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Encarta.» 3...»? fivefimfmhdm. was ,mmm: U“ “l “l \|\\\|\\\\H|\\ Igumwulu \fllfll LIBRARY " 3 1 O 1 O8 80 Michigan State University This is to certify that the thesis entitled The Low Temperature Cooperative Behavior of the Rare Earth Salts GdCl3 and PrCl3 presented by Jan Paul Hessler has been accepted towards fulfillment of the requirements for __Eh._D_.__degree in Phys ics £me Major professor Date 9 JuneL 1971 ABSTRACT THE LOW TEMPERATURE COOPERATIVE BEHAVIOR OF THE RARE EARTH SALTS GdCl3 and PrCl3 BY Jan Paul Hessler We have studied the three-dimensional ordered phases of GdCl3 and PrCl3 with Cl nuclear magnetic resonance in the temperature range 0.3 K to the transition temperature. The resonance transition frequencies were measured with a simple pulsed N.M.R. spectrometer to an accuracy of i 1 kHz. Simple 3He and 4He systems were used to obtain the necessary low temperatures. The absolute temperature was measured to an accuracy of i 2 mK. GdCl3 is an ionic ferromagnet with a Curie temperature of 2.2 K. In the ordered state the internal field at the chlorine site is along the principal X—axis of the electro- static field gradient tensor. The method of energy moments is used to determine the asymmetry parameter, n = 0.4265 i 0.0001. The nuclear quadrupole interaction Hamiltonian is diagonalized and a chi squared analysis is used to deter- mine the internal fields at the chlorine site, B(T). Both 35Cl and 37 C1 transition frequencies are observed and used in the analysis. In the critical region, T/Tc > 0.91, the internal field follows the relationship B(T) = A(TC-T)B where Jan Paul Hessler A = (4368.4 i 31.1) gauss/KB, Tc = (2.214 1 0.0016) K, and B = 0.3904 1 0.006. Below 1.0 K the internal field follows B(T) = BO - (AlT3/2 + A2T5/2)exp(-6/T) where B0 = (4950.8 1 2.9) gauss, A1 = (963.1 i 68.3) gauss/K3/2, A2 = -(90.7 i 41.6) gauss/KS/z, and e = (0.430 1 0.037) K. The measured temperature dependence of the internal field below 0.6 K is compared with the spin wave predict- ions based on the exchange parameters measured by pair spectra. There is a definite discrepancy. By comparing the internal field measurements to the bulk measurements of magnetization, we have calibrated the internal field in terms of the magnetization. This indicates an anomalously large zero-point magnetization defect. The magnetization measurements are also compared to the molecular field and Green function predictions. A possible mechanism and experiments to test the mechanism are put forth to explain the large zero-point magnetization defect. The low temperature phase of PrCl T - 0.4 K, 3’ critical— was studied to determine the nature and symmetry of the ordered state. In the paramagnetic region the asymmetry parameter was determined by applying an external field along the principal X-axis of the electric field gradient tensor and using the method of energy moments. n was found to be 0.4937 1 0.0001. In the ordered state, the local magnetic and electric field gradients were measured at the Cl site by applying an external field perpendicular to the high Jan Paul Hessler symmetry axis, C3, and studying the symmetry and behavior of the rotational spectrum. The Splitting of the pure quadrupole resonance line at 0.4 K is attributed to an effective crystallographic transition. to have no transition from P63/m asymmetric The application of a 10 Kgauss field appears effect on the temperature. to P6 or P3. line shape in zero field Splitting and on the The crystal space group is lowered An interpretation of the observed the ordered state is presented which implies that the three-fold symmetry is slightly distorted. THE LOW TEMPERATURE COOPERATIVE BEHAVIOR OF THE RARE EARTH SALTS GdCl3 AND prc13 by Jan Paul Hessler A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1971 DEDICATION To my parents, Ada and Harold Hessler, who taught me to never allow an educational institute to interfere with my education, and to Nancy, who lives so successfully in our ménage a trois: me, her, and physics, I dedicate this thesis. ii ACKNOWLEDGEMENTS My sincere thanks goes to Prof. Edward H. Carlson who suggested the tOpic for this thesis and gave me the freedom to pursue it on my own. For their many hours of conversation and stimulation I wish to thank Robert D. Spence and Thomas A. Kaplan. The numerous authors that have communicated their work prior to publication were a great help. A special thanks to David H. Current for his help in designing the apparatus and performing the experiments as well as being a patient listener for many of my wild and untested ideas. iii TABLE OF CONTENTS LIST OF TABLES I O O O O C O O I O O C 0 LIST OF FIGURES . . . . . . . . . . . . Chapter I. II. III. IV. INTRODUCTION . . . . . . . . . . EXPERIMENTAL TECHNIQUE . . . . . A. B. C. D. Crystal Preparation . . . . . Crystal Structure . . . . . . N.M.R. Measurements .'. . . . Temperature, Calibration and Measurement . . . . . . . . . GdCl THEORY AND BACKGROUND . . A. B. C. D. E. 3 Gd3+ Ion PrOperties . . . . . GdCl3 Bulk Properties . . . . Pair Spectra of Gd3+ . . . . Relation of Pair Exchange Constants to Bulk Properties Magnetization Calculations . Cl N.M.R. IN GdCl : RESULTS AND 3 DISCUSSION . . . . . . . . . . . A. B. N.M.R. Hamiltonian . . . . . Determine n by the Method of Moments . . . . . . . . . . . Determine the Magnitude of the Internal Field . . . . . . . Comparison of Temperature Dependence to Theory . . . . iv Page vi vii 10 12 17 17 18 18 19 20 32 32 35 39 42 Page E. Conclusion . . . . . . . . . . . 61 V. PrCl3 THEORY AND BACKGROUND . . . . 63 A. Pr3+ Ion Properties . . . . . . 63 B. Pr3+ and Pair Resonance . . . . 64 C. Bulk Properties . . . . . . . . 67 VI. Cl N.M.R. RESULTS FOR PrCl 69 3 . . . . A. Paramagnetic Phase . . . . . . . 69 B. Low Temperature Phase, T < 0.4 K 72 C. Conclusions . . . . . . . . . . 82 REFERENCES . . . . . . . . . . . . . . . . 83 APPENDIX A . . . . . . . . . . . . . . . . 87 APPENDIX B . . . . . . . . . . . . . . . . 93 APPENDIX C O O O O O O O O O O O I O O O O 109 Table 3.1 4.1 5.1 6.1 A.1 A.2 A.3 A.4 B.1 B.Z B.3 B.4 LIST OF TABLES Exchange Constants for GdCl3 Low Temperature Analysis . . Spin~Spin Interaction Parameters for Pr3+ Axial Pairs O O I 0 O O O O O O O O O O O O O I 0 Analysis of Possible Space Groups for PrCl3 Susceptibility Coil Calibration Data . Coefficients for 3He Solitron Ge Resistor Raw Data for Ge Resistor Calibration . . . R vs. T Calibration Data and Deviation . C1 Transition Frequencies in GdCl3 . . . . . . Results of X2 Analysis for B(T) Comparison of B(T) with Analytic Expression for T~'I‘ O O O O O O O O O O O O O O O O O O I O c Comparison of B(T) with Analytic Expression for T 0.91 . . . . . 50 3/2 4.7 Internal Field vs. T for T < 1.2 K . . . . . 54 4.8 Comparison of Low Temperature Data with Spin Wave Theory . . . . . . . . . . . . . . . . . . 56 4.9 Reduced Magnetization vs. Reduced Temperature . 60 5.1 Zero Field Splitting of Pure Quadrupole Line, PrC13 O O O C O I O I O O O O O O O O O O O O I 68 6.1 Transition Frequencies vs. Field . . . . . . . 70 6.2 Rotation Diagram, HJ.C High Temperature Phase 71 3' 6.3 Rotation Diagram, HICB, Low Temperature Phase 73 6.4 Transition Frequencies vs. Field, Low Temperature Phase . . . . . . . . . . . . . . . 74 6.5 High Field Behavior of Transition Frequencies, Low Temperature Phase . . . . . . . . . . . . . 76 vii Figure Page 6.6 Second Derivative of Absorption Curve, PrCl 79 3 O 6.7 Calculated Second Derivative of Absorption curve 0 O O O O O O O O O O O O O O O O O O O p O 8]- viii I. INTRODUCTION Magnetism is a cooperative phenomenon. This fact makes its study both interesting and challenging. Most attempts to describe the c00perative behavior in ionic compounds have assumed that we may use the concept of localized moments and the description of these local moments as a starting point. We then assume that a satisfactory approach to the understanding of magnetic phenomena will follow from: 1) a knowledge of the localized magnetic ions, 2) a know— ledge of the interactions between the localized ions, and 3) an accurate treatment of the statistical mechanics for the implications of the above two models for the behavior of the system. Parts one and two of this approach have motivated the idea of a spin Hamiltonian. The spin Hamiltonian is simply a mathematical model which is sufficiently general to account for all the experimental information observed in one and two. It is in this sense that the spin Hamiltonian is phenomenological. For a discussion of the derivation of spin Hamiltonians see Stevens (1963). Of paramount impor- tance is the assertion that the spin Hamiltonian describing the interaction between two isolated spins plus the princi- ples of statistical mechanics is sufficient to deduce the cooperative many body behavior of magnetism. A knowledge of the low lying states of the magnetic ion is obtained by studying the configuration terms of the l free ion by Optical methods (Judd, 1963). Paramagnetic resonance is then used to study the effect of the crystalline environment on the low lying levels (Hutchings, 1964). Figure 1.1 is a schematic representation of the 3+ Pr ion in the C symmetry of PrCl3 (Judd, 1957). 3h The interaction between the spins may also be studied by paramagnetic resonance, and generally may be described by a spin Hamiltonian of the form + _ + . + . - , 2 H12 — S1 012 32 + 3(§l 32) (1.1) For a rather complete discussion of exchange see Anderson (1963a, 1963b) and Wolf (1971). This form of the inter- action has proved very useful, eSpecially for the transition metal ions. However, in the rare earths the 4f electrons are shielded by the 552 and 5p6 shells, resulting in a significantly smaller exchange interaction due to the decreased overlap. Because of this reduction, competing effects from the crystal fields and the spin-orbit coupling contribution to the magnetic moment must also be considered. This complicates the form of the exchange interaction and allows other interactions to act as coupling mechanisms. The best known coupling mechanism is the dipole-dipole coupling, and it is the only one encountered classically. From magnetostatics the magnetic dipole energy is given by E |3 (Ml'M2-3(Ml°f1)(Mz'f2))/Irl-r2 (1.2) + + , + + where M1 and M2 are magnetic moments located at r1 and r2 respectively. For the special case of S state ions with 18 45140 «'1 I I I I 32 I / 1 I / 21430 cm- Ill, 11 x [I / / ’ 1 f2 4’ D 16920 en“ \ 0 \fi‘ 1 . \\ \ 0 8700 «'1 I \\ I 3 \\ I \\ \ 3r 6670 eun'1 [I], \ I,’ 11 131 cm'l \ _ I a' \ 3a,, 2430 cm 1 ’/1’ \\ -1 Coulod: + §\ \\ 12 97 cm Spin-Orbit \\ \ \\ \\ \\ 3 33 “.1 \ I \ \\ 12 o cm-1 can 0.x.r. Figure 1.1 ”3+ ion Incrgy Levels in LnCls. + (rl - f2) the axis of quantization, the spin Hamiltonian simplifies to _ 2 2 + .+ _ z z 3 312 — g “B(Sl 32 35152)/r12 . (1.3) The well known exchange interaction was introduced by Heisenberg (1926) and by Dirac (1929). Exchange effects are a direct consequence of the Pauli principle. Dirac showed that for the particular case of n electrons confined to Specific orthogonal orbits, the Splitting of the energy levels is the same as though we forgot about permutation degeneracy and used the potential vex = -igj(1/2 + ZSi'Sj)Jij ‘ (1.4) where J =fdrdrw+(r)w+(r)—£:—W(r)w (r) (15) ij 12ilj2+_+iZjl" r1 r2 and wi and wj are the orbital wave functions for the states i and j respectively. Extensions of this idea have led to equation (1.1). A third type of coupling is due to the aspherical charge distribution found in non S state ions. The moments may be coupled via electric multipole moment interactions. An example of this is the electric quadrupole—quadrupole interaction observed in CeCl3 (Birgeneau, Hutchings, and Rogers, 1968). A fourth coupling mechanism is virtual phonon exchange. This interaction is similar to the above and may be viewed in two ways. For non S state ions, the electrostatic interactions in the form of the crystal field and the interacting multipoles may be induced or modified by the phonons. In the case of the S state ions the exchange and dipole-dipole interaction coefficients Jij and aij' where aij a l/rij, are strong functions of distance and can couple to the phonons. Although this is generally considered a weak interaction, it is very impor— tant in the case of non-Kramers orbital degeneracy. This mechanism has been used to account for the antiferro- magnetism in UO (Allen, 1968). 2 An approach to studying magnetic systems is now straightforward. From symmetry and other considerations, we deduce the form and number of allowed interactions necessary to construct an interaction Hamiltonian for the system. We then experimentally determine the magnitude and behavior of the interaction coefficients in the Hamiltonian. With this information we apply the laws of statistical mechanics to deduce the c00perative behavior of the system. This behavior is measured, and we compare our results to the theoretical prediction. 