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QUANTUM TRANSITIONS IN ANTIFERROMAGNETS AND LIQUID HELIUM3 By BRIAN C. WATSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 ACKNOWLEDGEMENTS I would like to thank my thesis advisor, Professor Mark Meisel, for his guidance and encouragement over the past three and onehalf years. I gratefully acknowledge the members of my supervisory committee, Professors Art Hebard, Kevin Ingersent, Yasu Takano, and Dan Talham. In addition, I am grateful to Fred Sharifi for sharing his knowledge and experience with me. There are several additional people that have contributed to this thesis, and I am indepted to Dr. Naoto Masuhara for his enlightening physics conversations and Dr. Jiansheng Xia for his technical acumen. I am also grateful to Drs. Stephen Nagler and Garrett Granroth for their assistance with the neutron diffraction experiments. Garret Granroth also deserves thanks for teaching me the laboratory basics during the semester that we both worked together. Stephen Nagler has also contributed to this thesis by writing portions of the MATLAB fitting routines. Every member of the Department of Physics Instrument Shop has been extremely helpful. I would especially like to thank Bill Malphurs for his attention to detail and for noticing when my instrument designs were geometrically impossible. Dr. Valeri Kotov deserves thanks for his guidance during my foray into theoretical physics. I am grateful to Larry Frederick and Larry Phelps in the Department of Physics Electronics Shop for their support. Once again, I acknowledge invaluable input from Professor Dan Talham and the members of his research group, including Gail Fanucci, and Jonathan Woodward for operating the EPR spectrometer, as well as Melissa Petruska, Renal Backov, and Debbie Jensen for synthesis of the antiferromagnetic materials studied in this dissertation. The assistance from Dr. Donovan Hall during experiments at the National High Magnetic Field Laboratory was invaluable. I would also like to thank Professors Gary Ihas and Dwight Adams for loaning equipment and for their help during experiments at Microkelvin Laboratory. TABLE OF CONTENTS ACKNOWLEDGEMENTS . . ......... . ABSTRACT . . . .............. CHAPTERS 1 INTRODUCTION . . . . . . . . . 1.1 BPCB . . . . .................... 1.2 MCCL . . . . ................... . 1.3 Zero Sound Attenuation in Normal Liquid 3He . ....... . 1.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3HeB 2 EXPERIMENTAL TECHNIQUES 2.1 SQUID Magnetometer . . 2.2 Vibrating Sample Magnetometer * 2.3 AC Susceptibility . ...... . 2.4 Tunnel Diode Oscillator . . 2.5 Conductivity . ....... . 2.6 Neutron Scattering ..... . 2.7 Nuclear Magnetic Resonance 2.8 Electron Spin Resonance . . 2.9 Pulsed FT Acoustic Spectroscopy 3 THEORETICAL TECHNIQUES 3.1 Exact Diagonalization . 3.2 The XXZ Model . .... . 9 * 9 * . 10 . . 12 . . 16 . . 19 . . 24 * . 26 * . 28 . . 36 . . 36 4 STRUCTURE AND CHARACTERIZATION OF A NOVEL MAGNETIC SPIN LADDER MATERIAL ...... . 4.1 The Structure and Synthesis of BPCB ...... . 4.2 Low Field Susceptibility Measurements . ..... 4.3 Low Field Magnetization Measurements ..... . 4.4 High Field Magnetization Measurements ..... . 4.5 Universal Scaling . . . . . . 4.6 Neutron Scattering . . ............ . . . . 61 . . . 63 . . . 67 . . . 82 . . . 92 . . . 113 . . . 119 * 1 2 4 5 7 5 MAGNETIC STUDY OF A POSSIBLE ALTERNATING CHAIN MATERIAL . . . .................. . 5.1 Structure and Synthesis of MCCL . . ............ . 5.2 Electron Paramagnetic Resonance . . ............ . 5.3 Low Field Susceptibility Measurements . .......... 5.4 High Field Magnetization Measurements . . .......... . 6 ZERO SOUND ATTENUATION NEAR THE QUANTUM LIMIT IN NORMAL LIQUID sE CLOSE TO THE SUPERFLUID TRANSITION . . . . . . . . 6.1 Experimental Details . . . 6.2 Zero Sound . . . . . . . 6.3 First Sound . . . . 6.4 Error Analysis and Final Results . . ............. . 7 DIRECT MEASUREMENT OF THE ENERGY GAP OF SUPERFLUID 3HEB IN THE LOW TEMPERATURE LIMIT . . 7.1 Details of the FT Spectroscopy Technique . . .......... 7.2 Thermometry Issues . . . ................. . 7.3 Edge Effects . .. . . . . . . ... 7.4 Temperature Dependence . . . ............... . 7.5 Pressure Dependence . . . ................. . 7.6 Error Analysis . . . . ................... . 7.7 Absolute Attenuation . . . ................. . 8 SUMMARY AND FUTURE DIRECTIONS . . . 8.1 BPCB . . . . . . . 8.1.1 Summary . . . ................. . 8.1.2 Future Directions . . .............. . 8.2 M CCL . . . . . . . . 8.2.1 Summary . . . ................. . 8.2.2 Future Directions . . .............. . 8.3 Zero Sound Attenuation in 3He . . ............ . 8.3.1 Summary . . . ................. . 8.3.2 Future Directions . . .............. . 8.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3HeB  8.4.1 Summary . . . ................. . 8.4.2 Future Directions . . . .............. 8.5 Concluding Remarks . . . ................ . S 245 S 245 S 245 * 246 S 246 S 246 S 248 S 248 * 248 S 249 250 S 250 S 251 S 252 APPENDICES A LOW TEMPERATURE PROBE DRAWINGS B COIL FORMER DRAWINGS . ...... C PRESSURE CLAMP DRAWINGS . ...... . D NMR PROBE DRAWINGS . . . . 127 128 132 139 155 162 165 177 187 198 212 216 219 221 225 231 241 242 E POLYCARBONATE SAMPLE SPACE DRAWINGS . ....... . 333 F APPLESCRIPT ROUTINES . . . ................ 338 G ORIGIN SCRIPTS . . . ..................... 344 H MATLAB FITTING PROGRAMS . . .. ............. 347 I DATA SET PARAMETERS FOR THE LANDAU LIMIT EXPERIMENT . . . . . . . ... 350 LIST OF REFERENCES . . . . .................... 354 BIOGRAPHICAL SKETCH . . . .................. 362 Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy QUANTUM TRANSITIONS IN ANTIFERROMAGNETS AND LIQUID HELIUM3 By Brian C. Watson December 2000 Chairman: Mark W. Meisel Major Department: Physics Effects arising from quantum mechanics are increasingly common in new devices and applications. Two different, but related, topics, lowdimensional antiferromagnets and liquid 3He, have been studied to obtain a deeper understanding of the quantum mechanical properties that govern these systems. Low dimensional magnetism provides a means of investigating new quantum phenomena arising from magnetic interactions. Superfluid and normal liquid 3He exist in a very pure form and therefore allow severe tests of theoretical descriptions. More specifically, the magnetic properties of bis(piperidinium) tetrabromocuprate(II), (C5H12N)2CuBr4, otherwise known as BPCB, and catena(dimethylammoniumbis(j12chloro)chlorocuprate), (CH3)2NH2CuC13, otherwise known as MCCL, have been measured and are reported herein. Theoretically predicted scaling behavior has been observed, for the first time, in BPCB. In superfluid 3He, the pairbreaking edge has been measured at low temperature, thereby allowing for a measurement of 2A. These data indicate that the energy gap at low pressure is significantly less than predicted by BCS theory. Finally, subtle effects due to the attenuation of zero sound in normal liquid 3He have been measured. Evidence for the quantum correction to zero sound attenuation, predicted by Landau over 40 years ago, is presented herein. CHAPTER 1 INTRODUCTION Quantum mechanical properties of various systems are increasingly important both fundamentally and technologically as new materials and devices are being generated at the boundary between the classical and quantum worlds. This dissertation addresses the quantum mechanical properties of two apparently disparate systems, gapped antiferromagnets and 3He. In both cases, however, the quantum mechanical nature of these systems is apparent in their macroscopic properties. In fact, these two systems are models for the verification of various quantum mechanical predictions. The dissertation is arranged as follows. Chapters 2 and 3 detail the nine experimental and two main theoretical techniques that were used to collect and analyze the data presented herein. Chapter 4 reports the experimental results concerning the gapped antiferromagnetic material bis(piperidinium)tetrabromocuprate(II), (Cs5Hn2N)2CuBr4, otherwise referred to as BPCB, and Chapter 5 discusses the alternating chain material catena(dimethylammoniumbis(jt2chloro)chlorocuprate), (CH3)2NH2CuCI3, otherwise referred to as MCCL. These chapters include magnetization and neutron diffraction data from experiments at the National High Magnetic Field Laboratory (NHMFL) and Oak Ridge National Laboratory (ORNL), respectively. In addition, electron paramagnetic resonance (EPR) measurements performed by Professor Talham's research group in the Department of Chemistry at the University of Florida are included. The liquid 3He studies are presented in Chapters 6 and 7 which discuss the low temperature acoustic experiments performed in the University of Florida Microkelvin Laboratory. The first 3He experiment is an absolute measurement of zero sound attenuation in 3He above the superfluid transition temperature. The second 3He experiment uses Fourier transform techniques and is a measurement of the 2A pair breaking energy in superfluid 3HeB. The final chapter summarizes the experimental results and lists possible future experiments. 1.1 BPCB The objective of this work is to better understand quantum phase transitions in antiferromagnets. Low dimensional, gapped, insulating, antiferromagnetic materials are ideal candidate systems for the experimental realization of quantum phase transitions. These transitions are defined as phase transitions that occur in the low temperature limit (T 0), where quantum fluctuations have energies larger than thermal fluctuations (hco > kBT), and are driven by a change in some aspect of the system other than temperature. A current review of quantum phase transitions is given by Sondhi et al. [1]. When thermal and quantum fluctuations are equally important (hwo kBT), the state of the system is referred to as being in a quantum critical regime. Quantum critical behavior is important in two dimensional antiferromagnets, and the behavior of charge and spin density waves in the quantum critical regime of two dimensional doped antiferromagnets is observed to play a role in high Tc superconductivity [2]. To better understand the quantum critical behavior in two dimensional materials, we begin by studying quasitwo dimensional systems. The logical intermediate step between two dimensional planes and one dimensional chains are ladder materials. The long range order that occurs in a two dimensional plane of spins can be approximated by ladders of increasing width [3]. Ladders are formed by two or more one dimensional chains arranged in a ladder geometry with electronic spins at the vertices of the ladder interacting along the rungs of the ladder with exchange J1 and along the legs of the ladder with exchange J1. In order to further the analogy between the two dimensional cuprate high Tc superconductors and quasitwo dimensional ladders, we choose to study ladder systems with Cu2 S = 1/2 spins. Ladders with an even number of legs are expected to have a gap to magnetic excitations otherwise referred to as a spin gap, A [3]. A spin gap can be measured indirectly in nuclear magnetic resonance experiments or directly in neutron scattering experiments. In addition, a spin gap will manifest itself in magnetization studies at low temperature (T * 0) as a critical field, Hci, below which the magnetization is zero. Recent studies have revealed a connection between the spin gapped state in ladder materials and superconductivity [46]. Until now, the best experimental realization of a 2leg ladder was thought to be the material Cu2(l,4diazacycloheptane)2Ch4, otherwise known as Cu(Hp)Cl [714]. However, the low temperature properties of Cu(Hp)Cl have been recently debated [15 19]. Although quantum critical behavior has been preliminarily identified in Cu(Hp)Cl near H6, this assertion is based on the use of scaling parameters derived by fitting the data rather than the ones predicted theoretically. Clearly, additional physical systems are necessary to test theoretical predictions of 2leg S = 1/2 ladders including quantum critical behavior. Chapter 4 in this dissertation describes the investigation of the gapped antiferromagnetic S = 1/2 ladder material bis(piperidinium)tetrabromocuprate(II), (C5H12N)2CuBr4, otherwise referred to as BPCB. In 1990, the room temperature crystal structure of BPCB was determined, in an xray scattering study by Patyal et al. [20], to resemble a 2leg ladder. This crystal structure has been recently verified in neutron diffraction experiments. In addition, magnetization and EPR measurements have elucidated details of the magnetic exchange. Finally, evidence for quantum critical behavior in this material is presented. 1.2 MCCL The simplest antiferromagnetic low dimensional materials are electronic spins arranged in one dimensional chains with a single exchange constant between spins, J. An exact solution of the isotropic S = 1/2 one dimensional chain was provided by Bethe [21] in 1931 for the isotropic nearest neighbor case. In 1983, Haldane [22] predicted a gap in the spin excitation spectrum or spin gap for isotropic integer spin chains. The Haldane gap for both S = 1 [23,24] and S = 2 [25] systems has been experimentally observed. Spin gaps may also occur in half integer spin chains if the exchange between spins alternates between two values, Jd and J2, where, to leading order, 1J, J2 = A/kB. Chapter 5 presents the results concerning the alternating chain material catena(dimethylammonium bis(pt2chloro)chlorocuprate), (CH3)2NH2CuCl3, otherwise referred to as MCCL. The room temperature crystal structure of MCCL was determined in 1965 [26] to consist of S = 1/2 Cu2+ spins arranged in isolated zigzag chains with adjacent chains separated by (CH3)2NH2 groups. The distance between spins alternates between two values and the bond angle between spins is approximately 90 degrees. Consequently, the magnetic structure is expected to be an antiferromagnetic alternating chain with the exchange constant alternating between the values J\ and .J2. Preliminary neutron diffraction work at ORNL has verified the crystal structure. In addition, a structural transition has been observed at approximately 250 K and the possibility exists for a second structural transition occurring between 11 and 50 K. A description of the magnetic exchange is obtained by analyzing the results of magnetization and EPR experiments. 