Coherent laser studies of nonlinear and transient phenomena in Tb3+ activated solids by Guokui Liu A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Guokui Liu (1988) Abstract: Spectral and dynamical properties of the concentrated rare earth compound LiTbF4 and the dilute isostructural compound 1%ztb3+:LiYF4 have been investigated with photon echo, spectral hole burning, and other spectroscopic methods. Coherent dephasing has been measured as a function of magnetic field, excitation frequency, excitation intensity, and temperature for the transition between 7F6 Γ2 and 5D4 Γ1 levels in both compounds. Various interaction processes responsible for the line broadening and splittings of the rare earth ions have been determined. A possible transition from delocalized to localized states in the concentrated compound has been observed in the photon echo experiments. When the excitation frequency was varied across the inhomogeneous line, a sharp change in dephasing rate by a factor of six was measured as expected for an Anderson transition or mobility edge. Quasiresonant interactions and exciton band dispersion processes have also been considered for interpreting the frequency-dependent dephasing. Superhyperfine interactions (SHFS) between a Tb3+ ion and surrounding F nuclei have been studied through photon echo modulation for Tb3+:LiYF4. The theoretical coherent emission function has been derived by calculating the density matrix with a model Hamiltonian. That theory was in excellent agreement with the experiments, and both the field-dependent modulation frequencies and the SHFS parameters have been determined. Electron spin diffusion and instantaneous spectral diffusion have been observed in the dilute crystal. The echo decay time exhibited strong dependence on applied magnetic field and on the excitation intensity. These phenomena have both been interpreted as arising from the magnetic dipole-dipole interaction between Tb3+ ions. Hyperfine spectral hole burning has been observed in 1% Tb3+:LiYF4. The hole lifetime was a function of magnetic field and reached a value of 10 minutes with an external field of 40 kG. Crystal field eigenfunctions, derived from an analysis of the observed levels, provided an excellent description of the electronic Zeeman splittings and allowed accurate calculation of the hyperfine structure. The resulting Zeeman eigenfunctions have been used to qualitatively explain the hole burning process and to calculate the field-dependent echo modulation and both field and power-dependent coherence dephasing in the dilute compound.  COHERENT LASER STUDIES OF NONLINEAR AND TRANSIENT PHENOMENA IN Tb3+ ACTIVATED SOLIDS by Guokui Liu A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY Bozeman, Montana August 1988 ii ty* 1 1 APPROVAL of a thesis submitted by Guokui Liu This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. ate Chairperson, Graduate Committee Approved for the Major Department /Lr. /eff Date Head, MajpfDepartment Approved for the College of Graduate Studies tr V ----------- Date Graduate Dean ©1989 GUOKUI LIU All Rights Reserved iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a doctoral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this thesis is allowable only for scholarly purposes, consistent with "fair use" as prescribed in the U.S. Copyright Law. Requests for extensive copying or reproduction of this thesis should be referred to University Microfilms International, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted "the exclusive right to reproduce and distribute copies of the dissertation in and from microfilm and the right to reproduce and distribute by abstract in any format." Signature IV ACKNOWLEDGMENTS It is a pleasure to acknowledge the aid received from teachers and co-workers in carrying out the work performed for this thesis. Foremost, the author wishes to thank his advisor, Professor Rufus L. Cone, for his unlimited advice and assistance during all stages of this work. The stimulating discussions with Drs. James L. Skinner, Richard Si Meltzer, John Carlsten, V. Hugo Schmidt, Michael J.M. Leask, Bernard Jacquier, Marie F. Joubert, and Thomas W. Mossberg are greatly acknowledged. Thanks are due to Alfred Beldring for his help and expertise in building and maintaining electronic equipment; to Tony Knick in machining and building apparatus. Thanks are also due to his fellow graduate students Jin Huang for his help in running the experiments and David MacPherson for reading the manuscript. Finally, he wishes to extend warm thanks to his wife and parents for their encouragement and support over a number of years. VTABLE OF CONTENTS Page APPROVAL.................... ’ ..................................................................................... ii STATEMENT OF PERMISSION TO USE ........................................................ iii ACKNOWLEDGMENTS ......................................................................................... iv TABLE OF CONTENTS........................................................................................ v LIST OF TABLES ......................................................................................................viii LIST OF FIGURES . ix ABSTRACT ........................................................................................ xi 1. INTRODUCTION............................................................................................ I Spectroscopic Methods ........................................ Conventional Spectroscopy .................... ... . Nonlinear Laser Spectroscopy .................... Coherent Transient Spectroscopy — Photon Echoes Spectral Holeburning ........................................................................ 14 2. ENERGY LEVEL STRUCTURE OF Tb3+:LiYF4 ..................................... 16 Free Ion .................................................................................................... 16 Ciystal Field Analysis ............................................................................ 19 Symmetry and Selection Rules . . . ............................................ 19 Two Important Levels ........................................................................ 22 Observed Energy Levels .................................................................... 23 Calculations ........................................................................................ 28 Zeeman Effect . ......................................................................................... 31 Hyperfine Stmcture ................................................................................ 35 3. THEORY OF COHERENT EMISSION 40 Density Matrix and Optical Bloch Equation 41 Photon Echo Formation ........................................ 45 Optical Dephasing Theory for Two-Level Systems . . . . . . . . 52 Quasi-Two-Level Systems and Photon Echo Modulation .................... 54 4. EXPERIMENTAL APPARATUS AND TECHNIQUES 57 Experimental Setup for Photon Echoes ................................................ 57 in tn r- oo TABLE OF CONTENTS—Continued Page Lasers ................................................................................................ 57 Cryostat and Field Sweeping ............................................................ 59 Photon Echo Detector . . ................................................................. 62 Transient Signal Processing ............................................................ 63 Laser Frequency Stabilization 63 Time Delay Control .................................................................................... 66 5. INSTANTANEOUS DIFFUSION AND ELECTRON SPIN DIFFUSION IN Tb3+:LiYF4 70 Review of Echo Decay ................................................................................ 71 The Effect of Local Fields on Optical Dephasing ........................ 73 Instantaneous Diffusion ........................................................................ 75 Electron Spin Diffusion ........................................................................ 79 Experimental Details ................................................................................ 81 Summary .................................................................................................... 90 Stunulated Photon Echoes ........................................................................ 91 6. FREQUENCY-DEPENDENT DEPHASING IN LiTbF4 94 Summary of Results ................................................................................ 96 Anderson Transition ................................................................................ 106 Quasiresonant Interactions ................................................ .... . 109 Davydov Splitting and Exciton Dispersion 116 Exchange Splittings and the Search for Exciton Band Effect . . 120 7. SUPERHYPERFINE Tb3+-F" INTERACTIONS IN Tb3+=LiYF PROBED VIA PHOTON ECHO MODULATION 130 Hamiltonian for Superhypexfine Interactions 132 Photon Echo Modulation ........................................................................ 136 Experimental Results and Regression Analysis 138 8. HYPERFINE SPECTRAL HOLEBURNING FOR Tb3+=LiYF4 ................ 147 Holebummg Process ................................................................................ 148 Experimental Details ................................................................................ 149 Holeburning Spectrum and Mechanism ............................................ 149 Holeburning Rate and Quantum Efficiency .................................... 152 Hole Lifetime .................................................................................... 155 Experiments on LiTbF4 .................................... 155 Correlation of the Holebmning Phenomena and the Hyperfine Stmcture ........................................................................ 155 9. CONCLUSIONS ........................................................................................ 158 vi vii TABLE OF CONTENTS—Continued Page REFERENCES CITED . . . ................................................................. 166 APPENDIX----- Computer P rogram s............................................................... 174 DECHP1.C .......................................................... 175 SDEC5A.C ................................................................................................... 180 Viii LIST OF TABLES Table Page 1. Crystal-field splittings of J multiplets in S4 symmetry ........................................................................................ 20 2. Selection rules in S4 symmetry for electric and magnetic dipole transitions ............................ ... . 22 3. Eigenvalues and eigenfunctions of ^ D4 multiplet 26 4. Measured and calculated energy levels and effective g-faetors of 5 67Fj multiplets 30 5. Free ion and crystal field parameters for Tb3+:LiYF4 ................................ 32 6. Ground state exchange splittings in LiTbF4 128 7. Superhyperfine structure parameters of Tb3+: LiYF4 143 ix LIST OF FIGURES Figure Page 1. Energy level diagrmns for RE3+ .................................... 18 2. Lattice structure of LiYF4 21 3. Energy level structure of Tb3+:LiYF4 24 4. The Zeeman splittings of the fluorescence spectra of 5D4 F1 Io 7F5 F2 and F3 4 levels ............................ 27 5. Observed and calculated Zeeman effect of 5D4 F1 and F3 4 levels ........................................................................ 33 6. Hyperfine splittings of 7F6 F2 and 5D4 F1 levels ................................................ 37 7. Schematic description of photon echo 49 8. Setup for photon echo experiment ..................................................... 58 9. Flow charts of data-acquisition and control program for photon echo experiment ................................ 64 10. Time delay relations and photon echo detection . . . . . . . 68 11. Echo decay curve in Tb3+: LiYF4 . . . ......................................... 82 12. Excitation intensity dependence of dephasing rate of Tb3+: LiYF4 . ......................................................... 83 13. Power-dependence coefficient as a function of applied field 87 14. Homogeneous line broadening due to electron spin diffusion 88 15. Magnetic Field dependence of Echo Decay in Tb .LiYF4 .................................................................... 89 16. Stimulated photon echo decay .................... 92 17. Echo intensity as a function of excitation frequency with comparison of absorption line shape ................ ................... 98 18. Echo intensity as a function of excitation frequency and pulse separation in three-dimensional plot 99 LIST OF FIGURES—Continued Figure Page 19. Observed dephasing time T2 as a function of excitation frequency ................................................................................101 20. Comparison of experimental dephasing rate with absorption coefficient ............................................................................102 21. Echo decay time as a function of frequency and temperature .................... 104 22. Theoretical and experimental homogeneous line width for three different magnetic fields ........................ 112 23. Illustration of Davydov splitting and exeiton dispersion in LiTbF4 ................................................................................121 24. Absorption line shape of upper Zeeman component of the ground state to excited state with temperature as a parameter . . . % ............................................................................123 25. Excitation line shape monitored by the fluorescence of 5D4 F1 to 7F5 F3 4 with temperature as a parameter ................124 26. Theoretical and experimental echo modulation spectra for Tb3"1": LiYF4 at 11 kG 140 27. Fourier transformation of experimental echo modulation spectrum in Fig. 24 141 28. Theoretical and experimental SHFS splittings versus magnetic field . .................................................................. 145 29. Holebuming spectrum of Tb3^iLiYF4 with comparison of unperturbed absorption spectrum . .................... 151 30. Holebuming rate at the line center of the 7Ffi F2 to 5D4 F1 absorption 153 31. Hole lifetime as a function of applied field 156 32. Computer program---- DECHPLC................................................................175 33. Computer program---- SDEC5A.C................................................................180 X xi ABSTRACT Spectral and dynamical properties of the concentrated rare earth compound LiTbF. and the dilute isostructural compound l%Tb3+:LiYF4 have been investigated with photon echo, spectral hole burning, and other spectroscopic methods. Coherent dephasing has been measured as a function of magnetic field, excitation frequency, excitation intensity, and temperature for the transition between 7Ffi T2 and D4 F1 levels in both compounds. Various interaction processes responsible for the line broadening and splittings of the rare earth ions have been determined. A possible transition from delocalized to localized states in the concentrated compound has been observed in the photon echo experiments. When the excitation frequency was varied across the inhomogeneous line, a sharp change in dephasing rate by a factor of six was measured as expected for an Anderson transition or mobility edge. Quasiresonant interactions and exciton band dispersion processes have also been considered for interpreting the frequency-dependent dephasing. Superhyperfine interactions (SHFS) between a Tb3"1" ion and surrounding F nuclei have been studied through photon echo modulation for Tb3^ =LiYF4. The theoretical coherent emission function has been derived by calculating the density matrix with a model Hamiltonian. That theory was in excellent agreement with the experiments, and both the field-dependent modulation frequencies and the SHFS parameters have been detennined. Electron spin diffusion and instantaneous spectral diffusion have been observed in the dilute crystal. The echo decay time exhibited strong dependence on applied magnetic field and on the excitation intensity. These phenomena have both been interpreted as arising from the magnetic dipole-dipole interaction between Tb3+ ions. Hyperfine spectral hole burning has been observed in 1% Tb3+=LiYF4. The hole lifetime was a function of magnetic field and reached a value of 10 minutes with an external field of 40 kG. Crystal field eigenfunctions, derived from an analysis of the observed levels, provided an excellent description of the electronic Zeeman splittings and allowed accurate calculation of the hyperfine structure. The resulting Zeeman eigenfunctions have been used to qualitatively explain the hole burning process and to calculate the field-dependent echo modulation and both field and power-dependent coherence dephasing in the dilute compound. ICHAPTER I INTRODUCTION In solid state physics, various interaction mechanisms responsible for energy transfer have received much attention, from both theoretical and experimental perspectives.1'17 The basic ideas are relevant to rare earth compounds,10"15 transition metal compounds,4"7,16"17 organic molecular crystals,8,9 and biological systems. The efficiency of energy transfer in solids may not only be determined by the transfer mechanism but may also be strongly affected by whether the state of the donors (excited ions or molecules) is delocalized or localized. That . question involves a subtle interplay between energy transfer coupling and inhomogeneous broadening in real systems. The concept of an Anderson transition1 between the localized states and delocalized states and its extension in terms of mobility edges2 within an inhomogeneous, line have attracted wide attention for understanding energy transfer processes in such disordered systems. Recently, nonlinear spectroscopic methods such as time-resolved fluorescence line narrowing,3" 5 transient gratings,16'17 and photon echoes6 have been used for determining energy transfer mechanisms and searching for Anderson transitions or mobility edges in ruby and molecular crystals. However, no unambiguous demonstration of an Anderson transition or mobility edge has 2been found yet. The Anderson transition has thus become a very controversial topic for both theoreticians and experimentalists. In rare earth compounds, energy transfer processes and ion-ion interaction mechanisms have been studied for some time. Energy transfer processes in Gd(OH)3 and GdCl3 have been observed by Meltzer and Moos14 through the absorption line shapes of magnon-exciton transitions. By measuring the line shapes of band-to-band exciton fluorescence, Cone and Meltzer11 have determined the energy transfer mechanisms in Tb(OH)3. Both studies indicated that short range exchange interactions played a major role in energy dispersion in some exciton bands of those compounds. For another Tb3"1" compound TbF3,13 energy transfer processes have been studied via trapping dynamics, and again short range interactions were important. Evidence of energy transfer and short-range coupling mechanisms thus suggests that concentrated Tb3+ compounds may be potential systems for finding an Anderson transition. For that reason, emphasis has been placed on Tb3+ compounds in this thesis. In low concentration rare earth compounds, all the excited states are localized, since short-range coupling and energy transfer processes are inhibited. Therefore, such a system is ideal for studies of crystal field splittings, local field perturbations, and the effects of long-range coupling mechanisms such as dipole-dipole interactions on the dynamic properties of an isolated rare earth ion. Results from those studies are interesting on their own merit and also provide a detailed basis for analyzing the results from the concentrated systems. Indeed, they are the first optical coherent transient and holeburning results reported for any Tb3"1" compound. 