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AC relaxation in the Fe8 molecular magnet Rose, Geordie 2000-07-16

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AC RELAXATION IN THE Fe8 MOLECULAR MAGNET By Geordie Rose B. Eng. (Engineering Physics), McMaster University, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 2000 © Geordie Rose, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatively It isfc understood that copying or publication of this thesis for financial gain shall mot be allowed-wtthQut my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V6T 1Z1 Date: Abstract We investigate the low energy magnetic relaxation characteristics of the "iron eight" (Fe8) molecular magnet. Each molecule in this material contains a cluster of eight Fe3+ ions surrounded by organic ligands. The molecules arrange themselves into a regular lattice with triclinic symmetry. At sufficiently low energies, the electronic spins of the Fe3+ ions lock together into a "quantum rotator" with spin S = 10. We derive a low energy effective Hamiltonian for this system, valid for temperatures less than Tc ~ 360 mK, where Tc is the temperature at which the Fe8 system crosses over into a "quantum regime" where relaxation characteristics become temperature in dependent. We show that in this regime the dominant environmental coupling is to the environmental spin bath in the molecule. We show how to explicitly calculate these cou plings, given crystallographic information about the molecule, and do this for Fes- We use this information to calculate the linewidth, topological decoherence and orthogonal ity blocking parameters. All of these quantities are shown to exhibit an isotope effect. We demonstrate that orthogonality blocking in Fe8 is significant and suppresses coherent tunneling. We then use our low energy effective Hamiltonian to calculate the single-molecule relaxation rate in the presence of an external magnetic field with both AC and DC components by solving the Landau-Zener problem in the presence of a nuclear spin bath. Both sawtooth and sinusoidal AC fields are analyzed. This single-molecule relaxation rate is then used as input into a master equation in order to take into account the many-molecule nature of the full system. Our results are then compared to quantum regime relaxation experiments performed on the Feg system. ii Table of Contents Abstract ii Table of Contents iiList of Tables viii List of Figures ix Acknowledgements xi1 Introduction and Overview 1 1.1 An Introduction to Fe& 4 1.1.1 Giant Spins and Quantum Environments 8 1.1.2 A Related System is Characterized 9 .1.1.3 Magnetic Characterization of Fe8 13 1.1.4 Results in the Quantum Regime 4 1.2 An Introduction to Relaxation Experiments 18 1.2.1 DC Field Relaxation in Polarized Fe8 9 1.2.2 DC Relaxation of Annealed Crystals 11.2.3 Hole Digging and the Time-Dependent Internal Longitudinal Bias Distribution 21 1.2.4 AC Relaxation of Annealed Crystals 23 1.2.5 Extraction of Tunneling Matrix Elements1.3 Thesis Overview 27 iii 2 Effective Hamiltonians 29 2.1 The Fe3+ Free Ion Hamiltonian 30 2.2 The Effect of the Crystalline Environment 32 2.3 The Single Ion Effective Hamiltonian 4 2.3.1 First Order Perturbation Theory 35 2.3.2 Second Order Perturbation Theory 6 2.3.3 Higher Orders Perturbation Theory 37 2.4 The Single Molecule Effective Hamiltonian 9 2.4.1 Inclusion of Exchange and Superexchange Terms 40 2.4.2 "Offsite" Dipolar and Quadrupolar Contributions 40 2.4.3 Intra-Nuclear Spin Couplings 45 2.4.4 Couplings of the Nuclear Bath to External Magnetic Fields .... 45 2.4.5 Coupling to Phonons 42.4.6 Coupling to Photons 7 2.4.7 Bringing all the Terms Together-The Bare Fe$ Hamiltonian ... 47 2.5 Exchange/Superexchange and the Giant Spin Picture 49 2.6 Investigation of the Giant Spin Hamiltonian in the Absence of Environ mental Couplings 51 2.6.1 Exact Solution for Tunneling Matrix Elements via Diagonalization 52 2.6.2 Tunneling Matrix Elements via Perturbation Theory 59 2.6.3 Tunneling Matrix Elements via WKB Methods 60 2.6.4 Tunneling Matrix Elements via Instanton Techniques 60 2.6.5 Comparison of Approximate Methods to Exact Solutions 60 2.7 Back to the Full Hamiltonian-Separation of Tunneling Energy Scale Using an Instanton Technique 62 2.8 Off-Diagonal Terms and the Instanton Method 68 iv 2.8.1 Review of the Method of Tupitsyn et.al 69 2.8.2 The Tunneling Lagrangian 70 2.8.3 An Assumption is Made 2 2.8.4 Solution for the Free Instanton Trajectory 73 2.8.5 Inclusion of the External Magnetic Field and the Nuclear Spins . 75 2.9 The Final Single Molecule Effective Hamiltonian 77 3 Nuclear Spin Couplings in Fe$ and the Isotope Effect 78 3.1 Units and Constants 79 3.2 The Point Dipole Approximation 73.2.1 Magnetic Field at f due to a "Point Dipole" at 0 80 3.2.2 Magnetic Field at fv due to Eight "Point Dipoles" at fFei 8 .... 80 3.2.3 Isotopic Concentrations, Nuclear ^-factors and Quadrupolar Mo ments in Fe8 83.2.4 Definition and Evaluation of 7^, 7^, u^. and ufc 81 3.2.5 Contact Hyperfine Coupling Energies for 57Fe3+ 82 3.2.6 Calculation of wjj and u^r from Knowledge of Atomic Positions . . 84 3.2.7 Calculation of the Orthogonality Blocking Parameter K 85 3.2.8 Calculation of E0 93 3.2.9 Calculation of Topological Decoherence Parameters AkND and A . 97 3.3 Using Free Fe3+ Hartree-Fock Wavefunctions to Model Actual Spin Dis tributions 98 3.4 Tables of Nuclear Positions, Fields at Nuclei and Hyperfine Coupling En ergies 112 4 An Introduction to the Generalized Landau-Zener Problem 122 4.1 Introduction to and Exact Solution of the Landau-Zener Problem .... 122 v 4.1.1 Alternate Method of Solution for the Transition Probability I. All Orders Perturbation Expansion 126 4.1.2 Alternate Method of Solution for the Transition Probability II. Dychne's Formula 128 4.1.3 Analysis of the Transition Formula 130 4.1.4 Generalization of the Two-Level Landau-Zener Problem I. Exact Solution for A(t) ~ Vj|(t) 131 4.1.5 Generalization of the Two-Level Landau-Zener Problem II. Exact Solution by Mapping to Riemann's Differential Equation 137 5 The Landau-Zener Problem in the Presence of a Spin Bath 142 5.1 The Addition of an Environment to the Landau-Zener Problem: General Considerations 145.2 The Quantum Regime Effective Hamiltonian: Inclusion of a Spin Envi ronment 143 5.3 General AC Field Solution in Fast Passage 147 5.3.1 A List of Approximations Invoked in the Calculations That Follow 149 5.3.2 General Strategy for Calculating Relaxation Rates 151 5.3.3 Processing of the Transition Amplitude 155 5.3.4 Processing of the Transition Probability (i) The Formal Expression 159 5.3.5 Processing of the Transition Probability (ii) Averaging over the Randomly Fluctuating Ti Noise 161 5.4 Solution Without Spin Bath 163 5.5 Solution For a Spin Bath with no Quadrupolar Contribution 169 5.5.1 Pure Orthogonality Blocking 16vi 5.5.2 The General Case; Inclusion of Topological Decoherence 180 5.6 The General Single Molecule Relaxation Rate in Fe$ 194 5.6.1 Effect of the Nuclear Spin Environment on the Large A Single Molecule Relaxation Rate in Fe8 202 5.7 Summary and Discussion of Results 203 6 AC Relaxation in a Crystal of Molecular Magnets 206 6.1 Preamble 206.2 The Generalized Master Equation 209 6.3 Short Time Dynamics 210 6.3.1 Strongly Annealed Samples and the Large A Limit 212 6.3.2 General Solution Near the Nodes 214 7 Summary and Outlook 217 Appendices 221 A Bias Distribution in a Dilute Solution of Dipoles 222 B Time Evolution of Nuclear Spin States 228 Bibliography 231 vii List of Tables 2.1 Perturbation theory results for some simple Hamiltonians, from [70]. ... 59 3.1 Nuclear spin information for nuclei occuring in Fes- From [48] 81 3.2 Positions of the iron ions, units in Angstoms 86 3.3 Data for Hydrogen 113 3.4 Data for Hydrogen 4 3.5 Data for Hydrogen 115 3.6 Data for Hydrogen 6 3.7 Data for Bromine 117 3.8 Data for Nitrogen 8 3.9 Data for Carbon 119 3.10 Data for Iron 120 3.11 Data for Oxygen 1 5.1 Quantities coming from orthogonality and degeneracy blocking 195 5.2 Zero external field values for r0 and u0 for the three species shown. Units are in MHz 196 5.3 Quantities coming solely from topological decoherence effects 196 5.4 Zero field values of the topological decoherence terms for species in Fe$. . 197 5.5 Topological decoherence terms for three varieties of Fes 201 5.6 Quantities that come about due to interplay between orthogonality block ing, degeneracy blocking and topological decoherence effects 201 viii List of Figures 1.1 The number of atoms needed to represent one bit of information as a function of calendar year. Extrapolation of the trend suggests that the one atom per bit level is reached in about the year 2020. Adapted from [1]. 1 1.2 Clock speed (Hz) vs. calendar year. Adapted from [1] 2 1.3 Energy (pico-Joules) dissipated per logical operation as a function of cal endar year. The 1 kT level is indicated by a dashed line. Adapted from [1] 3 1.4 A 2-D projection view of the Fes unit cell onto the y — z plane. Distances shown are in Angstroms. Legend: Red, iron; Purple, bromine; Light Blue, oxygen; Green, nitrogen; Yellow, carbon; and Dark Blue Crosses, hydro gen. Note the central magnetic core, surrounded by a shield of organic species 5 1.5 A view of the Fe8 unit cell in the x — z plane 6 1.6 A view of the Fe& unit cell in the x — y plane. Here we are looking right down the "easy axis" of the molecule (see chapter 2) 7 1.7 Projection of the Mni2 unit cell onto the x — y plane. Here we are looking down the easy axis of the crystal. The axes scales are in Angstroms. Legend: Red, manganese; Purple, oxygen; Yellow, carbon; Dark Blue Crosses, hydrogen. Note the inner and outer "rings" of manganese ions. . 10 1.8 Projection of the Mni2 unit cell onto the x — z plane 11 1.9 Projection of the Mnyi unit cell onto the y — z plane 2 ix 1.10 Magnetization steps in the hysteresis curve of Mn12. From [17] 15 1.11 This data shows the log of the relaxation time vs. 1/T in Fe8. At high temperatures thermal activation is observed, while for T <~ 360 mK relaxation becomes temperature independent. Figure obtained from [14]. 15 1.12 Relaxation of the magnetization measured at H — 0 after first saturating in —* a field of % = 3.5 T z. As indicated in figure 1.11, the curves superimpose for T < 360 mK. Shown in the inset are relaxation characteristics in the quantum regime for some H ^ 0, applied along the easy (z) axis. Figure from [50] 20 1.13 Short time relaxation of a single crystal of Fe8, measured at 150mK. Here —* several different DC bias fields H were applied along the easy axis of the crystal. Note that the data is plotted against square root t. The inset shows the slope of each of these lines as functions of the DC bias field. Figure from [55] 21 1.14 Here we include some data from a different kind of molecular magnet, the Mnyi system. Here we again see the clear square-root relaxation character istic. However in this case the relaxation rates are temperature dependent. Figure from [57] 22 1.15 Here is data from an experiment on an Fe8 sample that was annealed in zero field, giving it zero initial magnetization. The sample was then exposed to longitudinal DC fields of various magnitudes. We see here relaxation away from M = 0, in the direction of the applied field, with the same square root temporal dependence as in the initially polarized case. From [49] 23 x 1.16 Field dependence of short time square root relaxation rates Tsqrt(Hz). The initial distribution is labelled with Min = —0.998 Ms whereas the others are distributions obtained by thermal annealing. The latter are distorted at higher fields by nearest neighbour lattice effects. Figure from [49]. . . 24 1.17 Quantum hole-digging. For each point, the sample was first saturated in a field of -1.4 T at a temperature of T ~ 2 K and then cooled to 40 mK. The sample was then allowed to relax for times t0. After this time had elapsed, a DC field Hz was applied, and rsgrt was measured. Note the rapid decrease in relaxation rate near Hz — 0. Figure from [49] 24 1.18 Quantum hole digging, as in figure 1.17, but now for a sample that has been annealed to Min = —0.2 Ms. The resulting evolution shows a very narrow hole (see inset). Near zero bias the hole develops very rapidly although the rest of the distribution hardly changes at all. Figure from [49]. 25 1.19 Here is plotted the difference between the relaxation rates at t = 0 (Tinit) and at t0 — 16s (T^), for several different amounts of annealing. Note that for \Min\ < 0.5 the hole width becomes independent of |M;„|, with an intrinsic width of ~ 0.8 mT. Figure from [49] 25 1.20 The quantity A here is related to the relaxation rate of the crystal's mag netization via (1.3). Here it is shown as a function of the magnitude of the transverse DC field \H\ = + H% for several orientations of this field f = tan-1 (Hy/Hx). In this case the longitudinal DC field was taken to be zero (Hz = 0). Figure from [51] 26 xi 1.21 The quantity A shown for ip = 0, as a function of \H\. Shown here are results for three different values of Hz. The lowest curve was obtained for Hz = 0; the middle curve for Hz = 0.22T, and the upper curve for Hz = 0.44T. In terms of the energy level structure of the Fe8 molecule's spin Hamiltonian presented in chapter 1, these applied fields correspond to resonance situations between | — S | + S >, | — £><->• 1 + 5 — 1 > and | — S >«-»• | + 5 — 2> respectively. Notice that a parity effect is observed. Figure from [51] 7 2.1 Exchange pathways in Fe8 in the isotropic model of Delfs et.al. [18]. Fits to susceptibility data give Ji2 ~ 35K, Ji3 ~ 180K, J15 ~ 22K and J35 ~ 52K, with all couplings antiferromagnetic 41 2.2 All of the unit cells (after Ashcroft and Mermin [138]). (i) Cubic, (ii) Tetragonal, (iii) Orthorhombic, (iv) Monoclinic, (v) Triclinic, (vi) Hexag onal and (vii) Trigonal 54 2.3 Variation of As-s with a^/D for four different |5| values (clockwise from top left, \S\ =2, 6, 10, and 14); tetragonal symmetry. On the x axis is plotted a±S2/D and on the y axis log10 As-s- Here we have taken the external field to be zero 55 2.4 Variation of As _s with Hx/D for aAS2/D = 0.25 for four different |5| val ues (clockwise from top left, |5| =2, 5, 10, and 15); tetragonal symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 A^-s 55 2.5 Variation of As-s with a2/D for four different \S\ values (clockwise from top left, \S\ =2, 5, 10, and 15); orthorhombic symmetry. On the x axis is plotted a2/D and on the y axis log10As_s. Here we have taken the external field to be zero 7 xii 2.6 Variation of As-s with Hx/D for a2/D = 0.25 for four different |5| values (clockwise from top left, |5| =2, 5, 10, and 15); orthorhombic symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 As,-s 57 2.7 Variation of AS-s with ae/D for four different \S\ values (clockwise from top left, |5| =2, 6, 10, and 14); hexagonal symmetry. On the x axis is plotted aeS4/D and on the y axis log10 As-s- Here we have taken the external field to be zero 58 2.8 Variation of As,-s with Hx/D for a6S4/D = 0.25 for four different \S\ values (clockwise from top left, \S\ —2, 5, 10, and 15); hexagonal symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 As-s 58 2.9 Comparison of perturbation theory, WKB results and instanton results to the exact solution for the tunneling splitting between the two lowest levels of the Hamiltonian of orthorhombic symmetry with S=10. Plotted on the horizontal axis is a2/D, and on the vertical axis log10 As,-s- Legend: Black, exact solution; Green, instanton solution; Red, perturbation theory and Blue, WKB 61 2.10 Comparison of perturbation theory and WKB results to the exact solution for the tunneling splitting between the two lowest levels of the Hamilto nian of tetragonal symmetry with S=10. Plotted on the horizontal axis is otiSP/D, and on the vertical axis log10As_s. Legend: Yellow, exact solution; Red, perturbation theory and Green, WKB 61 2.11 Z-projection of spin versus energy from HGS for the Fe8 system. The region of validity of the mapping to a two-state system is the region where excited states are forbidden (this region in shaded grey in the above). . . 64 xiii 2.12 Typical evolution of the projection of the excess spin S(t) along the easy-axis. We see two regimes; one where S evolves without tunneling (diagonal in f), and one where S tunnels from | + S | — S > (off-diagonal in f). Note the separation of scales; the time between tunneling events is much greater than the tunneling time 68 3.1 (Jl for all nuclei in Fes- Labeling is as indicated in the text. The dots represent values for 2H (labels 1..120), slBr (labels 121..128), and l5N (labels 129..146) 86 3.2 1H, emphasizing low end of the spectrum 87 3.3 1H, high end of the spectrum 83.4 2H, low end of spectrum 8 3.5 2H, entire spectrum3.6 79.Br, entire spectrum 89 3.7 81Br, entire spectrum3.8 14iV, entire spectrum 90 3.9 15N, entire spectrum3.10 57Fe, entire spectrum 1 3.11 13C, entire spectrum 93.12 17O, low end of spectrum 2 3.13 170, entire spectrum3.14 The orthogonality blocking parameter K over a large range of external fields applied in the x direction for the Fe™ax material 93 3.15 The parameter K for small values of external field applied in the x direction for the FeTx material. 94 xiv 3.16 Intrinsic linewidth W due to particular isotopes as a function of Hx for 100% concentrations of these isotopes. Note that u\ and therefore W drops slowly with field. This effect comes about because as the external field is raised, the two minima of the central spin complex are forced closer together (no longer are they antiparallel). The curve shown as "total" is the total result for a material containing 100% of the isotopes shown. . . 97 3.17 Intrinsic linewidth W as a function of Hx for Fe8*, Fe8D and 57Fe8. ... 98 3.18 Binned topological decoherence parameters \AkND\ for all nuclei, assuming 100% concentrations of (clockwise from bottom left) 1H, 79Br, UN, 170, 13C and 57Fe, using the point dipole approximation. The bin width here is 0.0001; plotted on the x axis is and on the y axis "number of nuclei". Note that the contribution to from 57Fe is almost entirely from the contact interaction 99 3.19 Hartree-Fock results for the free Fe3+ wavefunction 101 3.20 Comparison of point dipole and Hartree-Fock methods; zero field a>J| values in Fe™ax. The Hartree-Fock results are shown as dots 102 3.21 1H, Hartree-Fock, emphasizing low end of the spectrum 103 3.22 lH, Hartree-Fock, high end of the spectrum 103.23 2H, Hartree-Fock, low end of spectrum 104 3.24 2H, Hartree-Fock, entire spectrum3.25 79.Br, Hartree-Fock, entire spectrum 105 3.26 81Br, Hartree-Fock, entire spectrum3.27 UN, Hartree-Fock, entire spectrum 106 3.28 15N, Hartree-Fock, entire spectrum3.29 57Fe, Hartree-Fock, entire spectrum 107 3.30 13C, Hartree-Fock, entire spectrumxv 3.31 170, Hartree-Fock, low end of spectrum 108 3.32 170, Hartree-Fock, entire spectrum3.33 The orthogonality blocking parameter K as a function of Hx in the Hartree-Fock wavefunction picture 109 3.34 The orthogonality blocking parameter K as a function of Hx in the Hartree-Fock wavefunction picture, focusing on small fields 109 3.35 Intrinsic linewidth W as a function of Hx for Fe8», Fe$D and 57Fe8 in the Hartree-Fock picture Ill 3.36 Binned topological decoherence parameters |A^)£)| for all nuclei, assuming 100% concentrations of (clockwise from bottom left) 1H, 79Br, 14N, 170, 13C and 57Fe, using the Hartree Fock approximation. The bin width here is 0.0001; plotted on the x axis is lA^I and on the y axis "number of nuclei". Note that the contribution to |A^)£)| from 57Fe is almost entirely from the contact interaction Ill 4.1 Energy levels of the Landau-Zener Hamiltonian. Shown are both the eigenstates of az, which are linear in time, and the eigenstates of H(t), E±(t) = ±(A2 + vH2)1'2 123 4.2 Transition probability (4.10) as a function of t (in units of A) . Here we have taken A2/v — 0.63, 1 and 5 for the solid, dotted and dashed lines respectively 126 4.3 Transition probabilities for the pulse potential 135 4.4 Transition probabilities for the sinusoidal potential 136 4.5 Transition probability P^ for the pulse/ramp scenario. Plotted is ATT/U on the x axis and An/u on the y axis 140 xvi 4.6 Transition probability for the ramp scenario. Plotted on the x axis is A/u> and on the y axis ^ 141 5.1 Transition amplitude as a train of blips 156 5.2 Transition probability normalized to the standard Landau-Zener transi tion probability P^/Pff plotted against 2£/a; for 2A/u = 0.1. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation 166 5.3 Transition probability normalized to the standard Landau-Zener transi tion probability P^/Pff plotted against 2£/ui for 2A/UJ = 10. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation 167 5.4 Transition probability normalized to the standard Landau-Zener transi tion probability P^/Pff plotted against 2£/o> for 2A/u = 500. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation 168 5.5 Depiction of the precession of the kth nuclear spin during a "blip". The central spin is shown in black, with a schematic nuclear spin underneath. This nuclear spin feels a field 7^ for times t <ti, which we choose to be the axis of quantization. After the central spin flips at tx the nuclear spin feels a different field 7^ which contains in general transverse components. This causes a precession of the nuclear spin. After the central spin flips back, the nuclear spin will be in a state that has less than full overlap with its original state 178 5.6 pXil (dashed) and W(0/i) (solid) for p =x H 197 5.7 plfl (dashed) and W((LJ (solid) for p =79 Br 198 5.8 plfl (dashed) and W(0M) (solid) for p =14 N 195.9 plfl (dashed) and W(0M) (solid) for p =57 Fe 199 5.10 pllt (dashed) and W(0M) (solid) for p =17 0 19xvii 5.11 pXil (dashed) and W(0M) (solid) for (i =n C 200 5.12 Full width W for Fes* (dotted), Fe8D (dashed) and 57Fe8 (solid) 200 5.13 A3 (left) and A4 (right) for the three varieties Fe8* (dotted), Fe8D (dashed) and 57Fe8 (solid) 202 5.14 Presented here are (1) the bare result A0| cos<I>| (the lower curve) and (2) the large A result with nuclear spins A0| cos$| (the middle curve) plotted in units of Kelvin. Note the logarithmic vertical scale. The horizontal axis is Hx in Tesla-here we have ip = 0 (Hy = 0) 204 6.1 Top figure: Logw A2 vs. H°, for 6 = sin"1 (H%/H$) = 0. Curves are, from bottom to top, WD — 0,10, 20,30 and 50 mT. Bottom figure: Same, but with 6 = 1°. Note that these results are obtained from our relaxation rate which was derived assuming the instanton approximation for the tunneling amplitude 215 6.2 From [199]. Here the tunneling matrix element and its dependence on ex ternal field were extracted by exact diagonalization, with the nodal physics determined as discussed in the text. Note that while the general features of the exact result match those obtained by the instanton calculation they differ in detail 216 B.l Here we see a system with seven environmental spin states initially pre pared in one of them evolving via (B.3) 230 xviii Acknowledgements There are many people who have provided support during the completion of this work. Firstly I would like to thank Philip Stamp, both for introducing me to the complex and important world of the nuclear spin bath and for allowing me significant leeway in my approach to the difficult problems treated in this document. The education I received from him in the area of quantitative treatments of decoherence will be absolutely invaluable to the success of my post-doctoral pursuits. I hope that we will be working together for many years to come. I would also like to thank the other great teachers I have had the opportunity to learn from in my time at UBC, and in particular Ian Affleck, who has influenced my understanding of several aspects of condensed matter physics, and Haig Farris, who has become a good friend, mentor and business partner over the past year. As well I would like to thank colleagues and collaborators Mehrdad Sharifzadeh-Amin, Tim Duty, Sebastian Jaimungal, Alexandre Zagoskin, Jonathan Oppenheim, Suresh Pil-lai, Michel Olivier, Martin Dube, Stephanie Curnoe, Igor Tupitsyn, Nikolai Prokofiev and Jeff Sonier. Also I would like to thank Wolfgang Wernsdorfer for his amazing Fe8 data. The completion of this document would not have been possible without the support of my parents Dr. George and (soon to be Dr.) Sarah Rose, who I would like to dedicate this thesis to-I finally did it! I would like to thank Craig Thomas for trying to keep up with me on the beach volleyball circuit, RANDy for hours of revolting amusement, and Shawn Kenney, Don and Kamara Lucas and Camille Parent for putting up with me. xix Finally I would like to thank my wife Valerie, for believing that someday the re-write I was doing would be the last one. xx Chapter 1 Introduction and Overview Component sizes in commercially available semiconductor structures have been halving rather steadily every eighteen months or so since the early 1950s. This decrease in size, know as Moore's Law (named for Gordon Moore, one of the founders of Intel), is tracked by concommitant halvings in price and energy consumption and doublings of processor speed (see figures 1.1, 1.2 and 1.3) [1, 2, 3]. This continued shrinkage is producing much excitement and consternation in the high-tech world, for a very simple reason-it is clearly not sustainable. Naively one could say that this is because device sizes are limited to be larger than atomic length scales, Figure 1.1: The number of atoms needed to represent one bit of information as a function of calendar year. Extrapolation of the trend suggests that the one atom per bit level is reached in about the year 2020. Adapted from [1]. 1 Chapter 1. Introduction and Overview 2 which are on the order of Angstroms-at current shrinkage rates this barrier will be met in approximately 20 years. However it is not yet clear that the limitation on computing speed cannot be overcome in the near-term (10-30 years) with more efficient computer architecture-for example, stacking transistors horizontally [4], or even more exotic solu tions such as the recent IBM SMASH proposal [5]. Figure 1.2: Clock speed (Hz) vs. calendar year. Adapted from [1]. Physicists have long been thinking about what will happen when component sizes become mesoscopic; that is, much larger than atomic length scales, but small enough so that at least in part they must be treated quantum mechanically [6, 7, 8, 9]. This line of thought has produced many extremely startling predictions. It is now well known that standard models of computation, based on the universal computing model or Turing model [10], contain an implicit assumption. This assumption is that the physical system which encodes and manipulates information evolves according to the classical laws of physics. This assumption can break down when components become "small enough". For example, quantum effects in mesoscopic normal metal rings [11], superconducting structures [12, 13] and in molecular magnets [14, 15, 16, 17] have been observed. Chapter 1. Introduction and Overview 3 If one rewrites computer science with quantum mechanics implicitly included from the outset it turns out that the range of tasks that computing machines can perform is increased. The most famous example of this is the solution of the factoring problem [1, 19] using a "quantum computer" (which is a theoretical machine which has the capa bility of storing and manipulating coherent two level systems (quantum bits, or qubits)) in polynomial time, as opposed to superpolynomial time with classical computers. In addition to this, it is quite obvious that a quantum computer could in an analog fash ion solve many important quantum mechanical problems, some of which are exceedingly important (such as pharmaceutical design), which are completely unsolvable using even the fastest imaginable classical supercomputers. Year Figure 1.3: Energy (pico-Joules) dissipated per logical operation as a function of calendar year. The 1 kT level is indicated by a dashed line. Adapted from [l]. These theoretical musings are now coming face to face with some very real physical problems, as engineers undertake to build quantum devices. If one wants to construct a quantum computer, there are several aspects of mesoscale condensed matter physics that must be understood wholly and completely. The most important of these is the Chapter 1. Introduction and Overview 4 process known as decoherence, which involves the transferral of phase information from a quantum bit into an environment (such as nuclear spins [20] or phonons [21, 22]). Decoherence is anathema to quantum computation (and interesting also from a purely theoretical perspective), and yet how it works in practice, quantitatively, is still not satisfactorily understood. There exists a class of mesoscopic systems where an attempt can be made at a quan titative theory of decoherence. These are the so-called "molecular magnets". One of these materials, which we will refer to throughout as "iron-eight" (Fes), is particularly well suited to a quantitative study of decoherence due to localized environmental modes, and in particular nuclear spins. The development of this quantitative theory as a tool to be used in future investigations of mesoscale systems in the context of developing solid state quantum bits provides the main motivation for the work presented in this thesis. 