1.3 Zero Sound Attenuation in 3He In 1956, Landau advanced a theory based on the properties of normal Fermi liquids, and this description is commonly referred to as Fermi Liquid Theory [27,28]. In the 1960's, it was realized that 'He at low temperatures was a model system for verification of this theory. At this same time, the experimental apparatus needed to study 3He below 100 mK became available. Since then, this theory has afforded an extremely accurate description of the properties of 3He. Landau Fermi Liquid Theory describes a perfect Fermi gas, where the interactions between atoms are added as a perturbation. These interactions are included by considering elementary excitations with effective mass, m*, which are termed quasiparticles. There are two primary modes of sound propagation in 3He depending on the time between quasiparticle collisions, r oc lIT2, and the angular frequency of the sound, ca At high temperatures (an << 1), quasiparticle collisions provide the restoring force and the sound propagation is termed hydrodynamic or first sound. Consequently, the viscosity and therefore the attenuation of first sound decrease roughly with the square of temperature. At low temperatures (anr >> 1), quasiparticle collisions can no longer provide the necessary restoring force to propagate hydrodynamic sound. Instead, sound is transmitted, through quasiparticle interactions, as a collective mode by an oscillatory deformation of the Fermi sphere and is referred to as collisionless or zero sound. The attenuation in the zero sound regime increases as the square of temperature since the relaxation rate of this collective mode increases due to quasiparticle collisions. At temperatures well below the Fermi energy (T << TF), and above the superfluid transition temperature, (T> Tc), the attenuation of both first and zero sound are well described by Landau Fermi Liquid Theory. In the zero sound regime, the attenuation is dominated by scattering within a continuous band of quasiparticle energies near the Fermi energy, AE = EF + kBT. At high frequencies (kBT << h o << kBTF) collisions will scatter quasiparticles to unoccupied energy levels greater than ksT away from the Fermi energy. This quantum scattering produces a second term in the attenuation, and the attenuation of zero sound may be written as ao(o,T,P) =a(P)T2[h1 1+. (1.1) Because the second term is effectively temperature independent, determination of this term requires measurement of the absolute attenuation in the zero sound regime. Several attempts have been made to verify this second term [293 1], and the most recent effort was reported by Granroth et al. [32]. In the latest experiment, the temperature and pressure were held fixed while the frequency, f= d2ni, was swept from 8 to 50 MHz. To provide absolute attenuation, the received signals were calibrated against the attenuation in the first sound regime. The result of this measurement was that the frequency dependence of the quantum term was a factor of 5.6 + 1.2 greater than the prediction. However, the frequency range was limited by the polyvinylindene flouride (PVDF) transducers that were used so that fm. 50 MHz. By extending the experiment to higher frequencies, it should be possible to more accurately determine the quantum term. However, using the first sound regime as a means of calibration places an important restriction on the highest useful frequencies. For example, this type of calibration was not possible in most other reports [2931]. The objective of this work is to measure the absolute zero sound attenuation in He as a function of frequency. In this experiment, relatively lowQ, crystal LiNbO3 transducers were used to extend the frequency range to approximately fmax 110 MHz. Again, absolute calibration of the attenuation was determined using measurements in the first sound regime. For both the zero and first sound data, the temperature was held fixed while received signals were averaged at several discrete frequencies. Chapter 6 contains a complete description of the results and the analysis at the pressures of 1 and 5 bars. 1.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3HeB The pairing energy of Cooper pairs in the superfluid, 2A, can been estimated in the limit of weak coupling using BCS theory [33,34] as 3.5 kBTc, where Tc is the transition temperature from the normal to the superconducting state. Deviations from BCS theory have been introduced by Serene and Rainer [35] who used quasiclassical techniques to incorporate strong coupling corrections in a treatment known as weak coupling plus (WCP) theory. One of the first attempts to measure 2A(7) was performed by Adenwalla et al. [36] in 1989 who worked at T/Tc > 0.6 and between 2 and 28 bars. In 1990, Movshovich, Kim, and Lee [37] measured the 2A pairing energy over a range of pressures (6.0 to 29.6 bars) and temperatures (0.3 < T/Tc < 0.5). However, in both cases, experimental limitations required that the results were either dependent on a particular temperature scale or involved extrapolation to zero magnetic field. The measurement of the pairing energy in superfluid 3HeB using a novel acoustic Fourier transform technique [38,39] is described in Chapter 7. Both the temperature and pressure dependence of the 2A pair breaking energy are included. In addition, comparisons are made with the existing BCS and WCP plus theory as well as the results from previous experiments. CHAPTER 2 EXPERIMENTAL TECHNIQUES In this chapter, the experimental techniques employed to study both 3He and the antiferromagnetic materials are discussed. The first three Sections, 2.1 through 2.3, describe magnetic susceptibility measurements using a SQUID magnetometer, a vibrating sample magnetometer (VSM), and AC mutual inductance techniques. The vibrating sample magnetometer research was conducted at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, FL. The next two Sections, 2.4 through 2.5, discuss tunnel diode oscillator (TDO) and conductivity measurements, respectively. Section 2.6 describes neutron diffraction measurements that were carried out at Oak Ridge National Lab (ORNL), Oak Ridge, TN. Section 2.7 outlines the design of a nuclear magnetic resonance (NMR) probe as well as details of the spectrometer and superconducting magnet. Section 2.8 describes electron spin resonance (ESR) measurements which were performed by Dr. Talham's research group in the Department of Chemistry at the University of Florida. Two 3He acoustic spectroscopy experiments are described, and both were conducted in the University of Florida Microkelvin Laboratory. Section 2.9 highlights only the general experimental approach of both 3He experiments while leaving the details of each experiment to the relevant chapters. 2.1 SQUID Magnetometer The SQUID magnetometer used in our magnetization experiments (model MPMS5S) was from Quantum Design, Inc., San Diego, CA. The system is composed of a computer and two cabinets. The first cabinet houses the electronics and the second cabinet contains the liquid He dewar. The SQUID communicates with the computer over the IEEE488 general purpose interface bus (GPIB), and control of the measurement system is accomplished using software provided by Quantum Design. The temperature and magnetic field can be varied automatically via the computer software. The temperature is controlled by two heaters and the flow of cold He gas, and the useful range of operation is from 1.7 K to 300 K with an estimated error of less than 0.5% [40]. The temperature can be lowered from 4.5 K to 1.7 K by applying a vacuum over a small liquid He reservoir. For practical purposes, the minimum temperature is 1.8 K and typically 2.0 K was used to decrease the measurement time. The superconducting magnet provides reversible field operation over +/ 5.0 T. In this commercial device, the measurement is accomplished using a rf SQUID. A SQUID consists of a superconducting ring with a weak link or Josephson junction. The electrons in the ring form Cooper pairs, which must be described by a single wave function. The phase of the electron wave function on either side of the boundary is equivalent. Therefore, the flux through the loop is quantized and must be an integer of the flux quantum, h / 2e. A screening current will increase in the ring to enforce this criteria until each integer flux quantum is reached. Similarly, the voltage across the boundary will oscillate with a period of one flux quantum. Theoretically, the SQUID can measure magnetic flux with a resolution less than 1 flux quantum. However, practical design considerations make it impossible to measure the flux directly; e.g. the SQUID must only detect the flux due to the sample and not from the magnet. In a rf SQUID, the superconducting ring is shielded from the magnet and connected to the pickup coils with an isolation transformer. A rf signal is applied to an electromagnet so that the flux through the ring oscillates. A DC bias is also applied, using a feedback loop, so that the voltage across the link remains at the single flux quanta condition. This DC bias is proportional to the signal from the pickup coils and therefore the magnetization from the sample. The samples are mounted on the end of long stainless steel rods and lowered into the sample space. The magnetization of the sample is measured by moving the sample through the pickup coils using a microstepping controller. The pickup coils have been wound so that the voltage in the coils is proportional to the second derivative of the magnetization. The computer reads the voltage output as a function of position and compares it to a theoretical curve using a linear regression technique. This theoretical curve depends slightly on the geometry of the sample. The standard curve assumes a cylindrical sample. For all our experiments, 48 position steps were used over a 4.0 cm scan length. The output of the SQUID is given in units of "emu", which is an abbreviation for "electromagnetic units" but it is not really an actual unit. The manner in which "emu" is used to output the data has led to some confusion. In cgs units, the "emu" is equivalent to cm3 or erg/G2 [41,42]. Accordingly, the units of molar volume susceptibility can be derived from Curie's Law and may be written in unit form as nN 2, ( erg )2 S_ nNAg2P 2PB gauss erg emu cm3 V 3kBTV ks (erg/K)T(K)cm3 =gausscm3 cm3 cm3' (2.1) where n is the number of moles, g is the dimensionless Lande g factor, and NA = Avagadro's number. The units of magnetization can be obtained from a straightforward calculation of total spin as M = nNAMB =nNAPB( erg ) = emuG = cm3G. (2.2) gauss 2.2 Vibrating Sample Magnetometer High field (0 < H < 30 T) magnetization experiments were performed at the National High Magnetic Field Laboratory (NHMFL), Tallahassee. These measurements used a 30 T, 33 mm bore resistive magnet and a vibrating sample magnetometer (VSM). The general setup of the VSM is shown in Fig. 2.1. Powder and single crystal samples (m ; 100 mg) were packed into gelcaps and held in place at the end of a fiberglass sample rod with Kapton tape. The sample rod screws into the VSM head and is locked in place. To position the sample in the field center, the height of the VSM head is adjusted until the VSM signal is at a maximum. The VSM uses a pair of counter wound pickup coils (3500 turns/each, AWG 50). The sample is vibrated at 82 Hz in the center of the pickup coils to generate a signal. This signal from the VSM is sent through a 19 pin breakout box and then to a Lakeshore model 7300 VSM controller. The VSM controller VSM head V valve top plate Gas handing valm He level platUom heated  ~lSmsm Figure 2.1: Overview of the VSM setup [43]. does not have an IEEE interface and, therefore, a Keithley 2000 multimeter reads the EMU monitor on the VSM controller and communicates with the computer over the GPIB bus. Absolute signal calibration was not necessary during our measurements, because we were able to reach saturation magnetization. In addition, at saturation, we were also able to measure and subtract a small linear correction with negative slope that corresponds to the diamagnetic contribution from the gelcap as well as the diamagnetism from the sample. The VSM has a resolution of 103 emu and a maximum signal of104 emu. The largest sample signal was at least an order of magnitude below this limit at 30 T, so the VSM pickup coil response remained in the linear regime. The sample signal was greater than the minimum signal resolution of 103 emu at a magnetic field of approximately 1 T. Temperature control was achieved by varying the pumping speed on either a 4He or 3He bath. A heater was not used in our experiments. The resistance values of a calibrated cemrnox thermometer were measured and converted to temperature using a Conductus LTC20 Temperature Controller. The cemrnox resistor is calibrated only down to a temperature of 2 K. In addition, the cernox resistor has a field dependence that must be corrected using the results of Brandt et al. [44]. This thermometer is placed in a location directly adjoining the sample space (see Fig. 2.2). The sample space as well as the middle layer surrounding the sample space is filled with a small amount of 4He gas. When the temperature of the 3He bath falls below the 4He lambda transition temperature, some of the gas in the surrounding sample space becomes superfluid and the thermal connection with the bath is made. For this reason, above a temperature of 2.0 K, where 77 K I 1 I i sample rod rod 150 cm gelcap 4 He gas 3 He bath cernox 4He L thermometer superfluid 0  0.64 cm Figure 2.2: A sketch of the thermometer setup and thermal conduction mechanisms below the 4He lambda transition temperature in the NHMFL vibrating sample magnetometer. the thermometer is calibrated, the thermometer values were used directly. Below this value, the temperature is estimated from the He pressure. In our experiment, the lowest 3He bath temperature was 0.58 K. The actual sample temperature was warmer than this temperature. The main mechanism for thermal conduction inside the sample space was a small amount of 4He gas. At such a low temperature, an error in the temperature of 0.1 K becomes very important. By estimating the heat leak, we can determine the worst possible error in temperature from QH =Qc ,and (2.3) A AT AT Area K= Area A K (2.4) AL radius where QH and Qc are the rate of heat transfer into and out of the sample space and K is the thermal conductivity of the 4He gas. The area and radius characterize the inside of the AT 0.64 cm diameter tube. The factor is determined