3With the goal of studying the range of phenomena described above, spectral and dynamical properties of the concentrated (stoichiometric) crystalline ■ rare earth compound LiTbF4 and the dilute isostructural compound l%Tb3+:LiYF4 have been investigated with photon echo, spectral hole burning, and other spectroscopic methods. Coherent optical dephasing has been measured as a function of magnetic field, excitation frequency, excitation intensity, and temperature for the transition between the ground state 7F6 T2 and excited state 5D4 T1 in both compounds. Various interaction processes responsible for the homogeneous line broadening and static line splittings of the rare earth ions have been determined. A possible transition from delocalized to localized states in the concentrated compound has been observed in the photon echo experiments. When the excitation frequency was varied from the low energy side to the center of the inhomogeneous line, a sharp change in dephasing rate by a factor of six was measured as expected for an Anderson transition or mobility edge. Quasiresonant interactions and exciton band dispersion processes have also been considered for interpreting the frequency- dependent dephasing. Exchange coupling between neighboring Tb3+ ions was alternatively demonstrated by measuring the exchange splittings of the absorption from the upper Zeeman component of the ground state in this concentrated compound. Superhyperfine interactions (SHFS) between a Tb3+ ion and surrounding F" nuclei have been studied through photon echo modulation for the dilute compound. The theoretical coherent emission function has been derived by calculating the density matrix with a model Hamiltonian for the SHFS. That theory was in excellent agreement with the 4experiments, and both the field-dependent modulation frequencies and the SHFS parameters have been determined via regression analysis. Electron spin diffusion and instantaneous spectral diffusion have been observed in the photon echo experiments on the dilute crystal. The echo decay time exhibited strong dependence on applied magnetic field and on the excitation intensity of the second laser pulse. These phenomena have both been interpreted as arising from the magnetic dipole-dipole interaction between Tb3"1" ions. Dephasing is caused by the random change in the dipolar coupling between an echo ion and surrounding ions which occurs when the surrounding ions are either optically excited or flipped to the upper Zeeman component of the ground state. Long-lived stimulated photon echoes have been observed in this dilute compound. A population grating within the ground state hyperfine levels could be easily produced by two subsequent pulses, so that stimulated echoes could be read out at any time during the persistence of the population grating. Spectral hole burning via optical pumping of ground state hyperfine level populations has been observed in 1% Tb3+:LiYF4. The hole lifetime increased with applied magnetic field and reached a value of 10 minutes with an external field of 40 kG. Crystal field eigenfunctions, derived from an analysis of the observed levels, provided an excellent description of the electronic Zeeman splittings of the 5D4 F1 and F34 and 7Ffi F2 levels and allowed accurate calculation of the magnetic hyperfine structure. The resulting Zeeman eigenfunctions have been used to qualitatively explain the hole burning process and to calculate the field-dependent echo modulation and both field and power-dependent coherence dephasing in the dilute compound. 5Spectroscopic Methods With our emphasis on nonlinear and transient optical phenomena, the spectroscopic methods are an important part to be presented in this thesis. Before describing in detail the experimental and theoretical analyses, a brief introduction of the various spectroscopic techniques used in this work will be given along with some previous applications to rare earth compounds. Conventional Spectroscopy The classical spectroscopy of rare earth ions, in which the position and intensity of the lines are of fundamental interest, has been intensively studied in the last three decades.18"22 In trivalent rare-earth compounds, the rare earth ions have a unique configuration of 4fri5s25p6. The unpaired 4f electrons are in the optically active states, which are far below the valence band. In a free rare earth ion, the energy level structure of the 4fn electrons generally consists of degenerate J-multiplets due to the electron-electron repulsion and spin-orbit interaction. Since the 4f electrons are shielded from the crystalline environment by the outer 5s and 5p electrons which form two filled electronic shells with large radial extension, the crystal environment has only a moderate perturbation effect on the free rare-earth ion energy levels. This perturbation splits all or a part of the (21+1 !-degenerate states of the free ion. The solid state spectroscopic properties of rare earth compounds, such as sharp optical line widths, can be understood from a consideration of the weak 6crystal field. In turn, the wave functions of the free ion constitute a good zero order approximation for description of solid state properties. This is why rare earth ions are such a useful probe in solids, and why detailed studies of the interactions between the rare earth ions and their environment can be carried out. As an initial step in understanding the spectral structures and the dynamical characteristics of the systems which will be reported in this thesis, an analysis of the crystal-field splittings of the free-ion energy levels of trivalent terbium ions in a single crystal of lithium yttrium fluoride, Tb3+:LiYF4, was carried out. That work, which will be presented in Chapter 2, was based on a semi-empirical method developed by Judd,23 Wyboume,19 Judd et al,24 Crosswhite et al,25 and Camall et al.21 With this method, various interactions operating within the 4f-configuration and the crystal-field parameters can be identified by reproducing the observed spectral structure. The crystal field eigenfunctions resulting from this analysis provided a fundamental base for further studies of both spectral characteristics and dynamical behavior of the Tb3+ ions in the crystal.26 First of all, the crystal field eigenfunctions were used to calculate the electronic Zeeman interaction and the magnetic hyperfine splittings for the energy levels of the 7F60 and 5D4 multiplets. Then rediagonalization of the crystal-field, Zeeman, and hyperfine interaction Hamiltonian yielded a new set of eigenfunctions for a complete representation of individual Tb3+ ions in the presence of an external magnetic field. This has enabled an extensive understanding of the i o n - ion, and the ion—environment interactions. Using the eigenfunctions or the calculated expectation values of dynamical operators, various dynamical 7properties probed by photon echoes, such as instantaneous diffusion, electron spin diffusion, and superhyperfine interactions, can be quantitatively described. This also provided a qualitative description of the mechanisms and the efficiency for spectral holebuming processes. AU of these phenomena will be discussed in this thesis. Nonlinear Laser Spectroscopy In principle, the width of a spectral line yields information on the dephasing dynamics of the transition. Unfortunately, the spectral line shapes of electronic transitions in rare earth solids usually are determined not only by the ion-ion and ion-lattice interactions, but also by unavoidable crystal strains which may dominate the observed linewidth at low temperature. In spectroscopy, such physically different contributions to the spectral linewidth are classified into two categories: homogeneous line width and inhomogeneous line width. The homogeneous broadening of a spectral line arises from dynamical perturbations on the optical transition frequency due for example to lattice phonons or fluctuation of local magnetic fields. The inhomogeneous contribution can arise from static lattice strains or crystal defects. In rare earth crystals at low temperature, the inhomogeneous contribution is comparable to the homogeneous broadening for upper states in a J-multiplet but usually completely dominates the homogeneous linewidth of the lowest levels in a J-multiplet. The existence of undesired effects such as inhomogeneous broadening obscures the dynamical information carried by the homogeneous dephasing processes and also masks spectral structures such as hyperfine and 8superhyperfine splittings. Since the invention of lasers, nonlinear spectroscopy has developed into a significant subfield of physics. In solid state physics, its high resolution makes it possible to measure fine energy structures such as hyperfine and superhyperfine splittings in the frequency scale of kHz—MHz.^’^ The scale of resolution is usually limited to GHz in conventional spectroscopy by inhomogeneous broadening due to lattice strain and defects in solids. After the elimination of inhomogeneous broadening, the line width of an optical transition in a solid is determined by intrinsic interactions and fluctuations of the environment known as homogeneous broadening. Measurement of the homogeneous line width of an optical transition directly yields information about the dynamics of a physical system. Furthermore, very interesting nonlinear phenomena such as photon echoes32, and free induction decay34 have been exploited in the time domain. This branch of transient nonlinear spectroscopy has proven to be very powerful in studies of coherence dephasing, excitation transfer, and other fast processes.2 Various nonlinear spectroscopic methods developed with the application of tunable dye lasers have provided possibilities for increasing spectral resolution and extracting information which is obscured by the inhomogeneous broadening. For this purpose, time-domain photon echoes and frequency-domain spectral holeburning are the two major nonlinear spectroscopic methods used in this work. Coherent Transient Spectroscopy ------ Photon Echoes Since the pioneering work by Hahn35 on the discovery of spin 9echoes and by Torrey36 on the transient nutation effect, coherent transient phenomena in electronic and nuclear spin dynamics have been intensively studied in radio-frequency and microwave spectroscopy. By measuring the decay characteristics of coherent emission, various spin­ dephasing mechanisms could be examined.37"39 It became possible, after the invention of the laser, that coherent transient spectroscopy could be introduced to the optical region.40,32,34 This branch of nonlinear laser spectroscopy has provided unique ways for exploring dynamical interactions in optically excited atoms, molecules, and solids.26"31,40 From a physical point of view, optical electric-dipole transitions behave in the same way as the magnetic-dipole transition of spin systems. In either case, a collection of two-level quantum systems can be prepared coherently in superposition states in which all dipoles (electronic or magnetic) are in phase with each other so that they radiate coherently. This was first proved by Dicke.41 The equivalence of magnetic and electric dipole transitions was further shown by Feyman et al.42 who introduced a generalized treatment applicable to optical coherent transients. The dynamical response of any two-level system to a resonant excitation obeys a Bloch-type equation of motion43 the same as a magnetic spin-1/2 system does. This equation of motion is just the vector form of the Schrodinger equation.42 Therefore, the dynamics of an optical electric-dipole transition can be described in much the same way as those of a magnetic-dipole transition. This was not clear before the work of Dieke and Feyman et al. because in the case of electric dipole transitions the precession of the Bloch vectors takes place in an abstract space rather than in real space. 10 Photon echoes are the optical analog of the spin echoes in magnetic resonance. They were predicted and first observed in ruby by Hartmann et al.32,33 Since that time, photon echoes have been widely applied to studies of the coherent decay of optical transitions and are still the subject of active experimental and theoretical investigations.44"49 In the photon echo process, two short pulses of coherent light from one or two independent lasers separated by a time delay % are directed onto a sample. The first pulse creates a superradiant state which involves the coherent superposition of the ground and the excited state wavefunctions of the echo. ions. In the superradiant state, there is a macroscopic . oscillating dipole which is capable of emitting coherent radiation and which rapidly decays as the ions fan out of phase in the superradiant state due1 Io the inhomogeneous distribution of their resonance frequencies. The decay time is given by the inverse of the inhomogeneous line width or the laser line width, whichever is narrower. The second pulse has the effect of reversing the sign . of the accumulated phase for each given ion, so that the net phase shift is cancelled after an additional processing time r. This rephasing process leads to an additional burst of coherent radiation. This coherent radiation burst is called a photon echo. Inliomogeneous broadening effects are removed by the photon echo pulse sequence, and the decay of the photon echo amplitude or intensity reflects only homogeneous relaxation due to dynamical interactions.28,29 A description of coherent emission theory and the formation of photon echoes is given in Chapter 3; the experimental methods are described in Chapter 4. Since the photon echo technique can measure homogeneous line 11 widths in the presence of inhomogeneous broadening, it provides a powerful tool for studying the dynamical interactions of rare earth ions in solids. At liquid helium temperatures, thermally induced phonon contributions are often negligible. The dephasing in rare earth impurity compounds can be dominated by local perturbations affecting the rare earth ions. These perturbations include fluctuating magnetic fields due to magnetic dipole interactions with both ligand nuclear spins and neighboring electronic spins. Chapter 5 is devoted to discussion of these dephasing mechanisms along with the experimental results from the l%Tb3+:LiYF4 crystal. The intrinsic ion-ion interactions can dominate homogeneous dephasing and create many new phenomena as the concentration of optically active ions is increased. In rare earth solids, various ion-ion interaction mechanisms and their spectroscopic effects have recently been reviewed by Cone and Meltzer.10 In addition to the long range coupling mechanism of dipole-dipole interactions, electric multipolar coupling and electronic superexchange coupling between neighboring rare earth ions can also significantly contribute to the coherence dephasing. These interionic interactions may also affect the nature of the excited states of the rare earth ions leading to optical excitation transfer and diffusion or exciton band effects in the strong coupling limit.11"14 Since the photon echo decay is so sensitive to the nature of the excited state dynamics, it is possible to learn the nature of the energy transfer processes from photon echo experiments. = The Anderson transition is a theoretical model for describing the transition from localized to delocalized electronic energy states.1’2 Great efforts have been made in 12 the last two decades to experimentally search for the Anderson transition in optical resonant transitions. There is still no clear evidence to prove if the Anderson transition exists.3"17 The first photon echo measurement of coherent dephasing in a stoichiometric material was done by Shelby and Macfarlane on EuP5O15.50 The dephasing time was found to depend on the excitation frequency which was varied across the inhomogeneous line. This was interpreted as evidence for delocalization of the excitation due to energy transfer processes. This result was later fitted by Skinner et al49 with a theoretical model of quasiresonant interactions. That fit led to the conclusions that the frequency-dependent dephasing is due to a microscopic electric dipole-dipole coupling between neighboring resonant ions and that there is no energy-site correlation between the optically activated ions. Such a correlation is required in the Anderson model. In the concentrated compound LiTbF45 a strong frequency-dependence of the echo decay has been observed. The decay rate exhibited an abrupt change on the low energy side of the inhomogeneous line.51 Chapter 6 is devoted to analysis of the experimental results along with consideration of the Anderson model and the Skinner model. Related measurements on the dilute compound l%Tb3+:LiYF4 are described in Chapter 5. In addition to measuring the coherence dephasing or homogeneous line width of an optical transition via amplitude decay, a photon echo signal may carry another type of spectroscopic information via modulation within the decay envelope. Modulation phenomena are observed in echo decay when the ground and excited states of a two-level system are split into sublevels by hyperfine or superhyperfine interactions. Much of the 13 earlier experimental and theoretical work on photon echoes in solids was done in this area.52'60 If the sublevel splittings in a quasi-two-level system are less than the Fourier width of the excitation pulses, all substates in the system can be excited or occupied. A quantum interference can occur between coherences excited on two transitions which share a common level.55 As a result, an echo decay spectrum exhibits beats at frequencies determined by the ground and the excited state energy splittings and their sums and differences. These modulation frequencies can be obtained by a Fourier transform of the echo decay spectra in the time domain. Since echo modulation is a result of quantum interference, it is sensitive to both the eigenfunctions and the eigenfrequencies of the interaction Hamiltonian. Therefore, the deconvolution of echo modulation can provide a very useful tool for not only determining the energy splittings but also for testing the interaction Hamiltonian. By a regression analysis of the observed echo data, the interaction Hamiltonian can be experimentally determined. This sensitivity to phase is another advantage of photon echo spectroscopy over conventional spectroscopic techniques which • only measure energy splittings and which may not be sufficient for correctly testing the model Hamiltonian.57 A strong field-dependent photon echo modulation effect was observed in the dilute compound Tb3+:LiYF4. The experimental results and a theoretical calculation with regression analysis will be presented in Chapter 7. In the stimulated, or three pulse photon echo experiment, the second pulse puts ions with in-phase coherence into the exited state, and the 14 ions shift back into the ground state with an accumulation of TC phase. This accumulated phase information is then stored in the form of a population difference which is called a population grating in the ground state. This coherence can be read out by the third pulse until the /- population difference disappears.44,47,48 Therefore, the stimulated photon echo can in principle be used in the investigation of slow spectral diffusion.30,55 Furthermore, if the excited state population can relax to some long lived population reservoir such as metastable electronic states or hyperfine levels, an echo can be stimulated by a readout pulse so long as the population difference in the ground state persists. This could open a new technique for optical information storage in the future. In Chapter 5, . some initial results of the stimulated photon echoes from the Tb3+:LiYF4 will be discussed. Spectral Holebuming Spectral holebuming is a . very useful nonlinear spectroscopic technique to measure line splitting as well as line width in the presence of inhomogeneous broadening.30,31,61"65 The nonlinear spectroscopic information appears in the form of line structures obtained by selectively exciting a segment of the inhomogeneous line and then sweeping a probe laser across the inhomogeneous line. This was first demonstrated by Szabo in studies of ruby.61 The holebuming process becomes dramatic if a long- lived population reservoir can be used, such as a metastable excited electronic state or hyperfine levels. After the population is optically pumped from the ground state to a long-lived reservoir, narrow holes remain in the inhomogeneous line and can persist for a long time which is & 15 determined by spin relaxation if the reservoir is the hyperfine levels.62 Spectral holebuming in rare earth insulators has proved to be very useful for measuring the hyperfine and superhyperfine structure of both ground and excited states.61'65 Its resolution is limited by spectral diffusion or by the laser linewidth. Therefore, holebuming spectroscopy can also be an effective method to study spectral diffusion and relaxation processes in rare earth doped materials.65,66 Hyperfine spectral holebuming has been carried out on the dilute crystal Tb3+:LiYF4. The magnetic field-dependent holebuming efficiency and hole decay time seem to be explained by the Zeeman and hyperfine studies. Detailed analyses of the holebuming process in the dilute compound are given in Chapter 8. The experimental setups and various transient signal detection techniques used in the photon echo and hole burning experiments are described in Chapter 4. 