1.1 An Introduction to Fe8 Fes was first synthesized in 1984 by Wieghardt et.al. [23]. This material, with the rather imposing chemical formula {[(tocn)6Fe8(/i3 - 0)2(/*2 - OH)12]Br7 • H20}®[Br • 8H20f (1.1) where tacn = 1,4,7 triazacyclonane, was the first oligomer with greater than three Fe3+ ions per unit cell ever characterized. X-ray crystallography studies performed on this material indicated that six of the Fe3+ ions were bonded to amine ligands FeNsOs and the remaining two were surrounded by a distorted octahedral array of 6 O atoms. The iron ions are coupled via 12 \x2— hydroxo bridges and two /x3— oxo bridges. Three views of the unit cell are presented here, in figures 1.4, 1.5 and 1.6. The positions of the ions shown here were obtained from the Cambridge Crystallographic Database [24], which contains the original X-ray data obtained by Weighardt et.al. [23]. Chapter 1. Introduction and Overview 5 10 8-6 4-2 0 -2--6-+ + + + + + + + + + + + + + w + + + + + + + 1 1 1 i 1 1 1 1 i 1 1 1 ' i 1 • 1 1 i ' • • 1 i 1 ' 1 1 i > 1 • 1 i ' > 0 2 4 6 8 10 12 Figure 1.4: A 2-D projection view of the Fe$ unit cell onto the y — z plane. Distances shown are in Angstroms. Legend: Red, iron; Purple, bromine; Light Blue, oxygen; Green, nitrogen; Yellow, carbon; and Dark Blue Crosses, hydrogen. Note the central magnetic core, surrounded by a shield of organic species. Chapter 1. Introduction and Overview 6 Figure 1.5: A view of the Fe8 unit cell in the x — z plane. Chapter 1. Introduction and Overview 7 10- + s-G- •1-4-2-+ • # + •» • +t + + w+ * m * 0- • + + -2- • + # ** * + + + + , + <* *„ *+ •• -6-• + + + + + -G -4 -2 0 2 4 6 S Figure 1.6: A view of the Fe8 unit cell in the x — y plane. Here we are looking right down the "easy axis" of the molecule (see chapter 2). Chapter 1. Introduction and Overview 8 The amine groups are cyclic and hydrophobic. The Br atoms are bonded electro statically to NHe and OHe. The space group of the material is PI (triclinic), with lattice parameters a = 10.522, b = 14.05 and c = 15.00 Angstroms with unit cell angles a - 89.90, (3 = 109.65 and 7 = 109.27 [23]. The molecular weight of the substance per unit cell is 2250. The lattice structure is of the AB type, with the cation and anion in (1.1) occupying the A and B sites respectively (this is simply a distorted NaCl structure). 1.1.1 Giant Spins and Quantum Environments Experimental interest in the low temperature magnetic characterization of Fe8 surpris ingly did not arise until much later [18]. However in the experimental lull between 1984 and 1993 much theoretical work was being done that would lay the groundwork for un derstanding the low-energy magnetic nature of this substance. There are two basic themes that needed to be developed to understand the experi mental results that would later appear. The first of these is the understanding of the dynamics of "giant spins", that is systems that have large spin quantum number. One of the most obvious systems that can be thought of as a giant spin is a single isolated ferro magnetic grain, well below its Curie temperature. All the electronic spins lock together, and one could think of this object as a single degree of freedom, albeit with spin quantum number as large as S = 108. The interesting thing about these systems, and one of the main motivations for their early study, is that here we have an adjustable parameter (5) which as it is increased should cause the system to go from being quantum mechanical (say for S — 1/2) to being classical (say S = 108) in a way that we can study with some intimacy. Two papers that would prove to be important in this respect were those of Van Hemmen and Suto [25] and Enz and Shilling [26], both in 1986. These discussed the dynamics of giant spins in the WKB approximation. In particular, the problem of how Chapter 1. Introduction and Overview 9 large spin objects tunnel between energetic minima was treated. The second basic theme that needed to be developed was an understanding of the effects of "environments" on the dynamics of the degrees of freedom to which they couple. It was shown by Feynman and Vernon [21] that if there exist environmental couplings that are weak, in the sense that their effect can be treated in second-order perturbation theory, one can model their effects by coupling the interesting degree of freedom to an oscillator bath. In a solid at low energies, it is usually the case that delocalized modes, such as phonons and photons, can be mapped to oscillator baths. This is because low energy modes have long wavelengths and therefore their overlap with the localized interesting degree of freedom is small. Various aspects of the dynamics of a central degree of freedom coupled to oscillator baths have been investigated [22]. Perhaps the most famous of these is the treatment by Leggett et.al. of the so-called spin-boson problem [27]. In the spin-boson problem, the central degree of freedom is a two-state system (a spin). This is then coupled to an oscillator bath and the dynamics of the central spin are then extracted. It is possible to obtain analytic results for certain choices of parameters in the spin-boson Hamiltonian, but in general dynamical solutions are difficult to obtain. 1.1.2 A Related System is Characterized In 1991 a substance that would turn out to be related to Fe8 underwent low temperature magnetic characterization [28]. This substance is usually called Mni2-acetate, or simply Mni2, and has chemical formula [Mn12{CH3COO)16(H20)4012] • 2CH3COOH • 4H20 (1.2) Crystals of this substance were known [29, 30] to have tetragonal symmetry of spatial group 14 with unit cell parameters a = 17.3 Angstroms and b = 12.39 Angstroms. The Chapter 1. Introduction and Overview 10 total molecular weight per unit cell is 2060. X-ray crystallography was performed [30] on this substance, and the atomic positions stored in the Cambridge Crystallographic Database. This data is presented in figures 1.7, 1.8 and 1.9. We see that there are two "rings" of manganese atoms. The inner ring consists of four Mn4+ ions with spin S = 3/2, while the outer ring consists of eight Mn3+ ions with spin 5 = 2. It was proposed that all these ions couple to each other primarily via superexchange through mediating oxo-bridges [30]. Figure 1.7: Projection of the Mn-u unit cell onto the x — y plane. Here we are looking down the easy axis of the crystal. The axes scales are in Angstroms. Legend: Red, manganese; Purple, oxygen; Yellow, carbon; Dark Blue Crosses, hydrogen. Note the inner and outer "rings" of manganese ions. Chapter 1. Introduction and Overview Figure 1.8: Projection of the Mn12 unit cell onto the x — z plane. Chapter 1. Introduction and Overview Figure 1.9: Projection of the Mn12 unit cell onto the y — z plane. Chapter 1. Introduction and Overview 13 The 1991 paper of Caneschi et.al. described results of AC susceptibility, high field magnetization and EPR measurements on crystals of Mn12 which indicated that each Mn12 molecule had an S = 10 groundstate. A mechanism was described whereby the twelve manganese ions in each unit cell lock together via superexchange at low temper atures into a giant spin. In this material an easy axis was observed, and was explained as being due to crystal anisotropy and/or spin-orbit coupling effects. These results pro duced much excitement, as here we have what seems to be (in zero external magnetic field and in a low-energy limit) a two state system (15 = +10 > and \S — —10 >, corre-spondinging in a semi-classical picture to the giant spin pointing parallel/antiparallel to the easy axis respectively) whose dynamics should demonstrate quantum effects of some kind (as S is in the fuzzy mesoscale region). An analysis of what was known of Mnu and some other molecular magnets was performed in 1993 by Sessoli et.al. [31] and provides an excellent review. 1.1.3 Magnetic Characterization of Fes In 1993 a comprehensive characterization of the magnetic properties of Fe% at low tem peratures was performed by Delfs et.al. [18]. In this study, AC and DC susceptibility, magnetization as a function of external field and EPR studies were performed which indicated that Fe% had, like Mn12, a spin 10 groundstate. This paper described its results in terms of a unit cell containing eight S = 5/2 Fe3+ ions which couple to each other via exchange and superexchange via oxygen and hydroxo bridges. At temperatures less than T ~ 20 K these lock together into a spin complex with S = 10. Because of spin-orbit couplings and crystal field anisotropies the spectrum of the spin 10 rotator is split into 10 doublets and one singlet, with ms = ±10 being nearly degenerate lowest energy states corresponding in a semi-classical picture to the central spin object pointing "up" or "down" along the easy axis selected by the Chapter 1. Introduction and Overview 14 anisotropy in the spin Hamiltonian (a rough first approximation to this spin Hamiltonian is given by H = —DS2, where D is a measure of the strength of the various anisotropies in the crystal [32, 33, 34]-see chapter 2 for a comprehensive treatment of the low energy effective Hamiltonian for this system). The parameter D was reported to be on the order of D~0.3K [18]. 1.1.4 Results in the Quantum Regime The low temperature magnetic characterizations of Fes and Mnyi led to a flood of im portant results, both from theorists and from the experimental community. Theorists were presented with a truly mesoscopic problem with a growing number of experimental results, and experimentalists had access to systems where they could directly measure macroscopic quantum effects. The study of Mriu produced much work on the macro scopic quantum tunneling of the central spin of each molecule [35, 36, 37, 38]. Arguably the most important early experimental result was that of Thomas et.al. [17] in 1996, which presented clear-cut evidence for incoherent macroscopic tunneling of the magne tization in Mni2 (see figure 1.10). Even more astonishing were results obtained in 1997 on the Fes system which clearly demonstrates the existence of a regime where magnetic relaxation rates become temperature independent-the so-called quantum regime (see fig ure 1.11) [14]. The authors attribute relaxation below T ~ 360 mK to be due to purely quantum mechanical tunneling between groundstates of the Fes molecules in the crystals measured. Results of magnetic relaxation measurements on these systems for the most part was amenable to analysis within existing theoretical frameworks [40-48]. In particular a framework had been constructed by Prokofiev and Stamp [41, 45, 46, 47] to explain the physics of relaxation in molecular magnets. In the models of Prokofiev and Stamp, each molecular magnet is treated as a giant spin which couples to various environments, Chapter 1. Introduction and Overview 15 Figure 1.11: This data shows the log of the relaxation time vs. 1/T in Fe8. At high temperatures thermal activation is observed, while for T <~ 360 rnK relaxation becomes temperature independent. Figure obtained from [14]. Chapter 1. Introduction and Overview 16 the most important of which it is argued are localized modes, such as nuclear spins and magnetic impurities. Note that localized modes such as these can not in general be mapped to oscillator baths as the coupling strengths are not in general small (for example, contact hyperfine couplings in rare earths can be as large as 1 K [48]). They show that the relaxation characteristics of giant spins in condensed matter systems should be strongly influenced by these localized modes, and hardly influenced at all by oscillator baths (such as phonons). In the quantum regime demonstrated to exist in Fe8, the temperature independence is evidence that the phonon bath is not playing a significant role in relaxation. But here was a curious thing. The fact that the Fe8 crystals were relaxing at all in the quantum regime was quite strange, for the following reason. The bare tunneling amplitude between the groundstates of the Fe8 molecules was estimated (by exactly diagonalizing phenomenological spin Hamiltonians whose parameters were extracted from various experiments) to be A ~ 10-8 K [45]; and yet the scale of the dipolar interaction between different molecules can quite easily be estimated to be of the order of ~ 0.5 K. This means that in a crystal of molecules, only an extremely tiny fraction of molecules could ever be in resonance, and therefore their dynamics should be frozen. A resolution of this difficulty was proposed in 1997 by Prokofiev and Stamp, and the proposed mechanism involved the nuclear spins present in the Fe8 crystal [15]. At temperatures lower than ~ 360 mK, there exists only one source of dynamics in the system, and that is the nuclear spin bath. These typically do not freeze out until \iK temperatures and so at mK they are effectively in a high-temperature limit. The rate at which the nuclei in Fe8 perform so-called T2 flips [64, 65, 66], where the overall magnetization of a pair does not change, was estimated by these authors to be ~ 1 kHz — 1 MHz. The effect of these flips is to cause a time-varying magnetic field to be generated at each central spin, which was postulated to immensely increase the "resonance window" Chapter 1. Introduction and Overview 17 and therefore allow the crystal to relax. The mechanism works like this. At the beginning of relaxation, all molecules sit in a combination of the dipolar fields caused by all other molecules in the crystal and the time-varying magnetic fields caused by the nuclei present. There will be a small number of molecules that can be brought to resonance by the time-varying magnetic fields generated by the T2 flipping nuclei. When one of these molecules tunnels, it rearranges the dipolar field configuration in the sample. This can bring other molecules to resonance, and so the crystal relaxes. This theory contained certain testable predictions about the relaxation characteristics one should see if the hypotheses were correct. One of these was that the relaxation of Fe& in its quantum regime should be square root in time. Although relaxation data existed up to this point that was taken as a function of time, the presence of the square root temporal dependence was not realized until after Stamp and Prokofiev looked for confirmation of their theory. At this point it was real ized that earlier data did in fact follow a square root temporal relaxation. Subsequent measurements vindicated the idea, not only in Fe% but also in Mni2. The observation of the square root temporal dependence in Mni2 was not understood until quite recently [?]. The prediction of a square root relaxation rate depends quite clearly on the system that is relaxating being in a quantum regime-that is, the presence of thermally occupied higher levels destroys the square root. This seemed to contradict the fact that the relaxation characteristics in Mrii2 were clearly temperature dependent down to at least 60 mK. The resolution of this difficulty is that Mnu crystals contain "rogue" species of Mn-u that relax at different rates. Because of this it is possible that in certain temperature regimes one species of Mni2 is in its quantum regime and relaxes as a square root whereas the rest don't relax at all (see [63] for a discussion of this point). In the next section we shall take a close look at relaxation experiments, as they Chapter 1. Introduction and Overview 18 provide tantalizing glimpses of unresolved puzzles, some of which will be resolved in later chapters. 1.2 An Introduction to Relaxation Experiments The general strategy for performing a relaxation experiment on a molecular magnetic crystal is quite straightforward [49]. One takes a sample of the material and cools it down to some temperature T in some static field H over some time in- The initial magnetization of the sample is measured using a SQUID magnetometer array. The static —* field % is then abruptly changed to some new field, which is in general time dependent H(t), and the magnetization of the sample (M) is then measured as a function of time. —* —* This gives us a quantity M(T, t0, %, H(t), t) which tells us about how the crystal, initially —* —* prepared using \H, to}, relaxes in the presence of the field H(t) at the temperature T. Several such experiments have been performed of late [49-63]. We shall focus our at tention on those using the Fes system, as this material has generated a wealth of excellent experimental data deep into the so-called "quantum regime"-the region where relaxation characteristics become completely independent of temperature (this happens in Fe$ for temperatures lower than Tc ~ 360 mK; see figure 1.11) [14]. The other heavily studied molecular magnet (M7112) is similar in many ways in its relaxation characteristics. How ever, there is an important difference-the relaxation characteristics of Mni2 show clear temperature dependence down to the lowest temperatures investigated (T ~ 60 mK) [17]. In this thesis we have chosen to focus our efforts on understanding the physics of the quantum regime. Now as we have described the experiment above, there are four basic parameters that we can play with in order to customize a particular relaxation experiment-the tempera-—* ture at which it is performed (T), the static field in which it is cooled H, the time over Chapter 1. Introduction and Overview 19 which it is cooled t0 and the time-dependent field applied during relaxation H(t). Our strategy in this chapter will be to simply present the experimental situation in each par ticular case that we will review (that is, we will state what the parameters {T, to, H, H(t)} are) and then give the results. No theoretical justification or explanation will be presented here-we shall make this the task of the remainder of the thesis. 1.2.1 DC Field Relaxation in Polarized Fe8 The first class of experiment that we will review was historically the first to be performed, perhaps because it is the simplest [18, 50]. In these experiments, the Fe8 samples are cooled slowly from room temperature to the quantum regime in a large static bias field H, applied along the easy axis of the crystal (see chapter 2 for information about the crystal symmetry in Fe8). This has the effect of preparing the crystal in an initially polarized state. Once the initial magnetization has been measured, the large static bias field is removed, and a smaller DC bias field H in applied to the crystal along its easy axis. The magnetization as a function of time is then measured. Shown in figures 1.12 and 1.13 are results for M(t) for two different crystals in different experiments of this type. Note the unusual square-root relaxation characteristic. Also included here is a similar experiment performed on the Mn\i molecular magnet (figure 1.14) [58]. The relaxation for this material is also square-root in time, albeit with a strong temperature dependence. 1.2.2 DC Relaxation of Annealed Crystals A second class of experiments involve a different method of preparation of the sample under study. Instead of slowly lowering the temperature down into the quantum regime we may instead rapidly quench the temperature over a time to that is so small that the Chapter 1. Introduction and Overview 20 Figure 1.12: Relaxation of the magnetization measured at H = 0 after first saturating in a field of % — 3.5 T z. As indicated in figure 1.11, the curves superimpose for T < 360 mK. Shown in the inset are relaxation characteristics in the quantum regime for some H ^ 0, applied along the easy (z) axis. Figure from [50]. thermal distribution of molecular magnetization is frozen into the initial state of the crystal [49]. That is, the elevated temperature of the molecular ensemble before the quench enables a thermal distribution of the magnetic moments of the molecules. The state of the crystal after the quench retains this distribution initially, before it begins to relax. This procedure is called annealing. The initial magnetization of the sample can be arbitrarily chosen in this scenario, depending only on the pre-quench temperature and static field Ti. Note that the first type of experiment considered is a limiting case of this one. After the sample is quenched and the initial magnetization is measured, a longitudinal DC field is applied either against or in the direction of the magnetization, and the function M(t) is measured. Presented in figure 1.15 are results from this type of experiment. Chapter 1. Introduction and Overview 21 i i o a i « a sqrt(;j [sec"zj Figure 1.13: Short time relaxation of a single crystal of Fe8, measured at 150mA'. Here several different DC bias fields H were applied along the easy axis of the crystal. Note that the data is plotted against square root t. The inset shows the slope of each of these lines as functions of the DC bias field. Figure from [55]. 1.2.3 Hole Digging and the Time-Dependent Internal Longitudinal Bias Dis tribution —* It has been proposed that the DC relaxation of the magnetization M(H.t) can be related to the time-dependent distribution of internal bias fields in the sample [15, 51, 49]. In chapter 6 we shall review the current DC theory and supplant this with our AC results. The basic idea is that the relaxation characteristic M(H, t) is seen experimentally to be square root in time for short times. The "relaxation rate" Tsqrt(H), defined via M(H, t) ~ 1 — \JTsqrt(H)t can then be measured. The assumption then is that the internal distribution of longitudinal biases P(£, t) (where f is the longitudinal bias field) is proportional to the relaxation rate P(£,t) ~ Tsqrt(H) where the field H is applied at time t. Shown in figure 1.16 is the relaxation rate Tsqrt(Hz) of a sample that was prepared in Chapter 1. Introduction and Overview 22 11)0 150 20O 250 J0O Figure 1.14: Here we include some data from a different kind of molecular magnet, the Mn\2 system. Here we again see the clear square-root relaxation characteristic. However in this case the relaxation rates are temperature dependent. Figure from [57]. an initially polarized state. In figure 1.17 the evolution of these rates over time is shown. Note that as the sample evolves, there appears a "hole" in the relaxation rates near zero internal bias. Shown in figure 1.18 is the evolving relaxation rate spectrum of a sample that was initially annealed such that M(0) ~ 0.2. Again we see evidence of a hole being dug near zero bias in this distribution. This hole in the annealed samples has an interesting feature. If samples are annealed to |M(0) < 0.51 there is found an intrinsic hole width of approximately 0.8 mT (see figure 1.19). Stamp and Prokofiev suggested that this intrinsic linewidth was due to nuclear spins [20]. We present a framework in chapter 3 for calculating the linewidth due to the nuclei and demonstrate that our result agrees quantitatively with these experimental results, supporting the contention that nuclear broadening is responsible for this intrinsic hole. Chapter 1. Introduction and Overview 23 0.0005h 0.0015 M/M$ 0.001 h 0.002 0 _L_ 3 0 .^-2.2imT 0.56mT 1,12mT 2.80mT 3.36mT '3.92ml 1.68mT OmT 0 1 0 20 sqrt[t(s)] Figure 1.15: Here is data from an experiment on an Fe8 sample that was annealed in zero field, giving it zero initial magnetization. The sample was then exposed to longitudinal DC fields of various magnitudes. We see here relaxation away from M = 0, in the direction of the applied field, with the same square root temporal dependence as in the initially polarized case. From [49]. 1.2.4 AC Relaxation of Annealed Crystals The final kind of relaxation experiment that we shall review differs from the previous type in that the field applied during relaxation contains a periodic time-dependent component, applied in the direction of the easy-axis of the crystal [51]. The beauty of this type of experiment is that it is possible to measure extremely small relaxation rates, which presents us with a useful probe of much of the physics of these systems. In addition, there now exists a quantitative theory of how molecular magnets respond to this kind of perturbation, which we shall develop in the later chapters. 1.2.5 Extraction of Tunneling Matrix Elements Shown in figures 1.20 and 1.21 are quantities extracted from AC relaxation measure ments, which are related to the tunneling matrix elements of the single-molecule effective Hamiltonian of Fe8 (we shall show exactly how they are related in chapter 6). What is Chapter 1. Introduction and Overview 24 M0H(T) Figure 1.16: Field dependence of short time square root relaxation rates Tsqrt(Hz). The initial distribution is labelled with Min = —0.998 Ms whereas the others are distributions obtained by thermal annealing. The latter are distorted at higher fields by nearest neighbour lattice effects. Figure from [49]. -0.05 -0.025 0 0.025 0.05 fi0H(T) Figure 1.17: Quantum hole-digging. For each point, the sample was first saturated in a field of -1.4 T at a temperature of T ~ 2 K and then cooled to 40 mK. The sample was then allowed to relax for times t0. After this time had elapsed, a DC field Hz was applied, and rsqrt was measured. Note the rapid decrease in relaxation rate near Hz = 0. Figure from [49]. Chapter 1. Introduction and Overview 25 -0.04 -0.02 0 0.02 0.04 0.06 0.08 H0H(T) Figure 1.18: Quantum hole digging, as in figure 1.17, but now for a sample that has been annealed to Min = — 0.2 Ms. The resulting evolution shows a very narrow hole (see inset). Near zero bias the hole develops very rapidly although the rest of the distribution hardly changes at all. Figure from [49]. H0H(T> Figure 1.19: Here is plotted the difference between the relaxation rates at t = 0 (rinit) and at t0 = 16s (Tdig), for several different amounts of annealing. Note that for |Min| < 0.5 the hole width becomes independent of |M«J, with an intrinsic width of ~ 0.8 mT. Figure from [49]. Chapter 1. Introduction and Overview 20 actually measured here is simply the magnetization as a function of time, M(t) as per usual, in the presence of a longitudinal sawtooth AC field of amplitude A and frequency u>, a longitudinal DC bias field Hzz, and a static transverse DC field H = Hxx + Hyy. The authors find that M(t) ~ exp [—Ft], ie. the relaxation is exponential, with a rate F that is a function of the applied longitudinal AC field, the transverse DC field and the longitudinal DC field. They then define their quantity A in the figures shown via the following relation; Note that as advertised we are not going to try to justify this relation theoretically just yet-there will be much on this later. 11 t i i 1 1 1 1 0 0.2 0.4 0,6 0.8 1 1.2 1,4 rani (T) Figure 1.20: The quantity A here is related to the relaxation rate of the crystal's magne tization via (1.3). Here it is shown as a function of the magnitude of the transverse DC field \H\ = JW* + H2 for several orientations of this field ip = tan'1 (Hy/Hx). In this case the longitudinal DC field was taken to be zero (Hz = 0). Figure from [51]. Chapter 1. Introduction and Overview 27 r"""'"""r 11 i i r——-r 1 E I i f i r r i 1 1 -0.4*0.2 0 0.2 0.4 0.6 0,8 1 1.2 1.4 runs (T) Figure 1.21: The quantity A shown for ip = 0, as a function of \H\. Shown here are results for three different values of Hz. The lowest curve was obtained for Hz = 0; the middle curve for Hz = 0.22T, and the upper curve for Hz = 0.44T. In terms of the energy level structure of the Fe$ molecule's spin Hamiltonian presented in chapter 1, these applied fields correspond to resonance situations between | — 5 >+» | + 5 >, | — 5 >+»• 1 + 5—1 > and | 5 >o | + 5 — 2 > respectively. Notice that a parity effect is observed. Figure from [51]. 