by considering the dewar geometry. AL The liquid N2 bath temperature of 77 K is 150 cm above the sample. If we assume that the temperature gradient is a constant of 0.5 K/cm, then we arrive at a AT of 0.2 K. This calculation is obviously an overestimate; however it gives us a basis for determining the maximum possible error. Consequently, the lowest temperature in our experiment, originally reported as 0.58 K, was estimated to be 0.7 + 0.1 K. 2.3 AC Susceptibility The AC susceptibility measurement system is a standard mutual inductance technique that consists of a dewar, probe, and electronics. Detailed drawings of the probe are listed in Appendix A. Computer control of the instruments was accomplished using Labview and a GPIB interface. Five instruments were employed: a Picowatt AVS47 Resistance Bridge, HP3457a digital multimeter, HP6632 power supply, and two PAR 124A lockin amplifiers. A schematic of the susceptibility setup is shown in Fig. 2.3. The signal from the secondary coil is split into inputs A and B of both lockin amplifiers which PAR 124A Lockin D + 90' Signal RefOut A B Out 4.5 kQ Resistor box PAR 124A Lockin Signal' . Ref In A B Out S HP6632 Output HP3457a Input 0 Input 1 Dewar Sample leatet SThe"rm! primary secondary Lakeshore carbon glass resistor CGR11000 / *I "= = * 9 3 , * l: Figure 2.3: (A) Schematic diagram of the mutual inductance circuit used to measure AC susceptibility. (B) Overview of the copper sample plate indicating the position of the thermometer, heater, and susceptibility coil. Voltage AVS47 Current S Twisted pair Heater are operated in 'AB' mode. The inputs are filtered so that high frequency components (f > 1 kHz) are attenuated. At the start of the measurement, usually at the lowest temperature, the lockin amplifiers are adjusted so that the signal from one is a maximum and the phase difference between them is w 90 degrees. The adjustable reference output provides the primary excitation voltage (typically VREF = 5 Vpp). Because the resistance of the primary coil is small (50 Q), a 4.5 kM resistance box is placed in series with the primary so that the reference output behaves as a constant current source. The heater consisted of approximately 100 turns of manganin wire on a copper core, which was bolted to the copper sample support (Fig. 2.3 B). The resistance of the heater was approximately 50 C. Five watts of heater power was sufficient to bring the temperature from 4.2 K to 77 K. The Labview software controlled the heater to achieve an input drift rate, typically 0.2 K/minute. The temperature could be lowered from 4.2 K down to 1.7 K using the 1K pot. Temperature control in this range was achieved either by controlling the 1K pot pressure manually using the pumping valve or by opening the valve all the way and letting the computer control the temperature using the heater. The temperature was determined by measuring the resistance of a CGR11000 Lakeshore Cryotronics carbon glass resistor and converting to temperature. The resistor was wrapped in copper shim stock and bolted to the sample plate. The resistor wires were attached to the cold plate using GE varnish. The resistance was read using the AVS47 resistance bridge on the 2 kW range using a 1 mV excitation at 15 Hz. The susceptibility coil consists of two coil former, a primary (insert) and a secondary (outer). The coil former were manufactured from phenolic rod and the drawings are shown in Appendix B. The primary has 300 turns on two layers using copper AWG 40 wire. The secondary consists of two counter wound coils separated by a small gap with 1500 turns/side on 40 layers using the same wire. The insert is a tight fit into the secondary and the position of the insert is adjusted so that the signal from the secondary coil is a minimum. The sample is placed on one side of the insert so that any magnetic signal from the sample will unbalance the coil. The susceptibility coil was attached to a 16 pin connector using GE varnish and plugged into the sample plate. The sample plate is suspended below the 1K pot using two stainless steel tubes. 2.4 Tunnel Diode Oscillator The tunnel diode oscillator (TDO) technique uses a tank circuit that is sensitive to the inductance of a coil where the sample is located. The sample is placed inside the coil. As the inductance of the sample changes, the resonant frequency of the circuit also changes. In the case of a conducting sample, this inductance change is related to the skin depth. The TDO technique is particularly sensitive to superconducting transitions. For a discussion of the superconducting applications, please see the dissertation of Philippe Signore [45]. The circuit used for the TDO experiment is shown in Fig. 2.4. The tunnel diode (Germanium Power Devices, model number BD6) is the active element in the inductance circuit. This device has a negative IV curve when biased at the correct voltage and therefore behaves analogously to a negative resistor. A 5 pF capacitor is added in parallel with the tunnel diode to stabilize the oscillation. In series with the tunnel diode are the wire capacitance and the coil inductance. The wire capacitance is on the order of 300 pF. GPIB Computer GPIB Dewar Battery voltage (0 1.5 Volts) HP 5385A S Frequency Counter T Trontech RF Amplifier Twisted pair Coaxial cable i Wire cap (500 pF) BD6 TD 5 pF + . 15V HP E3630A DC Power Supply Coil Sample Figure 2.4: Schematic diagram of the tunnel diode oscillator setup. AVS47 V I liP 6632  + Lakeshore carbon glass resistor CGR I 1000 Additional capacitance can be added to change the resonant frequency which is typically 10 MHz. The coil is wound on a phenolic former fabricated in the University of Florida Instrument Shop. The drawings for various designs are listed in Appendix B. For 100 turns of AWG 40 copper wire, we expect a coil inductance of 5 gH. Power is supplied by two 1.5 V Mercury batteries at room temperature. The voltage bias is adjusted until the circuit begins to oscillate. It is useful to watch the signal output on an oscilloscope as the voltage bias is adjusted. The voltage where the signal appears most sinusoidal usually corresponds to the maximum voltage output. However, because this point usually occurs near the end of the voltage range for stable oscillation, it is not practical to balance the circuit at this point. It is more useful to balance near the center of the voltage bias curve so that changes in the sample inductance, due to a change in temperature, do not push the circuit out of its oscillation condition. After the DC component is removed, the signal voltage (50 mVp.p) is increased (1 Vpp) by a Trontech W500K rf amplifier. A HP5385A frequency counter reads the frequency and communicates with the computer using GPIB bus. Temperature control is identical to the method discussed for the AC susceptibility technique (Section 2.3). A major complication of this method is that the properties of the tunnel diode circuit are temperature dependent. Even without a sample, the resonant frequency of the circuit will change due to a change in the wire resistance, thermal contraction of the sample coil, and most importantly the thermal dependence of the tunnel diode. From 4.2 K to 77 K, this change is typically on the order of a few MHz. Frequency transitions related to the sample are at least an order of magnitude less, and a temperature dependent background must be subtracted. One possible way to decrease the background contribution is to mount only the tunnel diode on the underside of the 1K pot. The leads must be attached to the copper using GE varnish. As long as the 1K pot needle valve is open, the temperature of the 1K pot will remain stable and no background will need to be subtracted. However, at temperatures above 20 K, this method will boiloff a considerable amount of liquid He from the bath. To avoid this limitation, the wires connecting the tunnel diode to the circuit should have low thermal conductivity, such as a CuNi alloy. The increased resistance of the wires will lower the output signal slightly. It has been observed that the transition temperature of high Tc superconductors [46] and hole doped ladder materials [47] can be increased by the application of pressure. Two pressure cells have been constructed to study the effects of pressure on the superconducting transition temperature using the previously described tunnel diode technique. Both pressure clamps are based on the design of J.D. Thompson [48], which is an improvement over a previous design [49]. The second design is approximately twice as large as the first. The clamp bodies and sample cells are made from beryllium copper and Teflon, respectively, at the University of Florida Instrument Shop. The tungsten pushrods and wafers are made of tungsten carbide by Carbide Specialties, Waltham, MA. The larger cell is shown in Fig. 2.5. Complete drawings for this cell are given in Appendix C. The sample coil ( 1 mm diameter for the smaller cell, 100 turns, 4 layers of AWG 40 wire) is contained inside the Teflon sample cell. A second coil of the same size is usually included and contains lead wire for the pressure calibration since Tc(P) for 0 0 0 Oi LIII rn cz Figure 2.5: Overall drawing of beryllium copper pressure clamp. lead is well known [50]. The wires exit through the top of the Teflon cell and are secured to the brass lid with 2850 epoxy. The cell is filled with isopentane liquid to distribute the pressure. A retaining ring is placed on the bottom of the Teflon cell to prevent the cell from rupturing outward. The entire pressure clamp is placed in a press, and the tungsten carbide pushrods on either side of the cell transmit the pressure. As the pushrods are compressed, the top of the beryllium copper clamp (the side without the wires) is tightened. By using the surface area of the Teflon cell and the pressure applied it is possible to calculate the actual cell pressure. 2.5 Conductivity As part of a collaboration with Dr. Talham's research group (Department of Chemistry, University of Florida), several attempts were made to measure the conductivity of Langmuir Blodgett (LB) films deposited onto glass slides. These films are typically a few hundred angstroms thick and are expected to be semiconducting. Since the resistivity of these films was expected to be in the 10 Macm range, masks were engineered to optimize the conductivity measurements. The size and dimensions of the masks used are shown in Fig. 2.6. The masks were cut from stainless steel sheets using an electric discharge machine (EDM) in the Department of Physics Instrument Shop. The transport measurements were performed in the following manner. First, a mask was placed onto the LB film, the bottom of the glass slide was glued to the metal "puck" using rubber cement, and the whole assembly was placed in the evaporation chamber. During the evaporation, the sample is inverted. To prevent the mask from falling, the mask is held in place by gluing the sides of the mask (which hang over the sample) to additional glass slide pieces using rubber cement. The evaporation chamber was placed under vacuum using a combination mechanical/diffusion pump system. When the pressure reached 1 x 10.8 Torr (after approximately three hours), the evaporation would begin. The current source was set to 180 A, giving a gold evaporation rate of 2 A/s. Typically, 300 A of gold were deposited. After removal from the evaporation chamber, the glass slide was attached to a G 10 support using rubber cement. Using silver paint, gold wires were attached as current and voltage leads to the evaporated gold. These wires were also glued to flat copper contacts on each edge for strain relief The circuit diagram for the fourprobe technique is shown in Fig. 2.7. Measurements were done at 19 Hz and used an initial excitation voltage of 200 pV. If the result was infinite resistance, the excitation voltage was increased to 20 mV. In spite of considerable effort, no reasonable resistance ( < 100 MQ) was obtained. The samples were expected to be 0.010cm 0.010cm 010 cm .. 005cm  0.20 cm . . ... .. . E LC) E i :. :.. . 0 0: S." .' '" ,. . . i'.". ,,:" :: :. u , ," :,,, : :; ' : 1 ** ... 0 ..': '*...* '.'.: 0' .. L O C).. Figure 2.6: Stainless steel mask designs used for gold evaporation onto Langmuir Blodgett films. semiconducting, so warming the samples should have increased the carrier density and hence the conductivity. Each sample was placed under a lamp to increase the temperature up to 100 C above ambient. Efforts were also made to improve gold contact with the conducting layer of the LB sample through gold "scarring". To rule out the possibility that the contact resistance was unusually large, the contact resistance was measured using the mask in Fig. 2.6 (A). In every case, the contact resistance was on the order of 0.5 n, and the sample resistance was infinite. E 0 " 0 0 : " . ,. t .' '*' o0. 3( G10 Support copper shim stock gold super LR700 gold glue wire g Resistance Bridge Silver V I Paint LB Film glass slide evaporated gold (A) (B) Figure 2.7: Circuit diagram for fourprobe AC measurements on LB film samples. The figure in (A) is a magnified view of the LB film while (B) is an overview of the entire experimental setup. 2.6 Neutron Scattering Neutron scattering experiments were performed at the Oak Ridge National Laboratory High Flux Isotope Reactor (HFIR). When the reactor is operating at full power (85 MW), it produces a large thermal neutron flux of 1.5 x 10'5 (neutrons/cm2sec). This flux is accessed by four 10 cm diameter beam tubes that extend horizontally from the midpoint of the reactor core. The neutron flux passes through a sapphire filter to limit the amount of fast neutrons. At each of the access points, there is a triple axis spectrometer, labeled HB1 through HB4. All of the experiments listed in this dissertation utilized the HB3 spectrometer (see Fig. 2.8). The typical monochromatic neutron flux (resolution 1 meV) after collimation is 3 x 107 (neutrons/cm2 sec) [51]. The angle between the sample and the incident and reflected beams can be changed independently. Changing the 20M monochromator angle can continuously vary the incident energy. After interacting with the sample, the neutron beam is defracted by the analyzer which is usually the same material as the monochromator. It is also possible to operate without an analyzer, which essentially allows all final neutron energies. After the analyzer, a 3He detector registers the neutron flux. Both q scans at integrated final energy and AE scans at fixed q were performed. For both types of scans, the monochromator was pyrolitic graphite (PG)[002] with a fixed incident energy of 14.7 meV or 30.5 meV. The collimation was 20' at positions C2 and C3, before and after the sample. A collimation of 60' represents the open beam which has dimensions of 5 cm x 3.75 cm. A PG[002] filter was also used to remove second order reflections. For the constant AE scans at fixed q, a PG[002] analyzer was used to select a fixed final energy. With the collimation, the resolution was typically 1 meV. The BPCB sample was a single deuterated crystal with the approximate dimensions of 10 mm x 7 mm x 2 mm, while the MCCL sample was approximately 1 g of deuterated powder. The specimen was attached to a sample support which also served as a thermal anchor. Temperature control was accomplished by varying the pumping speed on an inhouse 4He cryogenic system. HB3 Spectrometer ( Collimator (C1). Collimator (C2) Shutter Sapphire Filter Monochromator Collimator (C3) Analyzer Crystal Collimator (C4) 3He Detector Figure 2.8: Overhead view of HB3 beamline at the HFIR facility at ORNL [51]. 