16 CHAPTER 2 ENERGY LEVEL STRUCTURE OF Tb3+:LiYF4 Free Ion Rare earth (RE) ions in solids have remarkable spectral features due to their sharp 4fn —> df1 transitions which have widths less than I cm"1. This has enabled detailed studies of the nature of the interactions between RE ions and their environment. The sharpness of the spectral lines is due to the fact that the RE ions, which are usually trivalent, have the special electronic configuration: 4 f15s25p6. The 4f electrons are shielded from the crystalline environment by the filled outer 5s and 5p shells. The ligand fields and the lattice phonons have only weak perturbation effects on the atomic energy levels, so that the spectra are still atomic-like. Therefore, the study of free ion energy level structure is of fundamental importance for understanding the energy level structure and the dynamics of RE ions in solids. Extensive work has been done in this area.18"21 Since all other spherical electron shells have the same effect on all the terms of a 4 f1 configuration, the energy levels of a free RE ion are usually calculated by considering only the interactions between the 4f electrons themselves. The interaction energies of the Coulomb repulsion and the spin-orbit interaction are of the same order of magnitude, so 17 intermediate coupling also has to be accounted for in the calculation of the energy structures. In addition, there are some perturbation terms such as configuration interactions and manybody interactions that must be included to calculate accurate energy levels. The basic structure of the free ion energy levels of all RE elements has been obtained in this way up to an energy of 40,000 cm"1. Figure I shows a very useful energy level diagram prepared by Dieke and Cfosswhite,67 and Camall et al.21 The general procedure used to evaluate the energy levels of RE ions involves a semi-empirical approach in which free-ion interaction parameters are fit to some actually observed energy levels. The free ion Hamiltonian Hf includes24'68 the usual electrostatic (Fk), spin-orbit (Qf), and configuration interaction (a, 13, j) terms plus spin-spin and spin-other- orbit interactions (Mk), two-body electrostatically correlated magnetic interactions (Pk), and three-body electrostatic terms (T1): Hf = E , Fkfk + E Q fIi1S. + aL(L+l) + BG(G2) + YF(R2) k=0,2,4,6 i=l,n + E Mk1Uk+ E P kPk + E T \. (2.1) k=0,2,4 k=2,4,6 i=2,3,4,6,7,8 In principle, all the parameters can be freely varied in the fitting program, but usually only the parameters in the first line of Eq. (2.1), which are the Slater parameters Fk, the spin-orbit parameters C, and the three configuration interaction parameters (a, B, y), are freely varied, and the parameters in the second line of Eq. (2.1) are usually not varied until a large number of levels are fitted. 18 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Fig. I. Energy level diagrams for RE1+ ions in crystals (Ref. 21 and 67). 19 This procedure also provides good free ion wave functions as linear combinations of Russell-Saunders representation states. These wave functions can then be used to describe the energy splittings in crystals and other properties such as the Zeeman effect and hyperfine structures. In the following part of this chapter, the calculated wave functions of the Tb3"1" (4f8) ion in the LiYF4 crystal field are used as the zeroth order approximation for calculating the Zeeman effect and hyperfine structures. Crystal Field Analysis For 4f ions in a crystal, as discussed above, the ion-lattice coupling is much weaker than the spin-orbit interaction. The energy levels are calculated by diagonalizing the free ion and crystal field Hamiltonian in an LS basis. The (2J+1 )-degenerate atomic energy levels of each J-multiplet will split corresponding to the crystal site symmetry. Due to the crystal field perturbation, J is not exactly a good quantum number. Slight J- mixing results from coupling by the crystal field. A crystal field analysis of the observed energy levels has been used to determine the field-dependent eigenvalues and eigenfunctions of all 7Fj states and 5D4 states. Details are given below. The energy levels of concentrated LiTbF4 are qualitatively the same and are discussed in Chapter 6. Symmetry and Selection Rules LiYF4 crystallizes in a tetragonal seheelite structure which is shown in Fig. 2. The compound Tb3^=LiYF4 is constructed by substituting Tb3"1" 20 ions for Y3+ ions at sites having S4 symmetry. The (2J+l>degenerate free ion energy levels of each J-multiplet are split into singlets and doublets which are characterized by the irreducible representations Tp F2, Fgj and F4.25 The F1 and F2 levels are singlets. The F3 and F4 representations are related by time-reversal symmetry, so their eigenvalues are degenerate. In the following they will be labeled F -.. The degenerate Fg 4 levels split into separate Fg and F4 levels in the concentrated compound at low temperature due to ferromagnetism which occurs below the critical temperature Tc = 2.87 K.69,70 The numbers of levels for each J multiplet are listed in Table I. Table I. Occurrence of each irreducible representation in the crystal- field splittings of J multiplets in S4 symmetry. J r i T2 r 3,4 G I 0 0 I I 0 I 2 I 2 I 3 I 2 2 4 3 2 2 5 3 2 3 6 3 4 3 21 O Li • Y O F 2. LiYF structure. Tlie six nearest neighbor sites of an Yttrium ion (0) are labelled by the index i=l to 6. The lattice constants are a=5.1668 A; c= 10.7330 A (Ref. 15). 22 In the presence of an applied electromagnetic radiation field, the two most significant terms in the interaction hamiltonian which lead to optical transitions between the F. levels are the electric and magnetic dipole interactions. Although the electric dipole transitions are parity forbidden within a pure single configuration, they do occur due to configuration mixing.18"20 In general for RE ions they are stronger than the magnetic dipole transitions. The selection rules for electric and magnetic dipole transitions in S4 symmetry are given in Table 2. Table 2. Selection rules in S4 symmetry. Ejt, and Ey are electric dipole transitions. M , and M are magnetic dipole transitions. The TC spectrum has the polarization parallel to the c axis and the a spectrum has the polarization perpendicular to the c axis. Tl r 2 r 3,4 r I M0 E* EaM7t T2 M0 E0M lt r 3,4 1V mK EjtM0 Two Important Levels The dynamical experiments, photon echoes and hole burning, were carried out for the ground state 7F6 F2 to .the excited state D4 F1 transition. Therefore, the study of the energy structures of these two levels is of fundamental importance for understanding the observed dynamical properties of Tb3+ ions. Figure 3 shows schematically the 23 energy level splittings of Tb3"1": LiYF4 due to the crystal field plus an externally applied magnetic field parallel to the uniaxial c axis. That was the field geometry used throughout the experiments. Laursen and Holmes70 have studied th e . ground state hyperfine splittings for Tb3^=LiYF4 by electron spin resonance. The electronic ground state in zero magnetic field consists of two F2 singlets separated by 0.9 cm'1. For each F2 state, the four hyperfine sublevels (nuclear spin I = 3/2) are grouped in two two-fold degenerate pairs. In the presence of an external magnetic field, however, the two electronic levels are admixed and rapidly split, becoming asymptotically Jz = ± 6 states. Each then has four equally spaced hyperfine sublevels with 0.1 cm"1 separation. The hyperfine level splitting is larger than the laser line width (= 0.05cm"1), so optical pumping of individual hyperfine sublevels is possible. (Small effects due to nuclear quadrupole coupling18,23,37 presumably modify this structure to a negligible degree for our present purposes.) The next electronic level is at over 100 cm"1, so it has no effect on the ground state properties. The lowest 5D4 level is the F1 singlet at 20,553.5 cm"1. As we shall see below, the Zeeman effect on this level is determined entirely by the field dependent coupling to another 5D4 F1 level which is 14.4 cm"1 higher in energy. The magnetic hyperfine splittings, for the 5D4 F1 levels are thus significantly smaller than those for the ground state. The complete 5D4 excited state level structure will be described in the following section. Observed Energy Levels To determine the crystal field splittings and identify the 7F06 and 5D4 energy levels, polarized absorption and excitation experiments were 24 r, 20568 r 20553 0 .0 cm'’ T X X Z - -L A=0.3 cm'' T ZEEMAN EFFECT HYPERFINE H = O H > 0 Fig. 3. Energy level structure OfTb34ZLiYF4. 25 made. The n and c polarized fluorescence lines from the lowest F1 level of 5D4 to the F2 and Fg 4 levels of the 7F0 to 7F6 multiplets are all in the visible spectral region. Those levels have been identified and the Zeeman effect of the F3 4 levels has also been studied. In an S4 symmetry crystal field, the 5D4 energy levels of the Tb3+ ion are split into three F1 singlets, two F2 singlets, and two F3 4 doublets. In zero magnetic field, the absorption spectra gave three TC polarized transitions to the F1 states and two G polarized transitions to the F3 4 states as expected for electric dipole transitions. The resulting energy levels are listed in Table 3. Thermally populated transitions from the F2 level at 0.9 cm"1 at zero magnetic field were well resolved and confirmed its location as inferred earlier from electron spin resonance experiments.70 The energy levels of 7F6 to 7F0 have been detected by fluorescence spectra at a temperature of 1.3 K. The observed fluorescence emissions from the lowest F1 level of 5D4 to the F2 and F3 4 levels of the 7Fj multiplets were not polarized. Each line had both TC and O components. This could be due to the emission of traps since a change in relative intensity of the two components was observed in time resolved fluorescence. The components which were not expected by the selection rules decreased as the delay time between the excitation pulse and signal gate was reduced. Further investigation of the trap emission may yield information about excitation transfer from donor ions to trap ions13 which is important for understanding aspects of the excited state dynamics as well as for determining the intrinsic energy levels. In order to identify the 7Fj energy levels, Zeeman experiments were 26 made. The fluorescence lines were identified assuming that the split lines are r 34’s and the non-split lines are F2’s. The relevant energy levels are listed in Table 4. Figure 4 shows the fluorescence lines and the Zeeman splittings of the lower levels of the 7F5 multiplet. AU the F34 lines are much stronger than the F2 lines. The observed fluorescence lines to the highest levels in the 7F6, 7F5 and 7F4 multiplets are strong and are thus assumed to be F3 4 levels even though they are so broad (> 30 cm"1) that the Zeeman splitting is obscured. These lines have side structures over an energy scale from decades to over one hundred wave numbers which probably are phonon sidebands. Table 3. Eigenvalues and eigenfunctions of 5D4 at 1.3 K. Each ket denotes the Mj of a 5D4 Mj state. State Absorption line ( cm"1 ) Calc. Eigenvalue ( cm'1 ) Calc. Eigenfunction r i 20644.8 TC 20644.2 0.909l0>+0.294(l4>+l-4>) r 3,4 20626.5 C 20626.1 -0.347l-3>-0.938ll> T2 20612.3 lW2(l2>+l-2>) r 2 20575.4 lW2(l-2>-!2>) Tl 20567.9 TC 20568.1 l/V2(l-4>-!4>) r 3,4 20558.8 C 20558.7 0.3471 l>-0.938l-3> Tl 20553.5 TC 20553.0 0.420l0>-0.642(l4>+l-4>) 27 10 kG H = 40 18.325 18.375 18.425 WAVENUMBERS ( 1000 1/cm ) Fig. 4. The Zeeman splittings of the fluorescence lines of D4 F1 to 7F5 F2 ami F levels. The effective g-factors for these two strong yellow lines are 11.6 and 3.9. The line shifts to the low energy side are due to the shift of the excited state 5D4 F1 in the applied Zeeman field parallel to the c axis. 28 Another unexpected observation is that all the apparent F2 levels are twins which are separated by 3.8 cm"1. This again is presumably due to traps or to another kind of rare earth impurity ions. Further study is required to resolve the general questions of traps versus intrinsic fluorescence. Calculations The combined free-ion and crystal field Hamiltonian matrix was diagonalized, so both intermediate coupling and J-mixing effects were considered. The total Hamiltonian is then H = Hf + H f, (2.2) where Hf is free ion Hamiltonian given in Eq.(2.1), and Hrf is the crystal field Hamiltonian detennined by the S4 site symmetry. Based on previous work by Christensen71 and others and for convenience in the calculation, the two imaginary crystal field parameters were set equal to zero. This gives Hrf an effective D2d symmetry: H=f = bOc? + B«C»+ B& + bOc S + B % . (2.3) The crystal field parameters B7, B^, B4, B®, B4 and all appropriate free ion parameters were automatically adjusted in the general calculation. The initial values of the free ion parameters which were used in the calculation are the free ion parameters for Tb3+:LaF3 listed in the Appendix II of Reference 21. As a first approach, the fitting procedure was carried out for 7Fj and 5D4 multiplets in the energy region 0 to 29 20,600 cm"1. The fit was first tried by varying the parameter Eav. Then the Slater parameters F2, F4, F6, and the spin-orbit parameter £ were simultaneously varied. After that step, the three parameters a, Ji, y for the configuration interaction were added, but those parameters were not sensitive to the energy levels of . the experimentally observed multiplets. The six T parameters were not varied. The multiple spin coupling parameters Mk and two-body eleetrostic parameters Pk were varied but the energy levels were not affected by those parameters. Then, the crystal field parameters were varied while the free ion parameters were held constant. Finally, both the crystal field parameters and the free ion parameters were varied simultaneously to achieve a best fit. The fitted eigenvalues of 7Fj are listed in Table 4. A standard deviation of 16 cm"1 was obtained in this fit. The relevant free ion and crystal field parameters resulted from the fit of the 7Fj and 5D4 multiplets are listed in Table 5. A second fit to only the 5D4 energy levels was obtained by varying only the Bk parameters, and it gave much better calculated eigenvalues and eigenfunctions for those levels which are listed in Table 3. The resulting eigenfunctions are needed for calculating , the Zeeman effect and hyperfine splittings which provide basic information on the spectral characteristics of those levels. The 5D4 multiplet is very isolated; it is 5,000 cm"1 lower than the 5D3 multiplet, and 14,000 cm'1 higher than the 7F0 multiplet. The mixing of other J-multiplets is thus very small and can be ignored. Therefore, the eigenfunctions listed in Table 3 involve only the mixing of Mj components of the pure J = 4 multiplet. 30 Table 4. Energy levels of the 7F multiplets at 1.3 K with errors, and the measured (g ) and the calculated (gcal) g-factors for the T3 4 states. J r Eexp (cm"1) ^cal (cm"1) A (cm1) e^xp Seal 6 2 0 0 0 2 0.9 0.9 0 3,4 98 111 -12 6.3 7.2 I 118 2 128 116 12 3,4 159 190 -31 5.4 10.0 I 236 3,4 354 354 0 2.3 I 356 2 355 5 2 2109 2123 -14 3,4 2126 2110 16 3.9 4.1 I 2072 I 2077 3,4 2196 2206 -10 11.57 11.39 I 2274 3,4 2391 2356 35 0.45 2 ' 2406 4 2 3333 3335 -2 I 3348 3,4 3374 3389 -15 5.7 4.69 I 3504 2 3408 3427 -19 . 3,4 3610 3597 13 0.45 I 3824 31 Table 4. continued J F Eexp (cm'1) ^cal (cm'1) A (cm"1) e^xp Seal 3 2 4354 4340 14 3,4 4426 4435 -9 3.63 3.44 I 4478 2 4498 4524 26 3,4 4541 4551 -10 1.49 1.67 2 I 5036 2 5064 3,4 5321 5306 15 2.62 2 5382 I I 5620 3,4 5691 5703 -12 2.81 0 I 5892 Zeeman Effect Since the z component of the magnetic dipole matrix element <[i> = 0 for the -F1 and T2 singlet states in zero field, there is no linear Zeeman effect for those levels. On the other hand, the F34 doublets are split into separate F3 and F4 levels by the linear Zeeman effect. All seven 5D4 levels are relatively close to each other, and the field- induced coupling between those levels can be strong. Hence, the combined crystal field and Zeeman Hamiltonian was diagonalized as a function of field for each irreducible representation F.. (The S4 symmetry is not 32 Table 5. Free ion and crystal field parameters for Tb3+:LiYF4, (cm"1). EAV 68576 M0 8.17 F2 89869 M2 4.58 F4 64412 M4 3.1 F6 46381 P2 832 a 20017 P4 624 13 -653 P6 416 Y 1445 ; 1604 T2 330 B0 832 T3 41.5 b S 353 T4 62 B: -988 rp6 -295 B0 1079 T7 360 B4 762 y8 310 changed by a parallel magnetic field.) As noted in the 5D4 energy calculation, the 5D4 multiplet is extremely isolated, so coupling to other manifolds is irrelevant. Within the 5D4 multiplet, the matrix elements were calculated using the zero field eigenfunctions IF> = Xa(Mj)IMp- given in Table 3. Since Jz transforms as Fj , non-zero matrix elements occur only between the three Fj levels, the two F34 pairs, and the two F2’s. The relevant eigenvalues are shown in Fig. 5. The calculation gave very good agreement with the experimental data at both low and high fields. The Fj levels vary quadratieally at low field and asymptotically approach a linear Zeeman effect of Mj = ±4 levels at high field. The F34 splittings are quite linear at all fields, because t h e EN ER GY O FF SE T ( c m "' ) - 10 .0 - 5. 0 0. 0 5. 0 10 .0 15 .0 20 .0 25 .0 33 I I I I I 0 .0 10.0 20.0 30.0 40 .0 50.0 FIELD ( kG ) Fig. 5. Observed and calculated Zeeman effect of the 5D4 F1 and F34 levels. The zero-field position of the lowest F1 level is" 20,553.5 cm"1. The experimental data are from the absorption spectrum with the ground state Zeeman shift subtracted. 