'If the effective Hamiltonian is correct and the statistical mechanics has been applied properly, we should expect agreement. Often this is not the case. We are then left with three alterna- tives: l) the effective Hamiltonian is not accurate enough and we must include additional information, 2) the statis- tical mechanical treatment was inadequate and needs improvement, or 3) the assumption that we can describe the COOperative phenomena in terms of simple two-Spin inter— actions is not valid. In this work we study the COOperative behavior of two rare earth salts, GdCl3 and PrCl3. We use the chlorine nuclear magnetic resonance as a microscopic probe. This will give us information about the magnetic field and the electric field gradient at the chlorine Site. In chapter II we discuss the crystal preparation and structure, along with the N.M.R. measuring apparatus and the low temperature apparatus. In chapter III we present the Single ion optical and paramagnetic resonance results for the Gd3+ ion. The pair Spectra in LaCl3 and EuCl3 along with the high temperature magnetic Specific heat results are discussed to arrive at an effective two-Spin interaction Hamiltonian. With these parameters we use the cluster expansion technique to derive the molecular field approximation and the two-spin correction term. We discuss the problems involved in evaluating the two—Spin correction term, and two approximate solutions. The Green function formalism and the spin wave approximation are also briefly discussed. In chapter IV we set up the nuclear quadrupole inter- action Hamiltonian and apply symmetry arguments to simplify the analysis of the observed spectrum. The method of energy moments is used to determine the asymmetry parameter, and an exact diagonalization is used with a chi squared analysis to determine the magnitude of the internal field as a function of the observed transition frequencies. Analytic expressions are found which describe the temper- ature dependence of the internal field in the critical region and in the spin wave region. The low temperature measurements, T < 0.6 K, are compared to numerical calcula- tions of the temperature dependence of the magnetization based on a spin wave calculation and the measured exchange parameters. Here there is a discrepancy which requires further investigation. The internal field measurements are also compared to saturation magnetization measurements to calibrate the internal field results in terms of saturation magnetization results. With this we detect an anomalously large zero- point magnetization defect. A possible mechanism to explain this defect is presented along with some experi— ments which should help our understanding of the low temper- ature behavior. We also compare our measurements to the molecular field and Green function calculations. Agreement with the molecular field calculation is relatively poor, and the Green function calculation is qualitatively correct. In chapter V we present the optical and paramagnetic resonance results for a single Pr3+ ion in LaClB. The axial pair spectra measurements are also reported, but no conclusive statement can be made about the interaction mechanism responsible for the three-dimensional ordering at 0.4 K. In chapter VI we study the low temperature phase of PrCl3 by applying an external magnetic field in the plane perpendicular to the symmetry axis, C The symmetry of 3. the rotation spectrum, and the behavior extrapolated to zero applied field indicate that the phase transition at 0.4 K is effectively a crystallographic phase transition. Studies in an applied field up to 10 Kgauss indicate that a magnetic field has no effect on the zero field splitting of the pure quadrupole resonance line nor on the transition temperature. Symmetry arguments are used to Show that the effective crystallographic space group is either P6 or P3. An analysis of the observed asymmetric line in the ordered state is presented which indicates that the 3-fold symmetry is also slightly distorted, although this is not verified by an analysis of our rotation Spectrum. The fact that PrCl3 has a non-Kramers orbitally degenerate ground state leads us to suspect that the ordering mechanism is dominated by the lattice vibrations. We review the litera- ture for evidence to this effect. II . EXPERIMENTAL TECHNIQUE A. Crystal Preparation The anhydrous rare earth trichloride from Lindsay was Slowly melted under vacuum in a vertical quartz tube. The polycrystalline sample was then transferred to a horizontal distillation apparatus and distilled in vacuum into a 17 mm diameter quartz tube. After distillation the tube was sealed under vacuum, detached, and placed in a gradient furnace. The lower tip of the tube was placed in the gradient and observed until a single seed crystal was produced. The tube was then slowly lowered through the gradient, producing a clear single crystal. Because the crystals are very hydroscopic, once they were removed from the tube they were stored in mineral oil when not in use and liberally coated with Apeizon N grease during use. No analysis of the stoichiometry or impurity content was attempted. The crystals are uniaxial with the axis easily recog- nized by observation of the striations that appear on the cleavage planes which are parallel to the axis. The crystals were cleaved, cut on a diamond saw, and ground with a grinding wheel to a convenient size and Shape. B. Crystal Structure Zachariasen (1948) has determined the structure for the rare earth trichloride series lanthanum to gadolinium and found it to be hexagonal With space group P63/m. There 9 10 are two molecules per unit cell. The locations of the rare earth ions are determined by symmetry as being i(%,%,%) while the physically inequivalent chlorines are at :(u, v, 1/4), (V, u-v, 1/4), (v-u, E, 1/4). Morosin (1968) has recently determined the parameters u and v and the cell dimensions for several of the anhydrous rare earth trichlorides. Figure 2.1 shows the crystal structure. The point symmetry for the rare earth ions is C3h' All the rare earths and chlorines lie in the mirror plane. C. N.M.R. Measurements All transition frequency measurements were made using a Simple pulsed N.M.R. spectrometer developed by S. Parks (1967). This Spectrometer compares the applied rf Signal from a cw oscillator to the induced rf signal from the spin system. The induced rf frequency is measured by displaying the beat pattern between the two frequencies on an oscilloscope and noting the frequency of the cw oscilla- tor at the zero beat. For accurate measurements five independent readings were taken for each frequency. The standard deviation of these readings was generally less than 1 kHz. To minimize sample heating at low temperatures a minimum amount of power was applied to the rf pulse by varying the voltage on the transmitter stage of the spectrometer. To assure thermal equilibrium, the line with the largest dv/dT was measured at 10 to 20 minute intervals while the temperature of the bath was maintained constant 11 Figure 2.1 Structure of the Rare-Barth Trichlorides La to Gd. 12 to within 1/4 mK. When two consecutive measurements of frequency agreed to within l/2 kHz the sample was assumed to be in thermal equilibrium with the bath. D. Temperature, Calibration and Measurement All of the interesting phase transitions in the rare earth trichlorides occur below 4 K. This necessitates the use of liquid 4He and 3He as refrigerants. Experimental techniques in this temperature range have been discussed by White (1968). For our experiments in the temperature range of 1.2 K to 4.2 K the sample was immersed directly in the 4He bath. A 200 ohm Manganin resistor was used as a heater for fine control of the temperature below the lambda point and as a stirrer above the lambda point. The 4He vapor pressure (Brichwedde, 1970) was used as an absolute measure of temperature throughout the entire temperature range. Above a vapor pressure of 100 Torr a standard U-tube mercury manometer was used to determine the vapor pressure. Below 100 Torr an MKS capacitance manometer Operated in the digital mode was used. This has a day-to-day reproduci— bility of 0.02 Torr + 0.05% of the pressure reading. Above 1.3 K this accuracy is well within the accuracy of. the 1958 4He Scale of Temperatures. Below the lambda point the temperature was regulated by monitoring the vapor pressure and keeping it censtant to within 0.002 Torr. This corresponds to a temperature l3 fluctuation of 0.45 mK at 1.2 K. The measured vapor pressure and temperature were strongly coupled as indicated by the fact that the resistance of a 33 ohm Ohmite carbon resistor located at the sample site tracked the vapor pressure with no visible lag in time. For the temperature range of 0.3 to 1.2 K a conven— tional 3He single shot cryostat was used. The design follows closely that of Walton (1966). Figure 2.2 Shows the low temperature section of the cryostat. Above 0.6 K the 3He vapor pressure was used to deter- mine the temperature (Sherman, Sydoriak, and Roberts, 1962). To measure the vapor pressure a 1/4" o.d. stainless tube was inserted inside the pumping line from the 1.2 K radiation trap to approximately 1 cm above the surface of the liquid. The tube was then increased to 3/8" o.d. and went independ- ently to an MKS capacitance manometer at room temperature. The day-to-day reproducibility when operated in the digital mode is 0.006 Torr + 0.05% of the pressure reading. To achieve a desired temperature, the vapor pressure was calcu- lated and the appropriate corrections for the manometer were employed. The digital dials were then set and the tempera- ture was obtained by monitoring the pumping Speed of the 3He gas with a series of valves. A pressure fluctuation corres- ponding to a temperature fluctuation of 0.1 mK was calcula- ted for each temperature and the pressure maintained constant within these limits. 14 I to CAPACITANCE C MANOMETER PUMPING LINE 5 \ y [r/ u 1. 2 K RADIATIONTRA 3He , J -—-— 1" 0.1). 3He PUMPING LINE ~— VAPOR PRESSURE TUBE 3 LIQUID He ——e OFHC COPPER Ge RESISTOR a- Carbon RESISTOR ——"—" --——HEATER 200 ohms Carbon RESISTOR ———-——fi- 352* SAMPLE Figure 2.2 Low Temperature Section of 3He Cryostat. 15 Below 0.6 K the thermomolecular correction becomes significant, even for our large tube (Freddi and Modena, 1968). Also dP/dT becomes small enough that pressure monitoring and measurement are not sensitive enough to keep temperature fluctuations on the order of 0.1 mK. Therefore, below 0.6 K, we used a germanium resistor for temperature measurement and a carbon resistor for tempera- ture fluctuation monitoring. Both resistances are measured independently by two Wheatstone bridges using a PAR lock—in detector as a source and as a null detector. A Triad G-lO "Geoformer" was used to isolate the unbalanced signal from the preamp and the PAR detector. To eliminate the effect of lead resistance for the germanium resistor a three lead Wheatstone system was used with an arm ratio of 1:1. With this method the lead resistance of 200 ohms at room temperature was effectively nulled to within 2 ohms. To calibrate the germanium resistor at low temperatures the magnetic temperature, T*, of ferric ammonium alum was used as a standard. The mutual inductance technique (Abel, Anderson, and Wheatley, 1964) was used to determine the susceptibility. The coils were calibrated using the vapor pressure of 3He as the temperature standard between 0.6 K and 1.2 K. The relationship between T* and T for T 1 0.2 K is T* - T = 0.00548/T (2.1) 16 to within 0.1 mK (Sydoriak and Roberts, 1957). The resistance vs. temperature data for the germanium resistor was then fitted to the equation ln(R) = a0 + alln(T) + az/T (2.2) from 0.3 K to 1.2 K. Four different sets of the coeffi- cients a0, a1 and a2 were necessary to fit the data to within 1 mK. At 0.7 K the resistance is 2239 ohms with a dR/dT of -8.1 ohms/mK. Our Wheatstone bridge has a sensitivity of approximately 1 ohm with a power level of -9 10 Watts, thus we are able to detect temperature fluctu- ations on the order of 0.2 mK at 0.7 K. The actual cali- bration data and analysis are discussed in Appendix A. III. GdCl3 THEORY AND BACKGROUND GdCl3 belongs to the small set of compounds which are both insulating and ferromagnetic; this is the primary reason for studying it so extensively. From the experi- mental point of view it belongs to a rather large series of isomorphic compoundswhich are easy to grow. This allows the experimentalist the opportunity to study the Gd3+ ion with many experimental techniques. A. Gd3+ Ion Properties The ground level of Gd3+ is 88 1 7/2 with the only (Piksis, Dieke, and Crosswhite, 1967). The first excited level of Gd3+ in is 6P at 32100 cm’l. The Gd3+ ions Show the structure of the order of 0.1 cm- LaCl3 7/2 least coupling with the crystal lattice of all the rare earth ions. Superimposed crystal vibrations are generally not observed in any lattice and the lines are reasonably sharp even at room temperature. Hutchinson Jr., Judd, and POpe (1957) and Hutchinson Jr. and Wong (1958) have measured the paramagnetic 3+ in LaCl and CeCl . They resonance absorption of Gd 3 3 find a spin Hamiltonian of the form H-g-s = g" “BHZSZ + IL “B(Hxsx + HySy) (3.1) 91 = 1.991 r 