2.7 Nuclear Magnetic Resonance The nuclear magnetic resonance system can be divided into four main sections: finger dewar, probe, spectrometer, and superconducting magnet. The drawings for the dewar, probe, and magnet are included in Appendix D. The finger dewar was purchased from Kadel Engineering, Danville, IN, and was designed specifically for our Oxford magnet. The overall length of the body is 49 inches and the tail portion is 33 inches long. It contains a liquid N2 shield with a 5.7 liter capacity designed to have a 2.5 day hold time and a liquid He capacity of 35 liters with a 13.2 day hold time. A stainless steel 0ring flange bolts to the top and necks down to a LF flange (A&N, part number LF100400SB). Six copper baffles are suspended below this flange along with three 440 stainless steel support rods. There are seven quick connects (A&N, part number QF16 075) arranged in circle 45 degrees apart for access to the He bath. The NMR probe was designed to be a versatile platform to study protons as well as other nuclei. It is based loosely on a design by A.P. Reyes and coworkers [52]. The main additional objective of this design was to allow the ability to tune the NMR capacitors from room temperature while the probe vacuum can was cold. In addition, the probe should allow for stable temperature control between 1.5 K and 300 K. The combination of these goals presents another difficulty, i.e. 4He gas exists as single atoms and therefore does not contain any molecular vibration modes. Therefore, He has a low ionization energy with a minimum at 1 x 103 Torr [53], which is incidentally the typical vacuum pressure produced by a mechanical pump. At high power (P > 100 W), current will arc between the terminals of each capacitor. To overcome this difficulty, the He vacuum pressure must be higher than 1 Torr or lower than 1 x 106 Torr. For the 1K pot to be effective, the lower pressure must be chosen. Although this pressure is easily reached using a turbo or diffusion pump, the lack of any exchange gas means that the sample must be thermally anchored by other means. We have chosen two ways to solve the capacitor arcing dilemma. First, the sample can be contained in a separate polycarbonate sample space (see Appendix E). Second, the capacitors can be contained in their own vacuum cans and pumped independently. The flexibility in the design allows us to switch between the two methods or use a combination of them. In the following paragraphs, I will discuss the notable features of the NMR probe design. Machining of the probe parts as well as the probe assembly took place at the Department of Physics Instrument Shop. The top of the probe is built on a LF flange that mates to the stainless steel adapter on the top of the dewar (Fig. 2.9). The electronic connections are made through the brass box at the top of the probe. The main pumping line is also connected to this box with a quick connect flange. There are five stainless steel rods that connect the room temperature flange to the vacuum can: the main pumping line, two capacitor pumping lines, the 1K pot pumping line, and the needle valve control rod. The horizontal position of the 1K pot pumping line is shifted near the top of the baffles, using a brass adapter, to prevent a direct line of sight from room temperature. The 1K pot pumping line also contains its own set of internal baffles. At the junction of the pumping line and the 1K pot, the design has been carefully chosen to prevent superfluid 4He from climbing the pumping line walls. The 1K pot is located near the bottom of the vacuum can and filled from a 0.050 inch OD capillary connected to the needle valve. The capacitor pumping lines also accommodate the capacitor adjustment rods. These capacitor control lines are fabricated from a combination of 1/8 inch diameter stainless steel and phenolic rod. Each rod is fitted with triangular phenolic spacers so that it remains in the center of its pumping line and ends with a stainless steel screwdriver tip designed to fit the variable capacitor. The phenolic portion of the rod is located at the bottom to prevent any increase in capacitance. Each control rod must be spring loaded because the height of the capacitor changes upon rotation. A double 0ring seal at the top of each rod allows adjustment of the capacitors without significant increase in vacuum pressure at 1 x 108 Torr. * a a a a C * 0 * 0 * a a a a a Baffle 3/8Cajun_ Needle Valve Control Rod Main Pumping Lin( Double 0ring Seal LF Flange 1K Pot Pumping Line Capacitor Adjustment Rods Baffle Sample Support^ tl r , Sample i Space L j 1K Pot I .69 inches Capacitor Can (A) (B) Figure 2.9: Detail of the NMR probe top (A) and bottom (B). The capacitors are 40 pF nonmagnetic trimmer capacitors from Voltronics, Denville, NJ (part number NMTM38GEK). The position of the capacitors has been chosen so that the distance from the NMR sample coil is minimized. The frequency range, over which the NMR circuit can be tuned, is limited by this length. Each capacitor is screwed into the bottom of its pumping line using a Vespel spacer to prevent any electrical connection with the stainless steel can. As mentioned earlier, each capacitor can be sealed in its own individual vacuum can using soft solder. The vacuum can is 28 inches long so that the vacuum can top flange is higher than the dewar neck. The vacuum seal is achieved using indium wire. The distance between the vacuum can outer wall and the dewar inner wall is only 0.055 inches. During a magnet quench or some other unforeseen event, the He below the probe would not be able to escape easily. A hard Styrofoam piece must be bolted to the vacuum can bottom to exclude any He liquid. The vacuum can contains its own set of six copper baffles to limit the radiation heat leak from the pumping lines. The CuNi twisted pair for the thermometer and heater is thermally connected to the top of the vacuum can and the 1K pot. The NMR signal is transmitted using semirigid coax which is thermally connected at those same points using hermetic feedthroughs (Johnson, part number 142000003). The use of hermetic feedthroughs insures that the inner conductor is also thermally anchored. Care has been taken at these points to insure that thermal contraction of the semirigid coax is permitted. The polycarbonate sample space has a separate 0.025 inch outer diameter capillary that runs to the top of the probe. The gas in this capillary is also thermally anchored using copper bobbins at the top of the vacuum can and the 1K pot. The spectrometer is a commercial instrument designed by Tecmag Inc., Houston, TX. The spectrometer can be divided into five main components: NMRKIT II, Libra, PTS 500, American Microwave Technology (AMT) Power Amplifier, and G4 PowerMac. A continuous wave rf signal is sent from the PTS500 frequency generator to the NMRKIT. The NMRKIT mixes the signal with an intermediate frequency of 10.7 MHz heterodynee detection) and uses only the lower side band. The signal is amplified by the AMT Power Amplifier for a maximum pulse power of 400 W. It is important to note that the AMT is limited to a maximum frequency of 300 MHz. The inphase and quadrature components are recorded by the Libra with a time resolution of 100 ns. The software allows the returned signal to be viewed in either time or frequency space and to perform data manipulation. The phase of the transmitter pulse is cycled to decrease the signal due to coherent noise. Due to the phase cycling, all sequences must contain a multiple of four pulses. The computer communicates with the Libra using a set of control lines connected through a ribbon cable to an internal Tecmag card. The computer is connected to the NMRKIT II using a MiniDin serial cable. Since the G4 PowerMac does not have a serial port (the spectrometer was originally designed for an older PowerMac version), a Keyspan (Richmond, CA) adapter card was purchased to meet that need. Tecmag has provided software that allows pulse sequences to be created. In addition, Applescripts [54] can be written to access the spectrometer software and perform automated data acquisition. A schematic of the pulsed NMR setup is shown in Fig. 2.10. The crossed diodes (Hewlett Packard Shottky, part number 1N734) have been chosen with a 2 ns response time to insure proper operation up to 300 MHz. The purpose of the crossed diodes in series after the power amplifier are to prevent unwanted noise before or after the pulse. Use of a directional coupler instead of a "magic tee" is preferred for pulsed applications. A directional coupler concentrates the losses to the input side, which can be overcome with additional power, while a magic tee divides the losses between the input and output [55]. The NMR circuit is balanced to 50 Q prior to connection with the directional coupler using a HP8712 Network Analyzer. The cable length between the directional PTS 500 I Main output Various Synthin interconnects TECMAG NMRKIT Probe in TECMAG Libra AMT Power Amp Rfin Rfout Shottky 1N734 crossed diodes in series Crossed diodes in parallel MiniCircuits 15542 Directional Coupler Semirigid SMA Dewar Lakeshore CX1030SD Cemrox resistor balanced to 50 ohms using the network analyzer (includes cables) Sample Figure 2.10: Schematic diagram of the pulsed NMR setup. u Serial' Line Rfout Miteq AU 114 preamp GPIB TS530A AVS47 Twisted pair I Kohm I1/4W Metal film chip resistor Ribbon ,jtlc coupler and the preamp should be X/4 (speed of signal 2c/3, where c = speed of light). The impedance of a k/4 transmission line obeys the following relationship: ZINP *ZOU =50 Q (2.5) When the pulse reaches the directional coupler, ZOUT = 0, due to the crossed diodes. The effective input impedance will be infinite along the preamp side and 50 Q along the NMR circuit side. Consequently, all of the power will be directed toward the NMR circuit. After return from the NMR coil, ZouTr = 50 Q, and the pulse will see a 50 K2 preamp circuit impedance. Temperature control is achieved using the Picowatt system AVS47 resistance bridge and TS530A temperature controller. The G4 PowerMac controls these instruments using the GPIB bus and Labview software. Thermometer resistance set points are sent to the temperature controller, which adjusts the heater power to achieve those set points according to the time constants set by the software. The temperature controller is limited to 1 W of heater power which is sufficient for most applications. The thermometer is a Lakeshore CX1030SD Cernox resistor calibrated from 1.4 K to 100 K. The heater is a 1 kQ, 1/4 W, metal film chip resistor. Both the heater and thermometer are attached to the polycarbonate sample space with Emerson & Cuming 2850 epoxy. The Oxford superconducting magnet has an 88 mm bore diameter and a maximum center field of 9 T located 570 mm below the top flange. When at the maximum field of 9 T at the center of the solenoid, the field at the top flange is 0.06 T. There are four shim coils to improve the field homogeneity. This magnetic field variation is less than 6 ppm in a 10 mm diameter spherical volume at the maximum field. The main magnet and the shim coils are energized using Oxford power supplies. The power supplies communicate with the G4 PowerMac computer over the GPIB bus. It is important to shunt or "dump" the shim coils when charging the magnet to full field. Failure to perform this step will result in current being trapped in the shim coils, and this excess field may cause a quench. 2.8 Electron Spin Resonance The electron spin resonance (ESR) measurements were carried out by Dr. Talham's research group at the Department of Chemistry at the University of Florida. The ESR spectrometer is a commercial Brueker Xband (9 GHz) spectrometer. Temperature control down to 4 K was achieved with an Oxford ESR 900 Flow Cryostat. Further details are available by reviewing the dissertation of Garrett Granroth [56]. The ESR measurements discussed in this dissertation were made on both powder and single crystal samples. Typical ESR spectra consisted of a single broad line with a width of approximately 500 G. Exact calibration of the Lande g factor for each material was made by comparison with the center frequency of the free radical standard DPPH. 2.9 Pulsed FT Acoustic Spectroscopy Pulsed Fourier Transform acoustic spectroscopy experiments on liquid 3'He were conducted at the Microkelvin Laboratory at the University of Florida. The sample cell was placed in a Ag tower mounted on a Cu plate attached to the top of the Cu demagnetization stage of Cryostat No. 2. Further details of this cryostat design are described by Xu et al. [57]. A Ag powder heat exchanger in the Ag tower provides thermal contact for cooling the liquid sample. Miniature coaxial cables with a superconducting core and a CuNi braid were used between the tower and the 1K pot. Stainless steel semirigid coaxial cables were used from the 1K pot to the room temperature connectors. Above 40 mIK, the temperature was measured using a calibrated ruthenium oxide (RuO2, Dale RC550) resistor which has approximately a 500 Q room temperature resistance [58]. The value of this Ru02 thermometer was measured using a Picowatt AC Resistance Bridge (Linear Research). A heater mounted on the nuclear stage supplied up to 1 jLW of thermal power. From 40 mK to 1 mK, the temperature was measured using a 3He melting curve thermometer. Below nominally 3 mnK, a Pt NMR thermometer (PLM3, Instruments of Technology, Finland) was used and calibrated against the 3He melting curve [59]. The pressure was determined using a strain gauge mounted next to the tower. Details of the Tecmag commercial spectrometer were discussed in the previous section. A cross sectional view of the sample cell, nominally a cylinder with a radius ofR = 0.3175 + 0.0010 cm, is shown in Fig. (2.11). In order to obtain a large frequency bandwidth, lowQ, coaxial LiNbO3 transducers were used and were separated by a 3.22 + 0.01 mm MACOR spacer and held in place with BeCu springs. The spacing was measured before the experiment and verified in situ by measuring the time delay between successive zero sound reflected pulses. The transducers had a fundamental resonant frequency of 21 MHz and were operated in four frequency windows: 812, 1625, 6070, and 105111 MHz. Figure 2.12 shows the power reflection coefficient vs. frequency for the Ag Cell Body ^ ^ ,,. BeCu Spring Upper Signal B S'rin Cable CuNi Shield  ..  