1.3 Thesis Overview Our goal will be to work up to a quantitative theory of AC relaxation in Fe8 crystals. In order to do this we shall need to develop several key concepts. We begin in chapter two with an analysis of the problem of deriving an effective Hamiltonian for a single Fe8 molecule. In chapter three we calculate the hyperfine couplings between a central spin object and all the nuclei in the molecule. Using this information we calculate all the decoherence parameters introduced by Prokofiev and Stamp in their theory of the spin bath. We show Chapter 1. Introduction and Overview 28 that there should exist measurable isotope effects in Fe% and give quantitative predictions of the linewidth due to nuclear spins in an Fe& crystal with arbitrary isotopic content. In chapter four we introduce and develop some of the machinery of the Landau-Zener problem. This involves using a time-dependent Hamiltonian to extract transition probabilities between states of the central object of interest. Chapter five is the heart of the thesis, and contains an extension of the Landau-Zener problem in which nuclear spins are included. We use the results of this calculation to find a single-molecule relaxation rate in the presence of an external magnetic field with both AC and DC components. Chapter six then uses this general single-molecule relaxation rate as the input to a master equation so as to model the temporal evolution of a crystal of Fe8 molecules. We extract time-dependent relaxation characteristics from our theory and compare these to experimental results. We conclude our analysis in chapter seven with a summary of results and the current outlook for our theory of AC relaxation. Chapter 2 Effective Hamiltonians In this chapter we derive a low energy effective Hamiltonian for a single isolated Feg molecule. We begin by listing all terms found in the Hamiltonian of a single free Fe3+ ion. We then describe how these will be modified by placing the Fe3+ ions into a crystalline environment, following the treatment of Abragam and Pryce [67] (see also [68, 69]). This leads to a single ion "spin Hamiltonian". We then build up the Fe$ Hamiltonian by introducing exchange/superexchange terms between the Fe3+ ions and terms coming from both the nuclear spin environment [20] and phonon [70] and photon [71] oscillator baths. This "bare" description of the Fe% molecule, containing eight single ion Fe3+ terms, exchange/superexchange couplings between these and the various environments is then investigated. The exchange/superexchange coupling energies are much larger than all other energy scales [72, 73]. This suggests the hypothesis that at low energies these couplings lock the electronic spins together into a "giant spin" [20]. We assume that this is the case and write down a "giant spin Hamiltonian" that we postulate could in principle be derived from the bare Hamiltonian in a similar manner to how the single ion spin Hamiltonians were derived from their bare descriptions, ie. by finding the "giant spin" ground state of the system and performing perturbation theory around it to eliminate all the electronic spin degrees of freedom but one. We then proceed to the investigation of the properties of general giant spin Hamilto nians in the absence of environmental couplings. We calculate tunneling matrix elements 29 Chapter 2. Effective Hamiltonians 30 for Hamiltonians with various symmetries using instanton [74, 75], WKB [25, 26], per turbation theory [70] and exact diagonalization methods. We conclude by returning to the specific case of the Fe8 giant spin Hamiltonian. The giant spin possesses two preferred directions due to crystal anisotropy which are identified, in zero external field, with the ±z directions (states \S = +10 > and \S = —10 >). At temperatures much lower than the difference in energies between | + 9 > and | + 10 > states (~ 5 K) only the | ± 10 > states have significant thermal populations. This allows us to derive a final effective description where the central spin object is treated as a two level (| ± 10 >) system, following the treatment of Tupitsyn et.al. [74]. 2.1 The Fe3+ Free Ion Hamiltonian We shall begin our analysis of the complicated Fes system (whose structure was shown in the introductory chapter) by concentrating our attention on the iron ions. We shall begin by studying a general Hamiltonian for a free Fe3+ ion. This treatment follows that of [69]. The dominant term in this description is the Coulomb interaction amongst the elec trons (here there are Ne of them) and between the electrons and nuclear charge Ze Ne /»•2 Ze2\ Ne e2 V'-£&-T)+&£ (21) The next most important term is the magnetic interaction between the orbital angular momentum lj and the electronic spin Sk VLS = ]C a3*h ' h + b3kl3 " $k + CjkSj • sk (2.2) 3,k where a^, bjk and Cjk are constants. Next comes the direct interaction between spins Vss = y *j' *k - 3(fJfc ' fffX^i* • (2 3) jk Tjk rjk Chapter 2. Effective Hamiltonians 31 Weaker still are the terms VN = 2gnjiBPn fc I rk rk \ o (2.4) where the term in curly brackets is the dipole-dipole interaction between the nuclear and electronic moments and the last term is the so-called anomalous hyperfine term which comes about from the overlap of the wavefunction of s electrons with the nucleus and Q e2Q 2/(2/ - 1) 1(1 + 1) 3(rk-I)2 (2.5) 'fc 'fc which represents the electrostatic interaction between the nuclear quadrupole moment Q and the gradient of the electric field due to the electrons. Interaction with an external magnetic field produces the terms and VH = '£^B(lk + 2sk)-H k Vh = -9nPnH • I (2.6) (2.7) corresponding to the interactions with the electrons and nucleus respectively. The total free ion Hamiltonian is now just the sum of these; H = VF + VLS + Vss + VN + VQ + VH + Vh (2.8) Orders of magnitude of these may be obtained from optical spectra and are, for Fe3+, VF ~ 5-105 K, VLS ~ 100-300 K, Vss ~1-2K,VN~ 1-200 mK and VQ ~ 1-2 mK [76]. We see that Vp is by far the dominant term in this expression. If we neglect all terms but this one, then L and 5 (the total angular momentum and spin of the ion) commute with Vp. This means that we may in this approximation label the states of Chapter 2. Effective Hamiltonians 32 the free ion with the quantum numbers L, Lz, 5, 52, J and Jz. Since the filled inner shells have 5 = L = J = 0we may describe the ion by referring only to the state of the partially filled outer 3d shell. It is known that the Vp term in Fe3+ leads to a groundstate that is an orbital singlet 6<S5/2 [77], where we use the standard notation that the superscript refers to the spin multiplicity 25 + 1, the capital script letter refers to the total angular momentum of the ion (S -+ 0, V —> 1, V —> 2, etc.) and the subscript refers to the total angular momentum J. This state may be obtained via the use of Hund's first two rules [78]-we first maximize the total spin by filling up five d orbitals with s = +1/2 electrons (giving total spin 5/2) and then maximize the orbital angular momentum (L = 2 + 1 + 0+ (-1) + (-2) = 0). Note that for all half-filled shells (here we have 5 d electrons out of a possible 10) we get an orbital singlet for the groundstate. 2.2 The Effect of the Crystalline Environment In general, when a transition metal ion is placed in a crystalline environment, the first question that must be resolved is the question of the nature of the bonding between the ion and the ligands. This is because the dominant new term that must be dealt with comes from the electrostatic interaction between the ion's d shell electrons and all the charged matter in the molecule. If this bonding is mostly ionic, then one can make the approximation that the ion sits in an electrostatic field coming predominantly from its nearest neighbours, which are treated as point charges. This is the so-called crystal field [79] approximation. Although it is crude, it is often a useful starting point for under standing the effect of the Coulombic interaction between the ion and its environment. A little more sophisticated is the ligand field [80] approximation. In this treatment al lowance is made for the direct overlap of the ion's d shell electrons with the ligands, ie. an attempt to deal with covalency is presented. Better yet are molecular orbital [81] Chapter 2. Effective Hamiltonians 33 methods, which use as their starting points the orbitals of each of the ligands and ions and the interactions between these. Now in our specific case we are interested only in the low energy properties of the Fe3+ ion. The free ion groundstate is, as mentioned, an orbital singlet 6«S5/2- What will be the effect of the inclusion of the Coulombic environment on this groundstate and the low-lying excited states? In order to answer this question we need only to know the relative strengths of the "on-site" Coulomb terms VF and the "off-site" Coulomb terms Vc- The strength of this coupling in several materials containing Fe3+ ions has been determined, with these ranging from Vc ~ 17000 — 23000 K [82]; however Vc has not been measured in Fe8. Fortunately there a way to know what the relative magnitude of these are without a direct measurement. If VF ^> Vc then the groundstate of the ion will remain 6<S5/2, as the electric field cannot split a singlet. If we are in the opposite limit Vc » VF then the Fe3+ ions will go into a "spin-paired" state with spin 5 = 1/2 (ie. Hund's rules are modified). Since it is known experimentally that the Fe3+ ions are in fact in a 5 — 5/2 state we infer that we are in the limit VF 3> Vc, which is in accord with the crystal field strengths reported for other materials with Fe3+ centers. We see that the experimentally observed fact that the Fe3+ ions have spin 5 = 5/2 simplifies our task tremendously. This is because this is prima facie evidence that the groundstate of the Fe3+ ions, even in the molecular environment, is 6«S5/2. More precisely, this is evidence that the groundstate of the Hamiltonian H = VF + Vc, where Vc includes all Coulombic terms coming from the interaction of the ion with the ligand environment, is an orbital singlet 6<5*5/2-Now the tack that we shall choose in what follows is this. Since the Coulombic environment does not split the groundstate here, but does affect the excited ionic states (the nearest state is a AQ state [165] in the free ion), and since we are not in the position to quantitatively account for its effects anyway (this would require a molecular orbital Chapter 2. Effective Hamiltonians 34 approach, and even these do not always work [83]), we shall adopt the crystal field paradigm in dealing with charges external to the ion. In this picture the excited states of the Fe3+ free ion are split by the crystal field, with some orbitals being favoured above others because of their spatial dependence and relative positioning in the molecule. In our case the Fe3+ ions all have six nearest neighbours arranged in a distorted octahedral shape. This allows us in principle to calculate the splittings of the excited states. We shall not do this however-as we shall see, the magnitude of these will only quantitatively change the results of the arguments to come. 2.3 The Single Ion Effective Hamiltonian We now want to consider the effect of the free ion terms VLS + Vss + VN + VQ + VH + Vh-What we shall do is follow the treatment of Abragam and Pryce [67], calculating their effect perturbatively on the 6<S5/2 groundstate of H = Vp + Vc- The first step in this treatment is to rewrite these perturbations in terms of total single ion spin and angular —* —* momentum operators 5 and L. This will be permissible as long as there is no significant chemical bonding between the iron ions and their surroundings, which as we saw in the preceding is supported by the 5 = 5/2 nature of the ions. This has been done; we repeat this process here (for a detailed explanation of the steps outlined here, see [68]). Our terms transform as follows; The effective spin-orbit coupling parameter A for free Fe atoms is measured to be \Z%K [84] and is expected to be slightly smaller for Fe3+ ions. The parameter p is difficult to VLS —> A L • 5 VSS -> P [(L • 5)2 + l-L • S - l-L{L + 1)5(5 + 1) p \\{Lal/ + L^La)SQSp - l-L{L + 1)5(5 + 1) (2.9) Chapter 2. Effective Hamiltonians 35 calculate [85] but is expected to be much smaller than A [86]. We also have VN gntipixn ^ [(L • 5) + £L(L + 1)(5 • /) - ^(L • S)(L • i) (s-i) v , ^Q 11 Q 2/(27-1) \r3 3,* 3(f-/)2 + |(Z.i)-L(L +1)7(7 + 1) where for the iron group (2/ - 1) - 45 , 77 = ±25e 5(2/- 1)(2/ + 3)(2L- 1) (the sign of n depends on whether the d shell is less or more than half filled) and h) = l* .Jtf(*0ls (2.10) (2.11) (2.12) ,r/ J r° where ip (f) is the spin density of the outer core d electrons. The factor K comes from the polarization of the core s-electrons due to the d electrons and will lead to the contact hyperfine interaction in 57Fe. The terms proportional to the external field are VH = (i(,H-{L + 2§) Vh = gnixnH-I (2.13) Putting all these together we get the perturbation term Hx = (A - \p)L -S-p(L- 5)2 + nPH • (L + 2S) +gnpnP |(L .T)-K§- I- 1-{L •S)(L-T)-l-{L- !)(L • 5)] + gnlinH • 7(2.14) where P = 2fip (^). 2.3.1 First Order Perturbation Theory We want to use (2.14) to perturb our groundstate out of the 655/2 state. In the first order perturbation we find that the only terms that contribute are ones that don't contain L, Chapter 2. Effective Hamiltonians 36 because < 0|L|0 >= 0 for an orbital singlet, except for the term < 0\LiLj + LjLi\0 >= \L(L + l)Sij + kj where lu = 0. As well, we have in this case that £ = 2/21 and rj — 0. Thus the terms we get from first order perturbation from &S§/2 are H2 = -gnPnP(^ + ^l^SJp + 2u.0H • S + gnlinH • I (2.15) 2.3.2 Second Order Perturbation Theory In the second order we have to calculate all the terms coming from ^ < 0|ffi|n >< n|#i|0 > ,01_. £ E(n)-E(o) (2-16) where n labels the excited orbital states. Where these excited states lie, ie. the exact values of E(n), are functions of the details of the crystal field splitting and as such we shall not attempt to calculate them exactly. We may obtain order of magnitude estimates for these by comparing to existing materials that contain Fe3+ centers. In these cases the energy of the first excited state E(l) is in the range 17000 — 23000 K [82], which is extremely large compared to the scale in which we are interested. Defining the tensors 0 <0\L*\nxn\I/l\0> E(n) - E(0) Q/3 _ _i€^_ v < 0\Ls\n >< n|L% + L7Zl|0 > U ~ 2 & E(n)-E(0) ' U T ~*ho E(n)-E(0) (2-17) gives, upon collecting terms, an effective Hamiltonian (2.18) Chapter 2. Effective Hamiltonians 37 where we have defined gaP = 2(5al) - AQ/3) , Da0 = -A2AQ/J - pla)3 Aa& - -P 7 7 , Rati = 1 - 2Pu.pAap (2.19) The meaning of these terms is as follows [87]. ga@ is the so-called "spectroscopic splitting factor". It is anisotropic in general, containing reference to the higher lying orbital states. DaP is a measure of the splitting of the ground state and contains reference to both the spin orbit coupling and the spin-spin contribution in an asymmetrical crystal field. Aa/} represents the hyperfine couplings between the nucleus and the electronic spin. This term is made up of contributions from overlap of s electrons with the nucleus («), an orbital contribution la/3 and a spin-orbit contribution XAa/3. In the expression for the coupling between the nuclear spin and an external field we see that there is an anisotropic component which can be of the same order of magnitude as the direct contribution. 2.3.3 Higher Orders Perturbation Theory It is clear that performing third and higher order perturbation theory will produce terms of higher spin multiplicity in the spin Hamiltonian [25]. Terms in the spin Hamiltonian up to 2Sth order in the spin operators Sx, Sy and Sz are in general possible. In the case of the Fe3+ ion this means that all terms up to fifth order must be considered in a general treatment; we can write this general fifth order spin Hamiltonian in the form HFe3+ = GQ1Q2Q3Q4Q5(#) SaiSa2Sa3Sa4Sa5 + g^^SaHp + gnu.n[Aal3SaIl3 + Raf3HaIp] (2.20) where the S{ai] can be any of Sx, Sy, Sz or the identity 1, the prefactor G can be a function of magnetic field, and we have explicitly separated out the Zeeman term. We Chapter 2. Effective Hamiltonians 38 —* only keep to first order in terms containing the nuclear spin I. The terms in the single ion spin Hamiltonian must inherit any symmetry that the crystalline electric field possesses, which reduces the total number of possible terms. Unfortunately in our case the Fe3+ ions do not sit in positions of high symmetry. However the symmetry is close to being cubic-the nearest neighbours of the Fe3+ are distributed in a distorted octahedral fashion. To give a concrete example of how the crystal field symmetry selects specific terms out of the general 2Sth order spin Hamiltonian, consider the case of an Fe3+ ion in a cubic crystal field. This case has been previously treated, with resulting spin Hamiltonian [77] (compare to (2.20). It is found that the constants gi, g2 and g3 are always very small and therefore the terms proportional to these are usually neglected [77]. Typical values for a range from 0.1 mK to 3 mK [88]. A Useful Approximation Measurements of ga/3 in other insulating materials containing Fe3+ support the following approximation [89]. We take the tensor ga^ to be isotropic and furthermore that ga/3 = g = 2, the spin-only value. The justification for doing this comes from the L = 0 nature of the groundstate-perturbations of the spin levels coming from crystal field and/or spin-orbit coupling will be very small and therefore a spin-only approximation for ga/3 is justified. H) rcubic Fe3+ — + + | (S4 + S4 + St) + ^HBSalp + 9i{S*Hx + S3Hy + S3HZ) g2(S5x + S5yHy + SlHz) + gnfxn [A^SJp + R^HJp] g3 [Sx(S4y + S4Z)HX + Sy(St + S$)Hy + SZ(S4X + S4y)Hy] (2.21) Chapter 2. Effective Hamiltonians 39 2.4 The Single Molecule Effective Hamiltonian We now want to consider the Hamiltonian of an entire Fes molecule. We label the spin Hamiltonian (2.20) Hp for the pth iron ion (there are eight of these). Now because each sits in a distinct crystal field, with different principle axes, it is very difficult (although in principle possible) to write down the specific terms for each ion. For example, if we were to make the approximation that the crystal field for each is exactly cubic and kept only the two dominant terms in (2.21), ie. H?& ~ ^(s4x + S4y + St)+gu.BH-S (2.22) then we immediately run into the following technical problem. The cLXGS j clS defined by the local crystal fields for each Fe3+, are different. This means that if we fix the axes such that for one of the ions (2.22) is correct, then in writing down all the other single ion spin Hamiltonians we have to rotate the axes through some (albeit known) angles from the "natural" basis picked out by the crystal field. This has the effect of hiding the symmetry explicit in (2.22), and bringing us back to a general type of description like (2.20). We shall therefore at this stage write the collection of eight iron single ion spin Hamiltonians as H = £ Hk>+ = £ [GriQlQiai(#) SS^SS^SS, + 9PBSp -H + gnpu.n [A^s*!* P=I p—i + ErpHaI%]] (2.23) where for all quantities the label p points to the pth iron ion. Chapter 2. Effective Hamiltonians 40 2.4.1 Inclusion of Exchange and Superexchange Terms It is known that there exist exchange and superexchange couplings between the iron ions. In general these couplings can be anisotropic, leading to a general expression of the form Hex = £ JgS>S} (2.24) p<q where the sums over p, q are over the ions 1..8 and the labels a, j3 refer to spatial directions x, y, z. Exact diagonalization studies on this term have been performed, with results compared to EPR and susceptibility measurements performed on Fe8 [18]. The model used in these calculations assumes that the couplings are isotropic, although it is not clear that this has to be the case (Dzyaloskinski-Morya interactions, for example, are excluded in the isotropic case [90]). The pathways, as well as the magnitude of the coupling strengths extracted from fits to experiment, are shown in figure 2.1. Note that the coupling energies extracted are far larger than the single-ion anisotropy terms-as we have seen, the terms in the single ion spin Hamiltonian are typically on the order of mK, while the energies extracted by Delfs et.al. [18] are in the tens to hundreds of Kelvin. 2.4.2 "Offsite" Dipolar and Quadrupolar Contributions The electronic state of each Fe3+ ion will couple via dipole-dipole interactions with all the other dipoles present in the molecule. This includes the other Fe3+ electronic spins and all the nuclear spins in the molecule. As well, all nuclei in the system with spins I > 1/2 will have an electric quadrupole moment which couples to inhomogeneous electric fields in the molecule. It will turn out that these contributions will be quite important in the chapters that follow. As such we shall go over the derivation of these terms carefully. The general interaction between the electronic and nuclear degrees of freedom may be split into two parts [64]. The first of these, the electrostatic interaction between nuclear and electronic charges, will be considered first. The second part, the magnetic coupling Chapter 2. Effective Hamiltonians 41 Figure 2.1: Exchange pathways in Fe8 in the isotropic model of Delfs et.al. [18]. Fits to susceptibility data give Ju ~ 3bK, Ji3 ~ 180K, J15 ~ 22K and J35 ~ 52K, with all couplings antiferromagnetic. between the moments of the nuclei and the magnetic fields generated by the spin and orbital currents of the electrons, will be dealt with later. The standard derivation [64] of the form of the electrostatic couplings begins by describing the nuclei and electronic clouds as classical charge distributions pn{rn) and Peife) with mutual electrostatic energy Pe(re)Pn{rn)dredrn E (2.25) This expression may be simplified by expanding the Coulomb potential in terms of spher ical harmonics. After some algebra one can show that this energy is the expectation value Chapter 2. Effective Hamiltonians 42 of the Hamiltonian Helec = £ A™B\ im* (2.26) l,m where BY1 = -e A? = e 21 + lf 21 + lf i=l i-l (2.27) (2.28) Here Nn and Ne are the number of protons per nucleus and electrons respectively, {Ri, Qi, <frj} and {r^, fy, 0j} are the polar coordinates of the ith proton and ith electron respectively and Y™ are the standard spherical harmonics. The general expression (2.26) may be further simplified by the following observations. It is well-known that stationary nuclear states have well-defined parities, implying that the expectation values of terms A™ odd in / are zero. That this is in fact true has been confirmed experimentally to a high degree of precision with the / = 1 (electric dipole) term [64]. This leaves us with a sum over even /, which can be manipulated further. The I = 0 (monopole) term can be seen to be constant, and can therefore be omitted from consideration. As well, it is seen experimentally that the strength of the terms in the series decrease rapidly with increasing I. In fact, direct contributions from the / = 4 term are so weak that they have never been seen in NMR studies in solids. We may therefore omit all terms with I > 2, leaving only the electric quadrupolar term HQ= J2 A?B™* (2.29) m=-2 The tensors appearing in (2.29) can be recast in a more useful form. One can show [64, 91] that the Hamiltonian (2.29) produces the same matrix elements in the nuclear Chapter 2. Effective Hamiltonians 43 spin subspace as the form N eQk 6/fc(2/fc-l) (2.30) where N is the total number of nuclei, Q2yk (2.31) dxadxP where Vk is the potential at the kth nucleus due to the charge distribution in the molecule, and Qk is the electric quadrupole moment of the kth nucleus, which is a measure of the nonspherical distribution of charge inside the nucleus. Qk is zero for all nuclear spins with Ik = 1/2 and is typically on the order of 1 - 10 • 10-24 cm2. In Fe8, the nuclei that have non-zero electric quadrupole moments are the two species of bromine 79Br and 81Br, 14JV and 160. The description (2.30) accounts for all electrostatic effects between electronic and nuclear charge distributions in the molecule. We now turn to the second part of the electron-nucleus interaction, the magnetic couplings. We have already used the fact that stationary nuclear states have a fixed parity to eliminate certain terms in the multipole expansion of the electrostatic coupling. We may perform a similar trick here by noting that the magnetic field feels the opposite parity effect of the electric field (since they are axial and polar vectors respectively). This means that all even orders of the multipole expansion of the magnetic structure of the nucleus have to be zero. Similarly to the electrostatic case, the strength of the contributions of higher order terms in the expansion fall off extremely rapidly with increasing /; as previously, no evidence of / = 3 (magnetic octopolar) terms has ever been directly seen by NMR in bulk matter. This makes our job considerably easier from the start, as we can treat the general nucleus as a magnetic dipole without loss of generality, and the interaction of the magnetic (2.32) Chapter 2. Effective Hamiltonians 44 field generated by the electrons and any magnetic dipole can easily be dealt with. One finds an interaction Hamiltonian of the form [64] .. _ 8 N+8 ^^EE^F1 [3 • h - 3$ • rIfc)(/* • rlk)] (2.33) 47r 1=1 fc=i rlk where we have used our isotropic g approximation, the sum over electronic spins is over the eight Fe3+ sites and the sum over nuclear sites is over all possible nuclear sites (N is the number of spins in the ligand bath; accounting for the possibility of up to eight 57Fe nuclei gives k = l..iV + 8). We do not include the "self-coupling" term here where the electronic spin interacts with a 57Fe nucleus on the same site, as this term is included in the single ion spin Hamiltonian derived earlier (it is a "contact" term). gnic and u.n are the nuclear p-factor of the kth nucleus and nuclear magneton respectively. It is possible to rewrite this term by defining a matrix Mlk{rlk) = ——-3-' Ik 1 ~ Sffkx -3flkxrlky -3flkxflkz -3rlkxrlky 1 - 3rfky -2>rikyrik z -3rikx?ikz -3fikyfikz 1 - 3ffkz This allows us to write (2.33) in the simple form 8 N+8 Hmag = ££ Qn^M^ SlJ* (2.35) 1=1 k=l where a and /3 are again spatial labels x, y or z. Note that one may calculate exactly the values of these terms, as they depend only on the relative locations of the nuclei and the iron ions which are known from crystallographic data (this will turn out to be quite important in later chapters, particularly chapters 3 and 5). Together with the term derived earlier (2.30) we have a complete description of the "offsite" (ie. not including on-site contact hyperfine interactions) electromagnetic inter action of all the nuclei with all the electronic spins in the Fe8 molecule. We may write (2.34) Chapter 2. Effective Hamiltonians 45 the final description of this nuclear-electronic coupling in the form (2.36) 2.4.3 Intra-Nuclear Spin Couplings A similar analysis may be performed on the electromagnetic couplings between the nuclei themselves. As was indicated in the previous section, by far the dominant contribution to this effect comes from dipole-dipole interactions of the form where gni and gnk are the nuclear ^-factors for the I and k species of nuclei. 