0.001. where gH = 17 18 B. GdC13_Bulk PrOpertieS Wolf, Leask, Mangum, and Wyatt (1961) have measured the susceptibility and magnetization of single crystals of GdCl3 and find that the substance orders ferromagnetically at 2.2 K. The specific heat (Wyatt, 1963) has a lambda- 1ike anomaly at 2.2 K, further substantiating the onset of long range order. C. Pair Spectra of Gd3+ Birgeneau, Hutchings, and Wolf (1967) and Hutchings, Birgeneau, and Wolf (1968) have measured the pair spectra 3+ of Gd pairs in LaCl and EuCl . They find the pair 3 3 Spectra are adequately described by a Hamiltonian of the form . . _ _ z z _ . . _ z z H(l,]) - guBH(si + sj) Jijsi sj + aij(§i sj 3sisj) + H (i) + H CEF CEF(j). (3.2) The axis of the pairs is the z-axis of quantization. J.. 1] is the isotrOpic exchange interaction and aij = gzug/rij. HCEF is the single ion crystal field Hamiltonian. Because the lattice parameters of LaCl and GdCl differ Signifi- 3 3 cantly, the variations of Jij and aij with respect to temperature were studied in the LaCl case. From this the 3 dependence of Jij on gij was inferred, and extrapolations to the GdCl3 lattice constants were made. The values of Jij for each case are Shown in Table 3.1. It is interesting to note that the nearest—neighbor exchange is antiferro- magnetic and weak while the next-nearest-neighbor exchange 19 is ferromagnetic and approximately four times larger than the nearest-neighbor exchange. There is no definitive explanation for this at the present time. D. Relation of Pair Exchange Constants to Bulk Properties There is no a priori reason why the exchange constants measured by pair spectra in a diamagnetic host should determine the magnetic behavior of the bulk system. In spite of this there is relatively good agreement between the bulk prOperties and the pair exchange constants. Marquard (1967) has developed a diagramatic technique for calculating the high-temperature expansion coefficients for magnetic systems with arbitrary symmetric tensor interactions between all pairs of Spins. He has calculated the first three coefficients for the specific case of GdC13. Clover and Wolf (1968) have performed high frequency susceptibility experiments to determine the magnetic Specific heat at 20.4 K and 77 K. Their results for the exchange constants along with the pair results are shown in Table 3.1. Table 3.1 Exchange Constants for GdCl 3 Specific Heat LaCl3a EuCl3 Jnn(K) -0.078 t 0.004 -0.033 t 0.004 -0.073 t 0.004 + (K) 0.096 t 0.004 0.105 t 0.004 0.091 _ 0.004 nnn aExtrapolated to GdCl3 lattice parameters 20 The relatively good agreement between the Specific heat results and the pair results for EuCl3 is encouraging. The discrepancy in the LaC13 measurements is not under- stood, although the extrapolation procedure may not be correct. E. Magnetization Calculations Now that we have a relatively good idea of the para- meters that go into a phenomenological Hamiltonian for the spin systems we can use these parameters to predict the behavior of the system. There are basically three approaches used in calculating magnetization vs. tempera- ture: the cluster series approximation, the double-time temperature—dependent Green function formalism, and the spin wave approximation. 1) Cluster Expansion (Molecular Field Approximation) The oldest approximation method of treating the Hamiltonian of a magnetic system is the molecular field approximation. This has also had much success in predict- ing the overall qualitative features of a magnetic system such as: the lambda-like discontinuity in the Specific heat, the dependence of magnetization on temperature, the magnetic susceptibility, and the existence of a critical point. The basic assumption in the molecular field approxima— tion replaces all spin-spin interactions with a spin- effective field interaction. This assumption therefore 21 eliminates any Spin-Spin correlation effects. An extension Of the "effective field" concept to include correlation effects was first used by Mayer and Mayer (1940) in treating the problem Of the non-ideal gas. This technique, commonly called the cluster expansion, was used by Streib, Callen, and Horwitz (1963) to derive a Similar series for the Heisenberg ferromagnet. This series has the molecular field approximation as the leading term. We Shall treat our Hamiltonian with this technique, thereby reproducing the standard molecular field results and Showing how higher order spin-spin correlations may be included, and the difficulties involved. Extending the pair Hamiltonian tO a sum over all pairs we have the Hamiltonian for the system: N H = ‘9“BH.2 5: ' .2. Jijgi'gj - 1=1 (1,3) g + § + 3( .'r..)( .‘r..) + (911B)2 .X. l3 {Si'Sj - 1 1%~ 11_.} . (1'3) rij rij (3.3) The first term is the Zeeman energy of the N gadolinium ions, the second is the isotrOpic exchange interaction, and the third is the dipole-dipole interaction. We assume that the external field, H, is applied along the z-axis. th Si is the standard spin Operator for the i ion. Because the ground level is 887/2, S = 7/2 in this case. J.. is 13 th and jth the isotropic exchange interaction between the i th ions and fij is the vector between the i and jth ions. 22 The summation (i,j) extends over all pairs Of ions in the lattice, each pair being counted once. First we transform the Spin Operators to the raising and lowering Operators, Si = s: i 13?. The Hamiltonian is then § 2 I z z 1 + - - + H=-gu H S. - {A..S.S. +—-B..(S.S. +S.S.) B i=1 1 (i,j) 13 1 j 2 13 1 j 1 j [cli (5637 + 5537) + cTi (3581 + sis?)] 1] l J J 13 l J 1 J - [off 5757 + of? SIST]} (3.4) 13 1 3 13 1 j where the coefficients are given by A.. = a.. + J.., (3.5a) 1] 13 13 Bij = Jij - aij/Z, (3.5b) _ 2 2 _ 3 aij — (guB) (3cos eij l)/rij' (3.5e) 1| = 3 2 l . £10.. Cij 2 (guB) :3— coseijs1n0ij e 13 , (3.5d) and l] :2 _ 3 2 l . 2 2120.. Dij - 4 (guB) :3— Sln eij e 13 . (3.5e) ij rij’ eij, and ¢ij are the standard spherical coord1nates. From the symmetry for a pair we have ¢ij = ¢ji and eji = n — eij. This g1ves aij = aji wh1ch leads to A.. = A.. and B.. = B... We also have CI; = -CIi and 13 31 13 31 13 11 4. 0;: = Dgi. Because the Gd ions lie on a mirror plane, for every ion above the plane at eij there exists an ion below the plane w1th eij. = n — eij, and w1th ¢ij = ¢ij'° 23 Therefore we have il 11 i1 :1 :1 C..= (C..+C.. = (C..-C.. =0. (1%) 13 E gj. 1) 13') Ej 1) 13) (3.6) All the ions on the mirror plane have Ci; = 0. Therefore our Hamiltonian reduces to N z z z 1 + - — + H = gu H X s. - { {A..S.S. + — B..(S.S. + s.s.)} B i=1 1 (ij) 1] 1 j 2 1] 1 j 1 j - X (of? 3737 + 07? sTsf}. (3.7) (ij) 1] 1 j 13 1 3 We now divide the Hamiltonian into a perturbed and unper- turbed part by introducing an expansion parameter oi, o. = S - S? . (3.8) The parameter S will be chosen tO minimize the free energy. This will somehow imbody the behavior Of the ions outside the cluster. The actual physical interpretation of S must wait until after the analysis is complete. Our Hamiltonian now becomes H = E0 + Ligloi + H1 = HO + H1 (3.9) where _ _ — _ —2 EO — guBHNS NAOS , (3.10a) L = ngH + 2AO§, (3.10b) Nil A = A.. , (3.10c) O j=l 13 and l + - - + H = - Z {A..o.o. + — B..(S.S. + 5.3.) .. 2 1. (1]) 13 1.] 13 1 3 3 + of? 5787 + D7? sis? } . (3.10d) 1] l J l] l 3 The total free energy, F, is given by -BF = 1n Tr exp[-B(HO + H1)] (3.11) where B = l/kT. The unperturbed free energy, F0, is given by —BFO = ln Tr exp[—BHO]. (3.12) We may introduce a correction tO the unperturbed free energy, F', by -8F' = -BF + BFO . (3.13) TO derive the molecular field approximation we simply assume —BF' = 0. Corrections to the molecular field approximation are Obtained by expanding -BF' in a cluster series. Before we carry out the cluster expansion, we will complete the derivation Of the molecular field approxima- tion. The unperturbed free energy is given by 2 ~8FO = -BNAOS + N 1n 01 (3.14) where _ z ¢i — Tr explBLSi]. (3.15) We calculate S'by minimizing Fo with respect tO S. Therefore a S = STEEY-ln Tr exp[BLS:] (3.16) or 25 S = s BS(SBL) (3.17) where Bs(x) is the Brillouin function defined by ZS+1 28 BS (x) = 76' {(28+l)coth( )x - coth(-2-)Si)}. (3.18) TO evaluate the magnetization we have _ _. _2 _ F. .1 z M — 8H — 8 8H 1n Tr exp[BLSi] (3.19) or _ 8 2 M —- NgUB m 1n Tr €Xp[BLSi] p (3.20) but from equation (3.16) we have M = NguBS . (3.21) Therefore we identify S with and have the standard molecular field results. In comparing our results with the standard molecular field equations, (Smart, 1966), we must remember that our effective exchange interaction, A0, is given by A = Z J.. + d.. O . 13 13 J = J + (gUB)2 Z —%—-(3 c0520.j- l) (3.22) j r.. where 13 J0 = zlJnn + ZZJnnn (3.23) 21 is the number Of nearest-neighbors and 22 the number Of next-nearest neighbors. Returning tO the correction term in the free energy, —BF', we see -BF' = ln Tr epr-BU!O + H1)] - 1n Tr exp[-BHO]. (3.24) 26 This is expanded in a cluster series of the form 'BF' = ln Tr p0 exp[ZQa] (3.25) a where po = expl-BH01/Tr exp[-BHO] (3.26) and the index a numbers the pairs (i,j) or "links" in the crystal. The expansion may then be written -BF' = ‘E‘I'BFgatl (3.27) where |a) denotes each topologically distinct cluster. The correction term can be computed to any desired order, and the parameter § is then chosen to minimize the total free energy. A criticism of this approach has been raised by Morita and Tanaka (1966). They point out that the condition of minimizing the free energy with respect to § is not justified from the basic principles of statistical mechanics. Using a variational technique, they show that this approach is valid for the pair approximation. We can therefore extend our expansion up to the two-spin approximation. A problem occurs when carrying the expansion to two spins due to the fact that Ho and H1 do not commute. However, this can be overcome by applying the theorem - B eBA e B(A+B’ = p exp[-f BAdA] (3.28) o where P is the Dyson ordering operator, and where BA 5 e B e . (3.29) 27 We have derived the two-Spin correction term and find -BF(2 = 2 1n Tr exp[-BQ..] + single ion terms ) (i .) 1] r] (3.30) where _ _, z z z 2 ..:-Q.. t. .... - Q13 13(S)( where we adopt the convention Isl,sz,+1/2,+1/2> ‘ |++> (3.33) and +1/2,-1/2> |+—> (3.34) 2’ Qij then has the form Isl,s |++> |--> |’+> |+‘> |++> F. Qij(S)-%»Aij —D;§ 0 0 - |--> -91“? Qij(S)--1; A1]. 0 0 |-+> 0 0 +%'Aij ’% Bi] ..-. o 0 +31]. +3211]. . (3.35) The coefficients Ai" B.., and Dij are all of the 3 1] same order of magnitude, therefore we cannot neglect the off-diagonal terms nor can we apply perturbation theory. All coefficients depend on r.., 6 13 and ¢ij' the Spherical ij coordinates of the (i,j) pair, and Qij also depends on §. Therefore we must diagonalize Qij for each pair in the crystal and for each §. We have carried out calculations in a very crude approximation by neglecting the off-diagonal terms and only considering the nearest-neighbor and next-nearest-neighbor interactions. These calculations predict a transition temperature slightly lower than the observed value, and a temperature dependence of the magnetization which is qualitatively correct. However, the crudity of this approximation renders it invalid, and indicates that the agreement is simply fortuitous. An improved approximation would be to correctly treat 29 the Operator Qij for the nearest-neighbor and next-nearest- neighbor pairs and include the effect of the other pairs only in A0. Although this is an improvement in terms of an accurate treatment of the neighboring Spins in the two- spin approximation, the two-spin cluster, even when treated correctly for all spins, does not satisfactorily treat the cooperative phenomena. This is especially true because of the antiferromagnetic nearest-neighbor interaction present in GdC13. For this reason this calculation was not pursued further. 2) Green Function A relatively recent approximation technique employs the double-time temperature-dependent Green function (Zubarev, 1960). The retarded double-time temperature- dependent Green function is defined as <> = - iB(t-t')<[A(t), B(t')]> (3.36) where B(t-t') is the unit step function. <> denotes the ensemble average and <<>> denotes the Green function. A(t) and B(t') are quantum mechanical operators. The equation of motion for the Green function is i 3%- <> = 6(t-t')<[A(t).B(t')1> + <<[A(t),H];B(t')>> . (3.37) Taking the Fourier transform over the time variable into the energy variable gives 30 N and w<>w = <[A,B]> + <<[A,H];B>>w (3.38) where h-= 1. Once the operators A and B have been chosen for the problem, the solution of the Green function <>w is nontrivial since it is given in terms of a higher order Green function <<[A1,H];B>>w which is also unknown. The general technique is to approximate the higher order Green function in terms of the lower order Green function. This is called decoupling the equation of motion. In magnetic systems the operators A and B are generally 5: and 5:. The higher order Green functions are generally of the form ; 21s» 339) £Sj' r w ' ( ' <>w £¢j <>w . (3.40a) <>w , (3.40b) Becker and Plischke (1970) have used the Hamiltonian (3.7) in the Green function formalism and solved the problem using the random phase approximation. We will compare our results to their calculation. 