34 other F3 4 doublet is 72 cm'1 higher, and its admixture is small. Since the F34 levels of all 7Fj multiplets are far away from each other (>70 cm'1), the magnetic field coupling between those levels is negligible, and their Zeeman splittings are all linear. They have been measured up to 40 kG by the fluorescence spectra of the 5D4 F1 to 7Fj F34 transitions. The measurements of the Zeeman splittings A(F34) allow determination of the effective g-factors g’ for these effective spin-1/2 levels by the relation A, (2-5) where gj=3/2 for both 7Fj and 5Dj multiplets, and =Eia2i(MJji. The effective g-factor then can be calculated by g’ = S^a2i(Mj)i. (2-6> The contribution of the 5D4 multiplet to the Fmixing in the 7Fj multiplets is small (a.<0.01). The sum i is then only over the F34 states of 7Fj multiplets. The measured and calculated g-factors for the F34 states of the 7Fj multiplets are listed in Table 4. As discussed in the subsection— Observed Energy Levels, the top F34 fluorescence lines of each 7Fj 35 multiple! are so broad that their Zeeman splittings are obscured. Hvperfine Structure The electron spin resonance study of Tb3+: LiYF4 by Laursen and Holmes70 provided a complete description of the ground electronic doublet. A later study of the LiTbF4 ground state by magnetic resonance73 gave similar results for it. Both of those results are completely consistent with our crystal field analysis. Pelletier-Allard and Pelletier73 have reported an optical fluorescence line narrowing study of Tb3+:LaG3 hyperfine splittings. They observed and analyzed the hyperfine splittings for the ground doublet in 7F6 and for several levels in 5D4. Their value ’ of the ground state magnetic hyperfine interaction constant was in excellent agreement with that for LiYF4. This is to be expected since the magnetic hyperfine interaction is a free ion effect not sensitive to the host lattice.18,23,37 Focusing our attention on the ground state, the two slightly separated F2 levels may be described by the following effective spin-1/2 Hamiltonian:70 Heff = S A h A + a V asA- (2 7) where g_ = 17.8, A/h = 27.98 GHz is the gap between the two F2 levels at zero field, and A/h = .6.26 GHz ( A = 0.209 cm'1 )is the magnetic hyperfine interaction parameter. Alternatively, the interactions may be expressed in terms of Jz.18,37,72 There is no perpendicular Zeeman term 36 for these states. The energy levels evaluated from this Hamiltonian are given by70 E(MsM1) = Ms[(g ^ H + AM1)2 + A2]1/2 . (2.8) Figure 6 (a) shows the hyperfine splittings of the ground state. The four hyperfine sublevels of each electronic Zeeman component quickly split; for Hz higher than 4.5 kG, they are equally separated with the separation A/2 = 0.1 cm"1. For the 5D4 F1 excited state, the hyperfine structure is different. As discussed in the section entitled Zeeman Effect, this level has no linear electronic Zeeman effect, and the next F1 state coupled to it is 15 cm"1 away. Compared to the ground state, the hyperfine splitting is a much slower function of field. Using the crystal-field eigenfunctions as unperturbed eigenfunctions, the hyperfine interactions can be treated as a perturbation using the following Hamiltonian:70,72 Hlrf= [3Jz2-J(J+1)][3Iz2-I(I+1)] 4J(2J-1)I(2I-1) (2.9) The first two terms represent magnetic dipole interactions between the nucleus and electrons, and the third tenn is the electric quadrupole interaction. In the absence of an applied magnetic field, the diagonal elements are zero for i = x, y, z, and the first order magnetic EN ER GY O FF SE T ( IO '1 cm '1 - 2. 0 - 1. 0 0. 0 1. 0 2. 0 - 1. 0 - 0. 5 0. 0 37 O 10.0 FIELD ( kG ) Fig. 6. Hyperfine splittings of (a) 7F5 F2 and electronic Zeeman shifts have been subtracted. 38 hyperfine interaction is zero. The only contribution to the hyperfine splitting in this case is due to the quadrupole interaction which partially removes the hyperfine degeneracy, giving two levels separated by B, since f B/2 for M1 = ±3/2 E h f = ' (2.10) I -B/2 CN+1IlSf S For Tb3+:LaCl3,73 B = 11 X IO'3 cm'1. The value for Tb3+:LiYF4 is unknown, but it would presumably be of the same general order of magnitude and would thus make only very small contributions to the level shifts. In the presence of an external magnetic field, there is strong coupling between the first two F1 states. Then, the field-dependent hyperfine splitting is mostly due to the first magnetic dipole term A JzIz since the matrix elements of A ( JxIx + JyIy ) between those states are zero. The hyperfine structure of the F1 level in a magnetic field can thus be evaluated by diagonalizing an 8 X 8 matrix using the following Hamiltonian: S j L1B h Zj Z + a j Z1 Z [3Jz2-J(J+1)][3Iz2-I(I+1)] 4J(2J-l)I(2I-l) (2.11) The perpendicular magnetic dipole term A(JxIx + JyIy) has non-zero 39 matrix elements between the T1 and the F34 states. Considering it as a second order perturbation, the hyperfine structures shift by k r i IA|2 Er - Er r 3,4 r I = 1.4 ■ A2 Er - Er 3,4 1 I (2.12) The average energy difference between F34 and F1 is 4 cm'1, and A = 16 x 10 cm . This gives 8 = 3 x 104 cm 1 which is two orders of magnitude smaller than the first order contribution of the Al I term Z Z Ignoring the perpendicular magnetic hyperfine term and the electric quadrupole term, the hyperfine splittings of 5D4 F1 versus magnetic field are shown in Fig. 6 (b). The splittings continuously increase with field up to 50 kG. 40 CHAPTER 3 THEORY OF COHERENT EMISSION A collection of independent quantum radiators (ions or molecules) can radiate in two different ways. First, if there is no particular phase relation between the radiators, the radiated field will be weak (»= N) and random. However, if a significant fraction of the radiators are in phase with each other, they will radiate collectively to form a coherent pulse at a rate « N2. The property of coherence in such a radiation system is the nature of the correlations between individual radiators. Coherent emission can be observed by various techniques of non-linear spectroscopy such as free induction decay (FTD)34,36,74 and photon echoes.32,33 A superradiant state, for which a large fraction of the oscillators are in phase, can be created after an excitation process in a two-level system.41 The dynamical properties of this superradiant state are described by a macroscopic oscillating electric dipole moment between the two levels of the quantum system. This macroscopic dipole moment is the ensemble average of the correlation between the oscillators. It exists until the system loses phase coherence or the population decays back to the ground state; then there is no more coherent emission. As discussed in the Introduction, the study of the coherence decay can yield dynamical information about ion-ion interactions and ion- 41 environment coupling. The. contribution of the dynamical interactions to the coherence decay, which is known as homogeneous dephasing, is usually obscured by inhomogeneous broadening arising from the static inhomogeneous crystal strains. Due to the existence of crystal strains and defects in rare earth solids, the decay time of the coherent emission is limited by inhomogeneous broadening. However, the dephasing due to this static inhomogeneous distribution can be removed by a particular pulse sequence so that the macroscopic dipole moment reappears after a rephasing process. In company with the reappearance of the macroscopic dipole moment, the system again radiates coherently. This coherent emission is called a photon echo. The decay of the coherent emission in the form of photon echoes is only due to the non-static or homogeneous dephasing by intrinsic interactions or fluctuation of the environment. The photon echo technique thus provides an effective way for measuring the homogeneous line width in the presence of inhomogeneous broadening and allows us to study the nature of the intrinsic interactions. In this chapter, the theories of optical dephasing and photon echo formation are briefly introduced. Density Matrix and Optical Bloch Equation When the information available about a quantum system is not sufficient to determine its state, the density matrix formalism is the most convenient method for calculating the expectation values of operators, which represent the physical quantities of the system. The density matrix is defined by75'77 42 p(t) = l'F(t)X'F(t)l, (3.D where IvP(I)) is the wave function of an isolated system. For an ensemble in which each individual ion can occupy any lattice site of the inhomogeneous distribution and where further statistical fluctuations affect each site, an ensemble average has to be taken. Then, the density matrix is defined as p(t) = II'Fi(t))pi<'Fi(t)l, (3-1’) where the summation over i is over all sites and all dynamical states, and p. is the normalized probability. For calculation of the time evolution of dynamical operators^ instead of solving the Schrodinger equation to find eigenstates, one solves the Liouville equation for the density matrix dp(t) dt i h [H,p(t)], (3.2) where p(t) is density operator and H is the Hamiltonian of the system. The formal solution of the Liouville equation (3.2) is p(t) = exp[-iHt/h] p(0)exp[iHt/h], (3-3) where p(0) is the initial density of the system. 43 Once the density matrix is known, then it is convenient to calculate the expectation values of dynamical operators by taking the trace of the matrix product (P*) = TrCpP*). (3l4) For coherent emission, P* represents a macroscopic dipole operator, and its statistical or ensemble average is known as the macroscopic polarization of the sample. The average is taken over all the states that the system may occupy. For a particular system, which is homogeneously and inhomogeneously broadened, the average must be taken over all the ions to sum both the inhomogeneous energy states and the homogeneous interaction couplings.49 Consider a two-level system with II) and 12) being the eigenstates of an unperturbed Hamiltonian Hq: HqII) = Oil), and 2) = hco0l2). The transition dipole matrix elements are assumed to be |i = (2l|i.Jl) = (lip. 12). (3-5) In the presence of an electromagnetic field E*(t)=xEx(t)+yEy(t), the interaction Hamiltonian for the two-level system, in the dipole approximation H1 = - (p+E > |iE +), (3.6) where E+=(Ex±Ey)/V2. The dynamic response of the two-level system to 44 the applied field, including the relaxation term, can be derived from the Liouville equation dp(t) i — ------= " — — [H0+HI+H elax,p(t)]. (3.7) dt h A semiclassical picture of this equation of motion has been used to explain the time evolution of the system in the applied field.28,33 It is useful to define the pseudo-dipole as = x(Px> + y(Py> + z(Pz), with (Px) = p(Pl2 + P2i)/V2 (Py) = P(Pl2 - P 2I ) / ^ 2) (Pz) = P(P22- P n ) (3.8) and an effective electric field as '^ eff = ^Ex + ?Ey + zE with a dc component zEz = - hco0/p. (3.9) The equation of motion Eq. (3.7) can be then written as28 45 !— = - — — (x = ---------- [xE + z(--------- ) ] X

>. (3.11) dt h M- In the rotating frame Eq. (3.11) becomes28 d dt M a a ----- [xE + z(- h h((00-(0) M )]X(P**> = EfXCF*), (3.12) where if* = z(co0-co) - xpEdr, and the effective field is then E ^eff= -hlf*/|i. The dynamic response of the pseudo-dipole, following Eq. (3.12), can therefore be described by its precession around E ^eff in the rotating frame. Before the initial excitation, all the ions in the two-level system 48 are in the ground state and the macroscopic dipole is pointing along the direction, as shown in Fig. 7 a. At 0 -> X < - T : r \ <- 2T W 2 ti t (b) (c) (d) (e) eF>/co)-(t3-F >2-r^/co) = (t2-E>2r >/to)-(t1-F>i r >/co), (3.16) where Ic^ 1, k*2, and F*e are, respectively, the wave vectors of the first pulse, second pulse, and the photon echo. The above equation yields the conditions for the photon echo: Vt3 = VtI and . (3.17) The second equation in Eq. (3.17) defines the direction of the echo propagation. Since the three E* vectors, are of equal magnitude, this requires that the excitation pulses be parallel to each other. If the excitation pulses are not parallel, the echo intensity is expected to be reduced. There will also be interference effects. These effects will be minimized in the direction of propagation satisfying the second equation in Eq. (3.17). For small angles <]) between k^2 and F^1, the echo is emitted at an angle - 2 . , (3.19) where a.+ and a. are creation and annihilation operators for the electronic excitation of ith ion. The summations are over all ion sites with the restriction i £ j. The first term is due to inhomogeneous broadening, where Si stands for the deviations of the resonant frequencies of individual ions from the central frequency (O0 in the inhomogeneous line. 54 The second term represents various interactions which cause the irreversible phase relaxation process. The V.. are the off-diagonal matrix elements of these interaction mechanisms. The dephasing mechanisms can arise in various ways by ion-ion and ion-lattice interactions. For rare earth materials, the ion-ion interactions can be:10 a. electric exchange, b. magnetic dipole-dipole interaction, c. electric multipole-multipole interaction, d. virtual phonon exchange. The magnetic dipole-dipole interaction is the leading dephasing mechanism in the l%Tb3+:LiYF4 in which the short range coupling mechanisms are absent. In the concentrated compound LiTbF4, the short range mechanisms such as electronic exchange must play an important role in the energy transfer and also in the optical dephasing processes. Results for both compounds will be discussed in the following chapters. Onasi-Two-Level Systems and Photon Echo Modulation The energy levels of the electronic states of ions or atoms in solids are generally split into sublevels due to weak couplings, such as the hyperfine interaction which was discussed in the last part of Chapter 2, or superhyperfine interaction between the optically activated ions and the nuclei of their host material which will be discussed in Chapter 7. A quasi-two-level system (QTLS) is a generalization of such a system where each of the single levels consists of several nearly degenerate sublevels. 55 If the ions in all the sublevels can be excited by the applied optical pulses in a photon echo sequence, the interference effect known as quantum beats or modulation is expected in the photon echo signal due to the coupling between the substates in both the ground state and the excited state. Important theoretical work on the modulation of coherent transient signals from QTLS has been performed by Lambert et al.,' Skinner et al.,49 and Grischkowsky et al.52 In the calculation of the density matrix after the echo excitation pulses, a unitary matrix can be used to diagonalize the Hamiltonian for the energy splittings in the ground state and the excited state. The matrix elements are time independent because the energy splittings are time independent, thus the modulation should not have an effect on the overall decay rate. The intensity of the photon echo after a two-pulse sequence with time delay X is given by52,60 I(2t) = I(0)ITrCI2exp(-4x/T2), (3.20) where TrC=E Py^cosKtV+OV-M klmn with Pkknn = ^ M WmnWk.* and 56 W = U U +, (3.21) where Ug and Ue are the unitary matrices that diagonalize the interaction Hamiltonian for the ground and excited states which have energy splittings Iicokm8 and hco^6, respectively. The summations are over all the energy substates. The modulation frequencies are obtained by the Fourier transfonnation of the echo profile. Not only the energy splittings in both the ground state and the excited state but also the interaction parameters in the Hamiltonian can be determined through the regression analysis. If the interaction Hamiltonian for the echo modulation is due to the coupling between the echo ions and their surrounding host ions via superhyperfine interactions, there will in general be a number of terms in the Hamiltonian for the couplings between an echo ion and each ligand. In this case each term can be diagonalized separately and Eq. (3.20) becomes I(2t) = I(0)inTrC.I2exp(-4T/r2), (3.22) where TrCj = I representation is ir i> = 0.42l0>-0.642(l4>+l-4>), (5 . 3 ) so there is no first order Zeeman effect. Due to the coupling to other higher F1 states in the same multiplet, the energy level varies quadratically at low field and asymptotically approaches a linear Zeeman effect at very high field. Therefore, in Eq (5.2) g2 is field dependent so that the fluctuation-induced frequency shift . is expected to depend on the applied magnetic field too, particularly at low fields. Fluctuations of local magnetic fields in Tb3+:LiYF4 can arise in several ways: (a) ion-lattice interactions; (b) electron spin diffusion between the two Zeeman components of the ground state; (c) spin-spin interactions among nuclear spins in the host material; (d) abrupt changes upon optical excitation and emission. At low temperature we neglect the effect of mechanism (a). The energy splitting between the two Zeeman components of the ground state is AE=17.8pgH, and when H>10 kG at T= 1.3 K, AE»kT, and the electron spin population in the upper component becomes extremely low. Mechanism (b) is thus negligible at high field, but it will substantially dominate the optical dephasing once the splitting AE is comparable with kT. Mechanism (c) is a slow spectral 75 diffusion source. It causes the stimulated echo decay which is discussed in the last section of this chapter and also affects the spectral holebuming process as will be seen in Chapter 8 . On the microsecond time scale of the observed echo decay, the contribution from (c) is a small constant part to the optical dephasing. As discussed above, the large change in magnetic dipole moment of Tb3+ ions due to optical excitations will cause significant changes in local fields, therefore, mechanism (d) will dominate the echo decay at low temperature and high field. Instantaneous Diffusion As mentioned in the previous section, optical excitations cause changes in the magnetic moment of the excited ions and thus lead to changes in the local fields of neighboring ions. In the dipole-dipole interaction approximation, the change in local field at site i after the excitation of a neighboring ion at site j is AHi = geff|J.B(l-3cos26ij)/rjj3, (5.4) where r.j is the distance between site i and j, and 6y is the angle between the z-direction and the line joining i and j. All the dipoles are parallel to the z-direction, so we do not consider the . effect of x-y components. The resonance frequency shift of i is then v i r gZ^ 2{il~3cos2di?