2.4.4 Couplings of the Nuclear Bath to External Magnetic Fields We shall assume that the nuclei in the ligand bath couple to an applied external magnetic field in the standard Zeeman way; that is, 2.4.5 Coupling to Phonons Both the super-exchange/exchange and nuclear spin couplings are local in the sense that their ranges do not extend outside of a given molecule; it is enough to treat their effects on a "per molecule" basis. In this section we shall, following the treatment given in [92], introduce couplings of a single molecule with phonon fields which at low energies have long wavelengths and as such owe their properties to the details of the crystal lattice. The presence of phonons in the crystal can be dealt with by introducing strain and rotation fields which are written respectively as [92] (2.37) Hext = 2\^9nkPnIk ' H( k ext (2.38) UJ, = - [daUj - d7itQ) Chapter 2. Effective Hamiltonians 46 where MO = £ kX h 1 1/2 <4A) [4A + O*A] are the displacements at f caused by a phonon field whose creation/annihilation operators are given by {a\.x,ak\} and is the magnitude of the polarization vector. A labels the branch and includes in general both optical and acoustic phonons. M is the mass of the unit cell and N is the total number of unit cells. These local strains and rotations cause there to be introduced into the Hamiltonian terms mediated by the phonons. For example, imagine the effect of a local rotation of a particular spin on the DijSiSj term in (2.20), and in particular on the DZZSZSZ component; SZ y SZ + U)ZXSX + U)ZySy (2.39) and DZZS2Z -> Dzz (s2 + u2zx + u2yuzx{Sz, Sx} + uzy{Sz, Sy}) (2.40) In general one can show [92] that the spin orbit interaction leads to an effective coupling between the ionic spins and the phonon fields of the form h 1/2 n 2NMuqX_ H (2.41) where VqX(St) = B%S?S? + (4 x q) • St (2.42) Here ^ is the polarization vector of the phonon involved. The terms can in principle be calculated from knowledge of the crystal symmetry and have been estimated to be ~ 0.01 mK for the Mn12 material [93]. One must keep in mind that in order to do Chapter 2. Effective Hamiltonians 47 this one must use the symmetries appropriate for the individual ions and not the full —# symmetry of the crystal. The term linear in S arises from treating the spin as being locked to the lattice due to spin-orbit effects. This term is likely to be much smaller than the Bap terms (see [92] for a discussion of this point). 2.4.6 Coupling to Photons The coupling to the photon field is taken to be of magnetic dipolar origin [94]. The magnetic field due to the photon field can be written B7 = Vxl=E 27^ 1/2 Vukx ^f(cw + Cy(Vx4) (2.43) where V is the volume of the sample, {c\x, ck\} are the creation/annihilation operators for the photon field and ^A is the polarization vector for the {k, A} mode. The coupling to the electronic spins is then straightforward and is given by Hsl = J2gpBB1-Sl (2.44) 2.4.7 Bringing all the Terms Together-The Bare Fe8 Hamiltonian The description of the systems under scrutiny given in the previous section contains seven terms, namely (2.23), (2.24), (2.36), (2.37), (2.38), (2.41) and (2.44). Explicitly we have, for our effective Hamiltonian, an expression of the form H = J2[G?a>a>a*'*(ii) S^a2Spa3S^a5+gpBSp-H + 9npPn[Apa0S^ + R^HaI^}]+ £ <tfS>S} p<q=l N+8 + E fc=i Chapter 2. Effective Hamiltonians 48 47r J<fc=l rife fc=l 8 9A l-l h 2NMugX 1/2 9 [4A^A(S<) - a*V&(3)] + £ g^B, • St (2.45) /=i Let us review what we know about each term in this expression. The first term is a sum over all the single ion Fe3+ spin Hamiltonians. In order to evaluate the coupling energies C7plQ2Q3Q4Q5 we could do the following. First, we identify the symmetry of the crystalline field surrounding each iron ion. This reduces the number of non-zero couplings (as we saw in the exactly cubic case in (2.22)). We then pick a set of axes, aligned so as to simplify the spin Hamiltonian of one of the ions. Next we determine the angles necessary to rotate each iron ion from the axes chosen by its local crystal field to our chosen basis and apply these rotations. This procedure allows us to approximately evaluate the G^1020^"40^ for Fes- We do not know for certain what the magnitudes of these couplings are, although as mentioned earlier for Fe3+ in a cubic field they are typically in the mK range [77]. We shall not explicitly perform this task for a reason that will soon be made clear. This term also contains couplings between the electronic spin and external field, which we have approximated as being isotropic (ga^ ~ g = 2), the "contact" interaction between the nuclear and electronic spin and the interaction between the external field and the nuclear spin. These two last will not in general be isotropic. The second term is the exchange/superexchange term coupling the electronic wave-functions of the iron ions. Extracting quantitative predictions about how large the are is a difficult task and has yet to be performed satisfactorily [18]. It is not known whether the anisotropicity here is important. As we mentioned earlier, preliminary in vestigations indicate that these couplings are in the tens to hundreds of Kelvin [18]. The third, fourth and fifth terms are the non-local coupling between nuclear spins and Fe3+ electronic spins, nuclear-nuclear dipole interactions and nuclear spin-external Chapter 2. Effective Hamiltonians 49 field coupling respectively. All of the quantities in these term are either known or can be calculated (we do this explicitly for all the nuclei in the molecule in chapter 3). The coupling energies here are found to be bounded above by ~ 5 mK. The sixth term is the phonon-electronic spin coupling. Here the only term that we do not know is the value of the Bap terms in (1.21), although these have been estimated to be on the order of 0.01 mK in Mn\2 [93]. The seventh and final term is the dipolar magneto-optical coupling. Here the photon field in the material will be changed to a renormalized (by optical phonons) "polariton" field and therefore the frequencies Cbkx are not known exactly, although methods to approximate these are available [95]. 2.5 Exchange/Superexchange and the Giant Spin Picture As was discussed in the introductory chapter, the low-energy phenomenology of the Fe8 system indicates that somehow the electronic spins "lock together" into some fixed-spin object. Because the exchange/superexchange coupling energies are much larger than all the other energy scales in our Hamiltonian (2.45), we are presented with a mechanism whereby we can understand how this can happen. In a numerical diagonalization study performed in [18], it was suggested that the groundstate of the Hamiltonian (2.24) is given by a state where six of the Fe3+ align parallel to each other while the other two align themselves anti-parallel, giving a "quantum rotator" with excess spin of S = 6 • 5/2 — 2 • 5/2 = 10. Because the jff are large, there exists a sizeable gap to excitations out of this ground state, whose magnitude these authors suggest is on the order of AE ~ 30K. This locking together of the electronic spins profoundly affects the ultimate form of the low-energy effective description. To begin with, as excitations from the ground state are energetically inaccessible as long as kBT <C AE (we will ultimately be interested in Chapter 2. Effective Hamiltonians 50 the mK range so this is reasonable), we may consider the term in our effective description (2.24) to be simply a constant which we henceforth remove from consideration. Note that this does not mean that the dynamics of the electronic spins are frozen-it is simply that the effective degree of freedom that they represent is, for ksT <^ AE, a single collective "quantum rotator" or "giant spin" which is still very much a dynamical quantity. Now it is quite a difficult matter to actually derive an effective description in terms of this new collective degree of freedom from the Hamiltonian (2.45). In order to do this one would have to first determine the ground state of the electronic spins and then perform perturbations out of this ground state in a similar way done for the single ion case. Instead what we shall do is, following [74], make the following hypothesis. We simply assume that the exchange/superexchange couplings lock the electronic spins together into a quantum rotator or "giant spin" S with S = 10, where six (two) of the electronic spins point parallel (antiparallel) to the direction of 5, as indicated by [18]. This we refer to as the giant spin hypothesis. We then, as a corollary to this hypothesis, rewrite (2.45) in the form H = saiiS^A.+S^'E^ + E^I^W + ^M P=I p=i N+8 r 8 „n 2 Af „ „ .  N 47r l<k=l Tlk fc=l 1/2 qX 1=1 h °UV*&) - • St (2.46) /=i 2NMuqX_ where Sai. — Sx, Sy, Sz or the identity. What we have done here is replace the sum over single-ion spin Hamiltonians and the exchange/superexchange terms with a giant spin Hamiltonian HGS — Gail-a{™SaiiSai2...SahQ. As we discussed in the single ion case, the spin Hamiltonian can contain in general terms up to 2Sth order in spin operators, which Chapter 2. Effective Hamiltonians 51 in this case is 20 order. We can however reduce the total number of terms significantly in the case of Fe8, because the giant spin Hamiltonian must contain the symmetry of the full crystalline lattice, which in the case of Fe8 is triclinic [23]. Note however that we are not going to derive any of these terms. As we saw in the introductory chapter much effort has been expended in trying to measure what the relevant couplings <5Qii -ai*> are. Unfortunately because there are so many of these, even after the crystal symmetry has been taken into account, most have been ignored as their magnitudes decrease quickly with increasing order in Sj. This is however quite dangerous as these small ignored terms can contribute greatly to the physics, in particular to the magnitude of tunneling amplitudes between different \ms > states of the giant spin (twenty one of which, ranging for S = 10 from | - 10 > to | + 10 >) [25]. The remaining terms are identical to their forms in (2.45). The only difference is that now we must remember that each individual electronic spin is locked to the giant spin. 2.6 Investigation of the Giant Spin Hamiltonian in the Absence of Environ mental Couplings If we completely neglect all environments in (2.46) and fix S = 53?= i Si we obtain H = r>i-^o SaiiSai2...Sai2Q + gpBH • S (2.47) We are going to digress somewhat at this point from our focus on Fe8. We will in what follows consider all possible crystal symmetries and not just the Fe8 triclinic symmetry. This we do because there are other similar molecular magnets (Mni2, for example) that may have giant spin descriptions of this sort that possess different crystal symmetries (for Mni2 this is tetragonal [28]) and for this reason it is worthwhile to say some general things about the description (2.47). Chapter 2. Effective Hamiltonians 52 Now as we discussed earlier, the spin Hamiltonian can contain terms up to 2Sth order in the spin variables. It is however difficult to keep all of these terms in our description for large S, and furthermore it is not clear that all of these terms can be directly measured anyway. What is usually done is that only the lowest order terms in S consistent with the crystal symmetry are kept, and all higher order terms are thrown away. In what follows we shall follow this tack. We emphasize however that even if the higher order terms are "small" they can still significantly affect the physics, in particular the amplitudes of the tunneling matrix elements between states of the giant spin. 2.6.1 Exact Solution for Tunneling Matrix Elements via Diagonalization We shall begin our study of (2.47) by exactly diagonalizing some particular subsets of it and thereby extracting tunneling matrix elements as functions of {G}, \S\ and H. Our plan of attack is as follows. We begin in each case by choosing one of the seven crystal symmetries so as to determine the allowed form of {G}. In the specific cases of the tetragonal, orthorhombic and hexagonal systems we then diagonalize a truncated version of the resultant Hamiltonian for a range of externally applied DC fields, for central spin values |5| = 1, 10 and 15. In each of these cases we assume the existence of an easy axis which we identify with the z axis. We calculate the tunnelling splittings between the two lowest lying states (| + S > and | — S >), corresponding to the giant spin pointing in the ±z directions, which we then plot as functions of the parameters in the bare Hamiltonian. The Cubic System A crystal with underlying cubic symmetry possesses a spin Hamiltonian obeying the symmetries [Sx —^Sy Sy —> — Sx] , [Sx Sz Sz —y —Sx] , [Sy Sz Sz —> ~Sy] Chapter 2. Effective Hamiltonians 53 This restricts the allowed terms in the spin Hamiltonian. If we only include the two lowest order terms obeying these symmetries the giant spin Hamiltonian can be written [96] H=-D (Sx + S4y + St) + E [St + S«y + S6Z + WS2xS2yS2) + gu.BS • H (2.48) The case of cubic crystal symmetry is somewhat anomalous in that it is the only case we shall encounter where an axis (easy or hard) is not singled out by the crystal field-as we see from the symmetry requirements, all three axes in the crystal are equivalent. We shall not say more about this crystal symmetry. However, it is worth noting that the physics of cubic molecular magnets should be particularly entertaining because of the lack of an easy/hard axis. The Tetragonal System In systems with tetragonal symmetry, the symmetries [Sx^Sy Sy^-Sx] ,[SZ^-SZ] (2.49) must be preserved (see figure 2.2). Keeping only the lowest order terms gives a spin Hamiltonian of the form H = -DS2 + a4 (Si + Si) + gu.BS • H (2.50) Shown in figures 2.3 and 2.4 are results of exact diagonalization of (2.50) for a variety of parameter regimes. Note that in zero external field, if \S\ is odd the tunneling splitting between the two lowest levels here is zero. This is known as Kramer's degeneracy [47, 97], and it arises here because there exists no path by which our Hamiltonian can connect the states \S > and | — S > if S is odd. Chapter 2. Effective Hamiltonians 54 (i) 171 rtVfl (ii) (iv) (v) (vi) O J^b a (iii) (vii) Figure 2.2: All of the unit cells (after Ashcroft and Mermin [138]). (i) Cubic, (ii) Tetrag onal, (iii) Orthorhombic, (iv) Monoclinic, (v) Triclinic, (vi) Hexagonal and (vii) Trigonal. The Orthorhombic System In systems with orthorhombic symmetry, the symmetries [SX y Sx Sy y Sy Sz y Sz (2.51) must be preserved (figure 2.2). Keeping only the lowest order terms leads to the Hamil tonian H = -DS2Z + a2 (52 + S2_) + gfxBS • H (2.52) Shown in figures 2.5 and 2.6 are results of exact diagonalization of (2.52) for a variety of parameter regimes. Chapter 2. Effective Hamiltonians 55 Figure 2.3: Variation of As,-s with a^/D for four different |S| values (clockwise from top left, |SI =2, 6, 10, and 14); tetragonal symmetry. On the x axis is plotted a±S2/D and on the y axis log10 &s,-s- Here we have taken the external field to be zero. -5.0 1 ' ' ' ' ' 1 -5.0 1 ' ' ' ' 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 | , , , 1 0.0 | 1 Figure 2.4: Variation of A5,-s with Hx/D for a4S2/D = 0.25 for four different |S| values (clockwise from top left, |S| =2, 5, 10, and 15); tetragonal symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 As,-s-Chapter 2. Effective Hamiltonians 56 The Monoclinic System Systems with monoclinic symmetry contain as a special case the orthorhombic symmetry. Because of this the spin Hamiltonian for these systems must contain the orthorhombic symmetries. In addition we shall have quartic terms, giving H = -DS2Z + a2 (S2 + S2_) + a^S* + a^St + gu.BS • H (2.53) The Triclinic System Triclinic symmetry is obtained via a distortion of monoclinic symmetry. As such the description of the triclinic case must contain the symmetry of the monoclinic case. In addition, we pick up a diagonal quartic spin term; H = -DS2Z - D0S4Z + a2 (S2+ + S2_) + oj+)Sj + a^Si + gn.BS • H (2.54) The Trigonal System In systems with trigonal symmetry, rotations around the body diagonal are three-fold symmetric. H = -DS2 + a3{Sz,Sl + S3_}+ gpBS • H (2.55) The Hexagonal System In systems with hexagonal symmetry, the symmetries Sx -> e^3Sx] , [Sy -> e^3Sy] , [Sz -> -Sz] (2.56) must be preserved. This implies, keeping only the lowest order spin terms, a Hamiltonian H = -DS2Z + a, (S« + S*) + g»BS • H (2.57) Shown in figures 2.7 and 2.8 are results of exact diagonalization of (2.57) for a variety of parameter regimes. Chapter 2. Effective Hamiltonians 57 ' 0.0 0.5 1.0" ' 0.0 0.5 1.0 Figure 2.5: Variation of As-s with a2/D for four different \S\ values (clockwise from top left, |5| =2, 5, 10, and 15); orthorhombic symmetry. On the x axis is plotted a2/D and on the y axis log10 A$-s- Here we have taken the external field to be zero. 0.0 i 1 1 1 1 1 1 0.0 Figure 2.6: Variation of As-s with Hx/D for a2/D = 0.25 for four different |5| values (clockwise from top left, \S\ =2, 5, 10, and 15); orthorhombic symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 As -s-Chapter 2. Effective Hamiltonians 58 0.0 I 1 1 , , , 1 0.0 Figure 2.7: Variation of As-s with a6/D for four different |5| values (clockwise from top left, \S\ —2, 6, 10, and 14); hexagonal symmetry. On the x axis is plotted a6Si/D and on the y axis log10 As,-s- Here we have taken the external field to be zero. Figure 2.8: Variation of AS-S with Hx/D for a6S4/D = 0.25 for four different \S\ values (clockwise from top left, \S\ —2, 5, 10, and 15); hexagonal symmetry. On the x axis is plotted HX/DS2 and on the y axis log10 As,-s-Chapter 2. Effective Hamiltonians 59 2.6.2 Tunneling Matrix Elements via Perturbation Theory The calculation of tunneling matrix elements via perturbation theory in the terms that break the easy axis symmetry (z <r+ —z) is quite straightforward as long as there is only one such term in the Hamiltonian. For these cases we shall just state the results, all of which have been previously calculated elsewhere [70]. Hamiltonian ASl-s -DS2 + a2 (S2+ + S2_) -DS2 + a4 (S+ + S4_) -DS2 + a6 (S% + St) -DS2 + Hx (5+ + S-) -DS2 + a3{Sz,(Sl + S3_)} «f(2S)! 4S-i£>s-i[(S_i)!]* af/2(2S)! 4S-2DS/2-l[(Sy2_1)l]2 af/3(2S)! 4S-sz)S/3-i[(s/3-l)!]'2 H*S(2S) £,2S-l[(2S_i)!] alS/3(2S)\ Z)2S/3-l62S/3-2[(5/3_1)!]2 Table 2.1: Perturbation theory results for some simple Hamiltonians, from [70]. When the number of symmetry breaking terms in the Hamiltonian is increased, the solution for A using perturbation theory becomes a little more complicated. This is because of the competition between these terms in deciding which are the preferred paths between the low lying states. Chapter 2. Effective Hamiltonians 60 2.6.3 Tunneling Matrix Elements via WKB Methods It has been shown [25, 26] that the Hamiltonian H = -DS2 + ak (Sk+ + Sk_) (2.58) leads to As,_s^p^!)2S" „59) as long as akSk <C DS2 and the ambient energy E is such that E <C DS2. 2.6.4 Tunneling Matrix Elements via Instanton Techniques The final approximate method of solution for tunneling matrix elements in spin Hamil tonians that we shall consider involves using instanton techniques. We will explicitly perform one such calculation in section (2.8), obtaining the following form for an or thorhombic (ie. easy-axis easy-plane) Hamiltonian (see (2.52)) ^s = ^S^2(D-2a2f4a12/4exp (2.60) V a2 . The same type of procedure may be followed in principle with any spin Hamiltonian that possesses well-defined semi-classical trajectories between its minima. However in practice one runs up against problems with all but the easiest quadratic spin terms. We believe that it is possible to derive forms similar to (2.60) but have left this task to future investigations [98]. 2.6.5 Comparison of Approximate Methods to Exact Solutions In order to give some idea about how effective these different approximation schemes actually are, we now compare the results of the preceding sections to the exact results for the tetragonal and orthorhombic crystal symmetries. Shown in figure 2.9 is the Chapter 2. Effective Hamiltonians 61 Figure 2.9: Comparison of perturbation theory, WKB results and instanton results to the exact solution for the tunneling splitting between the two lowest levels of the Hamiltonian of orthorhombic symmetry with S=10. Plotted on the horizontal axis is a2/D, and on the vertical axis log10 As-s- Legend: Black, exact solution; Green, instanton solution; Red, perturbation theory and Blue, WKB. Figure 2.10: Comparison of perturbation theory and WKB results to the exact solution for the tunneling splitting between the two lowest levels of the Hamiltonian of tetragonal symmetry with S=10. Plotted on the horizontal axis is a4S2/D, and on the vertical axis log10 As-s- Legend: Yellow, exact solution; Red, perturbation theory and Green, WKB. Chapter 2. Effective Hamiltonians 62 comparison for the orthorhombic case. Here we see that, as we could have expected, the instanton solution fails quite spectacularly if a2/D is small. This is simply because when we calculated our instanton action we assumed that the fluctuations around the semi-classical paths were small, which of course they aren't if a2/D is small. When a2/D is large, the instanton solution is quite good. The WKB solution we see is quite bad. It turns out that the functional form in a2/D is correct but the prefactor is not. Perturbation theory works quite well for the entire range of a2/D < 0.5 studied. We next turn our attention to the tetragonal case, shown in figure 2.10. Here we see a similar story. The WKB solution gets the functional form correct but again the prefactor is wrong. Perturbation theory works quite well in the small a±/D regime. 2.7 Back to the Full Hamiltonian-Separation of Tunneling Energy Scale Us ing an Instanton Technique As was discussed in the introduction and to a lesser degree in section (2.6), it is simply not feasible to measure all of the non-zero components of the tensor <5Qii -ai2o experimentally. Because of this what is usually done is that the lower order terms in Si allowed by crystal symmetry are kept and the rest of the terms thrown away. The coefficients of the kept terms are then used to fit experimental data. In the case of the Fe% spin Hamiltonian, the form that we shall adopt is HGS = Gaii~ai«>Sil...Si20 = -DS2Z + E(S2+ + S2_) + C{Si + S4+) (2.61) This form, with D = 0.292A", E = 0.046K and C = -2.1 • 10~5K, is good enough to accurately fit both the period of the "Aharonov-Bohm" oscillations and magnitude of the tunneling splitting [51] in recent experiments. Note however that it is clear that a large number of terms have simply been chopped off the "true" spin Hamiltonian (for example, even the quartic diagonal spin term has not been included). Chapter 2. Effective Hamiltonians 63 The effects of thermal phonons and polaritons at low enough temperatures can be completely neglected [20]. This is because processes involving these bosonic modes scale like their respective densities in the crystal, which are vanishingly small at low tempera tures. If we are in the "quantum regime" demonstrated by Sangregorio et.al. [14] we are at temperatures less than ~ 360 mK, and therefore kBT <C DS [20]. We therefore write the quantum regime Hamiltonian in the form H = -DS2Z+E(S2+ + S2_) + C(Si + Sl) + gu.BH-S + £ 9nPiin [A^SPJI + R/aPHQq] + f; 9nku.nH • ik P=I *;=i N+8 + £ A;=l 9nklinMlk ^0 + 6Ik(2Ik-l)V '"P .1=1 + ^ E ^[?i-?k-3{fi (2-62) 47r Kk=l Tlk There still remains one feature of (2.62) that we can take advantage of in order to simplify it. An examination of the relative strengths of the terms in (2.63) reveals that the largest term is the single-molecule anisotropy term DS2 in the spin Hamiltonian which is of the order of 29 K. All the other terms are small compared to this. We therefore see that if all ambient energies (primarily the lattice temperature and external field) are much less than the gap to the first excited level D(S2 — (S — l)2) the giant spin will only be able to access the two lowest energy levels SZ = ±5. At this low energy scale we shall define our final low energy effective description, within which the giant spin is mapped to a two-level system parametrized by a Pauli matrix r, where f2 = ±1 corresponds to SZ = ±5 respectively. This description will be valid in the quantum regime T < 360 mK [14]. There is another case of interest where this mapping may be performed. If an external longitudinal field Hz is applied to the system, the effect is to bias the wells drawn in Chapter 2. Effective Hamiltonians 64 Energy A E = 0 •10-9-8-7-6 -5-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 S. Figure 2.11: Z-projection of spin versus energy from HGs for the Fe8 system. The region of validity of the mapping to a two-state system is the region where excited states are forbidden (this region in shaded grey in the above). figure 2.11. If the applied field is strong enough, it can bring to resonance one of our original states and a higher-energy state on the other side of the barrier (for example, the application of a positive Hz could lead to a resonance condition between | +10 > and | — 9 >). In this case, the dominant tunneling dynamics in the system still involve only two levels. However, one must exercise caution here, as the lower-lying states (in this example, | — 10 >) are connected to the two primary states by several mechanisms, most importantly phonon emission taking | — 9 > to | — 10 >. Because of this complication, Chapter 2. Effective Hamiltonians 65 in what follows we shall explicitly consider only the case where the externally applied longitudinal field is such that Hz «C DS and only the two levels | ± S > are involved, keeping in mind that in certain circumstances we may be able to generalize our results to larger values of Hz. The dynamics of f in this regime are solely the product of tunneling from |5 >«-» | — S >. This observation leads to a natural separation of contributions coming from terms in (2.63) that are diagonal in Sz (and therefore are not involved in tunneling events) and terms that are not diagonal in Sz (and therefore coming into play only when the giant spin tunnels). This separation is helpful because it will turn out (we will show this) that the time scale for tunneling physics = A-1 is much smaller than the time between tunneling events = QQ 1 [20]. To see why this separation is helpful, consider the following argument. Let us imagine a likely trajectory for the excess spin S(t), assuming an initial condition corresponding to S(0) — +S z. We assume that the time between tunneling events is much longer than the time over which tunneling occurs. We therefore expect the central spin to evolve dynamically in a similar way to that shown in figure 2.12. This "separation of scales" allows one to consider the effective description separately in two different regimes; one in the regions between tunneling events and one during the tunneling. The regime between tunneling events we shall refer to as the "diagonal" region (as only terms diagonal in S enter into play) and the contribution of these terms to the final effective description we can simply read off our equation (2.63). We find under these circumstances that the diagonal contribution can be written JV+8 z p=l (±)P9npVn fc-i N N + gp,BSHzfz p=l fc=l k=l L eQk -\rkaf) jk 6/fc(2/fc - 1) Q/? Chapter 2. Effective Hamiltonians 66 + J2°^[lrIk-S(Irrlk)(Ik-rlk) 47r ttk rlk (2.63) where (±)p is shorthand for the direction in which the pth ionic spin is pointing (see figure 2.1). Our labelling system has (±)p = 1 for ionic spins {p-j-} = 1,2,4,5,6 and 8 and (±)p = — 1 for spins {p±} — 3 and 7 when the central spin is "up" (that is, tz — +1) with signs reversed if the central spin is "down". We may rewrite this in the form 'l + fz HZ, TP" • h + fc=i HD = N ETT^E^-/* + 'i-f2 N E7P2c)-4 + E7i2)-4 p=l fc=l N + gu.BSHzfz + E k=l eQk "64(24 - 1) «» 47r Kk rlk where (2.64) (2.65) (2.66) E <- E < + ^ Lpe{pt> pe{P4.} (2.67) -.(2) -.(2) 5 g 5 7fc = 7fcs + 2^nHp = -gnkVn • E E M$ + HP pe{pT} pe{p±} (2.68) The notation here is such that the f3th component of the expressions on the right corre sponds to the (5th component of the vector on the left. One can think of the expression (2.64) in the following way. There are two electronic spin configurations (1,2) that correspond to before (1) and after (2) the central spin Chapter 2. Effective Hamiltonians 67 complex flips. In each of these configurations, each nuclear spin in the system feels a magnetic field coming from the ionic spins and the externally applied field. In the above expression, these fields are represented by 7^, rfkc\ 7^ and 7fc2c^. The sign changes in going from 7^ —> 7^ and 7^lc^ —> come from the fact that the central spin object has reversed its direction along the easy axis. One may write (2.64) in a more transparent fashion by defining unit vectors -(1) , -(2) Tk +Tk \Tk +Tk I for k labelling the ligand spins k — 1..N and 4 = mk = -.