3) Spin Wave Calculation Unlike the cluster expansion and Green function approaches which apply to all temperatures, the spin wave 31 approach applies to a restricted temperature range. In the spin wave approximation we have sacrificed the large temperature range for an exact quantum mechanical treatment. The linear spin wave theory was first considered by Bloch (1930, 1932), and later by Holstein and Primakoff (1940). The spin operators in the Hamiltonian are replaced by creation and annihilation operators, a+ and a respectively, by the substitution s+ + (ZS)l/2a (3.41a) s" + (28)l/2a+ (3.4lb) S2 + S - a+a (3.4lc) Only terms quadratic in the creation and annihilation operators are retained. As long as the higher order terms are not significant, i.e. as long as multiple scattering processes are not important, the spin wave approximation is valid. Marquard and Stinchcombe (1967) have treated the Hamiltonian in the spin wave approximation by generalizing the interaction between the Spin operators to include any symmetric interaction. They have also treated the dipole- dipole interaction exactly by using the Ewald technique to evaluate the k-dependent dipole sums. IV. Cl N.M.R. IN GdCl3: RESULTS AND DISCUSSION A. N.M.R. Hamiltonian When a nucleus of spin I Z l is located in a lattice, the Hamiltonian describing the interactions between the nucleus and the local environments at the nuclear site due to the lattice may be written as +++ _). H = - u‘g - %6:§E . (4.1) The first term is the Zeeman interaction between the nuclear dipole moments, E, and the internal field E, The second term is the quadrupole interaction between the 4. nuclear quadrupole moment tensor, 5, and the crystalline ++ electric field gradient (E.F.G.) tensor, 6%. We use the XYZ coordinate system which diagonalizes the field gradient ++ + tensor, -(VE)ij — Vijéij' as our reference frame. The Zeeman interaction is then written ++ HM — - 9% I B __ _1_ +— —+ — thIsz + 2 (I B + I B )] (4.2) where Q = yBO , (4.3) Y is the nuclear gyromagnetic ratio. The quadrupole interaction may be written _ 2 _ 2 2 2 2 HQ - GAIBIZ I + n(IX + Iy)]/h (4.4) where A = eZQsz/4I(ZI+1) = eZQq/4I(ZI+1) (4.5) and n is the field gradient asymmetry parameter, 32 33 n = (Vxx - VYYVVzz (4.6) with the standard convention IV _.|V xxI i-lvyyl 22 The matrix elements of the Hamiltonian are given by = - 9m cose, (4.7a) Q . i¢ = - 5 /(I-m)(I+m+l) Sinee , (4.7b) 2 = A[3m - I(I+1)], (4.70) and = = ./(I+mflI—m+1) (I+m-1) (I—m+27 52"- n. (4.7a) The angles 0 and ¢ are the polar and aximuthal angles respectively of g in the XYZ principal axis system. From the analysis of a nuclear resonance spectrum we may determine: 1) the magnitude of the electric field gradient, q, 2) the asymmetry parameter, n, and 3) the direction and magnitude of an internal field H. All of these parameters are determined at the nuclear Site only. 35 Figure 4.1 Shows the Cl N.M.R. transition frequen- cies in GdCl3 as a function of temperature below the transition temperature of approximately 2.2 K. The facts that we see only three transition frequencies in the ordered state and that the chlorines lie on a mirror plane indicate that the internal field at the Cl site must be along C3. Also, because of the mirror plane one of the principal axes of the E.F.G. must be parallel to C3. In 34 TWL~ITITIIIITIIITTII [I 7.5)—- "Q- 7.0‘_ . o r 3"’0: N.M.R. ' . GdC|3 ‘ 6.5 - ”o = 53M kHz ‘ P ‘0 = 0.4265 ° H= O ' 6.0 -— ' v (MHZ) I 5.5t- '. 5.0‘— ...—m 4.5 — 1111111111111|l11||l11 0.5 LO I5 2.0 T(K) Figure h.l. 35Cl Transition Frequencies in GdCl3 35 all the rare earth trichlorides lanthanum to gadolinium we find this to be the X-axis. we now use the fact that the X-axis of the E.F.G. and the internal magnetic field coincide to simplify our analysis of the N.M.R. spectrum. B. Determine ngby the Method of Moments The asymmetry parameter is easily determined by the method of energy moments (Brown and Parker, 1955). They let the (21 + l) eigenvalues of the Hamiltonian be equal to A , n = l, 2, ... (21 + l). The moments of energy are defined as S1 = Z An , (4.8a) n s =sz (48b) 2 n ' ' n and S = X A3 (4 8c) 3 n n ’ ' The first moment is equal to zero since the representation is traceless. The second and third moments, S2 and S3, can be regarded as experimentally determined quantities if we can construct the energy level diagram from the observed transition frequencies. This is possible if each of the (ZI + 1) levels is implicated in at least one observed transition. Brown and Parker Show that it follows without approximation that 2 1 2 1V (4.9a) 2)+p and 36 _ 2 _ 2 2 S3 — P3C3 (l n ) + 3P2C3V X (3 c0320 - l + ncosZ¢Sin26). (4.9b) C3 is the pure quadrupole frequency interval with n = 0, c3 = 3 eZQq/ZI(ZI-1)h = 6A, (4.10) v is the Larmor frequency of K in the field H, v = pB/Ih. (4.11) The coefficients pi are polynomials in I as follows: p1 = 2I(I + 1)(21 + 1)/3!, (4.12a) p2 = 21(1 + 1)(2I — 1)(21 + l)(2I +3)/3(5!), (4.12b) and p3 = 21(I+l)(ZI-3)(ZI-l)(2I+1)(21+3)(2I+5)/3(7!). (4.12c) For our case of I = 3/2, the coefficients reduce to: p1 = 5, p2 = 1, and p3 = 0. For the case of n # O the observed pure quadrupole resonance frequency, V0, is given by 6ezoq(1 + § n2)l/2 VQ = . (4.13) 4I(21 - l)h Therefore we may express C3 as C3 = VQ/p (4.14) where 1 2)1/2 O = (1 + 3 n . (4.15) We now use the fact that the internal field, H, is parallel to the X-axis to reduce the expressions for the second and third moments. We find 37 + 5v (4.16a) and U) ll 10 (n-l) (4.16b) Since the magnitude of the internal field is a function of temperature and still unknown, we solve equation (4.16a) for v2 and substitute into equation (4.16b) to get a quadratic equation for n whose solutions are given by D t (02 — 4AC)1/2 n = 2A (4.17) where _ 2 _ 2 _ 2 A — 25 VQ (82 v0) 53/3, (4.18a) _ _ 2 2 c — A 3 s3 , (4.18b) and _ 2 2 D — 2A + -3- S3 . (4.18C) Figure 4.2 Shows the energy level scheme for the Cl nucleus in zero magnetic field and in a finite magnetic field. Because we see three lines in the ordered state, each going to the zero field pure quadrupole line at the transition temperature, we may uniquely determine the energy level diagram in terms of the observed frequencies f2, f3, and f4. The relationships between the energy levels and the transition frequencies are given by 411 = - f2 — 3f3 + 2f4, (4.19a) 412 = — £2 + 3f3 — 254, (4.19b) 413 = - £2 + £3 + 2E4, (4.19c) and 38 // * 4 >\4 / // 1 / i2 1 \\ \\\ \ X l 3 1’ Q 1) f2 f3 f4 ) 1 ”‘2 j: 1: B=O BséO Figure 4.2. Energy Levels for a Spin 3/2 Nucleus in a Crystal 39 4A4 = 3f2 + f3 - 2f4. (4.19d) We are now in a position to determine n at any temper- ature. All we require is a knowledge of the pure quadru- pole resonance transition frequency, which we assume remains constant through the phase transition. Table B.1 of Appendix B shows the average of the five independent frequency measurements for all the observed transitions and the standard deviation of the frequency measurements. From the low temperature results we find n = 0.4265 i 0.0001 . (4.20) C. Determine the Magnitude of the Internal Field We could apply the method of moments to determine the magnitude of the internal field as a function of temperature; however, we do not always have the three observed frequencies necessary to uniquely determine the energy levels. An alternate approach, which will allow us to determine the magnitude of the internal field with only a single observed frequency, is to calculate the transition frequencies in terms of the internal field magnitude and then solve for the field magnitude. The Hamiltonian written in matrix form from equation (4.7) is 4O 3A - % a o /3'An o 0 -3A — - (2 o f3'An /3 An 0 -3A + % Q o L o /'3' An 0 3A + -23-() . This is easily solved and gives _ _ g _ 1 2 2 _ ‘7 A1 - 2 2\/CQ + 4(9 QvQ/p) , (4.22a) _ _ g 1 2 2 _ *7 A2 — 2 + Z‘VAQ + 4(9 QvQ/o) , (4.22b) _ Q _ 1 2 2 ' A3 — ?- vaiQ + 4(9 + QVQ/p) I (4.22C) and _ Q 1 2 2 3 A4 - 2 + §\/QQ + 4(9 + QvQ/o) . (4.22a) The transition frequencies are now given by fl = A4 - A1 , (4.23a) f3 = 13 - A1 , (4.23c) and f4 = A3 - A2 . (4.23d) Since there are two isotOpes of chlorine which have a Spin of 3/2, 3Sc1, 75.4% abundant, and 37c1, 24.6% abundant, we observe two pure quadrupole resonance transitions in the paramagnetic state at 5314 kHz and 4188 kHz respectively. In the ordered state we see three lines for each isotope. 41 The energy levels of the different isotopes will be different because of the difference in the nuclear quadrupole moments, given by 35v 35Q Q - _ 57—— — 37— — 1.26878 (4.24) v0 Q and the differences in the nuclear gyromagnetic ratios 35 * 0.4172 kHz/gauss (4.25a) and 37 ¥ 0.3472 kHz/gauss (4.25b) To take advantage of the additional data from the two isotOpeS in determining the magnitude of the internal field, we employ a chi squared analysis. we define x2 by 1' _ 3' 2 x2 = (fi(T) fi(B)) 2| IH |'-' i,j Af(1w)2 + Afg(sd)2 (4'26) where 35 or 37, an isotOpe label, (.1. ll i = l, 2, ... (21 + l), a line label, N = the total number of observed lines at a given temperature fi(T) = the experimental average of five independent frequency measurements on line i of isotope j, at temperature T, fi(B) = the calculated frequency of line i, j and is a function of the internal field B only, Af(lw) = the inherent uncertainty in the frequency measurement due to the finite line width, assumed independent of isotope and line label, and 42 Afj(sd) = the standard deviation for the five independent frequency measurements. A simple computer program for the CDC 6500 was used to determine the value of B that minimizes x2. Because the line width is very dependent on temperature and not directly measured by our Spectrometer, we varied the line width in a systematic manner to force x2 to have a value no greater than 2.0. The uncertainty in the calculated field, B, was then determined by making a contour plot of x2 vs. B in the vicinity of Xiin' From this contour the uncertainty could be assigned by noting the value of B for which x2 equaled xiin + 1.0. Figure 4.3 shows the results of the internal field vs. temperature for GdClB. Table 3.2 of Appendix B tabulates the results of the x2 analysis. D. Comparison of Temperature Dependence to Theory 1) Calibration Nuclear magnetic resonance in the absence of an applied external field measures the local field at a nuclear Site. This field is given by + 3(<§j>‘§ij)fij <§.> Bi = (guB) Z { r5 - -—%— } 3 1 ij rij - ()rm1k)-1 Xij - <§j> . (4.27) The first term is the field due to the moments of the Gd ions, the second term is the transferred hyperfine field. + A is the transferred hyperfine interaction tensor between aollLJIIIIIIIIIITIIIIII . ~.. 4.5L 4.0r- .', 3.5 '. F BatClsite s GdCl3 3.0- B (kGouss) N on r N b 1 L5 )— .o — LO )— —: 0.5 r- 0,0llLlllllLlllllllllllll OK) (15 00 U5 25) T00 Figure h.3. Internal Field at Chlorine Site vs. Temperature for GdC13 44 the ith nucleus and the jth ion, and Sj is the spin of the jth Gd ion. The bracket, <>, denotes the thermal average. The time necessary for thermal averaging is of the order of the period of the phonon motion. This is many orders of magnitude smaller than our sampling time, therefore we may consider <§j> as a temperature dependent vector. If the lattice parameters are independent of tempera- ture and field, i.e. no thermal contraction or magneto- striction, the dipolar contribution to Bi is proportional to . Similarly, if i is temperature and field independ- ent, the total Bi is then directly proportional to <§>. Because we have no easy and direct way of measuring the temperature and field dependence of the lattice constant and the term i, we make the usual assumption that they are temperature and field independent. Magnetization measurements are very difficult and often inaccurate. Wyatt (1963) has measured the magnetiza- tion vs. temperature for GdCl3 by noting the temperature and field of the discontinuity in adiabatic isentropic magnetization measurements. Unfortunately his thermometry is only accurate to :4 mK, and he does not discuss the accuracy of his field measurements. From his curve of Specimen temperature vs. applied field during isentropic magnetization we can reasonably assume that the temperature of the kink is known to $4 mK and the field to ilO Oersteds. 