/Ti r (5.5) 76 The shifts due to the first pulse are present throughout the echo sequence (T1 » t ). This is equivalent to the normal static inhomogeneous broadening and will be removed by the echo sequence.28,33 Therefore, the shifts due to the first pulse have no effect on the echo rephasing. However, the shifts due to the second pulse are present for only a part of the echo sequence, so they will not be removed by the rephasing process and will thus prevent complete rephasing of the echoes. In the pseudo-dipole moment picture,28,49 if the amount of phase precession of individual dipoles in the dephasing period randomly differs from that in the rephasing period, the ensemble average of the moments will of course be reduced. To evaluate the dephasing rate due to the instantaneous diffusion, the ensemble average of the frequency shifts has to be carried out for the crystal lattice. The continuous medium approximation of the crystal lattice is applicable for low concentration and long range coupling mechanisms such as magnetic dipole-dipole interaction. With this approximation and using either the statistical theory of line broadening38,39 or the density matrix method of photon echo decay,49 the effect of the instantaneous diffusion is to reduce the echo intensity by I(t) = I0exp(-4t/T’2) (5.6) with the decay rate IfTj2 = 47t2Jne/9V3 77 = 2.53Jne, (5.7) where J=Seff2Iig2Zh and n. is the ion concentration of the sample in the case of an ideal k/2-k pulse sequence. When the pulsing condition is not the ideal k/2-k sequence, say 020» 5 , ET=h©0NT, where (O0 is the center of the inhomogeneous absorption line. This energy is absorbed from the laser pulse, thus we have hco0NT = tdAI2 {I -exp[-a()l2>AVV 0 the excitation is said to be localized. If AV=0 the excitation is said to be delocalized. Anderson’s conclusion is that if Vjj(Lj) falls off faster than 1/r3, then there is a 107 critical concentration of centers below which transport cannot take place and the excitation is localized. The criterion for localization is 5 — > ~ 2, V (6.1) where V is the average interaction between adjacent centers. V increases as the average separation between adjacent centers decreases. As the concentration of centers reaches a critical value above which the criterion no longer holds then the excitation becomes delocalized. Indeed, the excitation will undergo an abrupt change from a localized state to a delocalized state when the concentration increases through the critical concentration. Mott2 and others94,95 have extended Anderson’s idea to the case where the concentration is above the critical value. In an inhomogeneously broadened line, the site-energy distribution varies from low to high as the excitation frequency moves from the wings to the center of the inhomogeneous line. This frequency distribution may be considered as a "concentration distribution" as well. Excitations at the line center can be delocalized if short range interactions exist to quasi- resonantly transfer the excitation among the sites with that energy, and the excitation might be localized in the wings of the line because the effective concentration is below the critical value. There are bounds called mobility edges which separate the delocalized states in the center from the localized states in the wings. The demonstration of mobility edges would be indicative of an Anderson transition. 108 Combining the experimental results from the dilute and the concentrated compounds suggests that the observed fast dephasing is due to energy transfer. The sharpness of the variation suggests a possible Anderson transition from the delocalized states in the center to the localized states in the wings of the inhomogeneous line in LiTbF4. That is, the sharp change in T2 could indicate a mobility edge. As demonstrated recently,11"13 delocalization of optical excitations in terbium compounds is primarily due to short range interactions between Tb3+ ions. This leads to strongly delocalized states in the main region of the absorption line and consequently to a shorter value of T2. The position in the inhomogeneous line where T2 suddenly changes to a longer value perhaps corresponds to the frequency at which this delocalization is inhibited. This position would then be a mobility edge which separates the delocalized states at the center of the absorption line from localized states at the low energy wing. However, the above interpretation may not be consistent with the experimental observations on the high energy side. Phonon-assisted energy transfer is considered later in this chapter, but it connot quantitatively explain the dephasing on the high energy side. Another important feature which could certainly be involved in the echo decay behavior on the high energy side is that the absorption line shape is asymmetric. That feature is not yet understood either. The experimental line shape could possibly indicate that there are two overlapping lines rather than a single broadened line. Since there are two ions per unit cell, a Davydov splitting81 is expected for this compound, so the second line on the high energy side could be a weaker remnant of the second Davydov component 109 which will be discussed in a later section in this chapter. If this is true, the energy dispersion in the exciton bands could dramatically affect the coherence dephasing, and the non-radiative transition from the upper band to the lower band of the Davydov splitting could perhaps explain the short T2 on the high energy side. Before considering the details of the exciton band effects, an alternative interpretation of the observed frequency-dependent dephasing is presented in the following section. Ouasiresonant Interactions Root and Skinner,49 and others,45 have formulated a theory of photon-echo decay due to quasiresonant interactions in substitutionally disordered crystals. This theory can be alternatively used to interpret the frequency-dependent dephasing without directly addressing the Anderson transition. According to Skinner,96 the coherence decay information from photon echo measurements is probably not sensitive to Anderson localization since the Anderson model is usually taken to mean that excitations are localized or delocalized over macroscopic regions, and the dephasing only . depends on local environments or nearby interactions and can occur without any substantial population transfer. Nevertheless, coherence dephasing is quite sensitive to ion-ion interactions. Moreover, the quasiresonant model can test the neighboring site-energy correlation, and it predicts frequency-dependent echo decay. Analyzing the dephasing data along with this model can thus yield information about the strength of the intrinsic interactions and the nature of the inhomogeneous broadening. n o In the quasiresonant-interaction model, one assumes that the' dephasing is due to the interactions between the nearby ions which have the same or very close resonant frequencies. Two limiting cases of "macroscopic" and "microscopic" inhomogeneous broadening have been discussed by Root and Skinner.49 In the limit of macroscopic broadening the system consists of large domains of resonant ions; dephasing is produced by resonant interaction within domains. In the limit of microscopic broadening the site-energies are completely uncorrelated. In this case dephasing is due to quasiresonant interactions between nearby ions in both space and frequency. With the assumption of electric dipole- dipole coupling, the microscopic inhomogeneous broadening model has given a reasonable agreement with the experimental results from EuP5O1450 and Eu3"1": Y2O3,80 while the calculated dephasing rate resulting from the macroscopic inhomogeneous broadening model is independent of excitation frequency.49 The dephasing rate for the concentrated crystal LiTbF4 has been calculated by following previous work of Root and Skinner and fitted to the experimental data with the assistance of Skinner. Without considering phonon broadening, an assumption which is appropriate at low temperature, the interaction Hamiltonian can be written in terms of a diagonal inhomogeneous distribution and off-diagonal intrinsic interactions, H = Shoo-Iixil + S’VJixjl, (6.2) i 1 ij J where li> is the single-ion (ground or electronic excited) state of the ith site. The excitation frequency at site i is OOi=OO0^-S., where (O0 is the I l l central excitation frequency of the inhomogeneous line, and 5 . is the inhomogeneous deviation which is characterized by a distribution function P(5.). The interaction matrix element V„ between sites i and j is responsible for the homogeneous dephasing and the energy transfer processes. The summations are over all sites occupied by the optically active ions with i Vj. The basic restriction of this model is that the intrinsic interactions Vij are much smaller than the inhomogeneous broadening 5 , so the eigenstates in the system are assumed to be localized. Applying the Hamiltonian in Eq. (6.2) to solve the Liouville equation of Eq. (3.7) for the density matrix as a function of time, with the method of moments,38 the resulting echo intensity decays exponentially,49 I(t) = exp [-2tcP((0)Z ’ (Vjj)2^ ] , (6:3) with a dephasing rate constant (6.4) where the normalized distribution function P(co) is defined by a(co) P(co) = — ----------- Ja(co)dco (6.5) § and where of an excited state is defined such that all the ions are in the ground state except that ion j is in excited state (X . Thus, l | X > is an eigenstate of the single-ion Hamiltonian H. which includes all the intrinsic static interactions such as Couloumb interaction, spin-orbit coupling, crystalline field, and interaction with external fields. In the concentrated crystal LiTbF4, there are two ions in each unit cell, as seen in Fig. 2. Consider the crystal consisting of N unit cells, the single ion product states can be expressed as10 lV = lgIigW.............. n^j............... S m > > (6.11) where gnj and Pnj represent the jth ion on nth unit , cell are in the ground state and the excited state, respectively! Since all the sites are equivalent, there are 2N such states having the same energy. As the wave 117 functions of the crystal have to be consistent with the translational symmetry of the crystal, one can define the new crystal wave functions as linear combination of the single ion product states are still 2N degenerate states. This degeneracy is removed if one takes into account the off-diagonal ion-ion interactions. As defined in Eq. (6.2), the off-diagonal interactions V^. nl are responsible for the excitation transfer. The matrix elements between the states can be calculated with the Bloch wave functions. Note that the matrix elements between states differing in are zero because they belong to different irreducible representations.81 As a result, the matrix reduces to N 2-dimensional matrices, each of which has elements In principle the summation is over all unit cells, but if the interactions are short ranged, it is often adequate to include only nearest neighbor (6.12) where R* is the lattice translation from the origin to the nth unit cell, and £* is a vector in the reciprocal space. This is known as the Bloch wave function for the crystal. Since E* has N possible values and j=l,2, there (6.13) interactions. For each value of F \ therefore, there are two excitation energy values which can be calculated by the 2X2 matrix diagonalization10 118 I H11(Er^)-E H12(E^ )I I v . 1 = 0 (6.14) I H21A E-H22A I where Hn (E^)5 H22(E^) include interactions among ions on the same sublattices and H12(E^)j H21A ) involve interactions among ions on the two different sublattices. The equivalence of the two sublattices requires that Hn A)=H22A)> and H12A)=H*21A)- The resulting eigenvalues are EiA) = E0 + H11A) ± IH12A)!, (6.15) where E0 represents the single-ion energy of the exeiton band. For a large crystal, the adjacent values of E* differ little from one another, so the N values of the excitation energy form two quasi-eontinuous bands of excited states. The energy difference between the two bands at E*=0 is known as the Davydov splitting. In LiTbF4, a Tb3+ ion has four nearest neighbors (nn) which belong to a different sublattice and four next nearest neighbors (nnn) which are on the same sublattice. Including these nn and nnn ion interactions, the matrix elements in Eq. (6.15) can be written as H11A) = H22A) = 2V2[cos(kxa) + eos(ka)] (6.16) (H12A)I = 2Y1[2cos(kxa/2)eos(kya/2)cos(kze/2) + cos2(kxa/2) + cos2(kya/2)]1/2, (6.17) 119 where V1 and V2 are the nn and nnn interaction matrix elements, respectively, and a=5.2 A, c=10.89 A are the crystal lattice constants. Since the ion-ion interactions are strongly dependent on distance, the nnn interactions may be much weaker than those of the nn ions, and presumably the line shape of the optical transitions is determined primarily by the nn interactions, particularly for short range interactions. Unlike Tb(OH)3 in which the nn interaction element V1 gives the amplitude of diagonal terms Hn (E*) and H22CkV 1 the nn element of Tb3+ ions in LiTbF4 determines the Davydov splitting as well as the energy dispersion of the exciton bands. At = 0, the Davydov splitting is AE = ^H 12(O)I = SV1. (6.18) The possibility that the two-peak inhomogeneous absorption line, as seen in Fig. 19, could arise from transitions to the two bands of the excited states £* = 0 is intriguing. In that case the spectrum would directly reveal the Davydov splitting. It should be noted, however, that electric dipole transitions are restricted to only one band due to the crystal symmetry. • On the other hand, the inhomogeneity of the crystal may partially break this selection rule and allow a weak remnant of the transition to the other band. Figure 23 schematically shows the Davydov splitting for the hypothtetical case V1=0.025 cm"1 and V2=0.006 cm'1 and describes the energy dispersion E±V ) calculated using Eq.(6.16) and Eq.(6.17) along the (111) and (101) directions in the first Brillouin zone of the k-space. The excitons have dispersion along all directions in the k-space. In 120 a fluorescence experiment, all the energy states could be occupied after an initial E* = 0 excitation. Thus, in the fluorescence emission, a continuous band could be observed instead of a single line. A broad band fluorescence emission from the excited state to a 7F5 level would then be a direct demonstration of the energy dispersion.11 Alternatively, the exciton band could be probed via the E* V 0 absorption spectrum from the upper Zeeman component of the ground state. Both of these methods have been used in attempts to see the exciton dispersion. The fluorescence experiments yielded no conclusive evidence. The absorption experiments are discussed in the following section. Exchange Splittings and the Search for Exciton Band Effects In order to seek alternate evidence for exciton band States, the line shape of the transition from the upper component of the ground state has been compared to the IE* = 0 transition from the lower component. The so- called magnon band14 is about 5 cm"1 higher than the lower component due to the Zeeman splitting in the internal molecular field for this ferromagnetic material. For that band-to-band transition all IE* values are allowed; thus, the absorption line would exhibit the band structure. Due to the Zeeman splitting in the internal field* the upper component was 4 -5 cm"1 higher than the lower component, so the upper level was not well populated in this temperature region. The absorption experiment from the upper component of ,the ground state F6 to the excited state 5D4 T1 was thus made on a much thicker crystal with a thickness of 0.3 mm. The experiments covered the temperature region 1,2 EN ER GY (c m '1) 121 CS WAVE VECTORS (A ) Fig. 23. Illustration of Davydov splitting in the excited state 5D of LiTbF4 with given values of V^O.025 cm"1 and V =0.006 cm in the Eq. (6.15). The energy dispersion in the exciton bands along (111) and (101) directions in k-space is calculated from Eq. (6.16) and Eq. (6.17) with the given values of Vj and V2. 122 K to 2.3 K, without an external magnetic field. The absorption coefficient and the fluorescence intensity were synchronously recorded. The reflection from the back surface of the sample was detected as the transmitted beam to obtain a higher ratio for the absorption, as in this way the beam passed through the sample twice. Absorption from the excited magnon state was also monitored by recording the intensity of the yellow fluorescence of 5D4 F1 to 7F5 F3 4 at frequency 18442 cm"1. Figure 24 and Figure 25 show the observed absorption and emission, respectively, as a function of excitation frequency with temperature as a parameter. The two spectra yield similar information about the line shape which varies dramatically as a function of the temperature. The spectrum consists of a main line and several additional lines. Those lines are dramatically broadened at high temperatures and are inhibited at low temperatures. These effects have been attributed to coupling to the neighboring ions. The overall splitting is often known as the exchange splitting of the ground state. The extra lines arise from various excited spin clusters, which are thermally populated. These phenomena were first realized in rare earth compounds by Bleaney et al.98 and later by Prinz." They have been extensively discussed by Leask,100 Huftier,22,101 and Cone and Meltzer.10 The shift of the main line is due to the ferromagnetic ordering and approximately obeys the following equation from molecular field theory,102 AZA0 TZTc = tanh (6.19) A BS O RP TI O N C OE FF IC IE NT ( ar b. un its ) 123 T = 1.27 K T = 1.65 K T = 1.97 K T = 2.26 K ‘ 1-0 -2.0 -3 .0 -4 .0 -5.0 -6 .0 -7.0 RELATIVE ENERGY (cm"') Fig. 24. Absorption line shape for the magnon-exciton transition of F6 F2 to D4 F1 in LiTbF4 with temperature as a parameter at zero external field. All four spectra have the same vertical scale and the energy axis is measured relative to the transition energy from the lower component of the ground state. FL UO RE SC EN CE I NT EN SI TY ( a rb . u ni ts ) 124 T = 1.27 K T = 1.65 K T = 1.97 K T = 2.26 K RELATIVE ENERGY (cm"1) Fig. 25. Excitation line shape for the magnon-exciton transition of 7Ffi F2 to 5D4 F1 in LiTbF4 with temperature as a parameter at zero external field. The fluorescence is the emission from the excited state to 7F5 F34 at frequency of 18442 cm 1. All four spectra have the same vertical scale and the energy axis is measured relative to the transition energy from the lower component of the ground state. 