(1c) _ A2c) Ik Ik (2.69) (2.70) for k labelling the possible 57Fe nuclear spins k — N 4- l.-N + 8. We also define energies (2.71) for ligand spins k = 1..N and II if I I -.(lc) -.(2c) \ ± ,f\ i-*(lc) , -,(2c), uk = IhlgnMTk -7fc I , uk = \Ik\gnkMTk +% I for 57Fe spins k = N + 1..N + 8. With these we can write N+S —Ik • mk + —uUk • lk k=l N r E eQk T/fca/3 jk + + gu.BSHzfz POPN 9ni9nk (2.72) An Kk 'Ik h • Ik - • rlk)(Ik • flk)] (2.73) This form is similar to that derived by Prokofiev and Stamp [20]. It differs in two respects. Firstly, it shows explicitly what the energies uJk,uik and unit vectors mk,lk are in terms of parameters in the higher energy descriptions (and allows us to calculate these-we shall Chapter 2. Effective Hamiltonians 68 |-S> |+S> |+S> Figure 2.12: Typical evolution of the projection of the excess spin S(t) along the easy-axis. We see two regimes; one where S evolves without tunneling (diagonal in f), and one where S tunnels from | + S ><-*• | — S > (off-diagonal in f). Note the separation of scales; the time between tunneling events is much greater than the tunneling time. do this in the next chapter). Secondly, it includes the effect of quadrupolar couplings between higher spin nuclei in the ligand bath and electric field gradients in the molecule. The "off-diagonal" contribution to the effective Hamiltonian is somewhat harder to extract, and a different approach will be required. 2.8 Off-Diagonal Terms and the Instanton Method A method has been developed by Tupitsyn et.al. [74] that allows the extraction of the "off-diagonal" terms in the effective Hamiltonian, ie. those that act when the central spin object tunnels. Two objections have recently been raised which question the validity of this method [101]). In this section we shall review the method and point out the objections. Chapter 2. Effective Hamiltonians 69 2.8.1 Review of the Method of Tupitsyn et.al. In the effective description (2.63) we have reduced the Hilbert space of the central degree of freedom down to dimension D — 2. These states are collective objects of the form —* —• S = Yli Si; that is, they are formed of electronic spins that have locked together. In the approach to the Fe8 molecule that we have adopted, we have chosen a model where all the electronic spins lock together such that each lies parallel or antiparallel to the easy axis. We shall in what follows treat this collective state as a spin 10 quantum rotator, pointing out when this description must be modified because of the "true" eight spin nature of the object. —* —+ The states \a > and |/? > are collective states with a = \ + S > and /3 — \ — S >, ie. referring to the central spin "pointing in the up/down directions". The transition amplitude between these states can be defined to be Tapit) = 1^ D(9, <j>) exp (- j* dr (L0(r) + I^fr))) (2.74) where L0(T) and LN{T) are the Lagrangians corresponding to the bare spin Hamiltonian plus the external field term H0 = -DS] + E(S2+ + Si) + C(S* + Si) +gu,BS-H (2.75) and the contributions from the nuclear spins respectively. The spherical angles 9 and <f> are introduced so as to characterize S in the standard way; that is, Sx = \S\ cos <j) sin 6 , Sy = \S\sin <f> sin 9 , Sz = \S\ cos 9 (2.76) If we assume that the "bounce time" between minima fig 1 is much smaller than the time between transitions Agl, the evolution operator connecting the two minima will be given by {HND)afi = tfafl(t) , (a^/?,Q0-1«i«A0-1) (2.77) Chapter 2. Effective Hamiltonians 70 where H is the non-diagonal part of the effective Hamiltonian that we are looking for. This means that in order to calculate HND it suffices to calculate Tag(t), and this we can attempt to do by solving for the instanton (semiclassical) solutions of (2.74). 2.8.2 The Tunneling Lagrangian In the calculation that follows we shall explicitly use the easy-axis easy-plane spin Hamil tonian, vis. H0 — -(D - 2E)S2Z + AES2X (2.78) However, the tactics we employ here can be used for any spin Hamiltonian admitting clearly defined semiclassical paths between the minima \a > and \/3 >. Note that this spin Hamiltonian is equivalent to the one experimentally obtained for Fe8 if the quartic spin terms in this latter are neglected. Using the relationship S2 = S2 + S2 + S2 we can show that (2.78) is equivalent to H0 = -DS2Z + E(S2+ + S2_) (2.79) This truncation is performed simply for convenience, as inclusion of the quartic terms makes the analytic calculations that follow very difficult. As we discussed earlier in this chapter, the "true" spin Hamiltonian contains many terms that are inaccessible experimentally, and therefore dropping the quartic term simply emphasizes the point that in using a form like (2.78) we really are using a phenomenological description-the effect of dropping higher order terms on the amplitude of tunneling matrix elements can be drastic [164]. This being said, what is important in this case is the instanton trajectory, and this is not expected to be strongly affected by higher order terms [74]. For the easy-axis easy-plane model (2.78) one can show that the equations of motion Chapter 2. Effective Hamiltonians 71 are 2 0sin0-M£Ssin20sin2<£ = O (2.80) and (j) sin 9 - i4ES{ + cos2 0) sin 29 = 0 (2.81) Solutions of the classical equations of motion are simply 7r . , i E> — 2E ^ = J7__tsinh-M_^_ (2.82) and 0(t) = sind(t) = ——— , Q0 = 4£Ssinh w w cosh Q0t 2 sinh-1 ID-2E AE (2.83) where n = ± labels rotations clockwise and counter-clockwise in the easy plane. Now in the standard treatment, one assumes that the classical equations of motion are attractors in the sense that a small perturbation away from these costs action. If this is true then one can perform a gaussian integration over small fluctuations away from the classical equations of motion in the manner suggested by Tupitsyn et.al. [74]. However it was recently pointed out by Unruh [101] that if we perturb the variable <f> in the equation of motion (2.81) we obtain I r) OTP 8(f) = -d>ESsm(2smh.-1 \ ) cos 9 5(j> (2.84) V AE Now we see that as long as cos# > 0 any perturbation of d> is attracted to the classical solution. However we see from our solutions that cos# changes sign at 9 = ir/2. Thus it would appear that at this point the variable d> is pushed away from its stable point. If this is true it brings into question the validity of the gaussian integration technique. Chapter 2. Effective Hamiltonians 72 2.8.3 An Assumption is Made In what follows we shall make the assumption that we may perform gaussian integrations over small fluctuations in <f> around the solutions of the classical equations of motion. As Unruh has pointed out [101] it is not clear that this assumption holds even in the limit where <C 1-the case treated by Tupitsyn et.al. [74]. Furthermore in our case we have that = 2.67. That this number is large means the potential that holds (j> close to the classical solutions is not very steep, so that the assumption that gaussian integrations can be performed is questionable (however our results of comparing the exact solution to the instanton solution for the tunneling amplitude show good agreement in this case-see figures 2.9, 6.1 and 6.2). The Formal Calculation Explicitly the Lagrangians appearing in (2.74) are L0 = -iS(j)6 sind+ (D-2E)S2 sin2 9+ 4ES2 sin2 6 cos2 <f> — S (Hx sin 9 cos <f> + Hy sin 9 sin <j> + Hz cos 9) and LN = J2gnpfJin[Apa0SlIpp + Rp^HJ^+J29nkPnH-Ik p=l k=l N + £ fe=i + ^$:^[Ti-Tk-3(It • flk){Ik • flk)} (2.85) 47r l<k=\ Tlk We shall assume that the instanton bounce time Q.Q1 is such that Q,Q1 <C T2_1, where T2_1 is the time scale over which the nuclear-nuclear flip-flop processes mediated by the last term in the above expression occur. In this limit, the nuclear-nuclear term provides Chapter 2. Effective Hamiltonians 73 a static bias field acting on each nucleus and therefore we can rewrite our Langrangian in the form LN = E 9npPn P=I AT + E k=l k=l (2.86) where now the static field at each nucleus is given by the sum of the external field plus some contribution from the nuclear-nuclear term. In our model, each electronic spin is locked to the central spin. This means that —* (2.87) and therefore 8 N 8 E 9npPnA^SlIl + 9nkPnM^SPIkp p=l fc=lp=l 4 dripPn |_pe{pT} pe{pi} N E k=\ + E 9nkpJk3 E MS- E Kk Lpe{ptl pe{P4.} (2.88) 2.8.4 Solution for the Free Instanton Trajectory We assume that fluctuations in the variable <j> are small in our model. This will only strictly be true for 4E/(D — 2E) large (see figure 2.9). In our case we have seen that 4E/(D — 2E) ~ 0.37. Looking back to our comparison of the exact solution with the instanton solution for orthorhombic symmetry we see that in this regime the instanton solution is off by about a factor of two from the exact one in the determination of A0. Nevertheless we shall adopt this method in this case. Doing this allows us to perform a Gaussian integration over <j>. This will leave us with an effective description in terms Chapter 2. Effective Hamiltonians 74 of only one path-valued parameter 9(T) which is the angle between S and the z axis during the instanton trajectory. In the case where the giant spin is not coupled to the environmental spins and there is no external magnetic field it is known that this effective description reduces to (here we include only terms that affect the equations of motion) [75] <?2 . ~ Leff (6) — —92 + D sin2 9 (2.89) AE where we have defined E = AEJ2p\SP\2 and D = (D - 2E) £p |SP|2, following [75]. The classical equation of motion is readily found from (2.89) and is 9 = sin 9{T) = 1/ cosh(£V) , Vto = f \[hl5 (2.90) There are two things worth noting here. First is that the form of the instanton after the gaussian integrations have been performed is the same as the bare classical form with renormalized Q,0. Second is that the parameters D and E depend on the fact that we are really dealing with an eight spin object. In the case of a spin 10 object, E = 2ES2 = 200£, while for us E = 2E £p \SP\2 = E • 8 • 25/4 = 100£; likewise for D. Substituting this extremal trajectory into the effective Lagrangian and integrating over r gives for the instanton action ID - 2E Aefj = A0 + innS , A0 = 2S^—^— where n = ± corresponds to clockwise and counterclockwise paths respectively. Note that neither the Haldane phase TJTTS nor A0 depends on the eight spin nature of the collective central spin [75]. Chapter 2. Effective Hamiltonians 75 2.8.5 Inclusion of the External Magnetic Field and the Nuclear Spins We now make the assumption that the applied magnetic field and the couplings to the environmental spins are weak. Specifically we require that « l \H\ « 1 (2.91) Q,o Clo It is known [74] that the modifications to the instanton trajectory calculated above (2.90) coming from the external field and the spin bath first appear to second order in an expansion in powers of £/f20 where £ is one of u[\ uifc or the external field magnitude —* This means that if we are only interested in first-order corrections due to these effects (which we are) we can neglect these and use the trajectory (2.90) in the presence of the external field and the spin bath (as long, of course, as the conditions (2.91) hold). It must be noted here that the validity of this approach has been questioned [101]-in particular the assumption that the trajectory of the central spin responds to second order in the external fields. Substitution of the extremal trajectory into the general effective lagrangian and inte grating over r yields the following effective action, N+8 AEFF = A0 + irjTcS - ir,AH • H + n £ AKND • IK (2.92) k=l where AH is the contribution due to the external magnetic field H and the AND terms are the contribution due to the presence of the environmental spins. Explicitly these are irSgu,B„ .S2-ngnB -y-%- 2E -x (2.93) Sir 4O0 L b>e{PT} E <k ID-2E 4E E Kk E P€{p.|.} Mfk (2.94) Chapter 2. Effective Hamiltonians 76 for k = 1..N representing the ligand nuclear spins and In - OF .1 (2.95) 7»fc /_|_\ V 4E '4^o for A; = N + I..N + 8 representing the 57Fe ions. In both cases the f3th component of the tensors on the left hand sides are identified with the (3th component of the vectors on the right hand side. These expressions should be compared to equations (2.27) and (2.28) in [20], noting of course that here the hard direction is the x direction whilst in [20] it is the y direction. Aside from this the only differences here are that the terms due to the presence of the nuclear spins have been explicitly written in terms of parameters in a higher energy Hamiltonian and the nuclear spins can have arbitrary spin numbers. The tunneling splitting in zero field A0 is given in this instanton picture by A0 = ^0y|^exp(-Ao) (2.96) As we have seen earlier, this expression for A0 is off by approximately a factor of two from the exact solution for (D - 2E)/4E ~ 0.37 (figure 2.9). As discussed previously, for times fig 1 << t « AQ1, the relationship between the transition amplitude and the off-diagonal part of the effective Hamiltonian is HND = * (f_r4t(t) + h.c.) (2.97) where Tit(r) = itA0 £ exp(-Ae//) (2.98) and f_ is a Pauli lowering operator in the subspace of the two-level Neel vector. Using our expression (2.92) for Aeff and the criteria (2.91) allows us to write the off-diagonal part of the effective Hamiltonian as N+8 HND = 2A0f_ cos($ - i Y, AN,D • h) + h.c. (2.99) fc=i Chapter 2. Effective Hamiltonians 77 where $ = nS - An • H (2.100) 2.9 The Final Single Molecule Effective Hamiltonian As a result of all the above considerations we find that the form of the single molecule effective Hamiltonian is N+8 H = £ k=l N + E jt=i _L !~/fc • mk + Y^l'/fc • Ik eQk yka0 jk 6/fc(2Jfc-ir aP + gu.BSHzfz + 2A0f_ cos($ - i £ AkN>D • h) + h.c. k=l ^ %|n. ^ . /fc _ 3(j;. hk){fk . fjfc)] (2 1Q1) 4?r ra r/fc At this point we reiterate that objections as to the validity of the instanton calculation have been raised [101]. The term that is affected is the off-diagonal constribution. We shall find later on that the predictions that we obtain from the use of this term match both exact diagonalization and experimental results extremely well in the low field regime that we are considering. We treat this as evidence (but certainly not proof) that the approximations made in the instanton calculation are valid. Chapter 3 Nuclear Spin Couplings in Fe8 and the Isotope Effect Here we present our results for the quantities jkg and %g (for definitions see (2.67) and (2.68)) in Fes, which represent the dipolar fields due to the Fe3+ ions at the kth nuclear spin before and after the central spin complex tunnels respectively. We are going to do this using two different methods. The first will treat each Fe3+ ion in the molecule as a point magnetic dipole (and as such we call this the "point dipole approximation"). The second method we shall use is to model the actual spatial spin distribution near the iron ions by using previously calculated Hartree-Fock wavefunctions for free Fe3+ ions. This "spreading out" of the magnetic dipole changes the values for the fields at the nuclei. These fields (and therefore 7^ and 7^) are then used to calculate the dipolar coupling energies u>[' and uk. The conversion of field units to energy units is calculated using the dipole-dipole interaction using known nuclear ^-factors and assuming that the Fe3+ g factor is isotropic and equal to 2. We then use {u)k} and {ui^}, together with known contact hyperfine couplings due to the presence of 57Fe3+ ions, to calculate the orthogonality blocking parameter K, the topological decoherence parameter A and the full linewidth of the nuclear spins W for arbitrary isotopic concentration. We find that all these quantities show a significant isotope effect. 78 Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 79 3.1 Units and Constants We choose to work in the SI system of units. We therefore have that The Bohr Magneton is T- = 10"7 W 47T A2 HB = 0.9271203 • 10-23 ^ (3.2) and the nuclear magneton is fxn = 0.505038 • lO"26 ^ = 7.622462 (3.3) The proton and electronic iron g-factors are gpr = 5.58510 , gFe = 2 (3.4) Here we have assumed an isotropic spin-only g-factor for the electronic spin of the Fe3+ ions [?]. The unit conversion factors we shall use are 20.837 GHz = 1 K = 1.3807 • 10~23 J = 0.695045 cm-1 = 8.617 • 10"5 eV (3.5) 3.2 The Point Dipole Approximation Here we begin the problem of calculating the magnetic fields created by the iron ions. The tack we use here is to treat the iron ions as point dipoles. This is a "first order" approach which will only be useful if the spatial extent of the iron wavefunctions is much less than the distance between the iron ions and the protons. We will attempt a more careful treatment in the following section and compare its results to those obtained using the point dipole approximation. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 80 3.2.1 Magnetic Field at r due to a "Point Dipole" at 0 The field at f due to a point dipole at the origin is 7(0 =-^^l[^-3(m-f)f] (3.6) where m is the magnetic dipole moment of the dipole at the origin, which in our case is m = QFePBS = 9FePBSS (3.7) Therefore we find that l(r) = -^-^gFePBs[s-S(s-f)r} (3.8) which is equivalent to 7(f) = 4.636|^[3(5-r)r-s] T (3.9) where r is measured in Angstroms. 3.2.2 Magnetic Field at fp due to Eight "Point Dipoles" at fFei 8 Define the vector joining the pth nucleus and jth iron ion to be fPj = fp — fj. Then the total field at the pth nucleus due to the eight iron ions in the point dipole approximation is 7(rp) = 4.636 £ [3(Si • fpj)fpj - T (3.10) j=i \rpj\ where again distances are measured in Angstroms. 3.2.3 Isotopic Concentrations, Nuclear ^-factors and Quadrupolar Moments in Fe8 In table 3.1 we present information on the properties of the various nuclei that can be found in the Fe8 molecule. The nuclear magnetic moments are equal to gnn\I\ and are listed in units of nuclear magnetons. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 81 Species Concentration \f\ Nuclear Moment [//n] Quadrupole Moment Q [10 24cm2] 1H 99.98 1/2 2.79255 0 2H 0.02 1 0.857354 0.00273 98.88 0 0 0 13C 1.12 1/2 0.70225 0 UN 99.62 1 0.40365 0.02 15N 0.38 1/2 -0.2830 0 160 99.757 0 0 0 170 0.039 5/2 -1.8935 -0.005 1SQ 0.204 0 0 0 91.068 0 0 0 57Fe 2.20 1/2 0.05 0 50.56 3/2 2.10576 0.335 siBr 49.47 3/2 2.2696 0.280 Table 3.1: Nuclear spin information for nuclei occuring in Fes- From [48]. The isotopic concentrations shown in table 3.1 are the "naturally occuring" concen trations. It is quite possible to alter these concentrations and as such in what follows we shall treat the general case where the particular isotopic concentrations in the material may be varied. 3.2.4 Definition and Evaluation of , ui'kl and u£ 7^ and 7^ are the dipolar fields (in Tesla) at the point fk before/after the central spin flips respectively, for all nuclei k = 1..N + 8 (see (2.67) and (2.68)). Note that in the absence of an externally applied magnetic field we have 7^ = —7^ (the field due to the central spin cluster just flips its direction when the central object tunnels). Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 82 The energy of interaction of the nuclear dipole at rk with the external field is Uk - 9nPnh • Ik l+fz QnPri [gnpJk• ik]] + ^-y^ [gnPnh • fi2)] [^•(#+€))+U-(7i1)-#)] (3-11) As before we take -(1) -(2) -,(1) _ ^(2) - _ 7fc + Ik f _ 7fc Ik fi-\o\ *_l7i"+#l ' '"iTi""^! ' 1 ' 4'= I^WTI1'- €' I . 4 = l4l9«M»l7i1) + 7fl (3-13) which gives an interaction term of the form (compare to (2.73)) Uk = ^ik-mk + ^4h-ik (3.14) 3.2.5 Contact Hyperfine Coupling Energies for 57Fe3+ There is another contribution due to contact hyperfine interactions due to the presence of any 57Fe nuclei in the material. As was discussed in chapter 2, this coupling is of the form (see (2.61)) Ucp = gn^A^SVl (3.15) We are going to make the approximation in what follows that the off-diagonal elements of the tensor Apal3 are zero. This gives the form U; = ucpIp-Sp (3.16) where Sp and Ip are the electronic and nuclear spin of the pth 57 Fe ion respectively. Similarly to what we did with the dipolar term we write, with Sp^ and 5p2) the pth Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 83 electronic spin before and after the central spin flips respectively, u; = u;ip-sp f [TP • (S^ + Sf) + fzIp • (SM - Si2))] (3.17) Define 5(1) + 5(2) „ 5(1) _ 5(2) P ~ I 5(1) , 5(2) I ' ^-,5(1) 5(2) I ' i Op I I^P I 4c=\IPK\SU-SW\ , u,}* = \TP\U;\%» + (3.19) Then we may write the interaction term in the form Ucp = ^Ip • mcp + ^J\% • % (3.20) The value of the field at the nucleus of a free 57Fe3+ ion due to polarization of the s electrons by the outer shell 3d electrons has been previously calculated and was found to be Hc ~ 63 T [99]. If we take the nuclear ^-factor of the 57Fe ion to be g = 0.05 [166], this gives a zero field longitudinal contact hyperfine coupling of u;p|c~48 MHz Vp (3.21) These completely overwhelm the dipolar coupling energies (^k=N+i..N+s an<^ u)k=N+i..N+8^ as the dipolar fields at the iron nuclei are on the order of 300 — 800 gauss (see table 3.10). Because of this, it is easiest to think about the "diagonal" effects of the nuclear spins in the following way. All nuclear spins in the molecule k = 1..N + 8 are involved in dipole-dipole interactions via (3.14). However, the 57Fe nuclei k = N + 1..N + 8 also are involved in contact hyperfine interactions which absolutely swamp the dipole-dipole Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 84 interactions of the 57Fe nuclei. Therefore the interaction term in our Hamiltonian is (3.22) N+8 k=l where it is understood that the terms for k = 1..N are the dipole-dipole terms (3.14) but the terms for k = N + 1..N + 8 are contact terms from (3.20). 3.2.6 Calculation of uk and u)k from Knowledge of Atomic Positions In the Fes molecule, we know where all the atoms are in the molecule. This allows us to calculate what the parameters wk and u)^ are, for k = I..N, via the use of (3.10) and (3.13) together with the knowledge of the directions in which the central spin object points in the two lowest lying energy levels | +10 > and | — 10 >. This last is calculated as follows. The central spin Hamiltonian for Fe8, when truncated to terms of quartic order or —+ less in S, can be written in the presence of an external field H in the form H0 = -DS2Z +E(S2X- S2) + C(S* + Si) + 9iiBH • S (3.23) as discussed in chapter 2. Here we take D = 0.292/C, E = 0.046^ and C = -2.9 • 10~5K in keeping with the findings of Wernsdorfer et.al. [51]. If we write S in spherical coordinates Sx —>• S sin 9 sin 0, Sy —> S cos <j> sin 9 and Sz —> S cos 9 and substitute these into (3.23) it is then a simple exercise to find the angles (9, <j>) and therefore the quantities —• Sj for all the electronic spins in the molecule as functions of H. We assume that the individual electronic spins are locked to the direction in which the effective central spin is pointing, with relative signs given by taking | + 10 > and | — 10 > to correspond to the Fe3+ spins being {t, 1\ 1,1\ Ti t> I, T} and {I, I, t> l> l> 4> t> 4} respectively, where the labelling 1-4 8 corresponds to that in table 3.2. All coordinate positions are in Angstroms. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 85 We present information on the zero field values of the u>{! in the following manner. We label all the nuclei in the molecule such that the hydrogen atoms are tagged 1..120, the bromines 121..128, the nitrogens 129..146, the irons 147..154, the carbons 155..190 and the oxygens 191..213 (the positions of each of these are given in section 2.5). For each nucleus there are a variety of possible isotopes. Shown in figure (3.1) are the values for the zero field shown as functions of nucleus label for 1H, 79Br, UN, 57Fe, 13C and 170. This information is presented in a different manner in figures (3.2) through (3.13). In these figures we bin the hyperfine values, in each case assuming that the isotope in question represents 100% of the element in question (for example, in figure (3.2) we assume a 100% concentration of 1//). 3.2.7 Calculation of the Orthogonality Blocking Parameter K The Prokof'ev and Stamp theory [20] contains a parameter K which is defined to be AT+8 ncos^ Lfc=l (3.24) K = — In where cos2A = -7i1)-7i2) (3-25) Since we know what the fields 7^ and 7^ are, we may calculate the fa and then K. If the nuclear spin in question is a 57 Fe then because the contact hyperfine field is so much larger than the dipolar field (63 T and ~ 0.3 — 0.8 T respectively) we can take cos2& = -s£1).Sj2) (3.26) Note that for all nuclei fa is a strong function of the externally applied DC magnetic field. Shown in figures 3.14 and 3.15 is K as a function of a DC field applied in the x Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 86 Nuclear Label X y z Fel 5.737857 .064911 1.58115 Fe2 6.912112 -2.210346 3.04665 Fe3 6.656427 3.323106 .33915 Fe4 5.327499 2.408451 3.17265 Fe5 5.198078 -2.412666 -3.16665 Fe6 3.616621 2.196296 -2.98815 Fe7 3.888089 -3.321701 -.29265 Fe8 4.848748 -.043695 -1.54365 Table 3.2: Positions of the iron ions, units in Angstoms. 120.0 Figure 3.1: cJk for all nuclei in Fe8. Labeling is as indicated in the text. The dots represent values for 2H (labels 1..120), 81Br (labels 121..128), and 15A7 (labels 129..146). Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 87 6 |—, ——, , | , , , , , , r-, , 1 5 -4 -<D E 3 - I I z 2 - I I I I °0.0 5.0 10.0 15.0 20.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.2: 1H, emphasizing low end of the spectrum. 6 i • • 1 • • , • • 1 • • 1 5 -4 f 0.0 30.0 60.0 90.0 120.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.3: 1H, high end of the spectrum. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 88 6 i-5 -4 I 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.4: 2H, low end of spectrum. 6 | . . . , . . 5 -4 h w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.5: 2H, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 89 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.6: 79Br, entire spectrum. <D E 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.7: slBr, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 90 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.8: 14N, entire spectrum. 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.9: 15N, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 91 0.00 0.10 0.20 0.30 0.40 0.50 w_k|| [MHz], Binned in 0.001 MHz steps Figure 3.10: 57Fe, entire spectrum. <D £ 1 r 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps Figure 3.11: 13C, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 92 5.0 10.0 15.0 20.0 w_k||, [MHz], Binned in 0.1 MHz steps Figure 3.12: 170, low end of spectrum. 20.0 40.0 60.0 w_k||, [MHz], Binned in 0.1 MHz steps Figure 3.13: 170, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 93 direction for an Fe8 molecule with 100% concentrations of 1H, 81Br, UN, 57Fe, 13C and 170. We shall call this material Fe™ax. Note that the contribution to K from the presence of 57Fe is minimal because the minima of S don't change much as functions of field. 500.0 400.0 300.0 CO o. o. 200.0 100.0 0.0 0.0 0.5 1.0 H_x [Tesla] Figure 3.14: The orthogonality blocking parameter K over a large range of external fields applied in the x direction for the Fe™ai material. 3.2.8 Calculation of E0 The quantity E0 is related to the spread in energy space due to the presence of many nuclear spins. From (3.14) we see that in the absence of quadrupolar or contact in teractions and in zero external magnetic field the kth nuclear spin has 21 + 1 equally spaced energy levels between iwf/2. In the presence of an applied field the situation is similar. The variance in the distribution of the energy levels for this nucleus is defined to be a2 =< E2 > — < E >2 where E is the energy of the nucleus. If we make the approximation that the probability of each level being occupied is identical (effectively Total Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 94 10.0 cd Q. a 0.03 H_x [Tesla] Figure 3.15: The parameter K for small values of external field applied in the x direction for the FeT'ax material. an infinite spin temperature approximation) then < E >= 0 and |2 2/+1 cr2 = < E2 >= £ PiE2 = i=i i <4 21+ 1 4 2(2-1)' 2/ 7 + 1 |,2 (3.27) 12/ * Here £>i = ^f+i is the probability of the ith energy level of the nucleus being occupied. The total variance of the distribution of all nuclear levels is then given by the central limit theorem as N -°2 = zZ°. (3.28) fe=i where N is the total number of nuclear spins. In the case of Fe8 this gives = £(<K£(4/H£(4l.J) *2 ^79. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 95 + ^«:)+5E«2)+s«2) ksi Bt knN ki5N + E(4L/)+£EM + E( kl3c k57Fe + E UJ) (3-29) k57Fe The relation between the halfwidth of the distribution W, E0 and o is found via or El = Aa2 = 2W2 (3.31) The full width of the distribution is W = 2W = V2E0 (3.32) The width W is of course a function of the isotopic concentration in a particular Fe8 sample. For example, if we pick the easiest case where we have 100% concentrations of 1H, 79Br, UN, 160, 56Fe and l2C (we shall call this material Fe8*) then we find, in zero external field in the point dipole approximation, that 120 Y colk\H = 71887.545 [MH kiH=l z\2 \ E 47J = 70.27 [MHz]2 fc79Br=l o E "Lw = 181.8184 [MHz]2 (3.33) *J U . 1 which gives E0 = 268.6 MHz , W = 379.8 MHz = 18.23 mK (3.34) Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 96 What we have calculated here is an intrinsic linewidth due to spreading caused by the presence of the nuclear spins in the molecule. Note that these numbers are sensitive to which isotopic concentrations we choose for our molecule. This should lead to a clear isotope effect in this intrinsic linewidth which is easy to calculate using our formalism (it just amounts to changing the spin and nuclear moments for the new isotopes). For example, if we replace all the hydrogen nuclei by deuterium (we shall call this material Fesr)), which has 1 = 1 and g = 0.857354 /xn, we find that (3.35) which gives E0 = 69.06 MHz , W = 97.67 MHz = 4.687 mK (3.36) Note as well that because wjj is a function of external magnetic field (because the minima of S are), the linewidth EQ is also a function of external magnetic field. Shown in figure 3.16 is the intrinsic linewidth as a function of a field applied in the x direction due to specific isotopes for 100% concentrations of these isotopes (in other words, if the isotope is lH we are assuming that all the hydrogens are 1H; if the isotope is 81Br then we assume all the bromines are 81 Br, etc.). The addition of 57Fe to the mix significantly changes the value of the linewidth. This is because the contact hyperfine coupling energies are large. Let us define the material bl Fe% to be identical to Fe8* in every way except that every iron ion is a 57Fe ion; ie. 100% concentrations of 1H, 79Br, 15JV, 57Fe, l2C and 160. Then the contribution to the zero-field linewidth coming from the contact terms is, via (3.29), 8 8 £ 4'c2 ~ £ 2304 MHz2 = 18432 MHz2 (3.37) k=i k=i which is a significant fraction of the contribution from the protons (see (3.33)). Addition Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 97 0.5 1.0 H_x [Tesia] Figure 3.16: Intrinsic linewidth W due to particular isotopes as a function of Hx for 100% concentrations of these isotopes. Note that wf and therefore W drops slowly with field. This effect comes about because as the external field is raised, the two minima of the central spin complex are forced closer together (no longer are they antiparallel). The curve shown as "total" is the total result for a material containing 100% of the isotopes shown. of this term gives the linewidth £0 = 301.0 MHz , W = 425.6 MHz = 20.43 mK (3.38) which is 1.12 times the linewidth for Fe8*. 3.2.9 Calculation of Topological Decoherence Parameters AkND and A If we assume for the moment that there are no 57Fe nuclei in our molecule, then we can see from (2.89) that all parameters can be calculated in the expression for AkND. In addition, if we assume that the contact interaction is of the form (3.16) then we can calculate the general form for AkND. Furthermore, we see that these are not functions of the external magnetic field, unlike the orthogonality blocking parameters. Because the coupling energies are much less than the energy scale fin, all of the AkND turn out to 500.0 400.0 x — 300.0 5 200.0 100.0 •1 BramWe-81 Nitrogen-14 | Caroon-13 Oxyg.n-17 lron-67 - Total 0.0 0.0 Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 98 0.5 1.0 H_x [Tesla] Figure 3.17: Intrinsic linewidth W as a function of Hx for Fe8*, Fe8D and 57Fe8. be small (see figure 3.18). The topological decoherence parameter A [20] is given in this limit by * = ^E\An\2 (3-39) This is found to be A = 4.45 • 10~5, A = 8.87- HT5 and A = 1.38 • 10"5 for the Fe8„ 57Fe8 and Fe8o materials respectively. Note that these are extremely small! A is roughly the number of nuclear spins flipped per central spin tunneling event. 3.3 Using Free Fe3+ Hartree-Fock Wavefunctions to Model Actual Spin Dis tributions In this section we attempt to do a little better than the point dipole approximation. Here what we shall do is instead of treating the magnetic dipole nature of the Fe3+ ion as a point we shall assume that it is "spread out" in a way dictated by the spatial spread of the Fe3+ wavefunction. 500.0 400.0 I — 300.0 200.0 100.0 0.0 0.0 57-Fe Fe_8C Fe_8' Chapter 3. Nuclear Spi 99 0.000 0.001 0.002 0.003 0 0.001 0.002 0.003 Figure 3.18: Binned topological decoherence parameters \AkN D\ for all nuclei, assuming 100% concentrations of (clockwise from bottom left) 1H, 79Br, UN, 17O, 13C and 57Fe, using the point dipole approximation. The bin width here is 0.0001; plotted on the x axis is \A%D\ and on the y axis "number of nuclei". Note that the contribution to \AkNtD\ from 57Fe is almost entirely from the contact interaction. Specifically, we are interested in the five 3d electrons in the Fe3+ ion (its electronic configuration is of course [Ar]3d5). In a free Fe3+ ion, these five d electrons are spin-aligned due to the Hund's rule which asks for maximized spin angular momentum giving a total spin of 5/2. We can write down what the field at a point r is due to a Fe3+ ion at the origin; it is = ~ E/ j=i 3*T2-(™i ' [r-r]){f-r ) \r — T \2 (3.40) where m = gFel^BS = gFe^B^S (3.41) because the spin of each d electron is one half. Here the sum over j is over the five d electrons, and the integration r is over all space. We assume that the wavefunctions ^j{r) are properly normalized. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 100 Since the total angular momentum of the Fe3+ in its ground state is zero, the spin of the d electrons is distributed spherically and therefore we can approximate (3.40) by the expression 3 4rr J \r — r d m \f — r I2 (m • [f — r])(f' — r) (3.42) where ip(r) represents the spin distribution around the iron ion. The wavefunction ip(r) for a free Fe3+ ion has been previously calculated using a Hartree-Fock approach [100]. We can fit the numerical results of this calculation using the form ib(r) = (A + Br + Cr2)r2 exp (-r/d)1 (3.43) where A = -60.786097 , B = 68.94202-1 , C = -22.48757-L , d = 0.282745A (3.44) A A2 This is. the form we shall use in the following. Note that it is not exactly correct as immersion in the crystal will change the electronic distributions and therefore the spin distribution. However it is clear that using the free Hartree-Fock wavefunction here will give more realistic results than the point dipole approximation (it remains to see how different these are). The integrations in (3.42) are handled as follows. Instead of trying to do these ana lytically, we shall do them numerically using the following technique. We pick P points out of the \il>(r)\2 distribution to represent one iron ion wavefunction. This will be exact as P —> oo. Then 3 n} 4rrPptt \r-rp\3 fh _ (rh • [f- fp])(f- fp) T Tp | (3.45) Convergence is reached for P ~ 80 for all nuclei (the closer to an iron nucleus a proton is, the larger number of points are required for convergence). If the point dipole ap proximation were exact, then P = 1 would suffice (one point). Shown in figure (3.20) Chapter 3. Nuclear Spin Couplings in Fes and the Isotope Effect 101 24 15 05 / \ I \ I T-05 15 Figure 3.19: Hartree-Fock results for the free Fe3+ wavefunction. is a comparison of the Jjj. values obtained using the point dipole approximation and the Hartree-Fock method, using the Feg1"1 material (100% 1H, 81Br, UN, 170, 13C and 57Fe). We find that the lower energy nuclei are not affected by the change to the Hartree-Fock wavefunction. Only the higher energy nuclei are affected significantly. This however could be meaningful for several quantities of interest, primarily the intrinsic linewidth W which is sensitive to the higher energy couplings. Shown in figures (3.21) through (3.32) are the binned hyperfine values obtained using the Hartree-Fock wavefunction-these are the HF analogues of figures (3.2) through (3.13). We may recalculate K for the new fields generated in this approach. Shown in figures (3.33) and (3.34) are the analogues of figures (3.14) and (3.15), using the Hartree-Fock wavefunctions instead of the point dipole approximation. We may repeat our calcula tions for the intrinsic linewidth as well. With our new field values the numbers for Fe8* Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 102 120.0 i i . • i 110.0 100.0 -90.0 80.0 Nucleus Label Figure 3.20: Comparison of point dipole and Hartree-Fock methods; zero field wjl values in Fe™"1. The Hartree-Fock results are shown as dots. are, in zero external field using the Hartree-Fock wavefunction approximation, 120 2 £ UJIH = 71512.22[M#z]2 kH=i I £ 4B? = n.2l[MHzf y fcBr=i o 18 2 | £ 4N = imm[MHzf (3.46) 6 kN=l which gives EQ = 258.9 MHz , W = 366.2 MHz = 17.57 mK (3.47) Note that the values obtained are quite close to those obtained using the point dipole approximation. Note however that this did not have to be the case, as some of the larger w| increased and some decreased in going from the point dipole to Hartree-Fock approximations. Chapter 3. Nuclear Spin Couplings in Fes, and tie Isotope Effect 103 0.0 4.0 8.0 12.0 16.0 20.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.21: H, Hartree-Fock, emphasizing low end of the spectrum. 40.0 80.0 w_kll MHz, Binned in 0.1 MHz steps 120.0 Figure 3.22: 1H, Hartree-Fock, high end of the spectrum. Chapter 3. Nuclear Spin Couplings in Fes and the Isotope Effect 104 E 0 0.0 4.0 8.0 12.0 16.0 20.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.23: 2if, Hartree-Fock, low end of spectrum. .Q E 0.0 10.0 20.0 30.0 40.0 w_kll, MHz, Binned in 0.1 MHz steps 50.0 Figure 3.24: 2H, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe& and the Isotope Effect 105 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.25 : 79Br, Hartree-Fock, entire spectrum. 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.26: 81Br, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 106 2.0 4.0 6.0 8.0 w_kll, MHz, Binned in 0.1 MHz steps 10.0 Figure 3.27: N, Hartree-Fock, entire spectrum. 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.28: 15JV, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 107 0.0 0.1 0.2 0.3 0.4 0.5 w_kll, MHz, Binned in 0.001 MHz steps Figure 3.29 : 57Fe, Hartree-Fock, entire spectrum. 2.0 4.0 6.0 8.0 w_kll, MHz, Binned in 0.1 MHz steps 10.0 Figure 3.30: 13C, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 108 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.31: 17O, Hartree-Fock, low end of spectrum. 10.0 20.0 30.0 40.0 w_kll, MHz, Binned in 0.1 MHz steps 50.0 Figure 3.32: 170, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 109 500.0 400.0 300.0 200.0 100.0 Total Hydrogen Bromine Nitrogen Carbon Oxygen lron-57 Figure 3.33: The orthogonality blocking parameter function of Hx in the Hartree-Fock wavefunction picture. 0.00 0.01 0.02 0.03 H_x [Tesla] Figure 3.34: The orthogonality blocking parameter function of Hx in the Hartree-Fock wavefunction picture, focusing on small fields. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 110 As before, we can redo this calculation with any isotopic concentration. Here we treat the case where all protons are replaced by deuterium. This gives 120 2 2 ^ II 2 /0.857354\2 kD=i which gives || 2  ™ || 2 /0.857354\2 X>1'„ ^5 E4'„ (-j^J (3.48) £0 = 68.32 MHz , W = 96.62 MHz = 4.676 (3.49) which is very close to the point dipole result. Similarly to what we did in the point dipole case we compute the intrinsic linewidth W as a function of Hx for the three materials Fe8*, Fe8D and 57Fe8 and show the results in figure (3.35). We conclude by recalculating the topological decoherence parameters. Using our new field values we find that A = 4.23 • 10~5 , 8.73 • 10~5 , and 1.35 • 10~5 for the Fe8„, 57Fe8 and Fe8D materials respectively. Values for lA^^j are shown in figure 3.36. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 111 =• 300 0 S7-F« e Fa_8D 1 Fe 8-S 200.0 0.5 H_x [Testal Figure 3.35: Intrinsic linewidth W as a function of Hx for Fes*, Fe$D and 57Fe8 in the Hartree-Fock picture. 0 000 0.001 0.002 0.003 0 0.001 0 002 0 003 Figure 3.36: Binned topological decoherence parameters |A^D| for all nuclei, assuming 100% concentrations of (clockwise from bottom left) 1H, 79Br, 14AT, 17(9, 13C and 57Fe, using the Hartree Fock approximation. The bin width here is 0.0001; plotted on the x axis is |A^£,| and on the y axis "number of nuclei". Note that the contribution to |A^£,| from 57Fe is almost entirely from the contact interaction. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 112 3.4 Tables of Nuclear Positions, Fields at Nuclei and Hyperfine Coupling Energies In what follows we shall be considering only the case where the external magnetic field is zero for clarity of presentation. In these tables we indicate the locations of each ion in the molecule, presented in Cartesian coordinates (x,y,z) in Angstroms. As well we present the magnitude of the field at each nucleus due to the eight iron spins (note that in zero external field this magnitude is the same for both configurations of the central spin complex) both for the point dipole approximation and the Hartree-Fock approximation. We also present the hyperfine coupling energies u)k for both point dipole and Hartree-Fock cases (u^ is zero when the external field is zero). Here we have chosen the following isotopes in order to convert from field to energy units: XH, 79Br, 14N, 57Fe, nC, and 170. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 113 Nuclear X y z 17(1)IP1 a; II [MHz] IT(1)I m Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) HI 10.8 3.58 1.92 .0509 2.167 .0521 2.22 H2 9.38 2.12 1.38 .161 6.854 .152 6.476 H3 9.6 2.61 -.66 .112 4.757 .112 4.76 H4 10.3 4.3 -.173 .0422 1.797 .0415 1.772 H5 9.07 5.17 -1.88 .0978 4.168 .0943 4.016 H6 7.29 4.78 -2.37 .284 12.055 .298 12.676 H7 8.03 6.8 -1.13 .0556 2.365 .0577 2.46 H8 9.54 6.61 -.0732 .0242 1.026 .0246 1.052 H9 9.52 7.18 2.03 .0277 1.175 .0274 1.172 H10 8.24 6.26 2.61 .0988 4.209 .101 4.292 Hll 9.94 5.34 3.62 .054 2.301 .054 2.304 H12 10.6 5.61 2.36 .0347 1.477 .0345 1.472 H13 3.4 5.03 3.38 .15 6.383 .149 6.344 H14 3.68 4.34 4.85 .256 10.939 .251 10.708 H15 5.91 5.96 5.19 .106 4.532 .104 4.444 H16 5.79 5.65 3.57 .2 8.519 .2 8.532 H17 7.62 4.27 7.01 .0927 3.945 .0924 3.944 H18 5.88 4.07 6.59 .179 7.632 .183 7.824 H19 10.8 -.193 5.82 .0807 3.436 .0821 3.504 H20 10.2 -1.76 6.24 .109 4.626 .109 4.628 H21 11 -2.19 4.4 .0924 3.938 .0924 3.944 H22 10.3 -1.09 3.45 .137 5.832 .136 5.776 H23 6.01 1.8 7.08 .186 7.915 .189 8.056 H24 7.02 1.88 6.14 .249 10.634 .249 10.628 H25 2.87 .67 5.39 .195 8.311 .201 8.58 H26 3.72 2.35 5.84 .321 13.67 .326 13.88 H27 1.63 2.12 3.77 .11 4.68 .108 4.592 H28 2.01 .981 2.96 .109 4.656 .107 4.572 H29 6.79 -2.16 -5.9 .307 13.088 .326 13.94 H30 7.63 -.505 -5.25 .195 8.328 .192 8.196 Table 3.3: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 114 Nuclear X y z |7(1)I [T] Jl [MHz] l7(1)l [T] Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H31 8.61 -1.06 -2.96 .104 4.423 .101 4.316 H32 8.83 -2.19 -3.93 .118 5.023 .118 5.032 H33 6.52 -4.3 -5.18 .277 11.756 .277 11.784 H34 6.88 -5.19 -3.84 .153 6.517 .148 6.312 H35 4.57 -5.64 -3.71 .194 8.255 .19 8.1 H36 4.24 -5.96 -5.37 .0985 4.201 .0971 4.136 H37 4.52 -3.93 -6.69 .174 7.418 .178 7.568 H38 2.79 -4.09 -7.02 .0938 4.004 .0962 4.1 H39 3.43 -1.79 -6.2 .234 9.961 .232 9.884 H40 4.47 -1.73 -7.13 .179 7.632 .186 7.924 H41 4.21 5.46 -3 .163 6.943 .156 6.664 H42 3.44 5.64 -4.65 .131 5.581 .127 5.404 H43 .303 .862 -3.45 .145 6.198 .144 6.136 H44 -.515 1.95 -4.2 .0924 3.938 .0919 3.916 H45 .385 1.99 -6 .116 4.964 .115 4.916 H46 -.312 .338 -5.82 .0798 3.4 .0807 3.44 H47 1.19 -7.11 -2.18 .0331 1.409 .0326 1.392 H48 .406 -6.44 -1.32 .0274 1.169 .0272 1.16 H49 .0679 -5.37 -3.3 .042 1.785 .0425 1.812 H50 1.72 -5.16 -3.29 .124 5.287 .118 5.048 H51 1.75 -6.92 .851 .0333 1.423 .0321 1.372 H52 3.39 -5.77 1.73 .193 8.218 .18 7.676 H53 6.48 -3.54 6.51 .21 8.967 .221 9.436 H54 8.07 -3.5 6.48 .186 7.915 .199 8.492 H55 8.47 -1.16 7.11 .14 5.949 .138 5.904 H56 6.89 -1.24 5.94 .364 15.498 .368 15.716 H57 3.61 1.38 -5.96 .361 15.383 .387 16.488 H58 2.01 1.26 -7.1 .137 5.862 .14 5.96 H59 2.3 3.54 -6.3 .188 8.014 .184 7.856 H60 3.84 3.65 -6.53 .195 8.302 .192 8.188 Table 3.4: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect Nuclear X y z i7(l)i m Jl [MHz] i7(l)rm a;I' [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H61 .623 -2.88 .611 .08 3.407 .0718 3.056 H62 1.95 -1.71 .255 .251 10.708 .232 9.884 H63 1.75 -4.88 2.25 .129 5.505 .132 5.612 H64 .617 -5.23 .629 .0481 2.05 .0439 1.868 H65 8.72 -5.09 3.6 .137 5.85 .131 5.604 H66 9.24 -4.12 5.09 .165 7.021 .171 7.284 H67 6.92 -5.64 4.83 .131 5.57 .129 5.52 H68 6.26 -5.47 3.17 .163 6.965 .157 6.708 H69 1.1 4.07 -4.8 .16 6.83 .154 6.572 H70 1.72 4.94 -3.24 .142 6.053 .139 5.916 H71 .272 -2.64 -1.92 .0859 3.662 .0828 3.532 H72 -.208 -4.22 -1.48 .0453 1.927 .0446 1.904 H73 8.05 -.671 4.23 .622 26.468 .629 26.844 H74 7.98 -2.98 2.43 1.82 77.829 1.87 79.864 H75 5.76 -3.43 3.78 1.08 46.004 1.04 44.464 H76 7.86 3.23 1.98 1.12 47.703 1.16 49.468 H77 7.11 2.81 -1.02 2.44 103.95 2.65 113.492 H78 6.67 5.16 .275 .758 32.286 .779 33.22 H79 3.8 2.86 2.4 1.12 47.649 1.02 43.636 H80 7.07 3.95 4.32 .432 18.397 .439 18.688 H81 4.83 .935 4.13 .877 37.415 .823 35.084 H82 5.68 -.979 -4.05 .943 40.203 .833 35.512 H83 3.59 -3.76 -4.2 .579 24.676 .547 23.344 H84 6.76 -3.06 -2.37 .934 39.825 .903 38.492 H85 2.49 .701 -4.22 .647 27.551 .64 27.288 H86 4.9 3.6 -3.77 .765 32.557 .772 32.876 H87 2.54 2.85 -2.31 2.09 89.109 1.55 66.084 H88 3.73 -5.2 -.498 .72 30.655 .631 26.856 H89 2.7 -2.96 -1.82 1.24 53.006 1.34 57.224 H90 3.58 -2.99 1.29 2.06 87.885 2.74 116.536 Table 3.5: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect Nuclear X y z |7(1)I [T] Jl [MHz] |7(1)I [T] Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H91 8.78 .0423 5.01 .231 9.837 .242 10.316 H92 7.92 2.02 3.9 .403 17.188 .406 17.348 H93 5.34 -1.03 3.83 .542 23.124 .575 24.52 H94 6.2 -4.09 1.14 .371 15.776 .354 15.064 H95 8.76 3.34 2.78 .303 12.891 .293 12.54 H96 3.29 .719 1.14 .187 7.979 .182 7.756 H97 7.21 3.45 -1.76 .934 39.767 1.16 49.388 H98 7.9 -.733 1.32 .432 18.425 .415 17.656 H99 7.15 6.11 .465 .181 7.7 .172 7.32 H100 5.05 3.95 1.35 1.16 49.5 1.23 52.368 H101 2.81 3.16 2.39 .242 10.279 .246 10.492 H102 2.66 .832 -1.47 .556 23.686 .544 23.176 H103 5.85 1.45 -3.24 .638 27.153 .622 26.464 H104 2.57 -2.02 -3.9 .392 16.747 .38 16.208 H105 5.99 -.381 -4.76 .396 16.94 .42 17.892 H106 7.21 -.719 -1.14 .185 7.888 .193 8.208 H107 2.99 -4.75 -5.18 .15 6.398 .152 6.488 H108 5.45 -3.95 -1.35 1.17 49.671 1.19 50.76 H109 7.68 -3.16 -2.34 .249 10.567 .23 9.792 H110 4.29 4.05 -1.14 .378 16.101 .387 16.488 Hill 1.72 -.0423 -5.01 .227 9.663 .226 9.652 H112 7.97 7.92 7.83 .0219 .934 .0222 .948 H113 2.53 6.18 7.14 .0427 1.817 .0427 1.824 H114 6.65 6.66 9.57 .0265 1.133 .027 1.152 H115 3.35 -6.11 -.465 .182 7.755 .187 7.992 H116 7.55 5.74 10.6 .0242 1.033 .0246 1.048 H117 1.73 -3.34 -2.78 .291 12.401 .3 12.828 H118 2.95 8.36 4.44 .0227 .969 .0229 .976 H119 3.85 7.44 5.43 .0389 1.662 .0396 1.688 H120 9.69 -.874 .24 .0877 3.742 .0891 3.804 Table 3.6: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fes and the Isotope Effect 117 Nuclear X y z |7(1)| [T] Jl [MHz] |7(1)I [T] Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) Brl 10.2 2.65 5.67 .0521 2.215 .0537 2.293 Br2 3.37 -2.3 4.91 .137 5.856 .142 6.063 Br3 5.2 7.11 7.43 .0289 1.23 .0298 1.273 Br4 .265 -2.64 -5.66 .0516 2.201 .0535 2.279 Br5 7.09 2.28 -4.91 .14 5.957 .145 6.167 Br6 7.64 -4.75 .0737 .091 3.878 .0943 4.015 Br7 1.06 1.08 -.26 .0816 3.481 .0844 3.604 Br8 3.7 5.53 .2 .0734 3.132 .076 3.243 Table 3.7: Data for Bromine. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 118 Nuclear X y z |7(1)I [T] Jl [MHz] i7(i)im Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) NI 8.52 -1.05 5.1 .0547 2.332 .053 2.26 N2 8.73 -3.03 3.35 .0891 3.796 .0849 3.62 N3 6.48 -3.78 4.49 .112 4.765 .108 4.61 N4 8.87 3.76 1.97 .0486 2.073 .0479 2.04 N5 8.09 3.55 -.923 .138 5.892 .14 5.95 N6 7.75 5.71 .515 .0333 1.419 .0392 1.67 N7 3.45 3.02 3.18 .0917 3.908 .0807 3.44 N8 6.55 4.24 4.89 .0525 2.242 .0577 2.46 N9 4.83 1.44 4.98 .13 5.526 .132 5.63 . N10 5.7 -1.4 -4.92 .13 5.533 .136 5.78 Nil 3.96 -4.13 -4.86 .0582 2.476 .0525 2.24 N12 7 -3.14 -3.23 .0929 3.964 .0884 3.77 N13 2.01 1.12 -5.06 .0568 2.423 .0638 2.72 N14 4.07 3.86 -4.4 .1 4.283 .0896 3.82 N15 1.77 2.92 -3.2 .0927 3.947 .083 3.54 N16 2.73 -5.67 -.426 .0342 1.459 .045 1.92 N17 1.7 -3.67 -1.94 .0532 2.268 .0605 2.58 N18 2.48 -3.44 .983 .146 6.228 .145 6.18 Table 3.8: Data for Nitrogen. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect Nuclear X y z |7(1)I [T] a; II [MHz] i7(i)i m [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) CI 10 -1.15 5.49 .0352 1.499 .0357 1.52 C2 10.2 -1.83 4.1 .0422 1.797 .0441 1.88 C3 8.52 -4.37 4.1 .0659 2.807 .069 2.94 C4 6.95 -4.98 4.11 .0638 2.72 .0615 2.62 C5 7.21 -3.19 6.03 .0833 3.553 .0805 3.43 C6 7.76 -1.57 6.15 .0762 3.254 .0758 3.23 C7 9.72 3.16 1.34 .035 1.487 .035 1.49 C8 9.51 3.41 -.168 .035 1.491 .0342 1.46 C9 8.22 4.94 -1.55 .0633 2.703 .0615 2.62 CIO 8.46 6.13 -.569 .0217 .925 .0221 • .941 Cll 8.82 6.22 2 .0206 .878 .0213 .908 C12 9.69 5.27 2.58 .0194 .829 .0198 .844 C13 3.94 4.46 3.98 .0837 3.572 .0819 3.49 C14 5.59 5.22 4.4 .0598 2.553 .0615 2.62 C15 6.6 3.79 6.3 .0507 2.162 .0518 2.21 C16 6.18 2.13 6.23 .0812 3.456 .0821 3.5 C17 3.43 1.57 5.12 .0936 3.993 .0919 3.92 C18 2.45 1.85 3.65 .0521 2.224 .0495 2.11 C19 7.07 -1.45 -5.07 .0919 3.915 .088 3.75 C20 8.09 -1.88 -3.72 .0521 2.216 .0528 2.25 C21 6.36 -4.53 -4.28 .0851 3.631 .0821 3.5 C22 4.7 -5.22 -4.55 .0568 2.416 .0577 2.46 C23 3.83 -3.64 -6.35 .0507 2.162 .0488 2.08 C24 4.26 -2.07 -6.27 .0758 3.233 .0795 3.39 C25 2.73 1.66 -6.12 .0753 3.205 .0748 3.19 C26 3.2 3.26 -5.96 .0805 3.432 .0823 3.51 C27 3.5 4.96 -3.95 .0643 2.739 .0643 2.74 C28 .382 1.66 -4.01 .0432 1.835 .0418 1.78 C29 1.88 4.27 -3.84 .0666 2.837 .0687 2.93 C30 .49 1.26 -5.37 .0361 1.536 .038 1.62 C31 1.25 -6.23 -1.68 .0153 .652 .0147 .627 C32 1.13 -5.13 -2.69 .0253 1.081 .0249 1.06 C33 .645 -3.34 -1.37 .0298 1.273 .0307 1.31 C34 1.4 -2.74 .183 .0551 2.348 .0558 2.38 C35 1.68 -4.91 1.25 .0425 1.814 .0434 1.85 C36 2.44 -5.92 .938 .0333 1.424 .0331 1.41 Table 3.9: Data for Carbon. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 120 Nuclear X y z |7(1)I [T] Jl [MHz] |7(1)I [T] J' [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) FE1 5.72 .0651 1.58 .00729 .311 .00729 .311 FE2 6.9 -2.21 3.05 .00467 .199 .00467 .199 FE3 6.65 3.34 .339 .003 .128 .003 .128 FE4 5.31 2.41 3.18 .00793 .338 .00793 .338 FE5 5.19 -2.43 -3.17 .00779 .332 .00779 .332 FE6 3.61 2.2 -2.99 .00467 .199 .00467 .199 FE7 3.89 -3.33 -.293 .00293 .125 .00293 .125 FE8 4.84 -.0439 -1.55 .00737 .314 .00737 .314 Table 3.10: Data for Iron. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect Nuclear X y z |7(1)I [T] Jl [MHz] Jl [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) 01 6.9 1.68 3.29 .746 31.831 .699 29.8 02 5.4 -1.44 2.99 .798 34.036 .877 37.4 03 5.57 -3.6 1.31 .253 10.783 .249 10.6 04 4.14 .682 1.88 .544 23.202 .518 22.1 05 7.59 -.53 2.12 .608 25.926 .671 28.6 06 5.8 3.68 1.76 1.2 51.071 1.06 45 07 6.14 1.37 .185 .969 41.303 .931 39.7 08 4.5 -1.34 -.115 .973 41.493 .978 41.7 09 3 .503 -2.03 .633 26.96 .612 26.1 010 5.13 1.44 -2.94 .783 33.367 .887 37.8 Oil 3.65 -1.68 -3.24 .798 33.975 • .877 37.4 012 6.5 -.649 -1.85 .471 20.113 .483 20.6 013 4.74 -3.71 -1.76 1.14 48.459 1.15 49.2 014 4.94 3.55 -1.29 .239 10.154 .251 10.7 O100 8.98 8.16 7.79 .0121 .516 .012 .512 020 1.58 6.01 7.14 .0246 1.046 .0258 1.1 030 -1.02 6.85 5.61 .012 .51 .0116 .493 040 11.4 7.23 9.35 .00868 .37 .00901 .384 050 6.85 6.23 10.6 .0155 .662 .0153 .651 060 3.56 8.15 4.43 .0182 .775 .0174 .741 070 9.32 5.56 10.5 .0139 .593 .0134 .571 080 1.11 8.53 4.67 .0115 .492 .0116 .496 090 8.96 -1.17 .278 .101 4.302 .0973 4.15 Table 3.11: Data for Oxygen. Chapter 4 An Introduction to the Generalized Landau-Zener Problem An introduction to the standard Landau-Zener problem is presented. Three methods of solution are considered; one involves solving directly for the wavefunctions of the two-level system, the second is a perturbation expansion in the tunneling term and the third uses a complex analysis technique usually referred to as Dychne's formula. We demonstrate that it is possible to solve a more general version of the simple Landau-Zener Hamiltonian and present the solutions (see [102] for a complementary analysis). 4.1 Introduction to and Exact Solution of the Landau-Zener Problem Consider a two-level system (TLS) with time-dependent Hamiltonian H(t) = vtoz + AJx (4.1) where {a} are the Pauli matrices. This rather generic effective description was first considered by Landau [103], Zener [104] and Stuckelberg [105]. Landau and Stuckelberg used this Hamiltonian to model the evolution of two atoms scattering off each other, while Zener used it to model the evolution of the electronic states of a bi-atomic molecule. It has since been used in a large number of different contexts; chemical reaction kinetics [106], biophysics [107], examination of the solar neutrino puzzle [108], aspects of nuclear magnetic resonance [109], behaviour of atoms in photon fields [110], surface scattering [111], electric breakdown in solids [112] and many more. The reason for its wide usage is evident. There are many real physical systems that for one reason or another can be 122 Chapter 4. An Introduction to the Generalized Landau-Zener Problem 123 modeled as two level systems-either they really are two level systems (for example, the spin Hilbert space of a spin 1/2 particle) or they can be mapped to one (for example, the O(DS) low energy effective Hamiltonian derived in chapter 2). It is often useful to know how these systems respond to an externally applied time-dependent perturbation. It is clear that the simplest effective description of a two level system coupled to a time dependent perturbation is that given in (4.1). We have sketched the energy levels of this Hamiltonian as functions of time in figure 4.1. Figure 4.1: Energy levels of the Landau-Zener Hamiltonian. Shown are both the eigen states of oz, which are linear in time, and the eigenstates of H(t), E±(t) = ±(A2+v2t2)1/2. One of the most useful features of the description (4.1) is that one can solve it exactly, in the sense that one can solve for the wavefunctions explicitly as functions of time. In order get a feel for the model, we shall outline in the following how these wavefunctions are extracted. Insertion of (4.1) into the Schrodinger equation (we choose a system of units such Chapter 4. An Introduction to the Generalized Landau-Zener Problem 124 that h = l) .d i— dt [A) H(t) (4.2) yields I—— = Vtlpa + Alpb dt .dipb dt Atpa - Vtlpb eliminating ipa allows us to write ib\ + (y2t2 + A2 - iv) ipb = 0 (4.3) (4.4) where overdots denote derivatives with respect to t. This is the equation for a parabolic cylinder function. It has two solutions which may be written (i) _ -FH/I_,A£ I (ivt2) (2) /T 4 ' 4u 5 4 * ' (4.5) where W is the Whittaker function [113]. If we consider the asymptotic forms of these solutions as t —> — oo, we find that (1) 2 6 (2) |2 t—y — cx> 0 (4.6) This indicates that the choice of one of the two solutions is equivalent to the choice of an initial condition on the wavefunction ipb. The analysis of the other wavefunction ipa proceeds in an identical manner; solving 4>a + (y2t2 + A2 + iv) ipa = 0 (4.7) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 125 gives two solutions y/t 4 4u ' 4 7 (4.8) which have deep past asymptotics (1)|2 (4.9) This is consistent with our interpretation of the choice of one of these being equivalent to the choice of initial conditions. That is, preparation of the system in state tpa at time t = — oo requires that we use solutions ip^ and Similarly, preparation in the state ipb requires the use of ip^ and ip^. Finding the transition probabilities is now a straightforward exercise. The probability of finding the system in state ipa at time t given that it started in state ipa = | t> at time t = —oo is simply |2 Pt(t) = \^(t)\2 = \ft 4+ 4v '4 (4.10) This solution is plotted in figure 4.2. As t —>• +oo (4.10) asymptotically approaches Pt(t ->• oo) = 1 - e~^ (4.11) and therefore the probability to make a transition is Pn(t -> oo) = e'^ (4.12) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 126 Figure 4.2: Transition probability (4.10) as a function of t (in units of A) . Here we have taken A2/v = 0.63, 1 and 5 for the solid, dotted and dashed lines respectively. 