45 Using this as an estimate for the accuracy of his data, we have superimposed our N.M.R. results onto his magneti- zation results, using the relationship M(e.m.u./cm3) = KB(gauss) (4.28) where K = (0.1232 i 0.0027) emu/cmB-gauss. Figure 4.4 shows this comparison. Wyatt has also measured the magnetization in applied field using a vibrating sample magnetometer. Unfortunately at an applied field, H, of about (1/2) NMS, where N is the demagnetizing factor and MS is the spontaneous magnetiza— tion, the magnetization deviates from the expression M = H/N. If the law had held up to fields H = NMs then the point of departure would have been sudden and the value of the field would have been a measure of the Spontaneous magnetization. He estimates the spontaneous magnetization by plotting H against the internal field, Hi’ and extrapo- lates to Hi = 0. The results are Shown in Figure 4.5 along with our results using a calibration constant, K = (0.1137 2 0.0013) emu/cm3-gauss. Because of the uncertainty in the extrapolation procedure and because of the lack of agreement in the temperature dependence between our measurements we will use the isentrOpic magnetization measurements as a calibration. We must keep in mind the limitations of this calibra- tion. Whenever possible we will compare our results to the temperature dependent part of a theory, thereby not relying on the calibration. All fitting of our data to 46 CE. an n. o. no 0 . _ _ a o .. .8. av - loom a Eaflbemmfi «3.0 ...x 0.. u . e Em x "E: 8» 000 u ..d. 28 as: . 48¢ - o e. .o. 2.5 are; o .08 Go 0 o l o o o 0 L000 m5— I! u .02. _ _ _ _ (Emorn'w'am Comparison of N.M.R. and Isentropic Magnetization Data Figure 4.4. 47 l I 700 — _. )4— Ms 600— -a O flp‘. 1"’. 500- . o .. _ 0 Wyatt's Data 0%. 61 400- o — E 0 N. M.R. Data '00 ‘ 8 '. ~300~ ' - g M(T)= K B(T) ’3 o’ o 2 200 - K = O.||37 e.m.u. 3 '- cm3 gauss t 08 '00 " -1 0 J l o__1 a LG 2.0 T(K) Figure 4.5. Comparison of N.M.R. and Magnetization Data 48 analytic expressions will be carried out using the internal field data, and all coefficients will be quoted in the apprOpriate units. When it is necessary to compare our results to magnetization calculations, we will include the above assumed error in the calibration. One question which always arises when studying a ferromagnet is whether the resonances are from the center of the domains, or the domain walls. we have no conclusive evidence to answer this question. We infer that the resonance occurs in the domain rather than the domain wall. Because of the strong dipole-dipole interaction we would expect the domain resonance to be very broad, which we do not observe. Wyatt (1963) assumes the domains have a very small cross section with a wall thickness of one lattice Spacing. His magnetization measurements confirm this assumption. If this is true, the precise meaning of a domain wall resonance is doubtful. Measurements in an applied field may Shed some light on this question. 2) High Temperature Critical Behavior Domb and Sykes (1962) and Fisher (1967) have pointed out that magnetic systems may be characterized by a set of "critical exponents" in the vicinity of the critical temperature. For the magnetization one writes M(T) = A(Tc - mg (4.29) where TC is the critical temperature and B the critical exponent. 49 Figure 4.6 shows a plot of 1nB(T) vs. ln(TC-T) for T > 1.980 K. The very striking linearity indicates a behavior consistent with the critical exponent. Using a non-linear least squares fitting program we find B(T) = A(TC - '1')8 (4.30) where A =(4368 i 31.3) gauss/K , (4.31a) TC = (2.214 1 0.0016) K, (4.3lb) and B = 0.3904 1 0.006 . (4.310) 2 The x for the thirteen point fit between 1.980 K and 2.170 K is 0.710. Table B.3 of Appendix B compares the measured and calculated fields. Although we do not have any data available for T/TC > 0.985 this behavior is consistent with the Specific heat results of Landau (1971), which indicates that the critical region extends to T/Tc ~ 0.91. The value of B is consistent with measurements on other systems. It is definitely not equal to 0.5, the value predicted by both the molecular field theory and the Green function random phase approximation calculation. The small value of x2 indicates that better data are needed to actually detect a deviation from critical behavior predictions. It is not impossible to improve on our data. The most immediate improvement would be to keep temperature fluctuations below 10 uK and measure the 50 —'.‘3 I _IO. N I p I t? 5 _‘Q N I ~2 l 1 '1 l l l I I . . m. In to — IS F I~ N IS Is' fl N (Lleul Figure 4.6. 1nB(T) vs. ln(Tc - T) for T/Tc > 0.91 transition frequencies with a frequency modulated oscilla- tor and record the absorption line. This may also extend the data to a higher limit in T/Tc, but one should not be too Optimistic. As we approach the transition tempera- ture, the line width also follows a critical exponent behavior and becomes very large. It is this inherent limitation on the measurement of B(T) vs. T with N.M.R. that cannot be eliminated. we should also note that Landau (1971) has made high resolution Specific heat measurements on large Single crystals of GdCl He finds that the asymptotic form for 3. the specific heat is not singular. The transition is of the "diffuse" type (Pippard, 1957). He finds that the critical behavior of the Specific heat ends at T/TC ~ 0.999. If microscopic imperfections are severe enough, they may limit the maximum range of the correlations; the result could be an effective subdivision of the sample into an array of microcrystals. The microcrystals will not be identical and could have slightly different order- ing temperatures. In fitting his Specific heat results Landau has assumed a gaussian distribution for the fraction of subsystems that order at a given Tc' He finds a half- width of 1.5 mK will reproduce his experimental results exactly. This distribtuion of critical temperatures will greatly affect any measurements on B(T) in the vicinity of T . c 52 3) Low Temperature Behavior As we mentioned in the previous chapter, the low temperature behavior of the magnetization provides the best comparison of theory and experiment because the theoretical treatment of the Hamiltonian is exact over a finite temperature range. In their calculations of magnetization as a function of temperature, Marquard and Stinchcombe (1967) have shown that for very low T a! a T5/2 e‘e/T (4.32) 0 where AM 2 (Mo - AMO) - M(T). (4.33) M0 is the saturation magnetization and AMO is the zero point magnetization defect. The exponential term is due to the fact that there is a gap in the magnon disper- 5/2 behavior is unusual, and Sion curve at k = 0. The T it is unfortunate that the temperature necessary to observe this behavior is estimated to be below 50 mK. To observe the temperature dependence of the magnetization to the 5/2 behavior is almost accuracy necessary to establish a T impossible. Even with the high accuracy of the N.M.R. measurements the changes in B(T) are so small that it is not technically possible to measure and maintain the temperature of the sample to the required accuracy. Before comparing our results to the numerical calcula- tionS of Marquard and Stinchcombe for the magnetization vs. temperature we will fit our results to an analytic 53 3/2 for T < 1.2 K. expression. Figure 4.7 shows B vs. T This suggests that we may fit our results to the analytic expression B(T) = B — A TV2 e"e/T . (4.34) o 1 Since the spin wave approximation is a series expansion_ we have also fitted the data to 3/2 G/T + A2T5/2)e' . (4.35) B(T) = BO - (All' The results of these fits are shown in Table 4.1. The low value for the sum of the squares indicates that within our experimental accuracy we have a valid analytic expression for B(T). Table B.4 of Appendix B compares the measured and calculated field for both equations. In comparing our results with the numerical calcula— tions of Marquard and Stinchcombe (1967) we recall that our calibration of the magnetization is uncertain, there- fore we must compare the temperature dependence only. From their numerical calculations we see that at T = 0.300K they have M(T)/MS = 0.99460, where MS is the saturation magnetization. If we force our B(0.3 K)/C to equal 0.99460 we have C = 4940.8 gauss. Using this constant we can now compare the temperature dependence of the magnetization. Figure 4.8 shows Marquard and Stinchcombe's numerical results and our data normalized to agree at 0.3 K. The uncertainty in the data is the uncertainty in B(T) deduced from the x2 analysis. This shows there is a definite discrepancy between the theory and the 54 50W I I I I I I I I I 49 F .".. -( ' 3 - °, 8 at Cl site vs. TE 4 43 ' -, Gaol3 " 4.7 - ° d 45 - ° — 1; ' . 1 In 3 45- — (.9 o 8 ES t 3 4.4 — _- 4.3 - - 4.2 - - I I I I I l I I 4.I O.l LO T (K) .1 I I II I I_ I I l 0.0 0.2 0.4 0.6 08 Lo L4 Figure (4.7. T3 («3) Internal Field vs. T3/2 for T 1.2 K 55 Table 411 Low Temperature Analysis Eqn. 4.34 Eqn. 4.35 Number of Data Points 20 20 Temperature Range 0.310 - 1.000 0.310 - 1.000 Bo(gauss) 4957.5 1 2.4 4950.8 1 2.9 A1(gauss/K3/2) 811.0 a 9.9 963.1 a 68.3 A2(gauss/K5/2) - -90.7 i 41.6 0(K) 0.343 t 0.015 0.430 t 0.037 Sum of Squares 9.97 4.312 56 '-°°°[—I——I——I—I—I—b J. =-0.0242 K 0.995 .12: 0.0406 K _ M(T) 0.990 é 0.965)- C} 0.980 é 0.| 0.2 0.3 0.4 0.5 0.6 T(K) Figure 14.8. Comparison of Low Temperature Data with Spin Wave Theory 57 experimental results. Marquard and Stinchcombe have also estimated the zero point magnetization defect by taking an unweighted average over the whole Brillouin zone. Their result is 0.2%. Since the non-interacting Spin wave approximation is doubtful for large k, they feel, by analogy with the antiferromagnetic ground state problem, that the "true" value may be even smaller. Using the calibration from Wyatt's data and the T = 0 value of our analytic expression, B0 = (4950.8 i 2.9) gauss we have that Mo = (609.9 1 13.7) emu/cm3. For complete alignment of the 8 87/2 ground state we would expect 7 uB/ion or Mo = 668 emu/cm3. We therefore have a zero point Spin deviation of (8.7 i 2.1)8. Although this result is very tentative, it provides strong incentive for further investigation of the zero point spin deviation. 4) Molecular Field and Green Function Comparison From the derivation of the molecular field approxima- tion we recall that including the dipole-dipole interaction introduces an additional term into the expression of the exchange strength. We have 2 _ 2 1 _ Ao — JO + (guB) Z r.. (3cos eij 1) (4.36) J 13 where Jo = zlJnn + ZZJnnn , (4.37) 21 is the number of nearest-neighbors, and 22 the number of next-nearest-neighbors. we have calculated the dipole 58 sum term using the Ewald technique. For the lattice O 0 parameters a0 = 7.3663 A and c0 = 4.1059 A we have for a Spherical sample A l 2 _ 0-3 X -—-(3cos 8..- 1) - 0.03089 A (4.38) 3 rij 13 Taking the value of g as 2.0, the effective exchange due to the dipoles alone, Jdd’ is 0.0764 K. Using the EuCl exchange 3 constants, J = -(0.073 t 0.004) K and J = (0.091 i 0.004)K nn nnn we have A0 = (0.476 t 0.032) K. The molecular field Curie temperature, Tc’ iS given by TC = AOS(S+l)/3k . (4.39) Using our value for A0 we have Tc = (2.50 i 0.17) K. The calculation of magnetization vs. temperature is done by solving §§%.( %_,)x = Bs(x) (4.40) C self consistently, where Bs(x) is the Brillouin function, B ( ) _ 28+l ZS+l l s x _ _2S— _— °°th ‘28- X ‘ 28 coth 5% . (4.41) T/TC is the reduced temperature, often labeled Tr' The reduced magnetization, Mr = M(T)/M(0), is given by Mr = Bs(x). It is a simple matter to solve equation (4.40) for Bs(x) as a function of Tr on a computer. When comparing magnetization vs. temperature results, all theoretical predictions look approximately the same. Because the curve of Mr(Tr) is a universal curve for all results in the molecular field approximation, we will not compare the actual magnetization results, but the difference 59 between the reduced magnetization in question and the reduced magnetization in the molecular field approximation as a function of reduced temperature. We first normalize our experimental measurements of temperature to Tr by dividing T by the TC calculated in the critical point discussion. Therefore Tr = T(K)/2.214(K). In normalizing our magnetization measurements we must first convert our internal.field measurements to magnetization. We cannot Simply normalize our B(T) by dividing Bo because this would eliminate the zero point magnetization defect. We therefore convert our B(T) to M(T) by using K = (0.1232 i 0.0027) emu/cm3—gauss, and then calculate Mr by dividing by 668 emu/cm3. Thus we have our data in the form of Mr vs. Tr’ We then use the computer to calculate Mr(Tr) in the molecular field approximation and subtract this from our measured results. Figure 4.9 is a plot of Mr(data) - Mr(molecular field) vs. Tr' The error bars on the data points are due to the uncertainty in the calibration constant, K. AS we see, near Tc the magnetization rises faster than the molecular field prediction. As the temperature is lowered the experi- mental results fall below the molecular field prediction because the molecular field does not predict a zero point magnetization defect. In the Becker and Plischke (1970) Green function calcu- lation the predicted Curie temperature is (2.48 i 0.12) K- 60 J6- 14- [K .l0- Green Function REA. .08 - a N.M.R Data Calibrated by M(T)= KB(T) K= 0.