L F  Upper Crystal \ MACOR spacer Lower Crystal CuNi Shield $ Lower Signal $ Cable Figure 2.11: Cross section of the 3He acoustic cell. 0.40 0.35 0.30 0.25 0.20 0.15 0.10 104 106 108 110 112 Frequency (MHz) 114 116 Figure 2.12: Frequency response functions of the 5'h harmonic for two LiNbO3 transducers at 0.3 K [39]. 5th harmonic of each LiNbO3 transducer at 0.3 K [39]. The overlap of the transmitting and receiving transducer bandwidths determines the useful frequency range at each harmonic window. For the case shown in Fig. 2.12, this operational range is approximately 3 MHz centered at 108 MHz. Similar results were obtained for the other frequency windows. Due to the highly structured resonance peaks in the frequency response function of each transducer, the resulting signal is structured. A schematic circuit diagram of the pulsed FT acoustic technique is shown in Fig. 2.13. The strain gauge and the 3He melting curve thermometer (MCT) were both monitored using a capacitance bridge (General Radio Company, type 1615A) and the output was sent to a PAR 124A lockin. The balance of each lockin was determined by measuring the function output voltage using a HP34401A multimeter. The multimeters communicated with the G4 PowerMac over the GPIB bus. The pulse output of the Tecmag spectrometer was sent to an inline attenuator before reaching the connection points on the dewar. Although the pulse output of the spectrometer can be adjusted, it is easier to consistently set the voltage level of the input pulse using an attenuator. Because the output level of the NMRKIT II was controlled manually by a knob, for repeatability it was set to the maximum of 13 dBm at all times. The attenuator was set to either 10 or 20 dB. The received signals from the sample cell were amplified approximately 20 dB by a Miteq AU1114 preamp. The Pt NMR thermometer signal was monitored using a TDS430A digital oscilloscope. Each Pt NMR signal trace was sent to the PowerMac over the GPIB bus and recorded for later analysis. Labview software was used to communicate with the GPIB instruments. However, the spectrometer was operated using GPIB cable t) 0 PAR 124A Lockin Funcon A Out Detector GRC Capacitance Bridge Unknown PAR 124A Lockin Functi A out i HP34401A SInput HP34401A Detector GRC Capacitance Bridge Unknown PTS 500 NMRKIT II Mai oupWt Probe in Attenuator Various interconnects Miteq I DC power AU1114 power  PreampF Dewar . PLM3 NMR Thermometer Pt wire NMR therm Figure 2.13: Schematic diagram of the pulse FT acoustic 3He spectroscopy technique. Libra TDS430A MCT Strain gauge commercial software provided by Tecmag. The synchronization between Labview and the Tecmag software was accomplished using Applescript routines (Appendix F). Typically the spectrometer data at each frequency averaged the results of 128 pulses with a 4 s wait step between each pulse. Each transmitter pulse was 0.4 uts (Fourier transform spectroscopy) or 4 ts (amplitude/time of flight acoustic spectroscopy), depending on the experiment, and the pulse power at the sample cell was estimated to be approximately 20 dBm. The waiting time between different frequencies was at least 8 minutes. The real and imaginary components were separately digitized at 10 M samples/s for a 2048 samples. Spurious signals, resulting from electrical crosstalk, appeared in the first microsecond of data. Before taking the FT, this region of the data was blanked (set to zero). In addition, for some experiments, echoes were eliminated by blanking to avoid adding spurious structure in the frequency spectrum. This blanking and subsequent FT were accomplished using Origin scripts (Appendix G). Calibration of the MCT involved several steps. In the first step, the 3He MCT pressure was changed using a standard zeolite absorption He pressure bomb (i.e. "dipstick") and the resulting pressure was measured using a Digiquartz transducer. Figure 2.14 shows the result of a capacitance vs. pressure calibration of the melting curve thermometer using this method. This step should be accomplished at a relatively warm temperature (150 mK) to decrease the amount of time needed to reach equilibrium when changing pressures. The calibration of the strain gauge was accomplished using a similar technique. Once the capacitance vs. pressure calibration is complete, a predetermined temperature vs. pressure curve is required. The experiments in this dissertation used the 3He melting curve of Wenhai Ni (Fig. 2.15 [59]), which is consistent with the Greywall scale [60]. 32 xpernmenial roinis Polynomial Fit L 31 0. 0)O/ C, M 30 CL, o 29 28  28 29 30 31 32 33 34 35 Pressure (bars) Figure 2.14: The capacitance vs. pressure of the MCT at 150 mK. The solid line is a fifth order polynomial fit to the data. By combining the capacitance vs. pressure calibration with the pressure vs. temperature relationship, we can convert capacitance into temperature. However, it is still necessary to obtain an absolute calibration of the temperature curve using fixed temperature points. In this experiment, TA and TN, were used for this purpose. These fixed points, in the 3He phase diagram, can be easily identified as changes in slope when slowly (50 gK/hr) sweeping temperature (see Figs. 2.16 A and B). The absolute calibration results in the yaxis of the temperature vs. capacitance curve being adjusted by a constant value to match the fixed points. During the two 3He experiments listed in this dissertation, the vertical adjustment was typically 0.5 gK. The 34 Solid Phase :N AB A ^ 33  C,) 32 1 10 100 T(mK) Figure 2.15: Melting curve of 3He as determined by W. Ni [59]. The superfluid 3He ordering transitions, A (2.505 mK), AB (1.948 mK), and the solid ordering transition N (0.934 mK), are marked with arrows. 31.760 i i '' i' 31.755 A 31.750 CU 31.745 C. CU I 31.7450 TA = 31.745 + 0.002 pF 31.735 31.730  13000 14000 15000 16000 17000 Time (s) 31.790 ' < B 31.788 g LL S31.786 M 31.784 TN = 31.7873 + 0.0005 pF ". o ",'%. 31.782 31.780 ,,, , 4000 5000 6000 7000 Time (s) Figure 2.16: Identification of the capacitance value of the temperatures, TA and TN, from plots of the MCT capacitance vs. time while slowly warming. Solid lines have been added as guides to the eye. Both figures are different sections of the same data set. final temperature vs. capacitance relationship is shown in Fig. 2.17. The solid line is a fit to a 7th order polynomial. This polynomial was used to convert the MCT capacitance values into temperature above TN. The Pt NMR thermometer was calibrated using the MCT by sweeping the temperature from 3 mK to 0.5 mK and recording both the MCT and Pt NMR thermometer. There was a wait of at least 5 minutes between each Pt NMR pulse so that the nuclear spins could relax. The MCT temperature was recorded before and after each Pt NMR trace and averaged. The entire temperature sweep would take approximately 12 hours. The temperature scale was checked by comparing Tc(P) [61] with the temperature where there was a dramatic crossover from high to low attenuation in the zero sound signal. A graph of MCT temperature vs. Pt NMR integrated signal is shown in Fig. 2.18. The digitization rate of the TDS430A oscilloscope was faster than the PLM3, and therefore each Pt NMR trace was read by the oscilloscope and integrated after the experiment by the Labview software. The solid line represents a fit to the data from TN (0.934 mK) to 1.5 mK using A MT~c (2.6) Mp, +B where TMCT is the MCT temperature, Mpt is the Pt NMR integrated signal, and A and B are fitting parameters. The temperature range for the fit was chosen so that the Pt NMR integrated signal was at least a factor of 10 above the noise. The 3He melting curve determined by W. Ni and coworkers [59] did not extend much above 322 mK (the minimum in the melting curve). At this temperature, a separate calibration was used and was based on measurements by Grilly et al. [62]. The resultant temperature calibration from the Grilly et al. scale was adjusted by a constant to match the value of the 3He melting curve at the minimum given by Ni et al. [59]. 31.70 31.72 31.74 31.76 Capacitance (pF) 31.78 Figure 2.17: The temperature vs. capacitance curve generated by combining the capacitance vs. pressure relationship in Fig. 2.14 with the 3He melting curve in Fig. 2.15. The curve has been adjusted by a constant to match the fixed temperature points, TA and TN. The solid line is 7th order polynomial fit. 47 i  3.0 E 2.4 U) , 1.8 E 1.2 (D 0 0.6 0.0 6II, I I 0 5 10 15 20 25 Pt NMR Integrated Signal (a.u.) Figure 2.18: The MCT temperature vs. Pt NMR integrated signal. The solid line is a fit to Eq. 2.6 from TN to 1.5 mK. CHAPTER 3 THEORETICAL TECHNIQUES In this chapter, two theoretical models are presented that were used to investigate the magnetic behavior of the low dimensional magnetic materials reported in this dissertation. In each section, the relevant theory as well as details of the software are discussed. In addition, the advantages and restrictions of each model are considered. The first section discusses the exact diagonalization method, which is essentially the calculation of the partition function for a cluster of spins. Although, this approach, in principle, is an exact calculation, the number of spins included with this technique is limited by the computing power. This restriction places limits on the temperature range over which these systems can be accurately modeled. With respect to the Hamiltonian, this method is flexible and can be applied to any cluster of interacting spins with only minor changes to the software. The second section considers the mapping of the ladder Hamiltonian onto the XXZ model. Although the XXZ model is exactly solvable, the mapping approximates the magnetic behavior since it only includes the low energy states of the ladder Hamiltonian. Nevertheless, this method can model the magnetic behavior of ladders at temperatures significantly below the thermal energy represented by the magnetic interactions. 3.1 Exact Diagonalization The partition function for an arbitrary system of discrete states is written as Z = exp( E' (3.1) where E, are the energy states of the system and kB is the Boltzmann constant. For a cluster of spins, the magnetization can be calculated in a straightforward manner once the partition function is known; i.e. g93S, Sexp E, M =E (3.2) .exp E ' where S, represents the total spin of each state. A cluster of two Ising S = 1/2 spins that interact with a single exchange is the simplest cluster to model. This system can be represented by a Hamiltonian, where J is the magnetic exchange, given by Asng = J S, S,, g uH S,. (3.3) 1=1 j=1,2 In Eq. 3.3, the symbol g represents the Lande g factor and /a is the Bohr magneton. The first and second terms represent the interaction between spins and the interaction with the magnetic field, respectively. Using Eqs. 3.2 and 3.3, we arrive at the magnetization for n moles of spins that are arranged as interacting Ising pairs; namely M= (nNlt B exp(g kT p BH (3.4)I 2 xp B+exp B + 2expd S. kT2kT) where NA is Avagadro's number. It is easy to see that this equation will behave properly in the limit of high temperature or field. We can use a similar method to calculate the magnetization for a system of Heisenberg spins. Again, for simplicity, we consider a system with only two interacting spins and a single exchange constant J. The Hamiltonian resembles Eq. 3.3, except that the spin operator is now a vector, is = J S, S,+i, g9B SH g (3.5) 1=1 i=1,2 In the Ising case, the spin basis states were also the eigenvectors of the spin operator. For the Heisenberg case, this is not true. We explicitly write the spin basis eigenvectors that represent the electron wave functions with either spin up, a), or spin down, 10). The spin operator S, can be divided into its components S, = Si' + Si + Sz. The components are also operators that act on the spin basis function according to the rules given in Table 3.1. Table 3.1: The spin operators acting on the spin basis functions. SIcX) SJB) & Y 12I) Y210) /210) 210) We apply the Hamiltonian, Eq. 3.5, onto the basis function for two spins to obtain a matrix representation for the energy states of that system. Omitting the field interaction term, this matrix may be written as: (aal (ap I (/iaj (9P I (fta\ Iaa) la/ ) J/4 0 0 0 0 J/4 J/2 I\a) 0 J/2 J/4 ifp) 0 0 0 J/4 =0. (3.6) The matrix is blocked according to the total spin value, S = 1, S = 0, and S = 1. Each matrix corresponding to a particular spin can be diagonalized individually. For the case of only two spins, it would be just as easy to diagonalize the entire matrix at once. However, for large clusters of spins, a great reduction in the necessary computing power is achieved by diagonalizing each total spin matrix individually. After determining the eigenvalues, the field interaction term may be included. For two spins, we obtain the eigenvalues: 3J/4, J/4, J/4 gpsH, and J/4 + guBH, which correspond to the familiar singlet and triplet states for two interacting S = 1/2 spins. The triplet states are degenerate in zero magnetic field, and we can display this graphically using the diagram sketched in Fig. 3.1. E S = M s = 0 MH= S~l^   M^=O S=0 M = 0 Ms = .1 Figure 3.1: A graphical depiction of the energy eigenvalues for a system of two S = 1/2 spins in a magnetic field showing the singlet and triplet states. Once we have the eigenvalues, it is trivial to plug them into Eq. 3.2 to obtain the magnetization. Again, for practical purposes, we have assumed n moles of spins arranged as dimer pairs, and have [ n g rexp g.H 2 nNA9gUB k+T (3.7) 2 exp+PIr exp grexp 1 L k T )k T YkkTT In the low field limit, Eq. 3.7 becomes the BleaneyBlowers equation [42] ; which can be written as nNAg2 1 (3.8) A3kBT l+ exp(Jl/k) 3 At high temperatures, Eqs. 3.7 and 3.8 become the S = 1/2 Curie law, namely nN 2 22 SSI S Ag B _nNS+l (3.9) 4kBT 3kBT Figure 3.2 shows a graph of molar magnetic susceptibility vs. temperature in a magnetic field of 0.1 T produced using Eq. 3.7 with an exchange constant of J = 12 K. Below the peak temperature of approximately 7 K, there is an exponential decrease in susceptibility due to singlet formation. The peak in the curve corresponds to the thermal energy (gap) needed to form triplets. This type of curve is common to low dimensional gapped magnetic systems. Using this approach, we could have calculated other thermodynamic quantities as well. For instance, using the partition function, Z, and the energy eigenvalues, E,, for two Heisenberg spins, we write the entropy [63] of this system as: 1 ( E + I exp2( E2 )( o'=e k 'k+to +exp = +lnZ + Z ep kT ln Z \ kBT)[kBT (3.10) Sp E T In )ep( E E4 )( _E4T nZ Z kBT kT ,Z Bt, kTk Figure 3.3 shows the entropy vs. temperature for a system of two Heisenberg spins that interact with an exchange constant J in a magnetic field of 0.1 T. In the high temperature limit, the entropy approaches ln(number of energy states) = ln(4), and in the low temperature limit, it approaches zero. 25 20 E0 E 15 E E 10 5 0 10 20 30 40 50 60 70 T(K) Figure 3.2: The molar magnetic susceptibility vs. temperature in a magnetic field of 0.1 T for dimer pairs of Heisenberg spins that interact with an exchange constant of J = 12 K. The curve was produced using Eq. 3.7. 