125 with Tc=2.874 K, and Aq=S cm"1. There are two measurable lines on the high frequency side which have separations d1=0.6±0.05 cm"1 and d2=0.3±0.05 cm'1, respectively. The relative intensities of these lines were strongly dependent on temperature as expected by the Boltzman factor. In general, the effect of the magnetic moments of the ions in a crystal can be approximated by an average internal field. This gives a reasonable description of the energy splittings or shifts for magnetic ions in many stoichiometric compounds. However, the crystal is made up of distinct ions each with a magnetic moment. The individual interactions of the nearest and next nearest neighbors of course have the most significant effect on the energy shift. Any variation of the spin configuration of the near neighbors can lead to a resonant frequency shift of the optically excited ion. Since there are only a few probable ways to vary the neighborhood spin configurations, the line is split into multiple lines rather than a continuous distribution. At low temperature, the ground state of Tb+3 ions in the compound LiTbF4 is a very good approximation to an Ising system with gH=17.8 and gpO. AU the magnetic dipole moments are either parallel or antiparallel to each other along the crystal c-axis. The couplings of the Tb3+ ions include the magnetic dipole-dipole interactions (MDD) in addition to near neighbor exchange interactions. The MDD interaction is of course a long- range interionic interaction, which at high temperatures wiU tend to average to zero at any ion site. The short-range exchange interaction involves only nearby ions. The interaction Hamiltonian between site i and j can be written as 126 H = AijV jz + BijsizV (6.20) where Aij is the exchange coupling parameter, and , B.. is the MDD coupling parameter which can be exactly calculated for the known crystal structure of the compound, (l-3eos20ij)liB2g||2 (6 .21) where r.. is the distance between site i and j, Oij is the angle from the z axis to the line joining site i and j, and [Xb is the Bolir magneton. At very low temperature, an ion at site i in the upper component of the ground state with sz= l /2 is surrounded by neighbors all of which are in the lower component with sz=-l/2. This is the most probable case as the temperature is raised, and thus it is often called a single-ion transition. One thus observes a strong narrow line shifted by the combined effect of the neighbors. When one of its neighbors is also in a thermally populated spin-down state, the resonant frequency has a shift Aco. = (BljSjzH-AySjz)Asjz By+Ay 2 (6.22) 127 The excitation at the shifted frequency is called a pair transition. The population of the ion-pairs decreases much faster than the single-ion population as the temperature decreases, thus the relative intensity of the side lines decays faster than the main line. It is interesting to note that the shift has an negative sign if the transition is from the lower component. In that case all neighbors are in the spin-up state except one which is the spin-down state, so the frequency shift becomes By + Ay Am. = ---------- -------- . (6.23) The calculated MDD contribution to the splittings due to the first six types of neighbors sets are listed in Table 6 .103 The nn pairs and the 4th nn pairs have a negative effect on the energy splitting and should give side lines on the liigh frequency side. The three small splittings of the 3rd, 5th and 6 th neighbors would be obscured by the main line. The splitting due to the second nri pairs , has not been observed, since this splitting shifts the energy of the ion to the high energy side, and it is then not as favored by the Boltzman factor as are ’ the two observed side lines on the low energy side. It is important to note that the MDD splitting due to the 4th nn pair is in reasonable agreement with the observed value, and that the splitting due to the first nn pairs is 0.14+0.04 cm"1 larger than the observed value. The difference is then due to the exchange interaction, thus the exchange parameter A has the value of 0.28 cm'1. At low temperature, the excited spin configurations tended to be 128 Table 6 . Energy level splitting due to the neighboring ion-ion interactions in the ground state of LiTbF4. Type of impair number of pair calculated splitting MDD (cm"1) observed splitting (cm1) 1st 4 -0.74 -0.6+0.04 2nd 4 +0.50 3rd 8 +0 .01 4th 8 -0.26 -0.3+0.04 5th 4 +0.18 6 th 4 +0.08 inhibited. Only the main line for the single-ion transition remained. The width of the main line was about equal to the line width of the ground state E* = 0 absorption, so that the exciton band effect was not revealed. On the other hand, the exciton dispersion could be around 0.1 cm"1 and still not be observable in these line shape measurements. Thus, while exciton band effects are not completely ruled out by these experiments, they seem too small to explain the asymmetric absorption line shape or the frequency-dependent dephasing observed on the high energy side of the line. Another potential interpretation for the asymmetry of the line involves the magnetic interaction of neighboring Tb3+ ions in the ground state. The energy splittings due to the magnetic dipole-dipole interaction plus the nearest neighboring exchange interaction between the Tb3+ ions have been observed in transition from the upper ground state component. At low temperature the energy splittings saturated, and the line shape of single-ion excitation was asymmetric with a line shape which was mirror 129 reflection of the line shape for the lower component excitation. Even in the presence of a large external magnetic field, the exchange interaction and the spin excitation process could still broaden the absorption line. Further line shape calculations may fit the asymmetric line shape and yield a complete understanding of the fast dephasing on the high energy side. In summarizing this chapter, we have observed strongly frequency- dependent dephasing due to short range ion-ion interactions in the concentrated compound. To interpret our experimental results, the Anderson transition, the quasiresonant interaction model in addition to phonon-assisted energy transfer, and exciton band dispersion have been considered. No exclusive connection with any one of these theoretical models has been proved; however, the quasiresonant model indicates a dramatic sensitivity of photon echo measurement to energy transfer effects as small as I MHz in these compounds. Further coherence measurements on Tb(OH)3 are suggested for determining the effect of energy transfer processes on coherence dephasing since in that compound wthere the energy transfer processes and the interaction mechanisms are all previously determined. Echoes in Tb(OH)3 have already been observed in the two wings of the 7F6 to 5D4 transition in an optically thick (20 (Lim) sample. This sample was still so thick that the whole laser beam was absorbed at the center of the line. The observed echoes in the two wings exhibited similar decay properties. That means that the unexplained asymmetry present in LiTbF4 will not be a factor in this compound. Unfortunately, attempts to make a thin sample for completing the last step in this series of photon echo experiments have not yet succeeded. 130 CHAPTER 7 SUPERHYPERFINE INTERACTIONS IN Tb3+: LiYF4 PROBED VIA PHOTON ECHO MODULATION The previous two chapters have been devoted to discussions of the ion-ion interactions and the dynamical effects of the interactions between the echo ion and its environment. As discussed there, the photon echo decay rate directly measures the optical dephasing or homogeneous line width of the resonant optical transition. In this chapter another important feature of spectroscopy ---- static optical line splitting is presented through deconvolution of the modulation of the photon echo spectra.52" s s 60 Ag discussed in Chapter 3, echo modulation is observed in a quasi- two-level (QTL) system in which the ground and the excited states are split into sublevels by hyperfine or superhyperfine interactions.52"60 Since the photon echo signal is simultaneously sensitive to both the eigenfunctions and the eigenfrequeneies of the interaction Hamiltonian, the deconvolution of the echo modulation spectra may result in a more accurate and complete determination of the model Hamiltonian.55"60 This is an advantage of echo spectroscopy over conventional hyperfine spectroscopy techniques which usually measure only frequencies. Frequency information may not be sufficient to test the correctness of the model Hamiltonian. For instance, in the spin echo electron-nuclear- 131 double-resonance (ENDOR) studies of ruby, a more general Hamiltonian used by Liao and Hartmann59 resulted in only a 3% change of the Al resonance frequencies compared with a simpler Hamiltonian used by Laurance et al. ,104 but it produced large changes in the echo modulation. Superhyperfine interactions (SHFS) between the rare earth ions and the fluorine and lithium nuclei in rare-earth fluorides have been classically studied in NMR experiments by measuring line shifts of the nuclear magnetic resonance.105,106 These studies found that the superhyperfine interactions make the local field lower than the applied field. The Li shifts are dominated by dipole interactions, whereas the F shifts also have comparable contributions from the SHFS interactions between the rare earth ion and the F nuclei. Using the Tb3"1" eigenfunctions for the ground and the excited states, the interaction Hamiltonian, which contains crystal-field splittings, the electronic Zeeman interaction, and hyperfine interactions, the echo modulation spectra can be deeonvoluted with a regression-analysis procedure by treating the superhyperfine interactions between the echo ion and the surrounding nuclei as a perturbation. This regression analysis yields both eigenfunctions and eigenfrequencies as functions of the interaction parameters of the model Hamiltonian. These parameters are evaluated in the regression analysis and the correctness of the model Hamiltonian can be tested by doing the same regression analysis at different external fields. In the end, this gives a complete understanding of the observed echo modulation as a function of field and gives a quantitative description of the SHFS in this material. 132 Hamiltonian for Superhvperfine Interactions In the presence of an applied external magnetic field Hz parallel to the c axis, the electronic Zeeman energy is the largest term in the Hamiltonian for the energy shifts of the Tb3"1" ion. This has been discussed in Chapter 2. Since the energy of the Tb3+ ion is sensitive to the local magnetic fields because of its large magnetic moment, the magnetic dipole fields from nuclei of surrounding host ions play a role in the static energy splittings, as well as in the dynamical behavior of the Tb3+ ion such as the optical dephasing, as discussed in previous chapters. Considering the superhyperfine interactions (SHFS) between the Tb3+ ion and its surrounding host nuclei, the Hamiltonian can be generally written as H= H™+ a W (71) where the fust term is the interaction Hamiltonian for an isolated Tb3+ ; i ion in the crystal, and the second term, including the Zeeman interaction 1 of the nuclei and the coupling between the Tb3+ ion and the nuclei, is i responsible for the photon echo modulation. The hyperfine interactions (HFS) of Tb3+ included in the first term in Eq. (7.1) have a large effect on the energy splittings, as seen in Chapter 2. The HFS splittings are ; two orders of magnitude larger than those of the SHFS which are in the i MHz region. Therefore, the small superhyperfine interactions can be 5 treated as perturbations. | The deep and rather simple modulation spectra which were observed ' \ ' 'I 133 in the experiments suggest that only the nearest neighbors have significant effects on the echo modulation. The interaction Hamiltonian considered for the echo modulation thus includes the superhyperfine interactions between the Tb3+ ion and its four nearest surrounding F nuclei. For each type of pair of Tb3+ — F" ions, the Hamiltonian can be written as Hfclb - (7.2) The first term represents the Zeeman interaction of the F nuclear spin ( I = 1/2, y/h = 4.005 MHz/kG ) with the external field parallel to the e axis. The A term is associated with the exchange ( Fermi contact ) interaction, which measures the density of the Tb3+ electron clouds at the nuclear sites as transferred through the fluoride electron clouds. The B tenns are due to the magnetic dipole-dipole interaction. The transverse components of the spin operator are defined as I± = Ix ± ily, and

for the Tb3+ ions have been calculated and experimentally tested as discussed in Chapter 2 for both the ground stare (J = 6 ) and the excited state (J = 4). For the ground state g=- 135 5 .9 3 , which is slightly smaller than -6 due to a small admixture of My=±2 states.69,70 For the excited state, there is no first order Zeeman interaction for the singlet T1 level. The non-linear energy shift is due to weaker field-induced couplings between this F1 level and other F1 levels in the same 5D4 multiplet. Therefore, e is not a constant even at high field. As discussed in Chapter 2, the combined crystal field and Zeeman Hamiltonian was diagonalized as a function of magnetic field for the entire 5D4 multiplet. The eigenfunctions of the Zeeman Hamiltonian obtained from this diagonalization are given as mixtures of Mj components and are field dependent IF1) =L a(M pH)IMj). (7.4) Mj 1 z 1 Thus, the expectation values of the Zeeman Hamiltonian are also field dependent. By using these eigenfunctions, the value of e can be calculated as a function of magnetic field. It is linearly proportional to the applied magnetic field at low field and asymptotically approaches the value -4. Since l I is always much smaller than l I at low field, Z C - B .j the SHFS is much weaker when a Tb3"1" ion is in the excited state than in the ground state. This causes a variation of the SHFS as the Tb3"1" ion oscillates between the ground and the excited states and thus leads to a deep modulation in the echo signals. This will be discussed in detail in a general analogy to the modulation in ruby in . which the effects of the ! SHFS variation on echo modulation have been intensively studied for the Rj line between different levels.52,53,57,60 136 Photon Echo Modulation With a set of parameters and known values of , the SHFS Hamiltonian can be diagonalized in the nuclear spin space. According to echo modulation theory,52,53 as discussed in Chapter 3, the diagonalization can be carried out separately. for each of the Tb3+ — F coupling terms in the SHFS Hamiltonian. Therefore^ to calculate the echo modulation, one needs only to diagonalize Hrbp in Eq. (7.2) over the different Tb3"1" — F pairs. This is done by calculating a 2X2 matrix product. A unitary matrix U can be found to diagonalize the Hamiltonian, UHpxbU+ = E. The energy splittings due to a pair of coupled Tb3+ — F ions are obtained as E(Iz=±l/2) = ±(l/2)(ec2 + 132)1/2, (7.5) . where a = yhHz - (A+Bz) 13 = Cf >B . Z t The unitary matrix can be expressed as (7.6) I U= ---- -T ------------------ (((a2+132)1/2+a)2+132)1/2 f JBeltP (a2+J32)1/2+a ) Iv (a2+fi2)1/2+a -Be-ltP J 137 where the two columns in the right side of above equation are eigenfunctions for the spin-1/2 system. Applying Eq. (7.7) to Eq. (3.22), and assuming that the overall echo decay is a simple exponential, the echo intensity is of the form I(2t) = lyexpH t/T^rty lW / + IW2I4 +IW1l2IW2l2[2cos(ti)et) + 2cos(cogt) - C O S ( t O e t + t o g t ) - C O S ( C 0 e t - t o g t ) ] Ij2 (7,8) where j=l,2 for the two types of nearest F nuclei, T2 is the echo decay time constant, and W1 and W2 are defined below. The primary modulation frequencies are given by COg e = (l/h)(a2+B2)1/2ge. (7:9) This gives the SHFS splittings of the ground and excited states. The sum and difference of the SHFS splittings also appear in the echo modulation but their effects are weaker by a factor of four compared to the primary- frequency terms. This explains the Fourier spectra of the observed echo modulation in the Tb3+: LiYF4 compound and is consistent with previous observations on Pr3+ILaF3.55 Note that in Eq.(7.8), echo modulation will disappear if either CDg or COe is zero. This means that, at least in the ease of spin-1/2 , echo modulation depends on the co-existence of ground and excited state splittings. This has not been previously discussed in the 138 earlier echo modulation theory.54,53,55,60 In Eq.(7.8), the products of 8 transformation matrix elements of the 2X2 matrices W and W+, as defined in Eq. (3.21), have been represented by using two values of IW1I and IW2I which are defined as IW112 = [(hcog+ag)(hcoe+ae) + (htog- a g)(ha>e- a e) + 2fi BgIZh2OOeCOg IW2I2 = [(hcog+ag)(hcoe- a e) + (hcog- a g)(hcoe+ae) - 2BeBg]/h2ooecog (7.10) In Eq. (7.10) the quantities of OOg e , CCg e and Bge are not independent. They are functions of the external field Hz and the SHFS parameters (A+Bz) and Bf At a given field the echo modulation is completely determined by the values of the SHFS parameters. Experimental Results and Regression Analysis The photon echo experimental setup and techniques have been discussed in Chapter 4. The modulation spectra are recorded as a function of time delay between the two excitation pulses at various external fields. In order to minimize the fluctuation of the echo signal due to the fluctuation of the powers of the excitation pulses, 16 shots were averaged by a programmable transient digitizer for each data point, and time scanning was repeated several times at each field. Low excitation powers were used to reduce the effects of instantaneous diffusion which were 1 139 described in Chapter 6 . An observed echo modulation spectrum for the field of 11 kG is shown in Fig. 26 on a linear scale, and its Fourier transformation is shown in Fig. 27. In the Fourier spectrum, the four dominant modulation frequencies correspond to the excited and ground state SHFS splittings of the two sets of nearest F neighbors. This supports the suggested Hamiltonian Hjbp in Eq.(7.2) and confirms that Eq. (7.8) correctly describes the major modulations. Since the SHFS splittings are smaller than the pure magnetic Zeeman splittings of the F nuclei (4.005 MHz/kG), the parameter (A+Bz) must be negative for all the Tb3+— F pairs. The modulation strength was found to be sensitive to the applied field. At low field (< 15 kG), the modulation was very strong, but it became weak at high fields as shown in Fig. 11 for Hz = 35 kG. The reason for this can be understood from the field dependent eigenfunctions of the SHFS Hamiltonian which determine the modulation coefficients IW1I and IW2I, or it can be seen by defining an effective field in a way similar to that which has been used in the analysis of echo modulation in ruby.10 I f eff = (l/hy){[z(hYH-(A+Bz)] + ?[Bt] }, (7.11) A A where z and t are unit vectors parallel and perpendicular to the applied field, respectively. The F spins process around this effective field. Since both and (A+Bz) are negative as noted above, the z component of the effective field (Heff)z is always smaller than the applied field. In addition the expectation values of the electron spin operator are very different between the ground and excited states, with > 5 at low field. EC HO I NT EN SI TY (a rb . u ni ts) 140 THEORETICAL EXPERIMENTAL 0 .0 200 .0 4 00 .0 600 .0 TIME DELAY ( nsec ) Fig. 26. Theoretical and experimental photon echo modulation spectra for the 7F6 F7 to 5D4 F1 transition in T lr+:LiYF at H = 1 1 kG. The echo intensity is plotted on identical linear scales. The theoretical echo modulation function was fitted to the experimental echo modulation spectrum which has 176 data points, and MD is 0.02%. FO UR IE R AM PL IT UD E (a rb . u ni ts) 0. 0 25 .0 50 .0 75 .0 10 0. 0 141 100.0 125.0 FREQUENCY ( MHz ) Fig. 27. Fourier transformation of experimental echo modulation spectrum plotted in Fig. 26. The four strongest peaks in the center are the SHFS splittings in the ground and the excited states. Their sums and differences appear in the high and low frequency regions. Various frequencies in this spectrum arise from the products of the primary terms as described in Eq. (7.8). 142 The two terms in (Heff)z for the ground state are comparable at low field so that (Heff)z < (Heff)t, whereas the value of (Heff)z » Hz » (Heff)t for the excited state. The effective field varies in both direction and magnitude as the echo ion oscillates between the ground and the excited states, thus the modulation is maximized.60 At high field, Heff = Hz for both the ground and the excited states, and the F spins always precess around the applied field. This minimizes the variation of the coupling between the echo ion and the neighboring nuclei and thus minimizes the echo modulation. In comparing theory and experiment, Eq.