|232:I: 0.0027 e.mu/crn3-qauss '0 m l L=22MK b .Is I M. =668 e.m.u/cm3 '0 I0 Mr“ ) ‘ M(TvluoI. Ftetd l l l l I l 11 l l 0 0| 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 |.0 T/ 1;; Figure 4.9. Reduced Magnetization vs. Reduced Temperature 61 Using their preprint we have determined Mr(Tr) and subtracted the molecular field prediction. This curve is also Shown in Figure 4.8. The curve has an accuracy of only 1% because of the uncertainty in measuring the distances on the preprint graph. As we see, the Green function prediction is qualitatively correct, although the correction near TC is far too much. E. Conclusion The measurement of internal field vs. temperature at the Cl Site is very accurate. The accuracy of the field measurements ranges from 0.4% near TC to 0.05% at the low temperatures. The temperature measurements are within the accuracy of the present temperature scales. With this accuracy we conclude that our low temperature results do not agree with the Spin wave calculation. Whether this is due to assumptions made in the calculation or an insuffic- ient phenomenological Hamiltonian is not yet certain. The tentative discrepancy between the zero point magnetization defect predicted by the Spin wave theory and the experiment must be investigated further. We hope to measure this more accurately by doing N.M.R. in an applied field. If the large value for the zero point magnetization defect is indeed real, we must find a mechanism to explain 3+ this. Although Gd is an S state ion and the optical spectra Show no evidence of crystal field phonon interact- 62 ions, it is possible there may be a strong phonon-magnon coupling. Because the exchange integral is a strong function of distance and because of the long range dipole term we may expect significant phonon-magnon coupling. Rives and Walton (1968) have measured the field dependence of the thermal conductivity in MnC12°4H20 at temperatures well below the Néel temperature. Here the Mn2+ ion is also an 8 state ion, although the dipolar contribution to the exchange interaction is negligible. In the antiferromagnetic state they find a very strong dependence of the thermal conductivity on applied field, resulting in a kink at the Spin flop transition. Similar experiments can and Should be done on GdC13. These two experiments will hopefully provide the theorist with enough new information to again tackle the theoretical problem of the magnetic behavior of GdC13. V. PrCl3 THEORY AND BACKGROUND The Pr3+ ion and PrCl3 in particular have proven to be very interesting. The concentrated salt has two regions of c00perative behavior. The first is a linear chain magnetic behavior centered around 0.85 K, and the second is a phase transition to a three-dimensional ordered state at 0.4 K. we will be primarily concerned with the latter phase transi- tion. A. Pr3+ Ion Properties The absorption spectrum of the Pr3+ ion Shows the 3 ground level to be H4 (Judd, 1957). The point symmetry in LaCl3 is C3h’ which gives a non-Kramers degenerate ground state, cos(0)|4,i4>-+Sin(0)|4,i2>, with the notation IJ,MJ>. The first excited state is |4,3> and is 33.1 cm“1 above the ground state. Hutchinson Jr. and Wong (1958) have measured the para- 3+ in LaC13. Their results are summarized by the spin Hamiltonian magnetic resonance absorption of Pr H = 9N u stz xx yy B +g-LIIB(HS+HS)+ hcASzIz + AXSX + AySy. (5.1) The first two terms are the Zeeman interaction terms between the electron Spins and the applied field. The third term is the transferred hyperfine interaction between the 141Pr nucleus, I = 5/2, and the electron Spins. The ground state 63 64 of the C crystal field Should not give rise to paramagnetic 3h resonance Since 9 = 0 and the transition from one state of the doublet to the other does not contain either a AMJ = 0 or a AM = :1 transition. They have included the fourth J x y . _ 2 2 1/2 term, AXS + AyS With A — (AX + Ay) crystal strains and distortions, and thermal fluctuations. to account for Their results are g = 1.035 1 0.005, g = 0.1 i 0.15, 2 1 A = (5.02 t 0.03) x 10’ cm’l, and A = 0.02 cm— . Later Culvahouse, Pfortmiller, and Schinke (1968) have Shown that the microwave electric field is responsible for the magnetic field dependent absorption of Pr3+ ions. They replace the A term with a term of the form y(ExSx + Eysy) where 4 l). y = 6.0 x 10- (cm-1/statvo1t. cm- B. Pr3+ and Pair Resonance The Hamiltonian for two interacting ions, each with an effective spin of one-half, may be written in the form (Culvahouse, Schinke, and Pfortmiller, 1969) l l H 92 S2 mm' mm'( )Tlm(Il)Tlm' l (04") H — (fi' Amm'(2)T1m(Iz)Tlm'(SZ) + Jmm.(1.2)Tlm(Sl)T1m.(Sz). (5.2) The first two terms are the same as the Single ion terms discussed above. The last term is the interaction term between the electron spins. The existence of an inversion center between the Spins and the fact that the axis joining 65 the nearest—neighbor spins has a rotation symmetry of three or higher reduces the Hamiltonian to the form _ z z z x x x H — g" uBH (31 + $2) + gl.uB[H (81 + $2) + y y y z z z 2 H (81 + 52)] + A(Ils1 + 1252) + XX XX yy yy B(IlSl + 1252 + 1181 + 1232) + +,'* 222 K051 52 + K 5151 . (5.3) The interaction term may also be written 2 z _ x x y y where Pfortmiller (1970) has measured the pair Spectra of the O, LaES, nearest-neighbor pairs of Pr3+ in La2(C2HSSO4)6‘9H2 and in LaC13. His results are shown in Table 5.1. In estimating the actual interaction mechanism for the LaCl3 host lattice, Pfortmiller finds: 1) the error in JOO is too large to actually confirm a non-dipolar contribution to JO , 2) exchange is a negligible source for J based 1-1' on arguments regarding the size of the exchange, and 3) an O enhancement factor of 10 is necessary to account for Jl-l in terms of an electric quadrupole-quadrupole interaction alone. Finally he concludes, "Although large interactions 3+ 0 I O n.n. pairs, the dominant mechan1sm, exist between the Pr if indeed only one can be singled out, is not evident." He also did not find any pair resonances that could 66 .pmwwwum> maamucmsauoaxm me ofinmu ecu ca pmumfifl HIHh mo swam ecu "mooua cw taupe wuoz .HIHw pow umumeuma scauomumucw can paumasoamo ecu mo swam ecu uomawmu ban .mflawucoaaueaxm pmcfisumuwp comb uoa m>mn muaumaauwa cowuomumuca ecu mo «swam may 88..H moo.“H 38H 8o.“ S.+ S.+ S.+ S.+ 89H S.+ good 885 8.3. Ewe 3a.? 886+ 8a.? 20.? 8a.? 234+ :03 Soc.“ moo.“ 88H 89H So.“ 88.“ 89H 88H 8o.)H So.“ Hooo.o moo.o mamo. moo.H mos.0u «coo.oI 59¢.ot mnoo.oI «nq.oI nod.o+ mama < am < um and co HIM co m moauu mcofiuabfiuucoo muaumsmugm umaoawancoz kucaaauaaxm HI80 mo muss: mmumsm a>mn numuaawuaa :ofiuumuwucfi HH< muwmm Hwax< mum pow mumumsaumm c0wuomuwucH cwamIcHam .H.m manna 67 definitely be attributed to the off-axis next-nearest- neighbor pairs, although this does not preclude the existence of strong next-nearest-neighbor pair interactions. c. Bulk Properties The pair spectra results indicate an interaction between the Pr3+ ions along the chain. Colwell, Mangum and Utton (1969) have measured the Specific heat of PrCl3 from 0.2 K to 4.0 K. They find a broad peak in the specific heat centered at 0.85 K which is consistent with a linear chain interaction. They also find a sharp peak at 0.4 K indicating the onset of long range ordering. Because the pair Spectra results do not give any information on the interaction between the chains, and because the pure quadru- pole resonance 1ine Splits at 0.4 K, the three-dimensional critical temperature, we undertook an N.M.R. study in the three-dimensional ordered state to determine the type and symmetry of the low temperature phase. Figure 5.1 Shows 35 the zero field quadrupole frequency of Cl as a function of temperature (Colwell, Mangum and Utton, 1969). « 0.5 0.4 0.3 0.2 0.) I I I . I I I _ II- : '— 0. . 9. H- -' ' . 0 O .. : . : e P O . O O O ' e r I I I I I I N a) V’ «0. In m, <- 5 0 §45IQ Freq 44 4.3 4.2 4.) Figure I I O O) :3): It "'IZ3 YE H103 I I Mr. %%5 4 0 ° 0" I s. .4 "‘fl‘ % 17:: j H= 500 Oersted I .I I I I I I 60 40 20 6.3. Rotation Di agram, 0 20 40 60 80 l00 9 (Degrees) H .L C3, Low Tem erature Phase 5.4 - 5.3.. x HIIZ—Axis of E.F.G. Tensor“) o H 30° from Y-Axis of Tensors(2)ond(3) VQ=4600.I kHz V,=4532.9 kHz 5.l Frequency (MHz) 4.4 - 4.3 - 4.2 - 4.) r- I I I I I I I I I 0 .l .2 .3 .4 .5 .6 .7 .8 .9 LG H (KOersted) Figure 6.4. Transition Frequencies vs. Field, Low Temperature Phase 75 transition, therefore there is no internal magnetic field. The zero field splitting is due to an effective lowering of the crystal symmetry, thereby creating two non- equivalent chlorine sites. This conclusion is further substantiated by a calcula- tion of the observed Spectrum. The Hamiltonian of equation (4.7) can be diagonalized exactly by a computer for an arbitrary orientation and magnitude of B. We have calculated the Spectrum for a rotation identical to that of Figure 6.3. The pure quadrupole resonance frequency was assumed to be the zero field frequencies measured in the ordered state. The asymmetry parameter is the same as that measured in the high temperature phase, T > 0.4 K, and the magnitude of the internal field is just the applied field of 500 Oersteds. The calculated spectrum and observed spectrum agree to within the uncertainties in the measured frequencies. The measured and calculated frequencies are shown in Table C.l of Appendix C. To determine if the application of a magnetic field has any effect on the ordered phase, we applied an external magnetic field of up to 10 Kgauss along the Y-axis. The results are Shown in Figure 6.5. It appears that even a 10 Kgauss field does not alter the zero field Splitting. We have also been unable to detect any changes in the transition temperature with the application of a magnetic field. 76 03 FREQUENCY (MHZ) UI Figure 6.5. 2 3 4 5 6 7 8 9 l0 H (KlLO-OERSTED) High Field Behavior of Transition Frequencies, Low Temper- ature Phase 77 To determine the symmetry of the ordered phase we note that the 3-fold symmetry present in the high temperature phase is retained in the low temperature phase. The Space group of the high temperature phase is P63/m. This belongs to the class 6/m or C6h in Schoenflies notation. The phase transition lowers the symmetry, therefore the class of the low temperature phase must be either 6/m or one of its subclasses: 6, 6, 3, 3, 2/m, m, 2, I, or 1. The subclasses 2/m, 2, I, and 1 do not retain the 3-fold symmetry and may be eliminated as possible candidates. The possible Space groups for each of the other four classes are listed in Table 6.1 along with the reason for their being allowed or not allowed. The allowed Space group P6 removes the symmetry element of inversion and allows one set of three chlorines on a mirror plane to move out and the other to move in. The spaCe group P3 also removes the mirror plane, therefore allowing the Pr to move out of the plane of the chlorines. Although we have not definitely established the exist- ence of a real crystallographic phase transition, the actual ordering mechanism Should also remove these same symmetry elements. When the zero field lines are recorded with a cw marginal oscillator and second derivative detection using magnetic modulation we see the lines Shown in Figure 6.6. The upper line is from the high temperature phase and is 78 Table 6.1. Analysis of Possible Space Groups for PrCl3 Class Space Group ' Reason 6/m P63/m ImprOper number of C1 positions 6/m P6/m Both Pr either in a plane or on the same symmetry line. 6' P6' Allowed 6 P63 ImprOper number of Cl positions 6 P6h Improper number of Cl positions 6 P62 Impr0per number of Cl positions 6 P65 Impr0per number of Cl positions 6 P61 Improper number of C1 positions 6 P6 Both Pr in the same plane 3' R3' Improper number of Cl positions - P3. Improper number of Cl positions 3 R3 Improper number of Cl positions 3 P32 Improper number of 01 positions 3 P31 Improper number of C1 positions 3 P3 Allowed 79 JL. 4566.7 kHz T8 4.0 K L. 4528.3 kHz 4604.9 kHz T=O.32 K Figure 6.6 Second Derivative of Absorption Curve , 15:013. 80 symmetric down to the critical temperature. In the low temperature phase the lines are asymmetric and change shape as a function of temperature. This asymmetry is also observed when frequency modulation is used. Since the low temperature phase is of lower symmetry we have postulated that the observed line is not necessarily due to a single site. If we assume that the absorption curve is a gaussian of the form .1 1 _ (f—f°)2 A(f) = ——- —-e 2 (6.1) o 20 {Zn where f is the frequency of the detecting oscillator, f° is the resonance frequency of the line, and o is the line width, we find that the observed line shape is given by o 2 A"(f) = —l— {4(f-f°)2 - 2}.e'(f"f ) (6.2) /; where we assume 0 = —l;-. /5 We then assume that the observed line in the low temperature phase is given by 2 - - o 2 (f fi) N 1 {4(f-f3) - 2} e (6.3) An(f) = Z __ i=1 /F where N is the number of nearly equivalent sites. We can reproduce the observed asymmetry if we let N = 3, and let f3 = f2 = 0 and f1 = 1//§} or the line width. This calcula- tion is shown in Figure 6.7. From this we may conclude that the 3-fold symmetry is slightly removed. An attempt to 81 Figure 6.7 Calculated Second Derivative of Absorption Curve. 82 measure the deviation from 3-fold symmetry using the rotation spectrum was unsuccessful. C. Conclusions From the N.M.R. results, we conclude that the low temperature phase transition in PrCl3 is not to an anti- ferromagnetic phase as previously assumed, but instead is to an ordered state which effectively lowers the crystal symmetry. The actual mechanism responsible for the ordering can not be determined by N.M.R. Because the ground state of the Pr3+ ion is a non-Kramers doublet we may assume that a strong coupling to the lattice vibrations is present. If the exchange constants are of the proper magnitude this will allow a Jahn-Teller type distortion (Allen, 1968). 