,1988. Bibliography 52 [42] E. Oldfield, C. Loretsopouls, S. Yang, L. Reven and H.C. Lee, Phys. Rev. B, 40, 6832, 1989. [43] W.W. Waren et al., J. Appl. Phys., 64 6081, 1988. [44] C H . Pennington D.J. durand, C P . Slichter, J.P. Rice, E.D. Bukowski and D.M. Ginsberg, Phys. Rev.B, 39 2902. [45] F. van der Woude and G.A. Sawatzky, Phys. Rev. B, 4 3159, 1971. [46] J . Brewer, MSR group, TRIUMF, Vancouver, Canada. [47] N. Nishida et al., Dept. of Physics, Tokyo Institute of Technology, Ohokayma, Meguro-ku, Tokyo, 152. [48] G. Solt et al., Paul Scherrer Institute, CH-5232 Villigen PSI. Appendix A Programs To determine the candidate muon sites a calculation consisting of three steps was performed. The first step is to generate a cluster of lattice points by using XTLPLOT; the second step is to produce potential maps on the planes at equal separation along the z—axis in a crystal unit cell to decide possible muon stopping regions; and the last step is to calculate the magnetic field on the different planes to verify that those possible muon stopping regions have the correct magnetic field. This initial introduction to the programs will emphasize the relationships between the programs. Details about the procedures followed in each program can be found in the corresponding manuals. X T L P L O T provides information to another program ORTEP, which actually pro-duces the model cluster. One of 230 different space groups has to be specified by users in oder to generate a crystal structure. 10 space groups are available in the files USR2:[ALANA]*.SPG on the TRIUMF VAX cluster. A special space group can be generated by editing a *.SPG file in which the specific transformations are indicated. Different radii have been employed to distinguish different ions on the plot because of the difficulty of labeling the ions. In this study two kinds of units are used: unit cell length and actual length. Positions are in the unit cell length, radius are in actual length. POTENTMAP program generates electric potential maps of a crystal structure over the x — y plane at the fixed z values specified by the users, minima of which are likely muon sites. The potential maps are characterized by letters; each letter 53 Appendix A. Programs 54 represents potential levels, which can be decoded by the table in the documentation. POTENTMAP's preselection of candidate muon sites is very useful, because each cal-culation of the dipolar magnetic field consumes at least half an hour of CPU time on the VAX. From the potential maps it will be easy to decide where the magnetic field needs to be calculated. The Morse potential is used to calculate the interaction be-tween muon and ions. No consideration was given to any lattice relaxation or charge screening effects. The calculation involves more than 400 ions in the cluster. Some modifications of POTENMAP can be made to generate output in a format suitable for input to PLOTDATA which can produce beautiful contour graphs or density maps. Both methods were employed in this thesis. The lettered maps show pseudo-three-dimensional structure and the contour map gives more detail of the two dimensional potential at every z value. CLUSTSUM calculates the dipolar magnetic field caused by other magnetic dipole moments at a site. Because of the limit of the size of the cluster generated by XTLPLOT, the contribution from those outside the cluster will be replaced by an integral. The sum will vary with the radius of the cluster. POTENMAP and CLUSTSUM read in cluster files generated by XTLPLOT. Be-cause the cluster usually involves many more ions than on one plot, no plot should be drawn after one sees the prompt XTLPLOT). Also two kinds of clusters are optional: "unit-cell" shaped, to draw the entire structure of a unit cell; and "sphere", to look at the environment of a particular position. In the case of this thesis, the sphere is usually used. For POTENTMAP, a sphere centred on (0,0,0) is produced. For CLUSTSUM, a sphere centred on the candidate muon position is required. The cluster file created will be in the proper format to be read by the two programs POTENMAP and CLUSTSUM. The only thing that must be done before they are run is that the appopriate ion property be included in lieu of the "**•***" y O U w jU faA Appendix A. Programs 55 in the newly created file. For the program POTENMAP, this property would be the ion's hard core radius in your chosen length unit; for CLUSTSUM, it would be the ion's dipole moment, in Bohr magnetons or nuclear magnetons, depending upon which dominates the field. In the case of YBa 2 Cu 3 0 6 , Bohr magnetons are used; nuclear magnetons are used for YBa 2Cu 307. POSITION, PLOTPARAMETERS and XTLMAIN, subroutines of XTLPLOT, play the roles of reading in ion positions, plotting parameters and lattice constants respectively. It is possible to revise the programs for a batch job. One example is usr2: [qiang.thesis.program]li.for, which calculates the magnetic field at 100 evenly distributed points on the plane specified by the user in YBa 2Cu 306 with the cluster generated automatically. 

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