(7.8) was fitted to the observed modulation spectra. First the value of T2 was determined by the least-squares method. Then a regression program based on Marquardt’s algorithm1 1 was used to fit the observed echo spectra with five variable parameters — four independent SHFS parameters plus Iff The fitted spectrum for Hz = 11 kG is shown in Fig. 26. The mean deviation per point MD is 0.02% which is quite small. The value of MD is defined in Reference 60 as MD = (IZN)ZH -ItheIZItle X 100%, (7.12) N where N is the number of data points in the experimental observation. This regression procedure was carried out independently for three different modulation spectra at Hz = 9.5, 11, and 14.7 kG, respectively. The agreement between the experiments and the theory was excellent. The calculated SHFS parameters are listed in Table 7. The four SHFS parameters obtained from these three independent analyses have mutual 143 deviations smaller than 3%, which are smaller than experimental errors and are primarily due to variations of the observed spectra due to laser power fluctuation and other variations of the experimental conditions. Table 7. SHFS parameters for two types of nearest neighbor fluorines for a Tb3+ ion. TYPE I TYPE 2 (A + Bz) (MHz) 3.8 + 0.1 3.4 ± 0.1 to _ I 4.35 ± 0.15 4.4 + 0.1 The identical values of IBtI for the two types of nuclear pairs are expected by Eq. (7.3) due to the site symmetry. The sign of the parameter Bt cannot be determined in the deconvolution. This was indicated by Eq. (7.6) and Eq. (7.10). The regression analysis was initially carried out with 8 SHFS parameters with the assumption that the symmetry of F site may not be perfect so that the parameters could be different for the four nearest F nuclei. In that case, the differences between the related parameters of the four F nuclei were just the same as the deviations from fitting different experimental spectra. This indicates that, at the current resolution, in this sample the Sites of Tb3"1" and the four nearest F nuclei have good agreement with the expected structure. Figure 28 shows the theoretical and experimental SHFS splittings as a function of applied magnetic field parallel to the crystal c-axis. The 144 two upper lines represent the excited state energy splittings of the SHFS between the Tb3+ ion and its two types of nearest F neighbors. The linear relation between the SHFS energy splitting of the excited state and the applied field in this region can be understood from the expectation value of e as a function of the field. As indicated in the first section of this chapter, is linearly proportional to the applied field at low field. Therefore, by defining % (7.13) where T| = 0.12 kG"1 for the region of the field in Fig. 28, the SHFS splitting of the excited state in Eq. (7.9) can be rewritten as roC = Yeff1V (7 .1 4 ) where Yeff = [(Y-il(A+Bz)/h)2 + (T1BzZh)2Iiz2 »3.6MHz/kG. (7.15) Comparing that value to y = 4.005 MHz/kG for free fluorine, the F line will have a 10 % shift due to the SHFS. For the ground state, g is large and constant. This causes a stronger SHFS shift and leads to smaller modulation frequencies. In conclusion, the experimental and theoretical studies of photon FR EQ UE NC Y ( M H z ) 25 .0 35 .0 45 .0 55 .0 145 11.0 FIELD Fig. 28. Theoretical and experimental SHFS splittings versus the applied magnetic field parallel to the crystal c- axis. The points are obtained from the Fourier transformations of the experimental photon echo spectra. The continuous curves represent the theoretical values which are calculated by using Eq. (7.9) and Eq. (7.6) with the SHFS parameters from Table 7. 146 echo modulation in the dilute compound Tb3+:LiYF4 show that the existence of modulation in the echo amplitude is dependent on the energy splittings of the Tb3"1" ions in both the ground and excited states due to the SHFS interactions. The depth of the echo modulation is sensitive to the variation of the magnetic moment of the Tb3+ ion as it oscillates between the ground and the excited levels. This causes changes in the relative orientation of the local magnetic field and thus results in deep echo modulation. This is consistent with previously observed echo modulation in ruby due to the SHFS between Cr3+ and its surrounding Al ions.53,57,60 The rather simple modulation spectra indicate that, in this compound only the first nearest neighbors have significant contributions to the observed echo modulation. This is similar to the Mg=l/2 echo modulation in ruby. In contrast, recent studies of the Mg=-3/2 echo modulation in ruby indicated that as many as 375 Al ions have been included for a best fitting of experimental data.60 Futhermore, the resulting SHFS parameters from the regression analysis are identical for the ground state and the excited state, thus indicating that the echo modulation is solely due to the variation of the magnetic moment of the Tb3+ ions in the ground and excited states. The identical SHFS parameters for the . ground and . excited states are also expected for the higher crystal symmetry of this compound. Further PENDOR experiments may be worthwhile for checking the SHFS parameters for both the ground and excited states with greater accuracy. 147 CHAPTER 8 HYPERFINE SPECTRAL HOLEBURNING FOR Tb3+:LiYF4 Spectral holebuming in rare earth insulators has proved to be very useful for measuring the hyperfine and superhyperfine structure of both ground states and optically excited states even in the presence of inhomogeneous broadening.61"65 Its high resolution is particularly useful for studying the effects of external perturbations such as nuclear Zeeman ■ and electronic Stark splittings which can provide the kind of detailed structural infonnation available from nuclear magnetic resonance experiments. Holebuming spectroscopy is also an effective method to study spectral diffusion and relaxation processes in rare earth doped materials.66 Spectral holeburning via optical pumping of ground state hyperfine j level populations has been observed in 1% Tb3+: LiYF4 for the blue optical ; transition from 7F6 F2 to 5D4 Fr The hole lifetime increased with applied j I magnetic field and reached a value of 10 minutes with an external field of I ' ' j 40 kG applied along the c axis. The electronic and hyperfine levels that | I were discussed in Chapter 2 then provided a qualitative explanation for the j ' • i field-dependent holebuming process. These experiments demonstrate that ] ! the high resolution of holebuming spectroscopy can be readily extended to j Tb3+. I 1 148 In this chapter* hyperfine population holebuming is discussed for the lowest 7F6 to 5D4 transition in the visible spectrum of Tb3+:LiYF4, along with holebuming rate, quantum efficiency, and lifetime. Holebuming Process The observed 7F5 T2 to 5D4 T1 optical absorption line involves the superposition of all allowed hyperfine transitions summed over all ion sites. The inhomogeneous line broadening of 0.45 cm'1 in this dilute crystal is greater than the hyperfine splittings described and calculated in Chapter 2, so it obscures the individual optical transitions between hyperfine sublevels. Holebuming, however, provides a way to probe individual hyperfine transitions for a subgroup of Tb3+ ions even in the presence of this typical inhomogeneity. In the case of fixed-frequency excitation by a narrow band laser, a given ion undergoing optical transitions in the crystal is resonant with the laser on only one of the four allowed transitions between hyperfine sublevels of the ground and excited states (AIz = 0 for optical transitions). An excited ion, however, can relax to a different ground state hyperfine level than that from which it was excited, due to interactions with phonons and other neighboring ions or a small misalignment of the magnetic field. This optical pumping process leads to the redistribution of population among the hyperfine components of the ground state. At low ' temperature, the nuclear spin-spin and spin-lattice relaxation can be slow under appropriate conditions. Then, population changes in the ground state hyperfine components can accumulate and can last long enough for a 149 holebuming spectrum to be recorded by scanning a weak probe laser across the inhomogeneously broadened optical absorption line. Experimental Details In the experiments the holebuming spectrum, holebuming quantum efficiency, and hole lifetime have been measured as a function of magnetic field with two tunable nitrogen-laser-pumped dye lasers. The holebuming process is so strong that it can be easily detected by an absorption or excitation spectrum. Appropriate experimental power densities are noted below. Except for signal detection, the experimental setup for the holebuming is similar to that for photon echo experiments, which was shown in Fig. 8. The 0-60 kG magnetic field was supplied by the superconducting solenoid, and the crystal was immersed directly in liquid helium at 1.3 K. A photodiode monitored the laser intensity as the probe laser was tuned, and absorption was measured by a second photodiode after the sample. The PDP-11 computer which controlled the experiment calculated the ratio of the photodiode signals on a shot to shot basis, averaged it over ten shots, and displayed the resulting transmission data on the screen during a laser scan. The absorption coefficient was calculated directly from that data in the standard way. Fluorescence was detected by a Spex 14018 0.85 m double monochromator or via appropriate color glass filters and a photomultiplier. Holeburning Spectmm and Mechanism To observe the hole spectrally, a weak tunable "probe" laser was 150 sent through the sample at a 1° angle relative to the fixed frequency "puinp" laser. The probe beam was twenty times weaker than the pump beam, so it had a negligible effect on the populations. Both beams were overlapped in the sample, with the probe beam delayed by 10 nsec in early experiments. After it became obvious that the hole lifetime was generally measured in minutes rather than nanoseconds or microseconds, the timing was obviously not critical. Since each laser had a relatively broad line width (= 0.05 cm'1), the resolution of the hole spectrum was limited to about 0.1 cm'1. Figure 29 shows a representative hole spectrum for the Tt-polarized electric dipole transition from 7F6 F2 to 5D4 F1 recorded by absorption of the probe laser at 12 kG and 1.3 K. An unperturbed spectrum showing the equilibrium inhomogeneous line shape is given on the same scale for comparison. At both sides of the hole, the absorption is enhanced. This is due to the increased population in the other ground state hyperfine components which are higher or lower in energy than that from which the holebuming transition originated. Thus, in company with a hole, ahtiholes are also created. The presence of these antiholes provides strong evidence that the holebuming mechanism involves optical pumping of the hyperfine levels as outlined above. Holebuming could possibly proceed via the frozen-core process observed earlier for Pr3+: CaF2 by Burum, Shelby, and Macfarlane,63 but that process could not give antihole structure on the energy scale seen here. Unfortunately the laser resolution was not adequate to reveal stmcture in the side hole or antihole spectrum which would have allowed measurement of the individual hyperfine splittings. To determine those A BS O RP TI O N 0. 00 0. 25 0. 50 151 ENERGY OFFSET ( cm"’) Fig. 29. Holebuming spectrum for the transition Ffi F2 to D4 F at the magnetic field H = 46 kG (solid curve) compared with the unperturbed absorption spectrum (dashed curve). Both curves are displayed as absorption coefficients on the same scales. The full width at half maximum of the absorption is 0.45 cm"1. The observed hole width was instrumenlally limited, but, nevertheless. clear antiholes may be seen, implying optical plumping of hyperfine sublevels. 152 splittings, so that independent values of the hyperfine interaction parameters can be extracted from these experiments, a high resolution tunable laser—probably a single mode cw dye laser—will be required. The cw laser available in this laboratory, however, cannot presently operate in the blue region needed for Tb3+ transitions. Nevertheless, the dynamics and field dependence of the holebuming process still can be studied. The theoretical analysis of hyperfine interaction explains the observed field dependences and thus adds further evidence that the optical pumping of hyperfine sublevels is the dominant mechanism for the holebuming. It was noted that the upper electronic level of the ground state could also be a possible population reservoir for holebuming,112 especially at high field and low temperature. At 1.3 K and in fields up to 50 kG, however, no observable population storage has been found in that level. Holebuming Rate and Quantum Efficiency At line center for this transition, the 0.2 cm thick crystal absorbed 40 % of the laser energy, giving a peak absorption coefficient of 2.5 cm" 1. Upon exposure to the fixed frequency pump laser, the absorption quickly dropped to 15 %, due to the holebuming process. Figure 30 shows the corresponding decrease in the fluorescence intensity. Both quantities exhibited exponential decays with a time constant of 47 see. The magnetic field was 45 kG. The peak laser intensity at the crystal surface was about 100 MW/cm2, the pulse width was 5 nsec, and the pulse repetition rate was 6 per sec. Under these conditions, a hole depth of 10 % could be obtained FL UO RE SC EN CE I NT EN SI TY ( a rb . u ni ts ) 0. 00 0. 25 0. 50 0. 75 1, 00 153 150.0 200 .0 TIME ( sec ) Fig. 30. Holebuming rate at the line center of the 7F6 F2 to 5D4 F absorption as shown by the extinction of the 5D4 to 7F, fluorescence. The peak laser intensity was 100 MW/cm2, pulse width was 5 nsec, and pulse rate 6 Hz. The curve is exponential with a decay constant of 47 sec. 154 in 5 sec, with an incident energy per ion of 3.4 x IO"14 erg. From an alternate point of view, the average laser intensity at the crystal surface was 3 W/cm2. This value could be achieved readily with a typical unfocused cw dye laser. Moreover, the narrow laser line width would give one thousand times higher average spectral brightness. Based on these considerations and the long hole lifetimes described in the next subsection, holebtiming experiments on rare earth compounds containing Tb3+ or other ions with large ground state magnetic moments can be carried out easily. The quantum efficiency is an important parameter of the holebuming process. Since the hole growth rate is not constant, the quantum efficiency Tj has been defined by Moemer et al113 as the ratio of the initial rate of decrease of the number of absorbing ions per unit volume N(t) to the initial rate of absorption of photons. Thus,113 (dT(t)/dt) t=Q 11 ~ GTo(IZhv)(I-To-R) (7.1) where I is the incident laser intensity, T(t) is the time-varying sample transmission during the growth of the spectral hole, T0 is the initial sample transmission which is 0.6 at the inhomogeneous line center, R is the total reflection loss at the sample, h is Planck’s constant, V is the photon frequency, and G is the peak absorption cross section for the ions. Using the laser line width of 1.5 GHz, which is much broader than the homogeneous line width of 30 kHz measured in the photon echo experiments, as in Chapter 5, the cross section has been calculated G = 2 x IO"19 cm2 by following Ref. 113. This gives a quantum efficiency of Tj 155 Hole Lifetime Figure 31 summarizes the effect of an external field on the hole lifetime. These measurements were made by burning a hole with the pump laser, stopping that laser, waiting an appropriate interval, and then reading the hole depth with a single scan of the probe laser. Obviously, hole lifetime is decreasing toward small values at zero field. No hole could be observed at zero field in the current experiments due to lifetime considerations and to the laser line width. Details of the Zeeman effect and the hyperfine structure discussed in Chapter 2 qualitatively explain these observations. Experiments on LiTbF4 Similar experiments were carried out on an appropriately thin sample of the stoicliiometric compound LiTbF4. No evidence was found for holeburning in that concentrated magnetic system. Presumably the strong coupling of the Tb3+ ions leads to very much faster relaxation of the ground state hyperfine populations. Faster phase relaxation observed in the photon echo experiments on the stoichiometric compound is certainly consistent with this result. Correlation of the Holebuming Phenomena and the Hyperfine Structure = 0.02. The small values of the zero field ground state hyperfine splittings TI M E ( m in ) 156 O CN O 05 ~ O CO - O CO _ ■ Od 0.0 10.0 I I 20.0 30.0 FIELD ( kG ) 40.0 50.0 Fig. 31. Hole lifetime as a function of applied field, which is correlated qualitatively in the text with level splittings. 157 for small magnetic fields are less than the laser line width, and this certainly explains the difficulty of burning holes by optical pumping of the hyperfine populations for that case with the pulsed lasers used in these experiments. Relaxation processes affect the hole lifetime and thus affect the ease of holeburning. The fluctuating field due to the Tb3+ electronic spins is dramatically reduced in an applied magnetic field due to the exceptionally large ground state electronic g factor. As discussed in Chapter 5, the photon echo experiments on the same crystal have shown a dramatic reduction in electron spin diffusion with increasing magnetic field, which is just what one expects based on these ground state properties and the 1% Tb3"1" concentration. The Tb3"1" nuclear spin-spin relaxation will be slowed by the increasing magnetic field-induced separation of the ground state hyperfine sublevels. Those splittings become over two orders of magnitude larger than the nuclear Zeeman energies of the fluorine ligands. Strong "frozen core" effects also constrain the fluorine ligands near Tb3+ ions. All of these factors combine to explain the dramatic increase in hole lifetime with increasing magnetic field. 