1.50 ,' '' , 1.25 1.00 () G) > 0.75 0 L "C 0.50 0.25 J=12K 0.00 I I I I p I p p I p , 0 10 20 30 40 T(K) Figure 3.3: The entropy vs. temperature in a magnetic field of 0.1 T for a dimer pair of Heisenberg spins that interact with an exchange constant of J = 12 K. The curve was produced using Eq. 3.10. Calculating magnetization using the partition function is only trivial when the number of interacting spins, and hence the number of energy states, is small. This exact diagonalization method was used by Robert Weller [64] in 1980 to calculate the susceptibility for larger clusters (N > 12) of S = 1/2 magnetic spins. This method relies on mathematics that have long been understood, however, it was not a viable alternative until cheap computing power was available. Although this method can be easily scaled up to larger systems, the corresponding matrix size increases as the factorial of the number of spins. For N spins, the maximum matrix size of N or N choose N12, is such that w e N1 canobe mn/ ip y ta wN/2 when N = 12, the matrix size is 924 x 924. A matrix of this size can be manipulated by a desktop computer in a few minutes. For 20 spins, the maximum matrix size 184,756 x 184,756 and the computation quickly becomes impossible. At higher spin values, e.g. S = 1, the matrix size increases even faster. However, we have not utilized all possible symmetries of the problem. By considering geometric symmetry of the spins, we can reduce the problem computationally by several orders of magnitude. This method is referred to as the Lanczos algorithm [65,66]. Using this approach, the magnetization for a system of as many as 30 spins can be calculated. I did not use this method, and so I do not discuss the details here. Since the importance of the boundaries of a model system decrease with increasing system size, it is important to use as many spins as possible. In addition, a small number of spins can accurately describe a low dimensional material only as long as the correlation length does not exceed the total length of the system. At T = 0, for quasitwo dimensional systems, such as spin ladders, the correlation length can become infinitely long. For these reasons, this method will only give accurate results for temperatures that are the same order of magnitude as the exchange constants or higher, T > J. If the system size is too small, this method introduces erroneous plateaus in the magnetization curves as the temperature is lowered below the exchange constants. In this dissertation, the spins were arranged in either a ladder or alternating chain geometry. However, any arrangement consisting of interacting spins with exchange constants, J1, J2, ... JN, could have been used. Both the ladder and alternating chain model systems used 12 spins which were arranged in a ring to help alleviate the boundary problem. The programs were written in MATLAB and produced theoretical curves using the exact diagonalization method described above. I am grateful to Steve Nagler (ORNL) for his assistance in writing these programs, which have been included in Appendix H. Fitting the data involved three steps. First, the experimental curves were fit using a high order polynomial ( > 5). Second, the software would generate multiple curves over a preset parameter space. Each curve would be compared to the polynomial and the difference between the polynomial curve and the theoretical curve would be recorded as a chi2 value. The chi2 value is the sum of the square of the difference between each theoretical point and the polynomial curve. Finally, when the program was finished generating curves, the chi2 values would be searched to obtain the lowest value, hopefully corresponding to the best fit. It was beneficial to generate a curve using those final parameters to ensure that the theoretical curve matched the data. A typical parameter search generates approximately 300 curves and takes approximately 18 hours. It is possible to increase the efficiency of the process by allowing the software to choose the next parameters instead of blindly searching the whole parameter space. This procedure is described in the dissertation by Robert Weller [64]. However, this technique was abandoned as it tended to find local minima in the parameter space. 3.2 The XXZ Model During my investigation of low dimensional materials, it became necessary to produce low temperature (T << the lowest ladder exchange constant, e.g. J1 ; 4 K) magnetization curves for spins arranged in a ladder geometry. At the lowest experimental temperature of 0.7 K, there exists a feature in the data at half the saturation magnetization, Ms /2, that could not be modeled using the exact diagonalization method. As discussed in the previous section, the exact diagonalization procedure is increasingly inaccurate as the temperature is lowered below the exchange constants. In addition, the exact diagonalization method also introduced erroneous plateaus in the theoretical curves that resemble the feature at Ms /2. Therefore, another method was required to study the magnetization of ladder materials at low temperature. Chaboussant et al. [14] have previously created low temperature magnetization curves for the ladderlike material, Cu(Hp)Cl, by mapping the ladder Hamiltonian onto the XXZ model which was initially solved by H. Bethe [21]. The thermodynamics of the XXZ model have been completely described by Takahashi and Suzuki [67]. I begin with the ladder Hamiltonian including the field interaction term N12 N2 N Ladder =J 22 '92, +Jl 1, S,+2 + g91B S, (3.11) t=1 i=1 =1 We can consider only the restricted Hilbert space composed of a singlet S = 0, ms = 0) and the lowest energy triplet IS = 1,ms = 1) on each rung. These are the two lowest energy states (see Fig. 3.1) and therefore the most populated. This approximation is valid since we are interested in the critical region where the magnetic field is on the order of J. We can rewrite the effective Hamiltonian on this restricted Hilbert space as N/2 "r+1 + SrYl + Z. rZ)+Hff Sz, (3.12) r=1 2 r=l where the effective field is given by Heff =Ji + gBH' (3.13) and Sr now represents the total spin of rung r. It should be noticed that this Hamiltonian is completely symmetric around Heff = 0. Hence, any quantities computed from this Hamiltonian will also be symmetric around this point. This Hamiltonian (Eq. 3.12) can be identified as the effective S = 1/2 XXZ model. The thermodynamics of the XXZ model have been reduced to a set of nonlinear differential equations by Takahashi and Suzuki [67]; such that In r(x) = 33 Js(x) + s(x) ln(l + u(x)) (3.14) kBT u(x) = 2B(x) cosh 3BHff +K 2(x) (3.15) and ln t(x) = s(x) In(1 + (x)), (3.16) 1 I wheres(x)=sechIx, is the convolution product and q(x), u(x), and K(x) are 4 '2 parameters in the model. These equations must be solved iteratively from a known solution for each value of the temperature and magnetic field. In this case, the known solution was rq(x) = 3 and Kc(x) =2 for J11 = 0 and Heff = 0. The convolution products are calculated as discrete integrals using 200 points. Since, the hyperbolic secant function x and hence, s(x), decays quickly, was used instead of x in the argument to increase the 10 resolution. The convolution product must therefore be divided by 10 as well. Typically 10 iterations were sufficient to reach equilibrium with a 200 point resolution. Once a stable solution is reached, the free energy per spin can be calculated using F J F 11kBTInc(O). (3.17) N 2 dF The magnetization is proportional to M dH The curves produced must be normalized so that the maximum overall magnetization is 1. Curves generated this way using J11 = 0 were compared to the exact dimer results, Eq. 3.7, to ensure that the method was correct. It should be noted that there are three typographical mistakes in the treatment ofChabbousant et al. [14]: a sign error and a missing s(x) factor in Eq. 3.14 (or Eq. 30 as listed in the Chaboussant et al. paper) and a factor of 2 difference in Eq. 3.17 (or Eq. 32 as listed in the Chaboussant et al. paper). These integral equations were solved using MATLAB software. For reference, these programs are included in Appendix H. CHAPTER 4 STRUCTURE AND CHARACTERIZATION OF A NOVEL MAGNETIC SPIN LADDER MATERIAL Magnetic spin ladders are a class of low dimensional materials with structural and physical properties between those of 1D chains and 2D planes. In a spin ladder, the vertices possesses unpaired spins that interact along the legs via J11 and along the rungs via J/, but are isolated from equivalent sites on adjacent ladders, i.e. interladder J' << J\1, J. Recently, a considerable amount of attention has been given to the theoretical and experimental investigation of spin ladder systems as a result of the observation that the microscopic mechanisms in these systems may be related to the ones governing high temperature superconductivity [2,6]. The phase diagram of the antiferromagnetic spin ladder in the presence of a magnetic field is particularly interesting. At T = 0 with no external applied field, the ground state is a gapped, disordered quantum spin liquid. At a field Hcl, there is a transition to a gapless Luttinger liquid phase, with a further transition at Hc2 to a fully polarized state. Both Hci and Hc2 are quantum critical points [1]. Near Hci, the magnetization has been predicted to obey a universal scaling function [68]. Using a symmetry argument, this universal scaling can also be shown to be valid at Hc2. Until now, this behavior has not been observed experimentally. A number of solid state materials have been proposed as examples of spin ladder systems, and an extensive set of experiments have been performed on the compound Cu2(1,4diazacycloheptane)Cl4, Cu2(C5Hi2N2)2C4, referred to as Cu(Hp)Cl [7]. The initial work identified this material as a twoleg S = 1/2 spin ladder [714]. Although quantum critical behavior has been preliminarily identified in this system near Hci, this assertion is based on the use of scaling parameters identified from the experimental data rather than the ones predicted theoretically [13,14]. Furthermore, more recent work has debated the appropriate classification of the low temperature properties [1519]. Clearly, additional physical systems are necessary to experimentally test the predictions of the various theoretical treatments of twoleg S = 1/2 spin ladders. Herein, we report evidence that identifies bis(piperidinium)tetrabromocuprate(II), (CsHi2N)2CuBr4 [20,69], hereafter referred to as BPCB, as a twoleg S = 1/2 ladder that exists in the strong coupling limit, JI/J, > 1. Highfield, lowtemperature magnetization, M(H < 30 T, T > 0.7 K), data of single crystals and powder samples have been fit to obtain J = 13.3 K, J\ = 3.8 K, and A 9.5 K, i.e. at the lowest temperatures finite magnetization appears at Hci = 6.6 T and saturation is achieved at Hc2 = 14.6 T. An unambiguous inflection point in the magnetization, M(H,T = 0.7 K), and its derivative, dM/dH, is observed at half the saturation magnetization, Ms/2. This behavior has not been detected in Cu(Hp)Cl [810]. The Ms/2 feature cannot be explained by the presence of additional exchange interactions, e.g. diagonal frustration JF, but is well described by an effective XXZ chain, onto which the original spin ladder model (for strong coupling) can be mapped in the gapless regime HcI < H < Hc2. After determining Hci and with no additional adjustable parameters, the magnetization data are observed to obey a universal scaling function [68]. This observation further supports our identification of BPCB as a twoleg S = 1/2 Heisenberg spin ladder with J'<< J . This chapter is divided into six sections. In the first section of this chapter, I will discuss the structure and synthesis of BPCB. The second and third sections report the results of low field susceptibility and magnetization measurements, respectively. The fourth section presents the highfield magnetization work performed at the National High Magnetic Field Laboratory, while section five details the universal scaling behavior of BPCB. Results from the neutron scattering experiments, performed at Oak Ridge National Laboratory, are provided in section six. 4.1 Structure and Synthesis of BPCB The crystal structure of BPCB has been determined to be monoclinic with stacked pairs of S = 1/2 CuE ions forming magnetic dimer units [20]. The CuBr42 tetrahedra are cocrystallized along with the organic piperidinium cations so that the crystal structure resembles a twoleg ladder, Fig. 4.1. The rungs of the ladder are formed along the c*axis (the c*axis makes an angle of 23.4 with the ac plane and the projection of the c*axis in the ac plane makes an angle of 19.8 with the caxis) by adjacent flattened CuBr42 tetrahedra