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R., 1962, Report No. LAMS-2701, (unpublished). 86 Smart, J. 8., 1966, Effective Field Theories 9§_Magnetism, W. B. Saunders Co.,_Phi1adepphia and London. Stevens, K. W. H., 1963, in Magnetism, Vol. I, edited by G. T. Rado and H. Suhl, Academic Press, New York. Streib, B. Callen, H. B., and Horwitz, G., 1963, Phys. Rev. 338, 1798. Sydoriak, S. G. and Roberts, T. R., 1957, Phys. Rev., _88, 175. Walton, D., 1966, Rev. Sci. Inst., 33, 734. White, G. H., 1968, Experimental Techniques 32’Low-Tempera- ture Physics, Second Edition, Oxford Press. Williamson, J. H., 1968, Can. J. Phys., 38, 1845. Wolf, W. P., 1971, J. Physique, 33, 26. Wolf, W. P., Leask, M. J. M., Mangum, B., and Wyatt, A. F. G., 1961, J. Phys. Soc. Japan, 33, Suppl. 8-1, 487. Wyatt, A. F. G., 1963, thesis, Some Properties 93 Magnetic £222 32 Solids, Oxford University. Zachariasen, W. H., 1948, Acta Cryst., 3, 265. Zubarev, D. N., 1960, Usp. Fiz. Nauk., 33, 71. (English Translation: Soviet Phys.--Uspekhi, 3, 320). APPENDICES APPENDIX A CALIBRATION OF THE 3He SOLITRON GERMANIUM RESISTOR The susceptibility coil calibration data are given in Table A. 1. These data are fit to the equation D = A(T*-1) + B. Using a normal least squares analysis the result is l D = -(50824 i 81)T*- + (69692 1 101) (A.1) Because the normal least squares analysis weights all points equally, which is not our case as indicated by the variation in AT‘”.1 of Table A.1, we also fit the data using a least squares program of Williamson (1968) which weights the datum points with the inverse square of the standard deviation of each point, and allows for standard 1 deviations in both the T*- and D values of the datum. The result of this fit is 1 D = -(50527 i 76)T*- + (69382 x 75) (A.2) The low temperatures calculated using these two results agree to within the uncertainty in the calculated temperatures. We will use the latter result because our data do not fit the criteria for the normal least squares analysis. Table A. 3 shows the raw data for R vs. D for T < 0.6 K and the calculated temperatures based on the Williamson analysis. The uncertainty in the temperature is due to the uncertain- ties in the coefficients A and B. Table A.4 shows the 87 88 Table A.1. Susceptibility Coil Calibration Data T(K) T 14% AT *X lO-l‘a Regciiiaigb $216135; 1.200 0.8302 3.0 27h0h 20 1.121 0.8882 3.5 2h530 10 1.052 0.9h59 h.5 21582 10 0.990 1.00h5 5.0 18600 10 0.936 1.0617 5.9 15693 10 0.887 1.1196 7.0 12935 10 0.8h2 1.1785 9.7 09898 10 0.802 1.236h 12.8 06889 20 0.766 1.293h 1h.1 03976 15 0.701 1.h108 2h.3 -01936 10 0.673 1.h681 31.9 -0h900 10 0.6h7 1.5256 3h.0 .07836 05 0.623 1.5828 h5.1 -10808 05 0.600 1.6h17 56.1 -13873 10 a The uncertainty in T*‘1 arises from the uncertainty in T which is approximately constant for all temperatures. b This is the dial reading on a Cryogenics mutual inductance bridge and is directly pr0portional to the susceptibility. 89 calibration data for R vs. T for the temperature range 0.3 K to 1.2 K. Below 0.6 K the temperature is taken from Table A.3, above 0.6 K the temperature is determined by measuring the vapor pressure of the 3He hath. These data were fit to the equation ln(R) = a0 + a1 ln(T) + az/T. (A.3) Because the germanium resistor is so heavily doped we do not expect it to follow a single equation over a large temperature range. For this reason we have divided up the temperature range from 0.3 K to 1.2 K into four sections and fitted each section separately. The standard method of least squares (Mack, 1966) was used to determine the coefficients, a0, a1, and a2. The results of this fit are given in Table A.2. The last column in Table A.4 shows (R - Rmeas )/(dR/dT) to show the effective temperature calc. deviation between the calculated and measured values. Table A.2. Coefficients for 3He Solitron Ge Resistor Temp. Range (K) a a1 a2 1.200 to 0.700 5.769325 -0.875580 1.142215 0.699 to 0.550 6.378414 -1.859392 0.470550 0.549 to 0.415 6.139522 -1.639297 0.675030 0.414 to 0.300 6.076923 -l.717617 0.664094 90 Table 11.3. Raw Data for Ge Resistor Calibration Dial Reading T(K) AT(mK) B(ohms) _15097 0.589 1.5 3500 x 1 -16362 0.580 1.h 3650 x 1 -18232 0.567 1.h 3880 x 1 -19856 0.556, 1.h h090 x 1 -21407 0.5h7 1.3 A300 x 1 -23180 0.536 1.3 4550 x 1 -2h851 0.526 1.3 h800 x 1 —26700 0.515 1.2 5100 x 1 -28500 0.505 1.2 5h00 x 1 -30500 0.h95 1.2 5750 x 1 -32390 0.h85 1.1 6100 x 1 ~3hh37 0.h75 1.1 6500 x 1 -36613 0.h65 1.1 6950 x 1 -38900 0.h55 1.1 7h57 x 1 -h1555 0.hh3 1.0 8100 x 1 -hh000 0.h33 1.0 8700 x 1 -h6917 0.h21 1.0 9500 x 1 -50180 0.h09 0.9 1030 x 10 -5237h 0.h01 0.9 1100 x 10 -5h517 0.39h 0.9 1160 x 10 -58285 0.381 0.9 1300 x 10 -60812 0.373 0.8 1h00 X 10 -6h326 0.363 0.8 1550 x 10 -71293 0.3h3 0.8 1900 x 10 —7h810 0.33h 0.7 2100 x 10 -795h9 0.322 0.7 2&00 x 10 -8hh10 0.311 0.7 2750 x 10 -88900 0.301 0.7 3100 x 10 -93180 0.292 0.7 3507 x 10 91 Table A.4. R. vs. T Calibration Data and Deviation T AT R (Re-Rm)/(dR/dT) (K) (mK) (ohms) (mK) 1.172 1.0 738.4 -0.29 1.150 1.0 765.3 0.05 1.106 1.0 823.4 -0.24 1.100 1.0 832.5 0.11 1.050 1.0 911.1 0.12 1.047 1.0 916.1 0.04 1.000 1.0 1004.6 0.40 0.995 1.0 1014.4 0.19 0.903 1.0 1240.6 -0.10 0.900 1.0 1244.9 -1.63 0.850 1.0 1413.5 -0.61 0.827 1.0 1507.0 0.40 0.800 1.0 1623.2 -0.11 0.762 1.0 1818.0 -0.26 0.750 1.0 1891.5 0.30 0.707 1.0 2180.6 -0.32 0.700 1.0 2240.6 0.19 0.675 1.0 2453.8 -0.25 0.658 1.0 2615.6 -0.65 0.650 1.0 2712.7 0.60 0.625 1.0 3002.0 0.44 0.616 1.0 3110.4 -0.16 0.589 1.5 3500.0 -0.23 0.580 1.4 3650.0 -0.02 0.567 1.4 3880.0 0.06 0.556 1.4 4090.0 0.03 0.547 1.3 4300.0 0.73 0.536 1.3 4550.0 0.31 0.526 1.3 4800.0 0.06 0.515 1.2 5100.0 -0.19 0.505 1.2 5400.0 —0.34 0.495 1.2 5750.0 0.16 0.485 1.1 6100.0 -0.24 0.475 1.1 6500.0 -0.22 0.465 1.1 6950.0 0.01 0.455 1.1 7457.0 0.43 0.443 1.0 8100.0 0.23 0.433 1.0 8700.0 0.06 0.421 1.0 9500.0 -0.27 0.409 0.9 10300.0 0.45 0.401 0.9 11000.0 0.37 0.394 0.9 11600.0 -0.40 0.381 0.9 13000.0 -0.52 0.373 0.8 14000.0 -0.47 0.363 0.8 15500.0 0.17 92 Table A.4. (cont'd.) T AT R (Re-Rm)/(dR/dT) (K) (mK) (ohms) mm 0.343 0.8 19000.0 0.12 0.334 0.7 21000.0 0.32 0.322 0.7 24000.0 0.01 0.311 0.7 27500.0 0.29 0.301 0.7 31000.0 -0.29 0.292 0.7 35070.0 -0.04 APPENDIX B GdCl3 TABLES OF DATA AND COMPARISONS 35 Table B.1 shows the transition frequencies of C1 and 37C1 as a function of temperature in the ordered state.. The zero field paramagnetic state transition frequencies 35 37C1. The recorded are 5314 kHz for C1 and 4188 kHz for frequencies are the average of five independent readings and the standard deviation, SD, is the statistical standard deviation for the five readings. J is the isotope label and I the line label discussed in the text. The uncertainty in the temperature is the same as the uncertainty in the vapor pressure tables, 1 2mK. The table is in three sections, one for each different data taking run. Table B.2 shows the results of the chi squared analysis used to determine the magnitude of the internal field. Table B.3 shows the comparison between the measured internal field and the analytic expression for the critical behavior. Table B.4 shows the same comparison for the Spin wave region. 93 Table 8.1 Section 1 C1 Transition Frequencies in GdC13, 4He Data TEMP 2.170 2.164 2.149 2.139 2.120 2.110 94 J 35 35 35 37 37 35 35 35 37 37 35 37 35 37 #UNSUN UN6‘UN UNSUN UN¢UN #9 ##U FREQ. 5708.81 5120.00 4876.37 4521.57 4032.79 5735.58 5115.83 4859.37 4545.87 4029.43 5791.63 5108.43 4822.07 4590.04 4025.46 5824.74 5106.01 4803.96 4622.03 4023.26 3770.94 4770.45 3743.97 5106.14 4755.94 3732.77 5.0. .58 .41 1.37 .44 .20 .50 .07 1.19 .87 .30 .63 .21 .43 1.14 .46 .48 .31 .51 1.60 .44 .70 .31 .87 7.30 .24 .55 Table 8.1 Section 1 (cont'd.) TEMP 2.100 2.091 2.069 2.037 2.019 95 J 35 35 35 37 35 35 35 37 37 35 35 37 37 #UthJN #UéUN ##UN #UNSUN «FUNSUN FREQ. 5937.53 5102.83 4742.56 3722.28 5967.81 5105.14 4731.66 4025.45 3713.69 6021.19 5108.80 4708.90 4788.06 4032.38 3696.62 6097.52 5117.80 4680.30 4852.06 4041.11 3674.63 6136.14 5125.44 4667.28 4884.25 4046.93 3664.74 5.0. *.85 1.51 .41 .43 .50 .20 .15 1.44 .35 .48 .12 .26 3.24 .33 .69 .18 .15 .19 .67 .24 .63 .19 .25 .02 .64 .43 .42 Table B.1 Section 1 (cont'd.) TEMP 2.000 1.980 1.949 1.920 1.890 96 35 35 37 37 37 35 35 35 37 37 35 35 35 37 37 37 35 35 37 37 37 ouwaaw 9UN§UN #UNvtth-lN bUNl-‘UN #UNSUN FREQ. 6177.27 5130.16 4654.43 4920.35 4054.02 3655.33 6214.34 5137.65 4643.86 4952.56 4061.25 3646.49 6274.37 5150.35 4628.33 5002.42 4073.51 3636.03 6323.92 5162.42 4616.49 5045.42 4085.88 3626.73 6373.93 5175.80 4605.54 5088.00 4098.70 3619.07 5.0. .14 .11 .09 .24 .18 .29 .41 .20 .32 .35 .36 .56 .17 .08 .25 .20 .18 .21 .12 .09 .35 .45 .19 .13 .28 .16 .11 .40 .14 .16 Table B.1 Section 1 (cont'd.) TEMP 1.860 1.830 1.799 1.769 1.741 97 35 35 35 37 37 37 35 35 35 37 37 b1d¢Wth #00b4dh9 #LQN4>UIV #(dflhkhJN btd#%dhl FREQ. 6420.04 5188.85 4597.46 5127.30 4110.97 3613.73 6465.27 5202.75 4590.20 5165.76 4124.10 3608.33 6509.08 5216.83 4583.40 4137.77 3604.08 6549.63 5230.73 4578.00 4150.55 3600.92 6585.58 5243.51 4573.95 4162.66 3598.45 500. .53 .09 .19 .06 .11 .19 .20 .13 .40 .23 .10 .26 .28 .20 .21 .25 .37 .39 .05 .07 .09 .20 .29 .10 .11 .13 .70 Table B.1 Section 1 (cont'd.) TEMP 1.712 1.661 1.650 1.618 1.594 98 J 35 35 35 37 37 37 35 35 35 37 37 37 35 35 35 37 37 37 35 35 37 37 37 aumedw «PUN‘PUN PUNPUN FUN‘P‘UN éUNPUN FREQ. 6623.19 5257.48 4569.99 5300.02 4175.62 3596.13 6683.39 5280.71 4565.01 5351.50 4197.12 3593.56 6694.67 5285.10 4564.03 5360.99 4201.31 3593.38 6728.36 5298.77 4561.67 5389.72 4213.83 3592.17 6761.12 5312.84 4559.89 5417.87 4226.86 3591.52 5.0. .24 .12 .07 .19 .11 .50 .21 .06 .13 .34 .03 .18 .19 .08 .04 .32 .13 .30 .32 .07 .03 .17 .06 .22 .61 .10 .05 .22 .30 .30 Table B.1 Section 1 (cont'd.) TEMP 1.550 1.500 1.450 1.431 1.350 99 35 35 35 37 37 37 35 35 35 37 37 37 #UNPUN #UNPUN #UNFUN «PUNkUN PUNPUN FREQ. 6805.68 5331.88 4558.00 5455.80 4244.24 3591.02 6855.16 5354.04 4556.52 5497.99 4264.38 3591.33 6902.29 5375.83 4555.76 5537.98 4284.15 3592.19 6919.53 5383.91 4555.57 5552.33 4291.59 3592.32 6989.14 5417.01 4555.75 5611.45 4321.75 3594.38 5.0. .20 .19 .09 .43 .34 .56 .17 .12 .03 .24 .20 .38 .17 .12 .06 .31 .07 .55 .29 .11 .04 .42 .13 .24 .21 .05 .03 .33 .15 .19 100 Table B.1 Section 1 (cont'd.) TEMP J 1 FREQ. 5.0. 1.299 35 2 7029.47 .51 35 -3 5436.96 .05 35 4 4556.60 .08 37 2 5645.94 .15 37 3 4340.03 .28 37 4 3596.05 .18 1.251 35 2 7067.78 .27 35 3 5456.52 .08 35 4 4557.70 .06 37 2 5678.41 .09 37 3 4357.57 .07 37 4 3598.04 .23 1.2008 35 2 7105.74 .21 35 3 5475.79 .06 35 4 4559.07 .02 37 2 5710.45 .38 37 3 4374.84 .15 37 4 3600.15 .23 1.025‘El 35 3 5541.18 .22 35 2 7227.62 .39 37 2 5814.22 .18 37 3 4433.67 .05 37 4 3609.07 .14 a Net used because the uncertainty in the temperature measurement is too large. Table 8.1 Section 2 TEMP .599 .577 .530 .483 .450 .401 101 35 35 35 35 35 35 35 35 35 35 35 35 #UN PUN «PUN bum SUN bUN 3He Data of Feb 26, 1971 FREQ. 7458.51 5673.84 4587.21 7468.63 5680.46 4588.72 7488.42 5691.71 4590.37 7507.51 5702.93 4592.49 7519.05 5711.11 4593.95 7534.75 5720.68 4596.30 5.0. .31 .27 .25 .84 .45 .45 .34 .29 .64 .75 .43 .26 .59 .30 .41 .26 .51 .22 Table B.1. Section 3 TEMP 1.222 1.200 1.150 1.100 1.050 1.000 .950 .900 102 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 bUN PUN #UN #60") 50h) PUN #‘UN PUN FREQ. 7088.08 5466.97 4558.80 7103.62 5475.26 7140.45 5494.56 4559.10 7175.26 5512.88 4561.75 7208.37 5531.00 4564.69 7241.11 5548.96 4565.93 7272.01 5566.03 4567.88 7301.64 5582.90 4570.70 3 He Data of March 24, 25, and 26, 1971 5.0. .26 .17 .69 .25 .09 .38 .20 .11 .50 .13 .23 .50 .19 .09 .50 .23 .11 .40 .10 .07 .16 .09 .06 .50 V"I"——-r- ~— Table B.1 Section 3 (cont'd.) TEMP .850 .800 .750b .700b .650b .550 .500 .474 103 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 37 35 35 37 35 35 37 NUN MUN PUN PUN J-‘UN SUN bUN NUN FREQ. 7329.89 5599.16 4573.43 7357.90 5614.85 4576.40 7380.89 5628.17 4578.36 7407.10 5643.89 4581.06 7433.57 5659.61 4583.20 7479.04 5686.83 6027.83 7499.70 5699.20 6045.24 7509.74 5705.25 6053.61 5.0. .12 .17 .50 .24 .15 .50 .15 .11 .50 .12 .07 .50 .20 .08 .50 .12 .11 .17 .17 .41 .29 .17 .12 .20 104 Table 8.1 Section 3 (cont'd.) TEMP J I FREQ. S.D. .425 35 2 7526.88 .19 35 3 5715.72 .12 37 2 6068.65 .59 .400 35 2 7534.41 .15 35 3 5720.51 .16 37 2 6074.80 .29 .375 35 2 7542.10 .19 35 3 5725.30 .14 37 2 6081.30 .22 .350 35 2 7549.15 .14 35 3 5729.55 .14 37 2 6087.34 .09 .325 35 2 7554.64 .19 35 3 5733.05 .12 37 2 6092.26 .12 .310 35 2 7559.08 .06 35 3 5736.00 .03 37 2 6095.98 .22 .293c 35 2 7563.10 .04 35 3 5738.23 .17 37 2 6098.79 .31 Thermal equilibrium difficult to obtain, these points not used. c Not used because its the lowest temperature point. 