158 CHAPTER 9 CONCLUSIONS The concentrated crystalline rare earth compound LiTbF4 and the isostructural dilute compound 1% Tb3+:LiYF4 have been systematically studied with nonlinear spectroscopic methods such as photon echoes and holebuming. The effects of various interaction processes on the static energy structure and the dynamical behavior of the rare earth ions have been analyzed for both the ground and excited states. The experimental results from the dilute compound have been well understood, while the interpretation of the novel frequency-dependent dephasing in the concentrated compound involved several theoretical models, some of which are still controversial for describing the nature of the interactions in a real physical system. In the 1% dilute compound Tb3^iLiYF4, the Tb3"*" ions are relatively isolated. The ion-ion interactions are weak and the short-range coupling is absent. Since the Tb3"1" ion has a large magnetic moment 8.Pjxg in the doublet ground state, its energy is sensitive to the local magnetic field. The magnetic dipole-dipole interaction among the Tb3+ ions and between the Tb3"1" and the host F ions thus determines its spectral properties. At liquid helium temperature, the observed echo decay behavior and the hyperfine spectral holebuming process exhibited strong magnetic field 159 dependence. The observed homogeneous line broadening, energy splitting, and spectral diffusion can be described by magnetic dipolar interactions plus the exchange interaction between the Tb3+ ion and its nearest neighboring host nuclei. In the photon echo measurements of the dilute compound at a small external magnetic field, the echo decay time was dramatically shortened by the electron spin diffusion process in the ground state. The spin flip- flops of the Tb3"1" ions between the two ground Zeeman components cause fluctuations of the local field and thus randomly shift the resonant frequencies of the echo ions. Since the Zeeman splitting is strong, the two components are well separated at large magnetic field and the electronic spin diffusion is inhibited. The instantaneous spectral diffusion phenomenon has been dramatically observed in the photon echo experiments on this dilute compound. The measured dephasing rate depends linearly on the intensity of the second laser pulse and is independent of the intensity of the first laser pulse. Changes of greater than a factor of ten are readily observed for the dephasing rate. This effect is also magnetic field dependent. To interpret the experimental results, the magnetic dipole-dipole interaction between Tb3"1" ions was considered. The crystal field analysis and the Zeeman experiments on this material indicate that an optical transition of a Tb3"1" ion from the ground state 7F6 F2 to the excited state 5D4 F1 causes an instantaneous change in its magnetic moment which leads to a shift in the resonant frequency of its neighboring ions through the magnetic coupling. These random frequency shifts caused by the first pulse are removed by the echo sequence, but the shifts caused by the 160 second pulse are present for only part of the echo sequence. Since they cannot be removed by the rephasing process, thus lead to extra echo decay. The dephasing rate in the dilute compound has been calculated as a function of excitation intensity and magnetic field. The theoretical calculation is in reasonable agreement with the experimental measurements. Due to the existence of the instantaneous diffusion, the measured echo decay time T2 ’ is no longer the coherence dephasing time T2 of the related two-level system. This "apparent" homogeneous line broadening must be avoided when the photon echo measurement is used to probe the "real" homogeneous line width determined by the intrinsic ion-ion interactions without external disturbances. On the other hand, this experiment-induced dephasing process provides a way for testing the sensitivity of an optically activated ion to the perturbation of its environment. The instantaneous diffusion is significant only in those systems in which the resonant frequency of the optical ions is sensitive to changes in the magnetic moment and hence in ion-ion coupling during the excitation. Time-domain transient spectroscopy is not only sensitive to homogenous line broadening but also a very powerful tool for measuring small static energy splittings which are usually obscured by the inhomogeneous broadening. In the dilute compound superhyperfine coupling between the Tb3+ ion and F" neighbors has thus been obtained from the echo modulation phenomenon. The observed modulation characteristics confirmed that the magnetic dipole-dipole interaction and exchange interaction between an Tb3"1" ion and its first nearest neighboring F nuclei are responsible for the observed deep 161 echo modulation. A model Hamiltonian of the SHFS for this system has included both the magnetic and exchange interactions. Since the echo modulation is sensitive to both the eigenvalues and eigenfunctions of the splitting Hamiltonian, echo intensity as a function of time delay has been derived and tested by the experimental modulation spectra with the method of regression analysis. Both the modulation frequencies and the interaction parameters have been determined in the regression analysis. The excellent agreement between the theoretical calculation and the experimental data confirmed the correctness of the model Hamiltonian. This is the first investigation of the SHFS through photon echo modulation for rare earth materials. Spectral holebuming in Tb3+:LiYF4 has been demonstrated with tunable pulsed lasers, and the hole lifetime has been measured as a function of applied magnetic field. Optical pumping of the ground state hyperfine level populations is responsible for the observed hole burning process. In the presence of an applied magnetic field, the spectral diffusion process is slow in the ground state, so the hole lifetime increases with an applied magnetic field. The quantum efficiency for the hole burning process has been calculated in terms of the measured absorption decay rate. The observations and calculations indicate that experiments with a single mode cw dye laser are capable of yielding additional detailed information about both the ground state and excited hyperfine structure in Tb3+ compounds. Observation of side holes and antiholes or of optically detected nuclear resonance should allow accurate measurement of both the 162 magnetic hyperfine interactions and the nuclear quadmpole interactions. That in turn can yield detailed structural information about the Tb3"1" sites whether the ions are at intrinsic sites or are found near defects. The crystal field eigenfunctions, derived from the crystal field analysis of the observed energy levels, provided important information required to understand the various phenomena observed in this system. With the crystal field eigenfunctions, the electronic Zeeman effect, and the hyperfine interactions of the ground and excited states have been calculated. The resulting energy states and eigenfunctions provided an excellent description of the electronic Zeeman splittings of the energy levels in both 7F and 5D multiplets for fields up to 50 kG and allowed accurate calculation of the strongly field-dependent magnetic hyperfine strucmre in the ground and excited states. Based on these calculations, all the experimental results can be qualitatively explained. The eigenfunctions obtained as a function of magnetic field have been used to calculate the field-dependent echo modulation and both the field dependent and power-dependent dephasing rate in the dilute compound. Intense and long-lived stimulated echoes were observed in three-pulse photon echo experiments and could provide a probe for the study of spectral diffusion and coherent phase storage. They also suggest that accumulated grating echoes should be a practical method for studying faster dephasing at higher temperatures. The observation of strong frequency-dependent dephasing in the concentrated compound and its absence in the 1% dilute compound indicated that Tb3"1" ion-ion interactions dominated the dephasing processes and suggested the existence of energy transfer through short- 163 range interactions. The energy transfer processes could cause strongly delocalized states in the main region of the absorption line and consequently lead to a shorter value of T2. In the wings of the line, the effective concentration is low and the ions do not interact thus the excitations should become localized. This effect is just what is expected by the Anderson. models; however, correspondence of our observed result to the Anderson transition remains controversial. The apparent proportion of the dephasing rate to the absorption coefficient in the main region and on the low frequency side of the line suggests that the frequency-dependent dephasing is due to quasiresonant interactions among the Tb3+ ions. The dephasing rate in this part of the line has been fitted by the quasiresonant interaction model. The assumption of phonon-assisted energy transfer processes ' in the inhomogeneous line can qualitatively explain the faster dephasing on the high energy side but the calculation made with the one-phonon emission approximation does not agree with the experimental data. The line shape measurements for the transition from the upper component of the ground state to the excited state indicated no large exciton band effect in the excited state as observed in other Tb3+ compounds such as Tb3+(OH)3.11 Therefore, the exciton dispersion must be significantly weaker in this compound. The interpretation of the asymmetry of the absorption line as inhomogeneous broadening seems to oversimplify the nature of the line shape. The much faster dephasing rate on the high energy side could certainly be related to that line shape. A Davydov splitting is expected for this compound, because there are two ions per unit cell and the off- 164 diagonal interaction matrix elements responsible for the Davydov splitting are due to the nearest neighboring coupling. According to the calculation, a small interaction between the nearest neighboring ions could lead to the observed splitting between the two peaks in the absorption line. The validity of this explanation is questionable since the selection rules for the lc*=0 ground state which allow only one band of the Davydov splitting to be excited. Another potential interpretation for the asymmetry of the line involves the magnetic interaction of neighboring Tb3+ ions in the ground state. The energy splittings due to the magnetic dipole-dipole interaction plus the nearest neighboring exchange interaction between the Tb3+ ions have been observed in transition from the upper ground state component. 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' APPENDIX COMPUTER CONTROL PROGRAMS FOR PHOTON ECHO EXPERIMENT 175 Fig. 32 DECHP1.C /* PHOTON ECHO PROGRAM SCANNING TIME DELAY BETWEEN TWO LASERS, INTERFACING DIGITIZER, AND RECORDING THE ECHO INTENSITY AND POWERS OF THE LASERS */ #include ^include Crtl I .h> #include "da.c" #indude "ad.c" #include "Itc.c" main() i long INITT(),DRWABS(),MOVABS(); int x,y,xl,yl; int xO = 30; int yO =100; int xm ax = 660; int ymax = 760; int IBUPO; int write = 0; int read = I; int remote = 4; int local = 5; int digitizer = I; char *comdl; char *comd2; char *comd3; char *comd4; int lenstl; int lenst2; int lenst3; int lenst4; char strip[3]; unsigned char array [1024]; long sumai-ay[512]; int thi-ee = 3; int size = 1024; int . two = 2; int one = I; int bit9 = 256; int i,j,k,l,s; 176 char filnam I [20],filnam2[20]; FlO fio,fioo; int shots,ends,wid,sep; int charm,initial ,init; int range,delta,repts; int zero,zer,bin,ech,ezero; long pa[512],p0,p; long w[64],a0,a,b0,b,c0,c; long data[512]; char bell [2]; int baud = 1400; bell[0] = 07; bell[l] = 0; /* INITIALIZE IN/OUT */ putinitO; getinit(); for(i=0;i<512;i++) { sumaray[i] = (long)O; /* GIVE FILE NAMES */ putfmt("What is the name of the echo data fileTXn"); getfmt("%p\n",filnaml); if(! fcreate(&fio ,filnam 1,1)) { putfmt("Error: can’t open %p\n",filnamI); return; putfmt("What is the name of the power data fileTXn"); getfmt("%p\n",filnam2); if(! fcreate(&fioo ,filnam2,1)) ( putfmt("Error: can’t open %p\n",fihiam2); return; } fcall(INITT, I ,&baud); /* SET VOLTAGE FOR ZERO TIME DELAY */ putfmt("Type delay for simultaneous pulsesXn"); getfmt("%iNn",&zero); while(zero>=0) { zer = zero; putvolt(0,(int)(2047.0-204.8*(z;er/100.0))); putfmt("Type delay for simult pulsesNn"); getfmt("%iNn",&zero); } /* SET INITIAL TIME DELAY */ putfmt("Type initial delay in ns\n"); putfmt("Adjust echo location on digitizerVi"); 177 getfmt("%iW',&mitial); while(initial>=0) { hiit = initial; putvolt(0,(int)(2047.0-204.8*«init+zer)/100.0))); putfmt("Type initial delay in ns\n"); getfmt("%iNn" ,&initial); } /* SET ECHO SIGNAL LOCATION AND DURATION IN DIGITIZER’S TIME AXIS */ putfint("Which channel is the center of echoTW); getfmt("%f\n",&chann); putfmt("Type echo half width on digitizerNn"); getfmt("%i\n",&wid); /* SET NOISE-BACKGROUND LOCATION */ putfmt("Type seperation to sum baekgroundNn"); getfmt("%i\n",&sep); /* SET TIME DELAY RANGE */ putfmt("Type scan range in nsXn"); getfmt("%f\n",&range); /* SET SCAN INCREAMENT PER DATA POINT */ putfmt("Type delay increment in nsW); getfmt("%i\n",&delta); /* SET DATA AVERAGE NUMBER */ putfmt("How many shots do you wantTVi"); getfmt("%i\n",&shots); /* RECORD INITIAL VALUE OF LASER POWER */ w[0] = 0; for(j=I ;j<=shots;j-H-) { gettrgO; aO = getvok(l); bO = getvolt(4); cO = (Iong)(a0*b0/4000); putfmt("a0 =%l\n",a0); putfmt("b0 =%Ni",bO); wUJ = w|j-l]+cO; putfmt("w[j] =%lXn",w[j]); } pO = w[shots]; putfmtC'pO =%l\n",p0); /* SET DIGITIZER MODE */ comdl = "DIG SA,\0"; i = itob(comdl+7,shots,10); comdl [7+i] = ’\0 ’; comd2 = "READ SAND"; comd3 = "MOD TV,\0"; Comd4 = "MOD DIG\0"; lenstl =Ienstr (comdl); lenst2 = lenstr(eomd2); . IenstS = lenstr(eomdS); lenst4 = lenstr(comd4); 178 fcall(IBUP,2,&remote,&digitizer); fcall(IB UP,4, & write ,&digitizer, comd4,&lenst4); /* START */ putfmt("hit return key to startXn"); getfmt("\n"); /* DRAW AXES FOR PLOT */ fcall(lNlTT, I ,&baud); fcall(MOVABS,2,&xO,&ymax); fcall(DRWABS,2,&xO,&yO); fcgll(DRWABS,2,&xmax,&yO); fcall(MOVABS,2,&xO,&yO); x l= xO; yl = yO; /* START SCAN AND RECORD DATA */ for(i=init;i<=mit+range;i+=delta) putvolt(0,(int)(2047.0-204.8*(i+zer)/100.0)); for(k=l ;k<=50;k+=l) {} fcall(IBUP,4,&write,&digitizer,comdl,&lenstl); for(j=l ;j<=shots;j++) { gettrgO; a = getvolt(l); b = getvolt(4); c = (long)(a*b/10); w[j] = w[j-l]+c; p = w [shots]; bin = (i-init)/delta; pa[bin] = p/pO; /* SET DIGITIZER FOR SENDING DATA */ fcaU(IBUP,4,&write,&digitizer,comd2,&lenst2); fcall(IBUP,4,&read,&digitizer,strip,&three); fcall(IBUP,4,&read,&digitizer,array,&size); fcall(IBUP,4,&read,&digitizer,strip,&two); for(k=0;k<512;k++) { data[k]= (long) (array [2*k] *bit9 + array[2*k+l]*one); } ech = data[chann-wid]; ezero = data[ehann+sep-wid]; for(k=chann-wid+1 ;k<=ehann+wid;k+.+) ech += data[k]; ezero += data[k+sep]; } sumaray[bin] = (long)(eeh-ezero); sumarayEbin] =(long)((float)(suinaray[bm]/(shots*(wid+l)))); /* PLOT DATA ON SCREEN */ x =(int)((£loat)630*(i-mit)/range + 30); y = (int)((float)(0.75*sumaray[bin])+100); fcall(DRWABS,2,&x,&y); 179 fcall(M0VABS,2,&xl,&yl); x l =(mt)((float)630*(i-mit)/range + 30); yl = (int)((float)(0.75*pa[bm])+100); fcall(DRWABS,2,&xl,&yl); fcall(MOVABS,2,&x,&y); } ends = i - delta; fcall(IBUP,4,&write,&digitizer,comd3 ,&lenst3); fcall(IBUP,4,&write,&digitizer,comd3,&lenst3); fcall(IBUP,2,&local,&digitizer); fcall(MOVABS,2,&xmax,&yO); fcall(DRWABS,2,&xO,&y0); fcall(DRWABS,2,&xmax,&yO); fcall(MOVABS ,2,&x0,&y0); /* WRITE DATA INTO FILES */ putf(&fio,"%\n"); for(i=0;i<(l+range/delta/8);i++) { putf(&fio,"%+\06041 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061%+\06061%+\06061\n",(long)(8*i),sumaray[8*i+0],sumaray [8*i+l],s- umaray[8*i+2],sumaray[8*i+3],sumaray[8*i+4],sumaray[8*i+5],sumaray[8*i+6]- ,sumaray[8*i+7]); } fclose(&fio); putf(&fioo,"%Vi"); for(i=0;i<( I+range/delta/8);i++) putf(&fioo,"%+\06041 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061%+\06061%+\06061\n",(long)(8*i),pa[8*i+0],pa[8*i+l],pa[8*i+2],pa[8*i+- 3],pa[8*i+4],pa[8*i+5],pa[8*i+6],pa[8*i+7]); } fclose(&fioo); putfmt("%pV',bell); putfmt("done. Delay range =%i nsec\ti",ends); } 180 Fig. 33 SDEC5A.C /* PHOTON ECHO PROGRAM SWEEPING MAGNETIC FIELD ACROSS ABSORPTION LINE AND RECORDING ECHO INTENSITY AND ABSORPTIONS OF TWO LASERS */ #include #incl,ude #include "da.c" #include "ad.e" #include "Itc.c" main() { long INITT(),DRWABS(),MOVABS(); int x,y,xl,yl,x2,y2; int xO = 30; int yO = 100; int xmax = 660; int ymax = 760; int IBUP0; int write = 0; int read= I; int remote = 4; int local = 5; int digitizer = I; char * comdl ,*comd2,*comd3,*comd4; int I enst I ,Ienst2,lenst3 ,lenst4; char strip[3J; unsigned char array[1024]; long sumaray[512]; int three = 3; int size = 1024; int two = 2; int one= I; int bit9 = 256; int i,j,k,l,s; char filnam I [20],filnam2[20] ,filnam3 [20]; FIO fio,fioo,fiio; long z[64] ,pb[512],pl,al ,bl^cl; int wid,sep,ends,sran; int chmm,initial,Mt; int range,delta,delay; int zero,zer,shots; l&l long w[641,pa[512],p,a,b,c; long ech,ezero; long data[512]; long echo[50]; int bin,ne,nt,incr; float sweep; char bell[2]; int baud = 1400; bell[0] = 07; bell[l] = 0; /* INITIALIZE IN/OUT */. putinit(); getinit(); for(i=0;i<512;i++) { • sumaray [i] = (long)O; } /* GIVE FILE NAMES */ putfmt("What is the name of the echo data fileTXn"); getfmt("%pXn" ,filnam I); if(! fcreate(&fio,filnam 1,1)) { putfmtC'Error: can’t open %p\n",filnaml); return; } putfmt("What is the name of 11 absorption data fileTXn"); getfmt("%p\n",filnam2); if(! fcreate(&fioo,filnam2,1)) { putfmt("Error: can’t open %p\n",filnam2); return; } putfmt("What is the name of 12 absorption data fileTXn"); getfmt('' %pXn" ,filnamS); if(! fcreate(&fiio,filnam3,1)) { putfmt("Error: can’t open %pXn",filnam3); return; } fcall(INITT,l,&baud); /* SET VOLTAGE FOR ZERO TIME DELAY */ putfmt("Type delay for simultaneous pulsesXn"); getfmt("%iXh" ,&zero); while(zero>=0) { zer = zero; putvolt(0,(int)(2047.0-204.8*(zer/100.0))); putfmt("Type delay for simult pulsesXn"); 182 getfmt("%i\n",&zero); /* SET TIME DELAY */ putfmt("Type initial delay in nsNn"); getfmt("%iNn",&initial); while(initial>=0) { init = initial; putvolt(0,(int)(2047.0-204.8*((init+zer)/100.0))); putfmt("Type initial delay in ns\n"); getfmt( "%i\n" ,&initial); } /* SET ECHO SIGNAL LOCATION AND DURATION IN DIGITIZER’S TIME AXIS */ putfmt("Which channel is the center of echo?\n"); getfmt("%i\n",&chann); putfmt("Type echo half width on digitizerXn"); getfmt("%i\n",&wid); /* SET NOISE-BACKGROUND LOCATION */ putfmt("Type seperation to sum baekgroundXn"); getfmt("%i\n",&sep); /* SET DATA AVERAGING NUMBER */ putfmt("How many shots do you wantTNn"); getfmt("%i\n",&shots); /* SET FIELD SWEEP RANGE, INCREMENT, AND TIME DELAY BETWEEN INCREMENTS */ w[0] = 0; z[0] = 0; putfmt("Type field sweep range 0.0 to 10.0(0 to 2KG)Xn"); getfmt("%f\n",&sweep); putfmt("Type field sweep increment (int)\n"); getfmt("%iNn",&iner); putfmt("Type delay of field sweep (int)W); getfmt("%i\n",&delay); for(k=0;k<=(int)(204.8*sweep);k+=incr) { putvolt( I ,(int)(2047.0+k)); for(i=0;i<=2*delay;i++){} I /* SET DIGITIZER MODE */ comdl = "DIG SA,\0"; i = itob(comdl+7,shots,10); comdl[7+i] = \ 0 ’; comd2 = "READ SA\Q"; comd3 = "MOD TV,\0"; comd4 = "MOD DIG\0"; lenstl =lenstr(comdl); lenst2 = lenstr(comd2); lenst3 = lenstr(eomd3); lenst4 = lenstr(comd4); fcall(IBUP,2,&remote,&digitizer); fcall(IBUP,4,&write,&digitizer,comd4,&lenst4); 183 /* START */ putfmt("hit return key to startV); getfmt("\n"); /* DRAW AXES FOR PLOTS */ fcallCINTTT, I ,&baud); fcall(MOVABS ,2,&x0,&ymax); fcall(DRWABS ,2,&x0,&y0); fcall(DRWABS,2,&xinax,«&yO); feall(MOVABS ,2,&x0,&y0); sran = (mt)(409.6*sweep); x l = xO; yl = yO; x2 = xO; y2 = yO; /* START SCAN */ for(i=0;i<=sran;i+=incr) putvolt(0,(int)(2047.0-204.8* (init+zer)/l 00.0)); putvolt(l,(mt)(2047.0+204.8*sweep-i)); for(k=0;k<=delay;k++){} /* SET DIGITIZER FOR RECEIVING SIGNAL */ fcall(IBUP,4,&write,&digitizer,comdl,&lenstl); /* GET LASER ABSORPTIONS */ for(k=l ;k<=shots;k++) IgettrgO; a = getvolt(l); al = getvolt(2); bl = getvolt(3); b = getvolt(4); c = (long)(a*500/b); cl = (long)(al*500/bl); z[k] = z[k-l]+cl; w[k] = w[k-l]+c; } bin = (int)(i/iner); pa[bin] = w[shots]/shots; pb[bin] = z[shots]/shots; /* SET DIGITIZER FOR SENDING DATA */ fcall(IBUP,4,&write,&digitizer,comd2,&lenst2); fcall(IBUP,4,&read,&digitizer,strip,&three); fcall(IBUP,4,&read,&digitizer,array,&size); i fcall(IBUP,4,&read,&digitizer,strip,&two); for(k=0;k<512;k++) data[k]= (long)(array[2*k]*bit9 + array[2*k+l]*one); } ech = data[chann-wid]; ezero = data[chann+sep-wid]; for(k=chann-wid+1 ;k<=ehann+wid;k++) { ech += data[k]; 184 ezero += data[k+sep]; } echo [bin] = (long)(ech-ezero); sumarayfbin] = (long)(echo [bin]/(shots* (wid+1))); /* PLOT DATA */ x =(int)((float)630*i/sran + 30); y = (int)((float)(0.75*sumaray[bin])+100); fcall(DRWABS,2,&x,&y); fcall(MOYABS,2,&xl,&yl); x l =(int)((float)630*i/sran + 30); yl = (int)((float)(0.75*pa[bin])+100); fcall(DRWABS,2,&xl,&yl); fcall(MOVABS,2,&x2,&y2); x2 =(int)((float)630*i/sran + 30); y2 = (int)((float)(0.75*pb[bin])+100); fcall(DRWABS,2,&x2,&y2); fcall(MOYABS,2,&x,&y); /* RESET DIGITIZER AND PLOTER */ fcall(IBUP,4,&write,&digitizer,comd3,&lenst3); fcall(IBUP,4,&write,&digitizer,comd3,&lenst3); fcall(IBUP,2,&local,&digitizer); feall(MOYABS,2,&xniax,&yO); fcall(DRWABS,2,&xO,&yO); fcall(DRWABS ,2,&xmax,&y0); fcall(MOVABS,2,&xO,&yO); /* WRITE DATA INTO FILE */ putf(&fio,"%W'); for(i=0;i<=(int)(sran/incr/8);i++) putf(&fio,"%+\06041 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061 %+\0 6 0 6 1%+\0 6 0 6 1%+\0 6 0 6 1 \n",(long)(8 *i),sumaray[8 *i+0 ],sumaray[8 *i+l],s- umaray[8*i+2],sumaray[8*i+3],sumaray[8*i+4],sumaray[8*i+5],sumaray[8*i+- 6],sumaray[8*i+7]); } fclose(&fio); putf(&fioo,"%\n"); for(i=0;i<=(int)(sran/incr/8);i++) putf(&fioo,"%+\06041 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061%+\06061%+\06061V',(long)(8*i),pa[8*i+0],pa[8*i+l],pa[8*i+2],pa[8*i+- 3],pa[8*i+4],pa[8*i+5],pa[8*i+6],pa[8*i+7]); } fclose(&fioo); putf(&fiio,"%\n"); fdr(i=0;i<=(int)(sran/incr/8);i++) ; { putf(&fiio,"%+\06041 %+\06061 %+\06061 %+\06061 %+\06061 %+\06061 %+\0 6 0 6 1%+\06061%+\06061\n",(long>(8*i),pb[8 *i+0 ],pb[8*i+l],pb[8 *i+2 ],pb[8*i+- '! 3],pb[8*i+4],pb[8*i+5],pb[8*i+6],pb[8*i+7]); } fclose(&fiio); i . putfmt("%p\n",bell); 1I MONTANA STATE UNIVERSITY LIBRARIES I I I i I I I III I I I 111 I I I I 3 ' 762 O O in CD 18E CM