related by a center of inversion. The ladder extends along the aaxis with 6.934 A between Cu2' spins on the same rung and 8.597 A between rungs. The three dimensional crystal structure of BPCB, including the organic cations, viewed along the E axis is shown in Fig. 4.2. The atomic positions have been taken from the xray Figure 4.1: A schematic diagram of the crystal structure of BPCB viewed down the [010] axis as determined by Patyal et al. [20]. The magnetic exchange between S = 1/2 Cu2+ spins is mediated by nonbonding BrBr contact. The two primary exchange models considered were a ladder model, with parameters J and J\\, and alternating chain model, with parameters J1 and J2. In the ladder model it is possible to include a frustration exchange, J.a 0 a SCu ON O Br Q C 0 H Figure 4.2: The crystal structure of BPCB viewed along the c axis. The ladder direction is along the aaxis. The c*axis, rung direction, makes an angle of 23.4 with the ac plane and the projection of the c*axis in the ac plane makes an angle of 19.8 with the caxis. The solid lines indicate the unit cell. scattering data of Patyal et al. [20] and verified in the neutron scattering studies (see Section 4.6). The hydrogen positions have been calculated using symmetry arguments. The ladder structure is viewed edgewise (dark spheres) in Fig. 4.2, and it is apparent that the rungs of the ladder extend out of the ac plane. Adjacent ladders are separated by 12.380 A along the caxis and 8.613 A along the baxis. Although the baxis separation is approximately the same as the rung separation, it is unlikely that the organic cations provide significant superexchange between ladders along the baxis and hence the magnetic exchange between ladders is expected to be small (J' << J11). The magnetic exchange, Ji, between Cu2 spins on the same rung is mediated by the orbital overlap of Br ions on adjacent Cu sites. The exchange between the legs of the ladder, J1, is also mediated by somewhat longer nonbonding (Br Br) contacts and possibly augmented by hydrogen bonds to the organic cations. A diagonal exchange, JF, is possible, although it should be weak (JF<< J ), and since the diagonal distances (9.918 A vs. 12.066 A) are not equal, only one JF was considered in our analysis. Shiny, black crystals of BPCB were prepared by slow evaporation of solvent from a methanol solution of [(pipdH)Br] and CuBr2, and milling of the smallest crystals was used to produce the powder samples. The stochiometry was verified using carbon hydrogennitrogen analysis [70]. In addition, deuterated single crystal and powder samples were produced and used in neutron scattering studies performed at the High Flux Isotope Reactor at Oak Ridge National Laboratory. The protonated BPCB material has a molecular weight of 583.49 g/mol and a density of 2.07 g/cm3. The previous study by Patyal et al. [20] reported the Lande g factor along all three single crystal axes for BPCB as g(aaxis) = 2.063, g(baxis) = 2.188, and g(caxis) = 2.148. ESR measurements at a frequency of 9.272 GHz were performed on a powder sample of BPCB at room temperature and on a single crystal sample along the caxis from 20 to 300 K. The room temperature results were completely consistent with the previously reported data, i.e. g(powder) = 2.13 [20] and g(caxis) = 2.148. At all temperatures, the EPR signal consisted of a single broad line approximately 500 G wide. Figure 4.3 is a sample derivative trace, d//dH of the EPR signal intensity at 75 K. By plotting the area under the EPR intensity curve I(H) as a function of temperature, we obtain the graph shown in Fig. 4.4. This graph closely resembles the susceptibility curve of BPCB. The Lande g factor measured along the caxis decreases monotonically from 2.148 to 2.141 from 300 K to 20 K as shown in Fig. 4.5. This magnitude of change in the Lande g factor will not adversely affect the quality of the magnetization fits, which assumed g to be the temperature independent value of 2.148 along the caxis. 4.2 Low Field Susceptibility Measurements Although the crystal structure of BPCB resembles a ladder, other possible exchange pathways can produce similar results from macroscopic measurements [71,72]. Initially, an additional magnetic exchange model, i.e. alternating chain, was considered during the analysis of the magnetization data. Figure 4.1 shows the two primary exchange pathways considered, i.e. an alternating chain with exchange constants J\ and J2, and a ladder with exchange constants JL and J\. The Hamiltonians for N spins that interact with 20 15 10 5 *' 0 C6 5 10 15 20 2000 2500 3000 3500 4000 Field (G) Figure 4.3: The first derivative of the EPR signal intensity, dI/dH, single crystal (m = 18.6 mg) at a frequency of 9.272 MHz and 75 K. 30 CO S25 0) n 20 0 uJ 15 CL cQ. E 10 < "o 0) r.5 0) 0) CE 0 50 100 150 T (K) 4500 vs. field for a BPCB 200 250 Figure 4.4: The integrated EPR signal intensity vs. temperature for a BPCB single crystal (m = 18.6 mg) at a frequency of 9.272 MHz. I I I I I ' 0 H IIH caxis 0 0 0 0 o 0 0 0 000000 0 .O 0 0 0 00 I 0; 0* o ~ 0 100o 2.150 2.148 2.146 2.144 2.142  2.140 1 I I 1 I 0 50 100 150 T(K) 200 250 300 Figure 4.5: The Lande g factor along the caxis of a single crystal sample of BPCB (m =18.6 mg), determined by the EPR line center frequency at 9.272 MHz, vs. temperature. The room temperature value ofg agrees with the value reported earlier [20] (g = 2.148.). The temperature dependence of the Lande g factor is most likely due to the thermal contraction of the lattice. either a ladder or alternating chain exchange can be written as N12 N2 N 0,dr = JL j 2,, *s,2 + J1 g, S,+2 + gBLfH 1=1 i=1 1=1 N12 NI21 N * .n = JI g2,, .2, +J2 2S2'g21,+ +gu S, .Hi , I=] i=1 i=1 (4.1) (4.2) respectively. Low field (H < 5 T) magnetic measurements were performed using a Quantum Design SQUID Magnetometer. The low field, 0.1 T, magnetic susceptibility, X, of a BPCB powder sample (m = 166.7 mg) is shown as a function of temperature in Fig. 4.6. The general shape of the curve is typical of low dimensional magnetic systems, and more specifically, it possesses a rounded peak at approximately 8 K and an exponential dependence below the peak temperature. No evidence of long range order was observed at the minimum temperature of 2 K. A temperature independent diamagnetic contribution of Xdiam = 2.84 x 104 emu/mol was subtracted from the data in Fig. 4.6. The diamagnetic contribution is the sum of the core diamagnetism, estimated from Pascal's constants to be 2.64 x 104 emu/mol, and the background contribution of the sample holder. For all of the susceptibility data in this chapter, a diamagnetic contribution has been subtracted from the data and although no Curie impurity term was subtracted, in some cases a S = 1/2 Curie contribution was included in the fit. This Curie contribution is typically 2 % of the total number of S = 1/2 spins. In Fig. 4.6, the susceptibility data have been fit (solid line) using a high temperature expansion by Weihong et al. [11] based on the ladder Hamiltonian, Eq. 4.1, providing the exchange constants of J = 13.1 + 0.2 K and JH = 4.1 + 0.3 K. The first 14 terms of the expansion (up to fourth order in J/T) were used for the fitting procedure. These same data were also fit (Fig. 4.7) using the method of Chiara et al. [7], which assumes the alternating chain Hamiltonian, Eq. 4.2, providing the exchange constants of J1 = 13.74 + 0.03 K and J2 = 5.31 + 0.04 K. The fitting method of Chiara et al. [7] includes the data below the peak in the susceptibility and consequently is more accurate than the high temperature series expansion method of Weihong et al. [11]. However, although there are differences in the values of the two sets of exchange parameters, both cases provide physically plausible results. Therefore, using only the low field X(T) data, we are unable to distinguish between the ladder and alternating chain model. Similar results are obtained for BPCB single crystal samples. The magnetic susceptibility vs. temperature for BPCB single crystal (m = 46.9 mg) is shown in Fig. 4.8. The sample was zero field cooled to 2 K and then measured in a field of 0.1 T parallel to the aaxis. A small constant diamagnetic contribution of Xdiam = 3.16 x 104 emu/mol has been subtracted. Incidentally, the diamagnetic contribution for the single crystal samples is larger because more diamagnetic support material was used to ensure proper crystal alignment during the measurement. The solid line represents the best fit using a high temperature series expansion by Weihong et al. [11] with the parameters J1 = 12.9 + 0.3 K and J11 = 3.8 0.3 K. Figure 4.9 shows this same data fit using the method ofChiara et al. [7] with the results J, = 13.66 + 0.14 K and J2 = 5.57 + 0.12 K. Analogous to the ladder and alternating chain fits of the powder susceptibility data, both fitting methods generate plausible results. In addition, although the exchange constants from the single crystal and powder samples do not quite agree within uncertainty, the fitting results appear to be self consistent for both methods. The fitting results for powder and single crystal samples along all three axes have been summarized in Tables 4.1 and 4.2. The choice of 0.1 T as the applied field in the susceptibility measurements was not arbitrary. Figure 4.10 shows the molar magnetic susceptibility for BPCB single crystal with H 11 aaxis and applied fields of 1, 2, 3, 4, and 5 T. At high temperatures (T > A/kB), 25 1 1 1 1 1 1 1 1 0 O Experimental Data S  Ladder Fit: 20 SJQ j =13.1 + 0.2 K 0 J1= 4.1+0.3K E 15 8 ^sImpurity Conc. =1.2 0.2 % 0 0 o 10 :0 0 0 0 50 o BPCB Powder H H=0.1 T 0 1 1 I 0 20 40 60 80 100 T(K) Figure 4.6: The molar magnetic susceptibility vs. temperature for BPCB powder (m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T. A small constant diamagnetic contribution of Xdiam = 2.84 x 104 emu/mol has been subtracted. The solid line represents the best fit using a high temperature series expansion by Weihong et aL. [11] with parameters J = 13.1 + 0.2 K and JI = 4.1 + 0.3 K. 25 1 1111 0 Experimental Data Alternating Chain Fit: 20 J, = 13.74 0.03 K I J2 = 5.31 0.04 K E 15 Impurity Conc. = 0.9 0.1 % E 5 BPCB Powder H0 =0.1 T 0 I i ~ 0 20 40 60 80 100 T(K) Figure 4.7: The molar magnetic susceptibility vs. temperature for BPCB powder (m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T. A small constant diamagnetic contribution of Xdiam = 2.84 x 104 emu/mol has been subtracted. The solid line represents the best fit using the method of Chiara et al. [7] with parameters J, = 13.74 0.03 K and J2 = 5.31 0.04 K. 2 5 1 II I 1 I 11 1 1 0 Experimental Data  Ladder Fit using: 20 = J =12.9+0.3K ',J= 3.8+ 0.3K 5 BPCB Single Crystal ~ H=0.1 TIaaxis I 0 15I Impuit Ioc 3.I . E ?20 40 60 80 10010 5 BPCB Single Crystal H=0.1 TiI1 aaxis 20 40 60 80 100 T(K) Figure 4.8: The molar magnetic susceptibility vs. temperature for a BPCB single crystal (m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T parallel to the aaxis. A small constant diamagnetic contribution of Xdiam = 3.16 x 104 emu/mol has been subtracted. The solid line represents the best fit using a high temperature series expansion by Weihong et al. [11] with parameters J = 12.9 +0.3 K and J, = 3.8 + 0.3 K. 25 1 1 1 i i' 0 Experimental Data S Alternating Chain Fit: 20 % J,=13.660.14K 557= 5.57 0.12 K 15 Impurity Conc. = 3.3 0.4 % E (D 10 BPCB Single Crystal H =0.1 T jj aaxis 0 1 1 1 1 I 0 20 40 60 80 100 T(K) Figure 4.9: The molar magnetic susceptibility vs. temperature for a BPCB single crystal (m 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T. A small constant diamagnetic contribution of Xdiam = 3.16 x 104 emu/mol has been subtracted. The solid line represents the best fit using the method of Chiara et al. [7] with parameters J, = 13.66 + 0.14 Kand ,/2 5.57 + 0.12 K. 25 .. I ' 20 0 E 15 E ____1 Tesla 10 o 2 Tesla SaA 3 Tesla 5 v 4 Tesla o 5 Tesla H aaxis 0 i I I I a 0 5 10 15 20 T(K) Figure 4.10: The molar magnetic susceptibility vs. temperature for a BPCB single crystal (m = 14.2 mg). The sample was zero field cooled to 2 K and then measured in the fields of 1, 2, 3, 4 and 5 T. A small constant diamagnetic contribution of4.06 x 104 emu/mol has been subtracted. The data collapse onto a single curve at high temperatures (T> A/kB) indicating the approximately constant susceptibility at high temperature. Table 4.1: The alternating chain parameters, J, and J2, determined from fitting the susceptibility vs. temperature data using the method of Chiara et al. [7]. mass (mg) J, (K) J2 (K) Impurity Conc. (%) powder 166.7 13.74 + 0.03 5.31 + 0.04 0.9 + 0.1 aaxis 46.9 13.66+0.14 5.57+0.12 3.3+0.4 baxis 13.6 12.76 0.10 5.18 + 0.10 4.8 + 0.5 caxis 24.4 13.65+0.10 6.05+ 0.10 3.8+0.4 Table 4.2: The ladder parameters, J1 and J determined from fitting the susceptibility vs. temperature data using the high temperature expansion by Weihong et al. [11]. mass (mg) Jj_ (K) J11 (K) Impurity Conc. (%) powder 166.7 13.1 + 0.2 4.1 + 0.3 1.2 0.2 aaxis 46.9 12.9 0.3 3.8 + 0.3 3.3 0.4 baxis 13.6 13.4 0.3 3.7 + 0.2 7.5 1.0 caxis 24.4 13.3+0.4 3.8+0.5 3.0 1.0 all of the susceptibility data collapse onto a single curve demonstrating the approximately constant susceptibility. However, below the peak, the susceptibility curves begin to deviate. In addition, the peak temperature decreases with increasing field. At fields above the gap, A/gJIB 6.8 T, the peak in the susceptibility curve should disappear entirely. Although, a larger applied field would increase the signal to noise ratio of our measurements, we would measure the field and temperature dependence of the sample simultaneously, thus complicating our analysis. The inverse susceptibility as a function of temperature for BPCB powder (m= 166.7 mg) is shown in Fig. 4.11, and similarly, the inverse susceptibility vs. temperature for a BPCB single crystal with H 11 aaxis (m = 46.9 mg) is shown in Fig. 4.12. At temperatures above the spin gap, T >> A 8 K, the inverse molar susceptibility should be linear with temperature. The slope of this line can be determined by inverting the S = 1/2 Curie law, I =(T +e) 4kB (4.3) x NAg2J '2 where NA is Avagadro's number. The value of 0 is somewhat more difficult to calculate. Johnston et al. [41] have written a high temperature series expansion, by inverting a susceptibility expansion from Weihong et al. [11], for the inverse susceptibility in terms of J1 and J11 containing 42 nonzero terms. The first four terms of that series can be written as 1 B 1+(2J,, +J)2+(2J2 +j 