105 Table B.2. Results of')[2 Analysis for B(T). T(K) B AB3 7,2 Af(1-w) (Kelvins) (gauss) (gauss) (kHz) 2.170 1290.3 3.0 Av? -—- 2.164 1362.0 4.5 0.90 1.0 2.149 1505.0 3.5 1.04 0.7 2.139 1589.2 4.5 1.09 1.0 2.120 1735.8 4.5 1.27 1.0 2.110 1805.3 4.5 1.06 0.7 2.100 1872.4 5.0 0.72 1.0 2.091 1932.0 4.5 1.14 1.0 2.069 2054.0 2.5 0.80 0.5 2.037 2224.2 2.5 Av. --- 2.019 2308.0 4.0 1.28 1 0 2.000 2396.4 2.0 Av. --- 1.980 2475.9 2.5 A7. --- 1.949 2599.5 2.5 AV. --- 1.920 2702.0 2.0 AV. --- 1.890 2803.0 2.5 AV. --- 1.860 2894.5 2.0 0.93 0 5 1.830 2983.2 2.0 1.30 0.5 1.799 3067.8 2.5 Av. --- 1.769 3145.7 2.0 0.82 0 5 1.741 3214.0 1.5 AV. --- 1.712 3285.7 2.0 A7. --- 1.661 3398.4 1.5 0.92 0.5 1.650 3419.2 1.5 0.60 0 5 1.618 3481.5 1.5 0.92 0.5 1.594 3542.2 2.5 AV. --- 1.550 3622.9 2.0 0.87 0.5 1.500 3712.5 1.5 1.10 0.5 1.450 3796.7 1.5 0.93 0.5 1.431 3827.2 2.0 A7. --- 1.350 3949.8 1.5 0.79 0.5 1.299 4020.9 1.5 1.20 0.5 1.251 4087.3 1.5 1.10 0.5 1.222d 4121.4 0.5 0.55 0.0 1.200 4148.3 2.0 0.79 1.0 1.150 4211.5 2.0 0.69 1.0 1.100 4270.9 0.5 0.504 0.0 1.050 4327.6 0.5 1.215 0.0 1.000 4382.9 0.5 1.534 0.0 0.950 4434.8 1.5 A7. --- 0.900 4484.7 1.0 A7. --- 0.850 4532.0 1.0 A7. --- 0.800 4578.4 0.5 0.084c 0.0 0.750e 4616.5 0.5 0.326c 0.0 0.700e 4660.4 0.5 A7. --- 106 Table 8.2. (cont'd.) T(K) B Asa 2 Af(1-w) (Kelvins) (gauss) (gauss) x (kHz) 0.650e 4704.1 1.0 A7. --- 0.599 4744.6 1.0 A7. --- 0.577 4762.0 1.5 1.32 0.0 0.550 4780.1 2.0 1.30 1.0 0.530 4793.4 1.0 0.94 0.0 0.500 4813.7 2.0 1.17 1.0 0.483 4823.9 1.5 0.92 0.0 0.474 4830.0 1.5 1.061 1.0 0.450 4844.4 0.5 0.24c 0.0 0.425 4858.1 2.5 1.46 0.0 0.401 4869.7 0.5 0.94 0.0 0.400 4870.5 2.0 1.33 1.0 0.375 4883.1 2.0 1.31 1.0 0.350 4894.6 2.0 1.47 1.0 0.325 4903.8 2.5 1.81 1.0 0.310 4911.0 2.0 1.69 1.0 (3B is estimated to within 0.5 gauss. b Only values for £5f(l-w) of 0.0, 1.0, 0.7, and 0.5 were used. When the results for 0.0 and 1.0 differed by less than 0.5 gauss even though I 2 ranged from well below 1.0 to well above 1.0, an average between the two values was taken. 9 Increasing Af(l-w) will only make 12 smaller; this indicates a large spurious standard deviation error. d All temperatures 1.22 K and below used the 3He system. This value was not used in the analysis; thermal equilibrium.was difficult to achieve and the temperature was difficult to measure. 107 squared value is 0.710. Table B.3. Comparison of B(T) with Analytic Expression for TaNITc. Temperature B(T) lLB(T) 8m - B (Kelvins) (gauss) (gauss) (gauss 2.170 1290.3 2.5 -2.330 2.164 1362.0 4.5 3.520 2.149 1505.0 3.5 0.550 2.139 1589.2 4.5 -1.440 2.120 1735.8 4.5 -1.070 2.110 1805.3 4.5 —1.340 2.100 1872.4 5.0 -0.051 2.091 1932.0 4.5 3.270 2.069 2054.0 2.5 -2.510 2.037 2224.0 2.0 1.400 2.019 2308.0 4.0 -0.370 2.000 2396.4 2.0 2.780 1.980 2475.9 2.5 -2.610 The analytic expression is B(T)= A(Tc - T)8 where A " (4368.4 331.3) gauss/K9 , Tc= (2.214 :0.0016) K, and 140.3904 :0.006). The chi 108 Table 8.4. Comparison of B(T) with Analytic Expression for T <,1.0 K. T(K) B(T) AB(T) Bm - Bc Bm - 1:1c (Kelvins) (gauss) (gauss) Set A Set B 0.310 4911.0 2.0 -0.24 0.53 0.325 4903.8 2.5 -1.43 -0.91 0.350 4874.6 2.0 0.09 0.27 0.375 4883.1 2.0 0.18 0.10 0.400 4870.5 2.0 -0.01 -0.26 0.401 4869.7 0.5 -0.29 -0.55 0.425 4858.1 2.5 0.81 0.45 0.450 4844.4 0.5 1.09 0.69 0.474 4830.0 1.5 0.81 0.42 0.483 4823.9 1.5 0.17 -0.20 0.500 4813.7 2.0 0.54 0.21 0.530 4793.4 1.0 -0.33 -0.54 0.550 4780.1 2.0 -0.15 -0.25 0.577 4762.0 1.5 0.60 0.66 0.599 4744.6 1.0 -0.90 —0.69 0.800a 4578.4 0.5 -1.21 -0.14 0.850 4532.0 1.0 -1.06 -0.15 0.900 4484.7 1.0 -0.27 0.63 0.950 4434.8 1.5 0.58 0.37 1.000 4382.9 0.5 0.83 -0.42 Set A fitted to: B(T) = 4957.5 - 811.0 T3/2 exp(-0.343/T) Set B fitted to: B(T) = 4950.8 —(963.1 T3/2 - 90.7 T5/2) exp(-0.430/T) APPENDIX C PrCl3 DATA Table C.1 shows the measured frequencies as a function of angle for an external field of 500 Oersteds applied in the plane perpendicular to the C3 axis in the ordered state. The angles are in the laboratory reference system. The site label is arbitrary. LN is the line number which defines the transition, 1 being the highest in frequency and 4 lowest. The H and L refer to the High and Low zero field lines respectively. Because the lines are weak and were only measured once, the accuracy of the frequency measurement is estimated at i 5 kHz. The calculated frequency assumes H = 500 Oersteds, n = 0.4937, vQ(H) = 4598.6 kHz, and vQ(L) = 4535.5 kHz. The location of the Z-principal axis in the laboratory reference frame = 290°, and z = 50°. is assumed to be: Z = 350°, Z 3 1 2 109 Table C.1. 110 Cl Transition Frequencies in PrCl3 Angle Site LN LOG Freq. Calc. (deg) (kHz) (kHz) 5'. l 1 H 4901.9“ 4960.35 28*. 1 9 I 4ln3.70 4190.41 )4». y / H 47M/.7“ 4795.U1 246, w z 1 471H.20 47+K.9l 28:. a 1 H 4147.60 4404.52 2H,, / 3 l 4356.30 4341.44 2A», 3 r H 4615.00 4609.17 244, 3 2 L 4s/h.so uth./5 Ant. ,3 3 | 4S30.dh 453h.5n R W‘. l i .4 “A30;5.H(1 ‘49}h4.t+l #97. l l 1 4914.50 4925.4l 89-. 1 4 H 4210.60 4821.26 HIM. 9 K H 4800.HU 4908.63 ’”'- F 6 L 9741.50 4745.55 29:. g 4 H u}HH.Qh 4191.43 29., / s I 4337.00 4128.35 299. 3 r 4 4681.20 4673.58 Run. 1 a L 4627.70 4610.5H 29:. 3 s H 4528.70 4538.10 243,,. 3 3 L 414737.711 447501” $01. 1 1 H 5018.50 5007.44 +u~. l 1 I 4951.90 4944.82 3H4. ] u H QIQH.QU 4?01.1” sun. / 3 1 4353.60 4341.44 5M4. s r 1 4hh7.90 4678.11 1“ . 1 ‘ H 4464.00 447H.h5 14a. 4 i L 4407.90 4415.67 11w. r r 4 4755.20 476/.13 sl«. 2 r L 4692.80 4999-3“ 11‘. g g u 4465.10 4439.55 11’. 1 K H 4794.30 4788.91 113. 7 4 H 4411.30 4486.20 520. x r H 4/0l.90 4714.92 14». K n l 4fi4&.40 4651.H/ 427. a A H 449m.40 44HH.99 32». a i 1 4442.10 4425.94 121. 1 7- H quu./U 441n,va 3/:. i r‘ L «+757./H 44753.4/ 1)). ‘1 .4 H 4391./0 4399.17 12». i 3 I 4137.40 433h.8/ 111 Table C.1. (cont'd.) Angle Site LN LOC Freq. Calc. (deg) (kHz) (kHz) Ian. 1 a II 47hH.LH1 47h6.33 *1 . 1 K 1 4710.10 4h99.ah 11;. 1 ; 4 4455.40 4439.54 11'. / / H 4h47.50 4059.33 11 . K r I 45H5,H0 454h.a9 15 . g j H QHfiV.fiH 4§4’.ll 110. K i 1 4494.40 44H4.0H s4». 1 r H 479H.h0 47Qh.01 1+1. 1 K I 413H.HO 4738.91 14!. 1 5 H 4401.10 4404.5K 144. 1 s I 4144.40 4141.44 44,. 5 ( H qh/U.?H 4509.7/ 14;, 2 r L 4550.70 4540.15 341. x 5 H 4559.40 4599.56 140. d i L 4584.40 4530.35 141. I 1 H 4966.30 4960.35 14». I 1 1 4405.00 4897.35 151. 1 r 4 4805.50 4406.03 ifii. 1 r L 4744.90 4745.55 155. 1 1 H 43HH,HU 4141,41 «an. 1 1 I 4311.00 4134.35 ‘34. / / H hhfifiggn 457‘o5fl I51. ( t 1 4hd3.10 4410.54 151. x J 4 4524.90 455H.10 15‘. 2 5 I 44nd./0 4475.10 151. I I L 4941.00 4925.41 “A“‘o '5 "‘ H “2150““ “92.5085 350. I 4 1 4154.70 41hu.al 1mm, 1 r q 4751,!0 479h.01 1n1. 1 r L 47(7.60 473(.91 100. 1 5 L 4140.50 4441.44 150. H c L 4543.00 4575.11 3a,. 3 5 H QqhH.qO 447H.65 ,4”, 3 1 n 501r>.00 50111.04 1A0. 5 1 1 4950.00 4944.H/ MW". 9 “‘1 4198.7“ ‘0?”103“ 112 Table C.1. (cont'd.) Angle Site LN LOC Freq. Calc. (deg) (kHz) (kHz) 1'. 1 g H 471H.h0 4752.31 1!. 1 r L 4595.70 4599.Rh l". 1 j H 444h.30 4419.59 1“. 1 3 I 41H4.00 4375.4! 1 . K k 9 4797.10 4785.91 1 . g 3 H 441/.h0 4425.8“ 10. P J L 4555.90 4163.23 14. 5 1 L 4459.00 4054.40 1 . 1 4 w 41H4.70 4188.50 /0. 1 t H 4705.90 4714.92 24. 1 r L 4h45.00 4651.47 /4. 1 J H 449h.70 4408.99 (.1 . I .5 L 44 36. '41‘ 4425.94 21. x r H 441h.50 4Rlb.19 K0. / J H 4401.10 4399.19 (0. g 5 L 45J4.90 4336.2? 21. 1 1 H 50d5.h0 5018.99 21. 5 1 L 4954.50 4955.91 10. 1 r H 4h49.70 4659.31 11. 1 a L 4505.50 459h.29 10. 1 J H 4555.90 4547.11 11. l J L 4494.90 4484.05 10. 2 I I 4865.80 4Rn4.31 1;. d 4 9 4279.50 4?H7.51 1h. 5 5 H 4705.50 476d.3$ $0. 1 K L 4709.H0 4599.2h 4“. 1 r H 4hHl.H0 4409.77 .1. 1 f L 4557.70 4545.75 40. 1 A 4 4594.00 4599.31‘ 4). 1 3 L 4527.20 4530.15 44. (f 1 H 4967.00 49150.35 4'1. K 1 L 49071.70 41197.35 40. z 4 H 4241.10 4251.40 40. .1 r.’ ..1 4H01.H11 479mm 4w. 5 c I 4753.20 473/.91 4'. 1 5 H 940M.9U 4404.5/ 41. )1 1 | 4341.1H1 4141.40 113 Table C.1. (cont'd.) Angle Site LN LOC Freq. Calc. (deg) (kHz) {kHz} 3”. 1 K H 48Hh.4h 4673.5R W”, 1 f L “thgnU “61”05“ W”. l 5 H #865.”“ “RBHoln 11. 1 J I 4404.10 4475.10 5«, / 1 I 4969.70 4925.41 51. g 1 4 4214.20 4223.80 ‘31:. 2 4' L 41fi7.h11 41hU.// 01. 1 r H 4407.50 4ROH.h1 5.. 1 r 1 4741.20 4745.65 0 . 1 1 H 4395.90 4191.41 ~11. 1 A l_ ‘1151.1Hl ‘+3?H.'Vfi hi. 1 K H 474H.00 4775.1” h . 1 K L 45H4.h0 4572.11 "1. 1 1 H 4150.50 447H.h5 n . 2 1 H 5014.00 5007.H4 n». x 1 I 4944.50 4944.52 n0. / 4 H 4190.40 4801.30 40. 1 / H 4798.10 4796.01 M1. 1 r I 4751.90 473d.93 n1. 1 1 L 4149.40 4341.44 71. 1 r H 4797.90 4783.91 19. 1 1 H 4417.00 4426.20 70. 1 1 1 4154.40 41hJ.di 11. g 1 L 4954.50 9954-8” 70, g 4 H 4105.30 4IHH.nU 70. g 4 L 4185.50 4185.56 70. 1 K H 4755.10 4762.11 71. 1 r L 4hHH.10 4699.?5 In. 1 1 H 4456.50 4439.51 7”. 1 1 L 43HH.70 4176.47 H1. 1 c H 4H15.50 4916.1q HH. 1 r L 4754.HH 4751.4? wu, i 1 11 419/.fHI 4399.19 H0. 1 1 I 4313.50 4136.22 ~w. / 1 1 4944.40 4455.41 11.1. ,J 1.. I1 41H").(511 41744.91 00. x 4 L 4167.40 4121.H7 41, 1 / H 4704.40 4714.92 H1. 1 w I 4515.50 4651.“? A1. 1 1 H 4501.40 44HH.99 51. 1 1 I 4417.10 4425.94 114 Table C.1. (cont'd) Angle Site LN LOC Freq. Cale. (deg) (kHz) (kHz) 41. 1 / H 4704.40 47HH.91 ”I. 1 g 1 4704.00 4725.91 4.. 1 1 H 4417.40 44?h.?“ #1. 1 1 I 41/fi.40 4163.25 “I. 1 K H 4549.10 4h59.31 an, 5 a 1 4Hn1.40 4446.34 4.. 1 1- .4 4500.111 4847.11 40. 1 1 I 4497.50 44H4.0H /«H. 1 1 H 49H7.20 4475.14 2n». l 1 I 49¢].IH 401d.14 21“. 1 4 H 4a/H.H0 4?57.4H HAW. / r H 4HUb./0 4R05.3W 341. a ( I 4740.00 4745.31 243. / 5 14 4394.h0 4194.79 241. 2 1 L 41.14.70 4111.71 940. 1 r H 4051.70 4541.71 RHfi. 1 r L 45HU.H0 4§78.Id 8H5. J 1 L 4494.10 4905.7” 291. 1 1 H 5006.R0 4099.89 244. l l I 4440.40 4430.?H a4». 1 4 H 420£.h0 4P11.lh 244. a t H 4408.10 4005.39 99‘. K a 1 4753.20 4742.31 244. 1 a I 4h58.40 4441.8H 344. 1 J H 4446.30 4507.92 )9“. 3 J L 4433.40 4444.93 11». 1 1 H bOKh.10 5019.44 111. 1 1 L 495H.00 495h.39 11%. 1 4 H 41Hd.20 4185.70 111. H r H 4716.50 4739.94 11‘. 8 f I 4hh7.10 467h.8H 11%. r 1 H 4471.70 44hR.HI 11“. 1 K H 4H11.QU 480H.43 11w. 1 1 H 4402.10 4407.04 114. '1 A 1 4117.611 4144.07 115 Table C.1. (cont'd) Angle Site LN LOC Freq. Calc. (deg) (kHz) (kHz) ’34, 1 J L 4430. H! £w¥53.74 s2», 1 4 H ulflh.hfl 4180.01 $25. 1 4 L 4159.30 4122.95 x25. d a H 46H0.80 4687odH 32w. 3 a L 4614.90 4624.R4 1/w. d 3 H 4SHh.H0 4§l7.85 {54. z 3 L 44hH.Hn 4454.81 idw. J E A 4806.80 4RUH.43 saw. 3 8 L 4740.10 4745.45 ikfi. 3 J H 439H.%0 4407.04 355. 3 J L 434h.70 4144.07 345. 1 a H 47HV.QO 47Hl.%2 3R5. 1 8 L 47d1.fiO 4718.25 #35. l 3 H 4410.00 4419.7/ 35w. 1 5 L 4351.20 4356.70 335. d 6 H “61701“ “figqobq sjw. a K L 4555.H0 4Sho.hl *3“. 8 A H 4%HH.10 4878.09 5%». H 3 L 4Sd5.b0 4915.06 «5%. 3 J H 4453.10 4943.94 $4w. 3 4 H 4257.30 4270.5M $45. 1 a H 4006.00 4005.19 %4». 1 d I 4748.00 4748.31 504, 1 3 H 4392.00 4194.79 34w. 1 3 L 4334.90 4131.7l ‘41. k R H 4651.40 4541.73 ng. / / L 4R94,60 4S7H.7? 44». 2 J H 4554.10 45h4.7l 443. c 3 L 4445.00 4R05.70 $45, 3 1 H 49H3./0 4675.34 {41. 3 l I 4930.30 4918.54 +4». 5 4 H 4aa3.90 4PJY.4H +4., 1 x H 4403.70 4405.34 $fiw. l d L 473U.30 474d.$l 3‘31. 1 J H 4396.30 419%.?“ $5w. l ‘ I 43$H.?0 4131.7] ‘fi‘. 2 / q 4719.40 47U4.41 55). 6 K L 4hHH.h0 4541.Hfi 14». a A A 4495.h0 4H07.9K ‘54. g 5 L 4435.?“ 444uuq‘ *fiW. A l L 4941.40 4936.d/ ‘Hfi. 5 4 H 4200.40 4911.14 gfifi. J g | 414$.00 4148.18 116 Table C.1. (cont'd) Angle Site LN LOC Freq. Calc. (deg) (kHz) (kHz) . 3 1 H 5018.H0 5014.01 w. 4 1 L 4934.10 4H%0.9H 4. 1 4 H 41Hu.30 4193.7H 1w, 1 a H 4751.HU 4739.94 1). 1 c L whn7.VU 4h7h.H% 1:. 1 J H 44n4.30 44h2.H/ 1‘. 6 K N 4R14.90 4808.43 1*. K 7 I 47fi/.QU 4745.45 1». z s L 4134.00 4344.07 r~. 1 4 ‘4 41H0./0 4IHH./0 1». 1 w 1 41/1.10 4128.65 1». 1 r 4 4h17.H0 4629.h" 3». 1 r L 4H69.50 456h.hi 4., 1 3 H qHHU.hH 457H.04 1w. 1 3 1 45gb.00 4%15.07 14. g” 1 H 49519.5(! 494.5.Qh 1w. ,1 r H 4ln1.H0 47111.32 53. 3 / L 47d3.h0 471H.?H 3"). 1 w H 4414.50 4419.77 1». 3 A 1 4154.00 415h.70 q,. 1 r H 4hfi(.20 4641./fl 43, i K L H5HH.7U 43]“.7/ 4). l J H qg5%.h0 4%6H.71 4a, 1 5 1 4495.40 4605.!” +3. P 1 I 401H.80 4013.34 4w, H 4 H 9PKH.5U H?37.HH 1. r' 4 1 ulnfi.,-HI 1.41 (4.:11-3 +4. 1 g 4 4401,30 4405.14 1.4. ‘5 f" l 41911.51) 47442.31 ..n, 1 1 :1 4+1”1.~H' ¢+1H4./“’ .,.. s .1 | ¢+314. HI 4 311.11 ‘1“. 1 r' 1-1 (4h‘17,¢+1v u/I1t'+.H/ >\. 1 5 H 44~u..70 441,7.0/ a~, 1 5 L 4434.40 4444.”* a“. r 1 1 4046.70 443h.8H a». / 0 H 4)03./0 4211.15 1.4. / a 1 41cu..40 41Ln&.1H ~w», 1 d 14 4H1“).l(1 4%1H3.19 w‘. 1 r 1 4751.80 474d.31 a). 1 1 11 4100.10 4144.79 ~... 1 3 I 4-119.lt| 4 111.71 117 Table C.1. (cont'd.) Angle Site LN LOC Freq. Cale. (deg) (kHz) (kHz) In. 1 P H 4418.50 4000.43 71. 1 z 1 4747.00 4745.45 ("3. 1 1 1 41411.10 4344.07 74. 1 x H 473h.20 4739.94 In. 1 r 1 4hhh.40 4670.HH 7 . 1 1 H 44/1.h0 4468.14/ H . l H 1+»{11/.‘1(1 49071.41 “ - 1 - | 474d./0 4745.45 5-. 1 . q 4410.10 4407.04 W— . 1 s I- “1149.?“0 “.3440”, r1 . 1' c H 4615.9” ubH/.dH 5‘. 1 c L 4h18.40 4634.04 4 , 1 1 H 4557.70 4911.35 n-. 1 1 [ 44h7.d0 4454.21 9.. 1 1 L qAHM.hU 4HHU.9/ 9'. A d H 47/0.40 4705.54 q.“ 1 3 1_ 44113.00 43811.08 9‘1. 1 (' H 491111.00 4629.b‘9 9'. 1 d 1 4fifi4.h0 4Q66.Hl 41. 1 1 H 4449.80 4570.0u ‘91. i J L 4fi(’/./(1 “$15.11“ 104. 1 4 H 4244.h0 4251.40 101. 1 ( H 4AH4.10 4600.77 10». 1 K 1 4Hn4.H0 4844.74 10¥. 1 1 H 450%.H0 45u9.5l 10». 1 J L 4920.40 4H1h.15 "I1111111111111111111“