2) +(2J3+J13 +'"., (4.4) Z NAg2PB2 2 2 3 1 where x = . By comparing Eqs. 4.3 and 4.4, we can write the Curie temperature, 0, 2T to second order in T as Er(2J,+J) (2J,'+2 J ) (2Ji 3 + 3) (4.5) e= 4+... (4.5) 4 8T 24T2 0.8 1 1 1 1 1 1 1 1 1I o Experimental Data 0.7 Linear Fit (100 K to 300 K) 0 slope = 2.313 0.002 (mol/K emu) 0. E )=4.9 0.3 K E 0.5 o 0.4 0 0 E0.3 0 T 0.2 0.1 : BPCB Powder 0.1 =0.1T 0.0 1= 1 0 50 100 150 200 250 300 T(K) Figure 4.11: The inverse molar magnetic susceptibility vs. temperature for BPCB powder (m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T. A small constant diamagnetic contribution of Xdia.m = 2.84 x 104 emu/mol has been subtracted. The solid line is a linear fit over the temperature range from 100 K to 300 K giving a slope of 2.313 + 0.002 mol/(emu K) and 0 = 4.9 + 0.3 K. 0.8 1 o Experimental Data 0.7 Linear Fit (100 Kto 300 K)  slope = 2.50 0.01 (mol/emu K) 0.6 E=5.81.2K a E 0.5 a) Vo 0.4 0 E 0.3  0.2 0 FBPCB Single Crystal 01 H = 0.1 T 11 aaxis  0.0 1 1 1 1 1 0 50 100 150 200 250 300 T(K) Figure 4.12: The inverse molar magnetic susceptibility vs. temperature for a BPCB single crystal (m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T parallel to the aaxis. A small constant diamagnetic contribution of Xdiam = 3.16 x 104 emu/mol has been subtracted. The solid line is a linear fit over the temperature range from 100 K to 300 K with a slope of 2.50 + 0.01 mol/(emu K) and 0 = 5.8 +1.2 K. After plugging in nominal values for our exchange constants and comparing the magnitude of each term, it is clear that we only need to consider the first term as long as our linear fit begins above approximately 100 K. The solid lines in Figs. 4.11 and 4.12 represent linear fits over the temperature region from 100 K to 300 K. The slope and Curie constant for these two samples, as well as BPCB single crystal specimens along the b and caxis, are listed in Table 4.3 and compared to the theoretical result from Eq. 4.5. The exchange constants, J and J, used in Eq. 4.5 have been taken from Table 4.2. Table 4.3: The slope of the inverse molar susceptibility, l/X, vs. temperature and the Curie temperature, E, determined from a linear fit to the data from 100 K to 300 K. The theoretical slope, 4kB/g2 B2NA (Eq. 4.3) and Curie constant, (J + 2J,,)/4 (Eq. 4.5), are included for comparison. The parameters used in the calculation of the Curie constant were taken from Table 4.2. slope 4kBg2,UB2NA 9 (K) (J,1 + 2JI1)/4 (K) (mol/K emu) (mol/K emu) powder 2.312 + 0.002 2.35 + 0.02 4.9 0.3 5.3 0.3 aaxis 2.50 0.01 2.51 0.02 5.8 + 1.2 5.1 0.4 baxis 2.24 + 0.02 2.23 0.02 3.9 + 1.1 5.2 0.3 caxis 2.29 + 0.02 2.31 0.02 4.6 + 1.6 5.2 + 0.6 The molar magnetic susceptibility multiplied by temperature vs. temperature for BPCB powder (m = 166.7 mg) is shown in Fig. 4.13. The solid line is the theoretical high temperature Curie value for XT of 0.425 (emu K)/mol. At temperatures above approximately 100 K, XT approaches a horizontal line with the Curie value, indicating paramagnetic behavior. Below 100 K, the value of XT decreases when lowering the temperature, indicating antiferromagnetic behavior. If we had not subtracted the appropriate diamagnetic contribution, XT at high temperatures would have a nonzero slope. 0.5 0.4 0 E 0.3 :3 E S0.2 0.I1 0.1 0 50 100 150 200 250 300 T (K) Figure 4.13: The molar magnetic susceptibility multiplied by temperature vs. temperature for BPCB powder (m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of 0.1 T. A small constant diamagnetic contribution of Xdiam = 2.84 x 104 emu/mol has been subtracted. The solid line is the theoretical Curie value of 0.425 (emu K)/mol. 4.3 Low Field Magnetization Measurements The magnetization measurements were performed with a commercial SQUID magnetometer, which can apply a maximum field of 5.0 T. This limitation is particularly unfortunate in the case of BPCB, since the spin gap, expected from the susceptibility measurements, is approximately 7 T. The spin gap is calculated to first order as A/kB = JLJ11 (4.6) for the ladder exchange or A/kB = Ji J2 (4.7) when considering an alternating chain model [41]. For each measurement listed, the samples were zero field cooled from 300 K. The overall form of the low field magnetization measurements can be understood by examining the behavior of two Heisenberg S = 1/2 spins with a single exchange constant J. The molar magnetization of such a system can be calculated using Eq. 3.7. At low temperatures (T << A/kB), the magnetization will remain zero until the gap field is reached (H = A/g/iB) and then afterwards have a positive first derivative. At high temperatures (T > H and T >> J), Eq. 3.7 becomes approximately linear with applied field. Between these two temperature extremes, the magnetization will have a small positive first derivative (compared to the paramagnetic result of dM(H)/dH < NAg2JUB2/4kBT) and a positive second derivative. The molar magnetization vs. field for BPCB powder (m = 166.7 mg) at a temperature of 2 K is shown in Fig. 4.14. The solid line is a fit using Eq. 3.7 with an exchange constant ofJ = 12.6 + 0.1 K. The general shape of the curve matches the data commendably considering the simplicity of the model. This agreement is an indication that, regardless of which magnetic model is correct, BPCB exists in the strongly coupled limit, i.e. JJ/J\ >> or JI/J2 >>l. To facilitate fitting the magnetization data more accurately, we used the exact diagonalization technique discussed in Chapter 3. For all of the fits that are discussed, unless otherwise noted, the calculations used 12 spins arranged in a ring. The molar magnetization as a function of field for a BPCB single crystal (m = 166.7 mg) at the temperatures of 2, 5, and 8 K is shown in Fig 4.15. The solid line at 2 K represents the best fit using the 12 spin exact diagonalization procedure and an alternating chain Hamiltonian, Eq. 4.2. The magnetization curves at 5 K and 8 K were produced using the same best fit exchange parameters derived from the 2 K data, J1 = 13.20 + 0.05 K and J2 = 5.20 + 0.05 K. The experimental curve at 2 K is reproduced extremely well by this fitting technique. However, at higher temperatures, using the same exchange constants, the theoretical and experimental curves begin to deviate. The same fitting procedure can be applied to the data using the ladder Hamiltonian, Eq. 4.1. Figure 4.16 shows the same data fit using the exact diagonalization with a ladder Hamiltonian. The ladder best fit exchange parameters are JL = 12.75 + 0.05 K and J = 3.80 0.05 K. Contrary to the case for the alternating chain Hamiltonian, the higher temperature experimental and theoretical magnetization curves agree using the same exchange constants at higher temperatures. This agreement suggests that the data may be more accurately modeled using the ladder Hamiltonian. There are two reasons why the error in fitting the low field magnetization measurements is relatively large compared to the error in fitting the susceptibility measurements. First, a small discrepancy in the mass or temperature measurement will 400 1 1i i i 1i BPCB Powder T=2K K 300 30 0 Experimental Points  Dimer Fit ( J=12.6+0.1K S200 E 100 0 0 1 2 3 4 5 H(T) Figure 4.14: The molar magnetization vs. field for BPCB powder (m = 166.7 mg) at a temperature of 2 K. The solid line represents the best fit to Eq. 3.7, the molar magnetization for pairs of S = 1/2 Heisenberg spins with a single exchange constant of J 12.6 + 0.1 K. 1000 . , 0 T=2K A T=5K 800 V T=8K  Exact Diagonalization , E 0 J= 13.20 K "600 ir .9600 J2= 5.20 K , E 01 400 200 SH II aaxis " 0 0 1 2 3 4 5 H(T) Figure 4.15: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg) with H I aaxis at the temperatures of 2, 5, and 8 K. The solid lines are produced using the best fit parameters, J, = 13.20 0.05 K and J2 = 5.20 + 0.05 K, determined from the 2 K data using the 12 spin exact diagonalization procedure and an alternating chain Hamiltonian. At higher temperatures, using the same exchange constants, the theoretical and experimental curves begin to deviate. 1000 i 1 1 1 1 x 0 T=2K A T=5K 800 V T=8K " Exact Diagonalization 0 E 00 J= 12.75 K (.9 Jll= 3.80 K E S 400 200 LeIH aaxis  01 0 1 2 3 4 5 H(T) Figure 4.16: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg) with H II aaxis at the temperatures of 2, 5, and 8 K. The solid lines represent the best fit to the 2 K data, using the 12 spin exact diagonalization procedure and a ladder Hamiltonian. The ladder best fit exchange parameters are J, = 12.75 + 0.05 K and J: = 3.80 + 0.05 K. The higher temperature experimental and theoretical magnetization curves agree using the same exchange constants. result in a large difference in the best fit exchange constants. Experimentally, it is easier to hold the magnetic field constant than the temperature. Second, because we did not reach the saturation magnetization, or even the critical field Hci, absolute calibration of the mass or the critical fields, Hci and Hc2, is not possible. Determination of the exchange constants from the low field magnetization data relies on the absolute magnetization values. On the other hand, the susceptibility data contains a maximum with a unique temperature dependence that is sensitive to the values of the exchange constants. Figure 4.17 shows the 2 K magnetization data from the previous two figures. The solid and dotted lines represent the exact diagonalization fits extended to 20 T using the alternating chain and ladder Hamiltonians, respectively. At a temperature of 2 K, it should be possible to distinguish between the two models by continuing the magnetization measurements to high field. By lowering the temperature to 1 K, this difference will become more pronounced (see Fig. 4.18). Figure 4.19 shows the magnetization data for Cu(Hp)Cl at 0.42 K [14]. The solid line represents the best fit using the exact diagonalization procedure and an alternating chain Hamiltonian with exchange constants J1 = 13.20 0.05 K and J2 = 2.3 0.05 K. The first derivative of the data and theoretical prediction are provided in the inset. The asymmetry of the curve is obvious from Fig. 4.19 and is a result of the asymmetry in the magnetic exchange (see Fig. 4.19), i.e. J,11 J2. These results suggest that Cu(Hp)Cl, which has been considered a twoleg ladder material [14], is better described by an alternating chain model. 6000 . I i . 1 i . . 0 Experimental Data  Exact Diagonalization using / 5000 Alt. Chain Hamiltonian / J1= 13.20K // 75 4000 J2 = 520 K E  Exact Diagonalization using // 6 Ladder Hamiltonian // J 3000 J=12.75 K E J = 3.80 K / Q^/ 2000 1000 H II aaxis rT=2K 0 I . . I . . I , , 0 5 10 15 20 Field (T) Figure 4.17: The molar magnetization vs. field for a BPCB single crystal vs. field from 0 to 5 T. The solid and dotted lines represent the exact diagonalization fits extended to 20 T using the alternating chain and ladder Hamiltonians, respectively. At a temperature of 2 K, it should be possible to distinguish between the two models by continuing the magnetization measurements to high field. * Exac Alt. C J = 1 J2= 5 Exac Ladd J= 1 JII= 1000 [ t Diagonalization using / hain Hamiltonian / 13.20 K / 5.20 K / t Diagonalization using / er Hamiltonian // 12.75 K // 3.80 K H II aaxis  T=1 K 5 10 15 Field (T) Figure 4.18: The solid and dotted lines represent the exact diagonalization fits from the previous graph (Fig. 4.17) calculated at a temperature of 1 K. At this temperature, the difference between the curves becomes more pronounced. At a field of approximately 10.6 T, there appears to be an inflection in the predicted magnetization using the ladder Hamiltonian. 6000 5000 5 4000 E 0 , 3000 E 2000 1.0 % U nP_,J,'L T=0.42 K So Data 0.8 Alt. Chain Model 0.25 ............. JI = 12.85 K 020  S0.6 = 4.35K K f 1 0,15 0.4 0.10 o 00. .05 0.2 0.00 0 5 10 15 20 0.0 H (T) 0 5 10 15 20 25 H(T) Figure 4.19: The data from Fig. 7 in Reference 14 have been digitized. After interpolating to 200 equally spaced points, the 1st derivative was taken using 13 point smoothing (open circles in the inset). The theoretical curve was created with the exact diagonalization procedure using 12 spins arranged in a ring and an alternating chain Hamiltonian (solid lines). The exchange constants of J, = 12.85 K and J2 = 4.35 K were determined by varying the exchange constants to minimize the square of the distance between the theoretical and experimental curves. The theoretical curve consisted of 3000 points. A 250 point adjacent averaging procedure was applied to the first derivative of the theoretical curve (inset). A value of 2 was assumed for the Lande g factor. 4.4 High Field Magnetization Measurements The highfield, H < 30 T, magnetization, M, of a BPCB powder sample (m = 208.2 mg) normalized to its saturation value, Ms, is shown as a function of field and temperature in Fig. 4.20. Since the saturation magnetization was reached on our studies, we were able to measure and subtract a small, temperatureindependent contribution (Xdiam a 2.84 x 104 emu/mol), which is the same value obtained in the low field work (Section 4.2), by performing a linear fit to the data above 20 T. The data were acquired while ramping the field in both directions, and no hysteresis was observed. Although approximately 3000 points were acquired at each temperature, the data traces are limited to 150 points for clarity. The lines are fits using the 12 spin exact diagonalization and an alternating chain Hamiltonian, Eq. 4.2. The best fit exchange constants, which are listed in Table 4.4, have a systematic temperature dependence with J\ increasing and J2 decreasing with increasing temperature. The theoretical curves adequately reproduce the magnetization data at the two highest temperatures of 3.31 K and 4.47 K. However, at the temperature of 1.75 K, the exact diagonalization curve deviates significantly from the data at Hci =6.6 T and HC2 =14.6 T. Data were also taken at 0.7 K; however, the exact diagonalization technique fails to produce a reasonable curve for the reasons discussed in Chapter 3, and consequently, that theoretical curve is not shown. It should be noted that the exchange constants, J\ 13 K and J2 7.0 K, do not match the exchange constants obtained from the susceptibility data, .1\ 13.7 K and J2 5.5 K. Similar results are obtained for magnetization measurements of single crystal samples. The high field, H < 30 T, magnetization of a single crystal sample (m = 18.9 mg) with H 11 aaxis is shown in 
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