4.1.1 Alternate Method of Solution for the Transition Probability I. All Orders Perturbation Expansion It is also possible to solve for the transition probabilities without first finding the wave-functions [104]. Consider the amplitude Aa->b=<b\U(tf,ti)\a> (4.13) where U{tf,ti) = TeittH{T)*r (4.14) is the time evolution operator (T meaning "time ordered") and a, b can be either "up" or "down" (<Tz = ±1, respectively). Splitting the Hamiltonian into diagonal and off-diagonal (in o) parts H(t) = Hd(t) + allows us to rewrite the evolution operator in the more convenient form [71] Ufo^) = e-l%Hd{T)dTTe-l%"*dT (4.15) where 7-7 (+\ J It Hd(T)dT JJ -i f' Hd(r)dT . . H^(t) = e Jf* HAe Jti (4-16) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 127 Expansion of the time-ordered exponential yields A^b = f)(-t)B f dtn fn dtn_x... f2 dt, < b\ei^drvra'A&x 71=0 * * * , d,TVTdz „ i f'2 d,TVT&z . . i f*1 drvruz i ^ /, ,M e Jtn-i Aox...eJti Aoxe J-°° \a > (4-17) f) A2" ^ dt2n cft^-!... dtie- iv?T-il-W (4.18) Let us assume that a = b, and furthermore that \a >= | t> (the solution for \a >= | 1> is similar, differing only in an overall phase factor). Inspection of (4.17) shows that in this case only paths with n even contribute, and furthermore that the inner product in (T-space is easy to perform. Explicitly we find ftf rt2n T*2 dtzn I dtm-\ • • • N = Q J—ti Jti Jti where <p is an uninteresting phase. One sees that all of the time integrals may now be performed if we take U = — oo and tj = +oo; otherwise we are stuck. That is, it seems as though this approach gives us less information than the solution for the wavefunctions performed in the preceding section; this is somewhat strange, as usually if a problem is solvable in terms of known special functions in one representation it is usually solvable in all of them. In any case, we shall now take ti = —oo and tj = +oo. In this case the integrations over the time set {tj} can be performed explicitly, giving with subsequent probability Pn = 1 - \An\2 = (4.20) as before. Chapter 4. An Introduction to the Generalized Landau-Zener Problem i 128 4.1.2 Alternate Method of Solution for the Transition Probability II. Dychne's Formula The final method that we shall review for extracting results from (4.1) was first suggested by Landau [114] but later became known as Dychne's formula [?]. This method uses arguments from the theory of complex analysis in order to extract transition amplitudes from two level time dependent Hamiltonians such as the Landau Zener model (4.1). We shall not give a detailed analysis of this method, but just present its basic result. We begin by defining the standard rotation matrix cos6>/2 -sin 9/2 sin 9/2 cos (9/2 which, if 9 is a function of time, can diagonalize the general Hamiltonian Vji(t) A(t) R (4.21) H = V\\{t)crz + A(t)ox = A(t) -Vj|(t) (4.22) giving RHRj E-{t) 0 0 E+(t) (4.23) where E±(t) = ±JA*(t) + W(t) (4.24) are the adiabatic energy levels. In the case of the simple Landau Zener Hamiltonian, we find that E±{t) = ±VA2 + v2t2 (4.25) The result of Landau and Dychne states that the probability for making a transition from one eigenstate of oz to the other if the Hamiltonian is evolved over the range Chapter 4. An Introduction to the Generalized Landau-Zener Problem 129 —oo < t < oo is approximately Pa^b = e~2Im^ (4.26) where f(t) = 2 j1E{r)dT (4.27) and tc is the zero of E(t) in the upper half plane that is closest to the real axis. Note that this requires that we analytically continue the time variable throughout the complex plane. The error in this expression comes from the neglect of the contribution of all the other zeroes of E(t) in the upper half complex plane which are omitted here. Generally speaking, one can tell whether or not this method will give useful results by looking at the structure of the zeroes of E(t). If there are many closely spaced zeroes in E(t) off the real axis, then the contributions from subdominant terms will grow and this method will fail. For example, in the case of the simple Landau Zener model, there are only two points in the complex time plane where E(t) vanishes, namely tc = izA/y^, and only on of these is in the upper half plane. Because of this we expect that in this case the results obtained using Dychne's formula should be exact. This result is interesting as it provides insight into the reasons why this model is exactly solvable and many other similar models are not-for example, changing the time dependence from linear to say cubic in Vj|(t) causes the methods used in the preceding chapters to fail to produce exact results. This is most probably related to the existence of now two zeroes of E(t) in the upper half plane which introduce errors into the formula (4.26). It is quite likely that there is a deep connection here between the theory of special functions and issues in complex analysis. However, we do not choose to pursue this avenue at the present time-we shall only mention it in passing. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 130 Let us now apply Dychne's formula to the Landau Zener Hamiltonian. We find that f (*) = 21* VA2 + v2r2dr = WA2 + v2t2 + ^ In (yt + VA2 + v2t2) (4.28) and therefore A27T Im ate) = ^rj (4.29) and Pa^b = e'^T (4.30) which agrees with our previous exact results. 4.1.3 Analysis of the Transition Formula We now wish to step back from the preceding technical exercise and draw some conclu sions from this analysis. From the outset it should have been clear that the presence of the tunneling term Aox in the Hamiltonian would mix the two states that we are calling ipajt,- Furthermore one expects that the dimensionless parameter A2/v should be impor tant in the final transition expression. Both these suspicions, as we have seen (4.12), turn out to be justified. What else can we say about this solution? Perhaps the key point here is that as we are dealing with a two-level system the equa tions for the wavefunctions have to be second order homogeneous differential equations. For the specific Hamiltonian that we were working with (4.1) this differential equation turned out to be one that is a well-studied specific case of the hypergeometric equa tion. This allowed us to write down general solutions. Supplanting these solutions with information about the asymptotic behaviour of the system then gave us the specific tran sition probabilities that we were after. This realization suggests that there exists a class of two level time dependent Hamiltonians whose wavefunctions are obtainable in terms Chapter 4. An Introduction to the Generalized Landau-Zener Problem 131 of known special functions, and that this class is that whose Schrodinger equation can be mapped to the hypergeometric equation. The full determination of what is required of the Hamiltonian in order for it to belong to this class is a difficult task. We shall in the following demonstrate the method for some specific cases. 4.1.4 Generalization of the Two-Level Landau-Zener Problem I. Exact Solution for A(t) ~ Vj|(t) Let us now consider a more general case of (4.1), namely [102] H(t) = V\\(t)az + A{t)ax (4.31) We may perform the same type of analysis as we did in section 5.1. The Schrodinger equation . d i—-dt ( = H(t) yields t^ = A(t)^a - v{lmb eliminating x/ja allows us to write A - + (v~(\ + A2 -i Mi - Mi A ^6 = 0 (4.32) (4.33) (4.34) where again the overdot represents a time derivative. As before, we have to supplant this equation with a specification of the initial conditions; here we shall assume that \i)b{t= -00)|2 = 1 (4.35) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 132 ie. that the system has been prepared in state tpb in the deep past. The equation (4.34) cannot be solved with known special functions for general V\\(i) and A(t). However, it is possible to recast it in a form that allows solution for quite a few interesting specific cases. We map t -r z(t) (4.36) where the only requirement we have at this stage being that the map is onto, ie. 1 —> 1. With differentiation with respect to z denoted by primes, (4.34) becomes A + z A 7 ~ A" Vn + A2 i2 ^ii-Mix ^6 = 0 We now choose the mapping such that (4.37) i = A(t) (4.38) We then find that 1 + v2 ILL A2 i A /A A ^6 = 0 (4.39) Writing our equation in this form highlights the fact that if V\\ and A are constant multiples of each other, ie. Mi = KA (4.40) with K constant, then (4.39) reduces to ib'l + (l + K2) ^ = 0 (4.41) Define 72 = 1 + K,2; then ipb(z) = c0e 172 + cie i-yz (4.42) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 133 is the general solution of (4.41). In order to determine the normalization constants we require the following. Firstly, we want the solution for the probability to be bounded everywhere by 1; explicitly, |^6|2 = |Co|2 + |Cl|2 + c*Cle2iZ7 + clCoe-2jZ7 < 1 (4.43) As well, we know the solution for V\\ = 0 with our initial condition is simply |V;fc|2 = |c0|2 + |C!|2 + c*oCle2iz + c^cne"212 = cos2(z) (4.44) These fix the constants to be CQ = c\ = 1/2, and \Mz)\2 = cos2(7z) (4.45) In terms of our original parameters this transition probability is written \^b(t)\2 = cos2 [(1 + K2)1'2 f A(r)dr\ (4.46) This result has been derived previously by different methods [115]. We may pause now and ask how this exact solution compares to that obtained using Dychne's formula. In this particular case, we find that E±(t) = ±]/A2(t) + V2(t) = ±vT+^2|A(t)| (4.47) and that the zeroes of E±(t) are simply the zeroes of A(t) (where, of course, we analyti cally continue the time t throughout the entire complex plane). The transition probability from Dychne's formula is then Pn =<i \U(-oo, +oo)| t>~ e-4VWlmItc ^dT (4.48) where tc is the zero of A(t) closest to the real axis in the upper half plane. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 134 Let us solve for the transition probabilities for some specific potentials using our exact result. One that occurs quite regularly in this type of problem is a "pulse" potential that looks like A A(t) cosh cot The indefinite integral of A(t) is ft A / A(r)dr = —tan_1(sinha;£) + c2 J UJ (4.49) (4.50) To be consistent with our initial condition (4.35) we take c2 = AIT/2UJ. The transition probability from our exact result is then 12 - cos2 \Mt)\ The t —» oo asymptotic of (4.51) is Pn = cos2 ,A (1 + «:2)1/2— tan_1(sinhwt) + 7T UJ (4.51) An i i L u) (4.52) If we were to try to use Dychne's formula for this case, we would find that it fails. This is because since our energy levels are asymptotically approaching zero as t —>• oo, there is no zero of E(t) that is closest to the real axis and therefore one cannot find a unique tc. This breakdown of Dychne's formula in this regime has been noted previously [102]. Our exact solution is quite interesting. We see that the t —> oo transition probability may be varied between zero and one by altering an external parameter (A or u). A train of such pulses with these varied could be used as "quantum logic gates". The reason that this particular application is interesting is that there is absolutely no source of decoherence present in this system-phase coherence is sustained throughout the evolution of the system simply because there is nothing coupled to the two level system that can remove it. We shall not say anything further about this potential application; some embellishment of this basic idea may be found in recent reviews [102]. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 135" Figure 4.3: Transition probabilities for the pulse potential. Another potential of interest is the following one; A{t) = 0 , |*| > t0 A(t) = Asmu(t + t0) , \t\<t0 where we fix t0 = mr/2u. Here we find that for \t\ < t0 (4.53) |<M*)|2 = cos2 (1 + ^1/2- [cosu;(t + to) — 1] (4.54) with dynamics frozen for \t\ > to- This solution is shown in figure (4.4). Note that here Chapter 4. An Introduction to the Generalized Landau-Zener Problem 136 we also have a tunable switching effect, this time also depending on the value of n that we pick. Figure 4.4: Transition probabilities for the sinusoidal potential. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 137 4.1.5 Generalization of the Two-Level Landau-Zener Problem II. Exact Solution by Mapping to Riemann's Differential Equation There is a similar approach to the solution of (4.31) that generates solutions for a different class of functions V\\(t) and A(t). Let us take the mapping Then (4.31) becomes A + l l - + A z z-1 A A + z(t) = - [tanhwi + 1] Li Vn + A2 (4.55) z(z-l) W\\ ~ VWA A A = 0(4.56) 4to2z(z - 1) 2u which is suspiciously close to Riemann's differential equation (RDE) [113]. We have been able to map this equation into the RDE in two specific cases, which we shall review. Let us take V\\(t) = Atanhwi , A(t) = (A2 + A2)1'2 1 cosh cot (4.57) As functions of z, these are V\\(z) = A(2z - 1) , A(z) = (A2 + A2)1^z(l-z) (4.58) This models some scenario where the tunneling between the two states is externally enhanced near t = 0 and killed for \t\ >> 1 while the two levels are crossed. Insertion of these into (4.56) yields A + n I 1 !- + 7 .Z Z — 1. A l z(z-l) k k z z — 1 A = o (4.59) where Chapter 4. An Introduction to the Generalized Landau-Zener Problem 138 This is a specific case of the RDE. Its solutions are ^b = za(z- iy [CF[a, b; c; z] + Dzl~cF[l + a - c, 1 + b - c; 2 - c; z]\ (4.61) where C and D are constants chosen such that \ipb(z = 0)|2 = 0. In order to determine what the constants a, 7, a, b and c are, we refer to Abramowitz and Stegun [113]. Using their nomenclature, 1-a-a'= 1/2 , 1-7-7=1/2 aa = k , /?/?' = -A\ 77 = , a + b = \ + a — a + 7 — 7 c = 1 + a — a In terms of our original parameters we find that a=^(l + 2A0i) , y = -A0i a= Ai 2 and our general solution becomes ^ = z(l+2A0i)/2^_1yA0i b=\ + A1 1 + 1 + AAQI CF[^-A1,^ + A1;^ + 2A0i;z] + Dz-{l+AAol)l2F[-2AQi - Au -2A0i + 2A0i; z] (4.62) (4.63) (4.64) The condition |^ = 0)|2 = 0 (4.65) gives D = 0. The solution that we require is the t —» 00 asymptotic; this corresponds to the z —> 1 limit. Near z = 1 we find that r(| + 2Aoz)r(| + 2A02) ^(2 = l) = e^°C r(l + 2A0i + 4i)r(l + 2A0i - Ax) (4.66) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 139 Using the reflection formulae we find that \AY = 2 C2e2*A° (1 + 16 A20) 4(4A2 + A2) COS2 7rAi cosh2 2A0TY (4.68) As we require that the probability be bounded above by one we take C2= 4(^ + ^> (4 69) - e2^°(l + 16A2) 1 ' which gives, in terms of our original parameters, the final transition probability cos2 ^ |^(t-+oo)|2 = l- —^ (4.70) We emphasize that this result is exact. To the best of our knowledge, this result has not been published previously. This solution is sketched in figure (4.5). If we try to compare this exact result to that obtained from Dychne's formula, we encounter another interesting conundrum. In order to obtain closed form solutions for Dychne's formula, one has to be able to obtain the indefinite integral of E(t) = \JV\\(t) + A2(t). Now it is evident that the presence of the square root will make the set of V\\(t) and A(t) that can be dealt with quite small. The present case is an example of a Hamiltonian that produces time-dependent energy eigenstates that do not produce closed form solutions to this integration. This line of thought is quite interesting, as it seems to relate the integrability of square root functions to the solution of second order differential equations. It is plausible that progress could be made in the study of these kinds of indefinite integrals via the connection that has been established here to the hypergeometric equation. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 14P Figure 4.5: Transition probability P\± for the pulse/ramp scenario. Plotted is ATT/UJ on the x axis and ATT/U on the y axis. There is another case where we can map exactly to the RDE, and that is for the parameter set V[|(<) = Atanhu/t , A(t) = A (4.71) This is similar to the previous example, except that the tunneling term is constant for all time. The procedure is identical to that done explicitly in the preceeding; we shall Chapter 4. An Introduction to the Generalized Landau-Zener Problem 14 j Figure 4.6: Transition probability PT| for the ramp scenario. Plotted on the x axis is A/ui and on the y axis -just state the result. We find that sinh2 4* l^^°°)|2=sinh^;A,)./2 <4-72> Note that for A/A « 1, this reduces to the standard Landau-Zener result (4.12). This is another new exact result. This solution is shown in graphical form in figure (4.6). Chapter 5 The Landau-Zener Problem in the Presence of a Spin Bath At temperatures T < 360 mK thermal occupation of all but the two lowest lying levels in the Feg molecular magnet is vanishingly small (see figure 1.11 for experimental evidence of this and chapter 2 for theoretical justification). This means, as discussed in chapter 2, that in this "quantum regime" the central spin complex of the Fe8 molecular magnet can be described by a two level system. In the presence of an externally applied time-dependent magnetic field, the Hamiltonian of an isolated molecule is of the generalized Landau-Zener form (4.31) with the added complication that the central two level degree of freedom is coupled to an environmental spin bath. In this chapter we solve for the transition probability between fz eigenstates of the central spin object in the presence of an external AC field. We use this general result to calculate the one-molecule relaxation rate for any system with a Hamiltonian of the form (5.6). We then calculate the one-molecule relaxation rates for the specific case of the Fe% system. 5.1 The Addition of an Environment to the Landau-Zener Problem: General Considerations Whenever we study a condensed matter system, there is of necessity a distinction made between a (perhaps collective) degree of freedom and "everything else". When we write down an evolution operator (such as (4.31)) for this interesting degree of freedom without considering the "everything else" of the system we run the risk of misunderstanding what is actually happening inside the material. While the exact solution of toy models such 142 Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 143 as (4.31) may perhaps be pleasing to the aesthetic sense, it does little to further physical understanding. It is not even clear that questions of principle in quantum mechanics may be settled via the use of toy models such as these. As we discussed at length in chapter 2, there exist standard ways of obtaining realistic effective Hamiltonians for some condensed matter systems. These effective descriptions, no matter what the material under study, always require us to specify a subset of the total information contained in them to be the "degrees of freedom of interest", while relegating all the "uninteresting degrees of freedom" to an environment [20, 21, 22, 74, 75]. This split, while not necessary in principle, is usually necessary to make the study of the system tractable (for, as we have seen, realistic effective Hamiltonians are complicated objects!). The consequences of this (artificial) distinction between system and environment are still not fully understood. Some of the ramifications, both physical and philosophical, were addressed in previous chapters and in earlier classic works [74, 20, 21, 22]. In this section, however, we shall consider some of the technical problems that arise due to the inclusion of an environment. 5.2 The Quantum Regime Effective Hamiltonian: Inclusion of a Spin Envi ronment The form of the low energy effective Hamiltonian for the Fe8 molecular magnet was derived in chapter 2. We may ask how this Hamiltonian is changed by the application of an external time-dependent magnetic field. Looking back to (2.101) we see that there are four terms that will be affected; one due to direct interaction of the field with the nuclear spins (uj^rhk) ; one due to the field acting on the nuclear spins coming from the central spin whose minima are functions of the external field (w['4); and two due Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 144 to the interaction of the field with the central spin (the diagonal term ~ fzHz and the off-diagonal terms in the phase $). The first of these is uimk = \Ik\gnkpn{^kl) + of) = \Ik\gnkixn{fkl)s + 7$ + H(t) + H(t + fin"1)) (5.1) where we have defined the fields -(1,2) _ IkS — 9nkPn E sf'2) - E M°SS«W (5.2) to be those due to the central spin before/after the central spin flips respectively at the kth nucleus, t is the time at which the flip occured and fin"1 1S the tunneling time for the central spin complex. If we assume that fig 1 is much less than the timescale over which the externally applied field changes, then we find that u^rhkit) « Ihlgn^nd^ + IkS + 2H{t)) (5.3) This assumption also leads to ujklk being time independent, since u>KlK = \Ik\gnkpn(t] - of) = \h\9nMikS ~ iks + H{t) - H(t + fi^1)) « uKlk (5.4) Also we find that in general the phase $ becomes time dependent if the external field contains transverse components that vary with time. In the particular case of the easy-axis easy-plane Hamiltonian in the limit studied in chapter 2 (that is, the DC bias field \H\ -C fi0 and AE j(D — 2E) not too small (so that the instanton calculation is valid, see chapter 2)) we find that $ = 7rS- nS2gnB nSg^B „ -l — tlT H — tin. 2E fin ->• $(t) = TTS + 1 2~ Hx(t) ^—Hy{t) (5.5) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 145 Together with the obvious diagonal coupling to the external field we find that in the presence of a general external time-dependent field the one-molecule effective Hamiltonian is N+8 H(t) = £ fc=l ^-4 • mk{t) + ^Jk% • lk + gnBSHz(t)fz ( N+8 _ _\ + 2A0f_ cos \^(t) - i £ AN,D • hj + h.c. N _yko.fi jk + S [6/fc(2/fc - 1) " la0\ + Mo/^n 9ni9nk E 47r rZk rik Ti-Tk-s(Ti- rik) (h • hk) (5.6) where the notation is the same as that introduced in chapter 2. From this point onwards we shall drop all g-factors and magnetons for reasons of notational simplicity. Whenever confusion may arise we shall explicitly include them. This expression is still quite complicated; however, one notes that it has the form of a generalized Landau-Zener Hamiltonian which has been coupled to a spin environment. Note that the phonon bath does not enter into the quantum regime effective description, for reasons detailed in chapter 2 (although of course the oscillator bath is present when relaxation becomes temperature dependent-a proper treatment of these baths is under construction. Note that there are several (often conflicting) treatments of oscillator baths in this context [116, 117]). We shall now simplify the expression (5.6) by restricting our attention to the following two kinds of externally applied field. In all that follows we shall be considering this external field to be either a longitudinal sinusoidal field plus a static component in an arbitrary direction Hcos(t) = (A cos cot + 0 z + Hx x + Hy y (5.7) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 146 or a longitudinally applied sawtooth field plus a static component in an arbitrary direction Hsaw(t) = 2Aco 7T £ e 2n+ 1 cu t)e t-2n-l + Hx x + Hy y (5.8) Note that we have chosen to use the notation that the longitudinal component of the externally applied field is denoted by Hz —> £. Now because neither of these two fields have time-dependent transverse components, we find that the phase $ remains time independent. This simplifies our expression somewhat. To emphasize the relation of our effective Hamiltonian to the simpler Landau-Zener models studied previously, we rewrite it in the form H(t) = g(t) + V\\(t)fz + Af_ + h.c. (5.9) where m = at) + ^Y,^[rrh-3(ll^flk)(rk.flk)} 47F Kk  rlk (5.10) Vj|(t) = f + Hz(t) / N+8 \ A = 2A0cos ($ - t E A\>D • Ikj and we have defined N+8 N+8 k=l k=l  Z k=\ N eQk yka/3 jk 6/fc(2/fc - 1) a\ and N+8 N+8 f - E Tfc - E -ylk • h k=l k=l  Z (5.11) (5.12) (5.13) (5.14) Note that g(t), V\\(t), A and AkND are all operators in the {Ik} (environmental spin) subspaces. r Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 147 5.3 General AC Field Solution in Fast Passage We understand from looking at the effective description (5.9) that it will not be possible to compute the transition probabilities from the solution of a simple second-order differential equation. This is because all the environmental spins are dynamic and coupled to the central degree of freedom, vastly increasing the Hilbert space, and requiring the solution of (for a bath of N spin / nuclei) a (21 + \)Nth order differential equation which proves to be quite impossible in practice. We therefore must look for a different method of solution. Consider the perturbation expansion presented in section 4.1.1. We saw in the simple case presented there that there exists a natural perturbation parameter A2/v, which is small if the time it takes the external field to sweep through the resonance is much smaller than the "bounce time" A"1. We see explicitly in this case that if the sweep velocity is large enough, it is sufficient to calculate only to first order in this quantity. Let us set up this perturbation expansion for our effective description. As we indicated earlier, our externally applied field will be either sinusoidal or a sawtooth function. Often we shall find that analytic results are easier to obtain with the latter. Whenever possible we shall solve for interesting quantities for both shapes. Before we proceed with our technical investigation, let us pause and consider what the dynamics of the central spin complex should look like. When calculating the time evolution of this collective degree of freedom we can either calculate the evolution am plitude from our initial time (say, t = 0) to our final time t or we can choose a "coarse graining" time tc < t and only evolve the system from t = 0 to tc. After this shorter-time evolution, we calculate probabilities and piece together a series of these, assuming that each piece is independent of the other pieces. Now in the usual case of a model like (4.31) it is clearly not permissible to do this, as the dynamics of the central degree of freedom Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 148 at any particular time interfere (in the quantum mechanical sense) with dynamics at all other times. Another way of saying this is that it is necessary to calculate the evolution amplitude over the entire time range of interest-there is no decoherence (loss of phase information) in this system. However, the coupling to the environmental spin bath in (5.9) changes this story. Be cause the spin bath is dynamically active, both because of its internal dynamics (coming from the nuclear dipole-dipole term in the Hamiltonian) and its response to the dynamics of the central spin moments, it absorbs phase information from the central spin complex [64, 167]. This has the effect of decorrelating successive sweeps of the AC field through resonance-while the evolution of the entire system is of course unitary, the evolution of the central spin degree of freedom is not. One can understand why this should be so from the following argument. During each sweep of the AC field through resonance, the central spin configuration can flip, and in so doing it can flip or rearrange the distribution of the environmental spins. This rearranges the density matrix of the spin environment, which can in general contain off-diagonal terms. Now as the sweeping field moves off-resonance, the nuclear spin-spin relaxation mechanism tries to equilibrate the environmental spin set. If the sweeping frequency is larger than ~ r0, where To is the energy range over which the nuclei sweep due to nuclear dipole-dipole interactions, by the time the sweeping field comes back to a resonance the off-diagonal spin bath elements will still be present (note that the energy scale r0 is roughly r0 ~ \fNT2~1 for an ensemble of N nuclei, where T2_1 is the standard spin-spin relaxation time). This means that for high enough sweeping frequencies one cannot neglect quantum correlations between successive passes of the field. In this case one could invoke methods from the so-called Floquet theory [118]. However, if T0 is much larger than u, then the spin bath does have time to equilibrate itself between passes of the field, and we can therefore neglect all quantum correlations between successive passes, treating each sweep as being "decorrelated from its history". Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 149 This consideration presents us with a natural "coarse graining time", the time it takes for one full sweep through a resonance by the AC field (which is tc = TT/LU for both the sinusoidal and sawtooth fields). If the timescale for the readjustment of the density matrix of the system (which is tb « l/r0) is such that tb/tc < 1 then this coarse graining procedure is justified. Typically in magnetic insulators at mK temperatures T2_1 ~ 104 — 106Hz [48, 64]. In the specific case of Fe% we can calculate r0 (see table 5.2); we find that r0 ~ 3 — 13MHz, depending on the nuclear spin isotopes present. This gives us an approximate gauge of the highest frequencies that we can apply before the coarse graining approximation breaks down. For a sinusoidal sweep, tb/tc « TTTQ/UJ, which gives co < TITO <~ 10 MHz (5.15) In the experiments that have been performed to date on the Fe8 and Mn12 materials, the sweeping frequencies are much less than this; in the Fes experiments these frequencies were in the range 0.01 — 5 Hz [51]. We shall therefore use the coarse graining approx imation in what follows. It is worth noting here that this point gives ample warning that temporal quantum coherence of the central degree of freedom in this system on timescales of the order of tc will be difficult to maintain, as the nuclear spin bath has ample time to both absorb phase information from the central spin and bias the system between sweeps of the AC field-and therefore processes mediated by the AC field will most likely be incoherent in the sense that each sweep through resonance is decorrelated from all other such sweeps. 5.3.1 A List of Approximations Invoked in the Calculations That Follow For the sake of clarity we shall list in this subsection all the approximations that we shall be making in the sections that follow. Whenever one of these is invoked, we shall refer Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 150 the reader to the following list. (i) The coarse graining approximation. This approximation is described in some detail in the preceding section. To recap, whenever the condition UJ < -KTQ is met, one can neglect quantum mechanical correlations between successive sweeps of the AC field. This is because the internal dynamics of the spin bath mediated by nuclear-nuclear terms in the Hamiltonian carry phase information away from the central spin. (ii) The fast passage limit. The justification for taking this limit was presented in the preceding section. Formally in this limit we consider only the regime where A2 AUJ where A and UJ are the amplitude and frequency of the sweeping field respectively. (iii) Nuclear spins with a given polarization group M are in thermal equilibrium at a polarization group temperature 0M = 0. The "polarization group" of our set of N + 8 nuclear spins is defined to be [20] N+8 M = £ Ikz k=\ We note that this quantity explicitly depends on which axes of quantization we pick for the nuclear spins. What we choose to do in this work is to pick the axes of quantization of the nuclear spins to be such that zk = 7^; that is, the z direction for the kth nuclear spin corresponds to the direction of the field at the spin when the central spin complex is in its +TZ eigenstate. Because the internal field due to the iron spins is quite inhomogeneous, these axes will only be mutually aligned in the case where the external field H is much larger than the internal fields (which we have seen in chapter 3 are in the 0.01 — 0.26 Tesla range). Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 151 5.3.2 General Strategy for Calculating Relaxation Rates The core of the calculation of the one-molecule relaxation rate is the calculation of the transition probability between the fz eigenstates of the central spin complex during one sweep of the AC field. Before we begin to attack this problem, we shall explicitly describe the general strategy that shall be used to calculate these probabilities and hence the one-molecule relaxation rates. Qualitatively the situation that we have to deal with is similar to that examined in the previous sections dealing with the simple Landau-Zener transition. The big difference here is that the Hilbert space of the system now contains not one two level system but a two level system plus the full Hilbert space of the environment. This means that when we calculate the transition amplitudes (and then the probabilities) between the fz eigenstates, we also have to explicitly include the effects of the other N + 8 systems. Formally we can see how this is done by noting the following. For the Hamiltonian (5.6) the amplitude to go from some initial fz eigenstate \a > to some final fz eigenstate \/3 > during a single sweep of the AC field can be written where we have defined \P > and \P > to be the initial and final states of the spin bath respectively. Let us take a moment and explicitly detail the formalism that we shall use in order to describe the spin bath states. In a general molecular magnet there can be many species of environmental spins included in the effective Hamiltonian. For example in Fe$ we have 120 protons, 18 nitrogens, 8 bromines, 8 irons, 36 carbons and 23 oxygens. Our formalism must be able to handle these different species. To this end we shall define the following notation. The symbol A% =< 0\ < lf\e^oCH^dT\p >\a> (5.16) (5.17) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 152 represents the number of environmental spins of spin s and species JJL. The kth spin of the subset N£ is labeled = l..Njf. The total number of spins in the spin bath is N = J2fj.N^. As an example, consider the Fe8* material. Here we have that A^2 = 120 , iVf = 18 , N3% = 8 (5.18) With this notation the spin bath states \P > and \P > can be written in the form f N> 1 \r> = n nK>kK>®i4>-«^>} » (^=i j i^> = n(ii K>UnK><>-®i/ir>} The transition probability is then simply P% = A^A% (5.19) This expression for the "per sweep" transition probability contains explicit reference to the initial and final states of the spin bath. Now in any actual experiment performed on these materials, what is actually measured is a relaxation rate. In the calculation of this rate we will have to specify the distribution and characteristics of the initial and final spin bath states during a sweep of the external AC field. Now because of reasons elaborated in the preceding section, the strategy that we shall use in specifying the form of the initial states of the spin bath is the following. We shall define the set of states |MM > to be all the allowed spin bath states with polarization group in the pth subset of nuclear spins. That is, we explicitly treat each species of nuclear spin separately, defining a polarization group for each. We shall see later that in this calculation all the environmental spin subspaces separate; this means that it is permissible here to treat each species as being independent of all the other species. We define the number of states in the polarization group to be CM^- We have then Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 153 p = I.-CM^ states with which we label \K>=\i\ i7i>l (5-2°) U/x = l J Because this notation may at first sight appear confusing, let us explain here again what each of the symbols represent. The state |M£ > is a particular state which is a member of the set of states \M^ >. The states |MM > have particular polarization groups fixed for each species. For example, imagine a fictitious material with 2 protons and 2 nitrogen atoms. We begin by fixing (say) MH = 0 and MN — 1- Then the states |MM > are the set of particular states with polarization group MM. So in this example we could label our states \MH >= | + 1/2,-1/2 > , \M2H >= | - 1/2,+1/2 > and |i\4>=|i,o> , |M£>=|O,I> the former of which constitute the set |M# > and the latter the set \M_y >. Because of our approximation (iii) in the previous section we may define a density matrix for the spin state set \M^ > of the form 1 -e-e^Ml X M?\ 9 )Q AMI >< MP\ (5.21) with partition function ZM^M,) = j:e-^EP^CM^ (5.22) i=i Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 154 These definitions allow us to write down the transition probability for the central spin complex in the form Pa3=J2U K^t] (5-23) APG /i where the sum is over all possible configurations of All Polarization Groups (APG) and P$=Tr^Y*%} (5-24) where we have summed over all final spin bath states for our transition probability (any state of the spin bath is acceptable as a final state). Here the ensemble average over all possible configurations of all polarization group states APG is normalized such that £n^ = i APG ii-For a single species spin 1/2 bath, WM = ^\ .„ .„ I (5-25) 2 The way this works is as follows. Each particular molecule begins its quantum mechanical evolution in one of the nuclear spin states fl^ 1-^ >• The probability of its making a transition from central spin state \a > to central spin state |/3 > is then W^P^p. The relaxation rate which we are after will involve contributions from all molecules in the crystal, which will of course be in different polarization groups. Therefore in order to extract the quantity of interest we need to perform an ensemble average over all these contributions. This ensemble average is given by (5.23). We can write the expression (5.24) in a simpler form by explicitly summing over all final spin bath states, using the completeness relation J2\if> \I^ >< = 1. Performing the sum gives p$ = Tr |pMM < a\e-l^H^)dT\(3 >< ftjfc H^dT\a >} (5.26) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 155 In order to finish our job, we need to translate this transition probability into a relaxation rate. We can do this by noting that since this is a "per sweep" transition probability, the relaxation rate can be obtained by dividing this quantity by the time for one sweep (which is of course simply the coarse-graining time tc). Furthermore, we may without loss of generality here choose \ct >= \ t> and \(3 >= \ i>. Therefore the one-molecule relaxation rate, with the molecule's polarization group state initially >, is = lTr{pM» <t |e-*J>(^| i><! |C'J>M*-| T>} (5.27) and the final ensemble averaged relaxation rate is, from (5.23), ^=EI1 [WMSMI] (5-28) APG f-We now turn to the explicit evaluation of this quantity. 5.3.3 Processing of the Transition Amplitude In this subsection we shall recast the transition amplitude (5.16) in a new form that is easier to deal with. We begin with our original definition A% =< p\ < If\e^o C HW*\r >\a> (5.29) This form is correct to all orders in A. As it is our wish to do perturbation theory in this flipping term we may rewrite this in the form A% . E f *. r «*.-.... r *. < a < I'^^HJ^ n=0 J° J° J° * HAeJ^ drHi[T)...H^&d^\P > \a > (5.30) where H_t(t) contains all terms in the Hamiltonian diagonal in the f representation Hd{t) =g{t) + V\\{t)Tz (5.31) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 156 and contains all off-diagonal terms #A = Ar_ + h.c. (5.32) This amplitude can be thought of as a sum over diagrams containing n "blips" (see figure (5.1a)). n r~~Ln_ 'i >i h '« h h (!>) Figure 5.1: Transition amplitude as a train of blips. Taking \a >= | t> and \P >— I i> we find that the leading order diagram contains only one blip (see figure (5.1b). This leading order term is simply 4i = i/fedt1 <l\<lf\eifi*Hd{T)H*ei£*H<V\Ii>\l> (5.33) J 0 Now it turns out that the presence of the nuclear dipole-dipole term in Hd{t) introduces unnecessary complications into our calculation. This is because it couples the subspaces of all the environmental spins; that is, one cannot consider each environmental spin subspace separately if this term is explicitly present. However we may use the following physical argument to rewrite this term in a different way. The nuclear dipole-dipole term has the effect of causing a time-varying random field acting on the central spin, coming from the fast nuclear-nuclear dynamics in the bath. The transverse components of this field may be absorbed into the already existing transverse terms in the Hamiltonian. As Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 157 long as these latter are much larger than the amplitude of the nuclear-nuclear noise ~ r0 we may neglect them-we shall assume this is the case for now and then explicitly show in section 5.6 that this holds. The longitudinal terms, however, may not be so absorbed. This is because the energy scale that we must compare these to is A0, which is much smaller than r0. Therefore we shall replace our microscopic and deterministic description with an equivalent stochastic form, following [20]. In this stochastic version the internal dynamics of the spin bath are replaced by a randomly varying time-dependent field acting on the central spin complex which models the dynamics generated by the original nuclear dipole-dipole term and each environmental spin subspace is assumed to be decoupled from all others. In terms of our Hamiltonian this entails replacing the intra-nuclear term in (5.10) with a term 8£(t)fz, where 8£(t) represents the longitudinal component of the fluctuating nuclear-nuclear bias field. Explicitly we have that H{t) = $(t) + Vj|(t)f2 + Af_ + h.c. -> Hit) = C(£) + (V\\(t) + 6£(t)) fz + Af-+ h.c. (5.34) We have denoted this new stochastic Hamiltonian via an overbar; that is, we write Hit) for the original Hamiltonian and H(t) for the stochastic approximation. Because our new Hamiltonian now contains this stochastic term, we shall use a slightly different notation for the amplitude, writing A\{ = ifjdh <1 | < lf\eifi*SdiT)HAeif?dTR<V\Ii > | t> . with the overbar reminding us that we are now using H(t) to evolve the system, and not Hit). With this choice, the inner product in the f subspace can be taken, yielding A\{ = i fj dh < If\e1^ ^'^A^fo1 «MC(r)+i*(T)} |r > (535) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 158 where we have defined F(r) = V»(T)+5Z(T) = t + AcosujT + £ + 6Z(T) (5.36) Here we have chosen for the sake of concreteness the sinusoidal form for the sweeping field. For the sake of notational simplicity we shall separate terms in £ that are time-dependent from those that aren't, writing N+8 C(r) = £ [4 + AcOSUJTlzk (5.37) k=l where (f£ + Hxx + Hyy) N eQk \rkafi jk 6Ik(2Ik - 1) N+8 c = EG fc=i (5.38) (5.39) and the time-dependent piece is the Zeeman interaction between the external AC field and the nuclear spin. We find that, after performing the r integrals in the exponents, n Jo £.e*[tf+t+«*i+e^^+Zo1 )r > (5>40) We have defined the initial and final polarization groups of the \ith species to be N^ = E LW , Mfll = E i, S (5.41) Because we are always in the limit that SgHB » 1 Ngnpn we can drop the direct field-nucleus interaction ~ A term from our expression. (5.42) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 159 Because all the environmental spin spaces are independent in H(t) we may separate each contribution. Rewriting the cosine coming from the flip term as a sum of exponen tials / W+8 \ N+8 A = 2A0 cos U - i £ A\iD • Ik) = A0 £ <?m II EHAKN'D'H \ k=l 1 h=±l k=l we find that A*f A • A* 4^SinWtl-X?dT^/W+/oldr^/W+^-*1)l /i^ — £±01J oX\e L 1 J N+8 Y EIH* II <//|ei{^"Tfc}(tc"il)e/l^.°'/*:ei{^+tA:}tl|4 > (5.43) h=±l k=l where \Pk > (\Ik >) is the initial (final) state of the kth nuclear spin and r5£j/ is the (time dependent) nuclear-nuclear longitudinal bias field acting during the transition from \P > to \P >. This is the form of the transition amplitude that is best suited for using as input into the calculation of the transition probability. 5.3.4 Processing of the Transition Probability (i) The Formal Expression The next step on our journey towards the final one-molecule relaxation rate is the com putation of the transition probability from the result (5.43). As we noted in our intro duction of the problem, the transition probability from the state | j> <S)\P > to the state | l> <g>\lf > with Hamiltonian H(t) is simply PH = A\"A\{ (5.44) Now in our case we are working with the transformed Hamiltonian H(t). Does this change this expression? Looking at the form of our transition amplitude (5.43) we realize that it contains explicit mention of the time-varying random field 5£(i). This field has to be Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 160 averaged over in the final expression for the transition probability. This means that if we use H(t), the expression for the transition probability becomes P;( = J V[S^f(t)}V[S^f(t)}A^A\{ (5.45) where we have explicitly included a functional average over the fluctuating bias field. Using our result (5.43) we find that this transition probability is p;{ = Al £ * fj tofvmit)] PK„(i)]e'[-(,i"""-A"'!|-2^ Nj~* ei(W-m**) JJ < ^e-i{a+rk}t2ernA%^DJke-i{Ck-rk}(tc-t2)^jf > h,m=±l k=l < lJ_\ei^k-^k}(tc-n)ehA%DTkei{ik+yk}ti |ji > (5.46) To translate this into the form P^ we sum over all final states of the spin bath (using the completeness relation) and restrict the set of initial spin bath states to be those with polarization group M^, giving P^f = A20 fj dt, fj ^2e^(si"-^-s>"^)+«^-^)] £ ei(fc*-m**> I"^[^(t)]P[^(t)]e-2l^d^(r) £ ~ h,m=±\ 7 peMj K Y[ < Mpfc|e~l{^+Tk}t2emA~k"'D'he~l{^~T'^-^ ehA»'D'Ik el{^k+r k}tl \Mpk > (5.47) where \Mpk > is the initial state of the k^ nuclear spin in the pth spin bath state having polarization group M;i and S^M^ is the fluctuating longitudinal bias term coming from the nuclear spins in the MM polarization group. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 161 5.3.5 Processing of the Transition Probability (ii) Averaging over the Randomly Fluctuating T2 Noise We see from the preceding that the contributions from the effective nuclear-nuclear term in our Hamiltonian H(t) coming from the pth species are wholly described by the expres sion /V V[S^(t)]e-2lKdT5^{T) (5.48) In order to explicitly evaluate this contribution we first need to specify the probability functional P[8£,M^(t)]- We assume that this random process is gaussian and therefore take 7> [<5£M/i(*)] = ^ dsi ^ dS2S^M^Sl)KM^Sl~S2)5^M^S2) (5.49) where the quantity KM,,, can be understood in terms of the following. The probability functional is defined so that the average of any operator over it is simply < A^MM >= /V[5^(t)]A[8^(t)]V[5^(t)} (5.50) It follows therefore that the autocorrelation function of the noise is < s^(Sl)s^(S2) >= j 2>[(&£MM(t)]<w*i)<y^ (5.51) The formal solution to this equation is [119] < 6ZMM)SZM,{S2) >= KMM ~ S2) (5-52) where KMl is defined by Jds'KMll(si - s)KMl^s - s2) =6(si- s2) (5.53) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 162 In our case, we are interested in a noise functional S£,M^(t) which changes by roughly SUJM^ in a time T2/N£ (this quantity is related to the spread in energy space of the Mff polarization group -simply YMy. = VNs5tuM^)- It then diffuses in energy space with diffusion constant = - KJt ^ r2 where we have defined AM = -^L. This now allows us to find out what KMl(s\ — s2) is for our problem. This is because < (<S£M„(SI) - 5iM,{s2))2 > = < (5£MM))2 > + < (^M,(S2))2 > -2 < ^Mfi(Sl)S^Mfi(s2) > = 2DSiM^\si-s2\ = AMii\si-s2\ (5.55) and therefore A3 < ^M,(SI)^M,(S2) >= KM\{Sl -S2) = -^- (\Sl\ + \s2\ - \Sl - s2\) + 5f^(0) (5.56) We now introduce the characteristic functional $[QMJ dual to our probability functional V[KMM = I^O^We"* F*0^W^(T)<&[QM|1(*)] (5.57) For a gaussian probability functional it follows that $[QM (t)] = e^^^^M^e'^*81^82® (5.58) The quantity that we are trying to evaluate (5.48) is simply $[—2]. Because we know what KMl(s\ — s2) is from physical grounds, we may now write fv V[6£Mll(t)]e-*%dT6iM'>lT) = $[-2] Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 163 Performing the s integrals gives 3 /V [S^(t)\ ViSUM^1^^^ = e-^(t2-tl)3-r-,(i2-tl)2 (5.60) We find therefore that A3 pM» = A2 ftc df /"*cdt i[^(sinU*1-sinWt2)+«*1-*2)]e--^(t2-ti)3-r^(t2-t1)2 n °Jo Jo ^2 gj(/i*-"i**) >p ^ h,m=±l peM? ^Mf> Nt JJ < Mpfc|e_i{^+i}t2em^V^e^^^ > (5.61) Note that since AMFI «C TMF_ (that is, TMFI ^> T2~L) for the systems under consideration, we can always drop the cubic term in (ti — t2) in the above expression. We therefore find that P^ = AQ dti A*0 rft2E4^(SINWTL~SINWI2)+^*1_I2)]E~R^^2-IL^ £ ei(/.*-m*-) _1_ J° J° h,m=±l VZMICM» Nt JJ < j\^Pfc|e-i{Cfc+tfc}i2emA^D-4e-i{Cfc-ffc}(t1-t2)e/i^^ > (5.62) 5.4 Solution Without Spin Bath The expression (5.62) is still quite opaque. In order to see how one extracts meaningful results, we shall see how to resolve it in the situation where the spin bath is absent. Here the Hamiltonians H(t) and H(t) are identical and (k = TK = AND = TM = 0. Since Uk=i < Mlk\Ml >= 1 and P$ -> Pn, we may write (5.62) as Pn = A2 dtx rCrft2e^(sin-ti-sina;t2)+£r(«1-i2)] V- j[h*-m**) (5 gg) J° h h,m=±l Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 164 The sums over {h,m} collapse, and we have Pn = 2A2| cos$|2 ftc [tc ^2E4^(sin^-sin^)+C(ti-*2)] (564) Jo Jo These integrals can now be performed, with result A27T2 ^^Hcosc&l2 (5.65) where J and £ are the Anger and Weber functions respectively [113]. This result, normalized to the transition probability for the standard Landau-Zener process P(o) _ A2?R _ AO|COS$|2TT IS PA n p(0) n LO Aui (5.66) u, \ ui 2A\ (5.67) This quantity is a function only of A/ui and ^/ui. We have plotted it as a function oi^/ui for three fixed values of A/u> in the figures that follow. Note that similar transition amplitudes have been calculated previously for related problems [115, 120, 121]. Perhaps the most closely related problem for which a published solution exists is for the Hamiltonian [115] H = £ —A cos uit An Ao -e (5.68) The published result is the transition probability over one entire cycle of the field, which is a time which is double that of the coarse-graining time we are using, and it is given by P A'in2 ^ /2A 12 LO1 u V ui (5.69) where J^(x) is a fj, order Bessel function. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 165 We may also solve this problem for the case of a sawtooth field. In this case we have that Pn = 4A2|cos$|2 /W2 dt, fC'2 dfee'IW-^fc-*)] (5.70) J—tc/2 J—tc/2 These integrals may be performed, giving error functions [113] which can be easily plotted. This result, again normalized to P$, is compared to the results obtained from the sinusoidal perturbation in the figures that follow. Note that what we have calculated here is the transition probability for one sweep. Because we do not have the spin bath to absorb phase information from the central spin we cannot really claim to have calculated a "relaxation rate", as each sweep of the field here is correlated (in the quantum mechanical sense) and this calculation implicitly assumes a decorrelation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 166 0.2 n 0.18-; 0.16-: 0.14 \ 0.12: \ 0.1-0.08 0.06^ \ 0.04 0.02 \ 0 2 4 6 8 10 0.1-, 0.08-0.06-0.04-0.02-0 2 4 6 8 10 12 14 16 18 20 Figure 5.2: Transition probability normalized to the standard Landau-Zener transition probability P^/Pff plotted against 2^/ui for 2A/to = 0.1. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 167 3 2.8 2.6 2.4 22. 2 1.8 1.6 lAi i 0.8 0.6 0.4 02 0 2 4 6 8 10 12 14 16 18 20 2n 1.8-1.6-1.4: 1.2^ / \ 1- \ 0.8 0.6 \ 0.4^ v~\ 0.2^ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Figure 5.3: Transition probability normalized to the standard Landau-Zener transition probability P^/pff plotted against 2£/tu for 2A/u> = 10. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 168 5-3-2-1 i-..^»,,-..,w,-»»l«'V: 100 200 300 400 500 600 12.A 0.6 0.4 A 02. A 100 200 300 400 500 600 Figure 5.4: Transition probability normalized to the standard Landau-Zener transition probability P^/pff plotted against 2£/UJ for 2A/u = 500. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 169 5.5 Solution For a Spin Bath with no Quadrupolar Contribution If the spin bath contains only spin 1/2 nuclei, or if the contribution due to the electric quadrupolar term is omitted from the Hamiltonian, then the expression (5.62) simplifies considerably. In this section we shall calculate the full general relaxation rate produced by the effective Hamiltonian (5.6) in the limit where the terms representing the electric quadrupolar effects are taken to zero. Formally what we shall do is take It shall turn out in what follows that our choice of a sinusoidal sweeping field intro duces technical difficulties in evaluating integrals when the spin bath is included. For this reason we shall in this section consider only the sawtooth perturbation (5.8). With this applied field (5.62) can be written dti / T/2U J-K/2U, h,m=±X =A2 r/2w dt, r/2u dt2ei[^(ti-t2)2+«ti-*2)]e-r^(ti-t2)2 E e^*-m*-) J—7T/2UI J—TT/2UJ u I 1 £ II < Mpk\e~l^''k»t2emA»'^ (5.71) 5.5.1 Pure Orthogonality Blocking In order to get a feel for how this calculation will go in the general case, let's start off with a particularly interesting limit-that of pure orthogonality blocking (for the definitions of orthogonality blocking, topological decoherence and degeneracy blocking we refer the reader to [20]). In this case we take the topological decoherence terms Ak^D to be zero. The reason that this limit is particularly simple is that without these terms, we may take the axes of quantization of each of the nuclear spins to be such that 7^ = z without Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 170 having to worry about the effect that these rotations have on the form of the AtfD. We find in this limit that p^f"=A2,i cos $i2 rl7u dtl r/2uj ^2e4^^-^2^-^]e-r^^^)2 J-TT/2UJ J-TT/2UJ J_ £ n e<)l/^^-^)<Mf|e-<)^(tl-t2)|Mf > (5.72) Cm» PeMl fcM=i where we have used notation such that IkfiZ means the z state of the k1^ spin, normalized such that —1 < Ikz < 1- Note that we must be careful here, as the conversion from field to energy units contains a factor of \Ik\. We now choose, without loss of generality, 7^ = cQz + Clx (5.73) where co = -|7£)|cos2^ , Cl = |7g)|sin2^ (5.74) wi ith Changing variables to cos 2/3fc|, =-7^.7^ (5.75) X = u>{tx-t2) , Y = J-±±^ (5.76) allows us to write |2 U CO 2 J-n/2 J-00 r/2 l N* ^l1 >i ^ -— E fl J-^1"*** < Mlk\e-l~^-'^x\Mlk > (5.77) Here we have extended the limits on the X integral. This will be permissible as long as rM„ 3> to (which we assume here), as the gaussian term cuts off the large X contributions. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 171 Similarly if < ^/NTTM^ (which we shall see is also the case-see table 5.2 and tables 2.3-2.11) we may expand the inner product to quadratic order in X, giving < Mf \e-i^"I^x\MPk >t -(2) '-(2) Tk l-iX< Mf\^ • Ik\M? > —— < Mf\ [^•IkA \Mf > (5.78) IT12)I x2\fk 1 + %XIkJ-^- cos 2ft, -to 2uil 2 1^(2) 12 a^M1 2 + cos22/?fc/i/2 x2l7<2)l2 4W2 • sin2 2/3, 1 = e (5.79) Insertion of this into (5.72) yields i%» = A°lc°f{2 r dY r dXe^x^Ae-^ CO' T/2 l^[2)l c M, E n A? 2zXIkaZ^cos*pk J— sin2 2/3, 1 (5.80) where we have used the fact that |7^| = ITI^I (ie- the magnitude of the field acting on the k1^ nuclear spin before/after the central spin complex flips is the same). Defining the quantities TV," = 2£ Vlffc1J|cos2ft k»=l we can write this as 1 £ l7i1}|2sin22^ 1 j2 • z (5.81) pMii = A2|coscfr|2 r/2 CJ2 7-7T/2 / dy / J-n/2 J- CM, (5.82) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 172 The Sum Over Initial States IM* > We are now going to perform the sum over the initial states \M? >. Before we begin, let us take a close look at the expression (5.82). We see that both e% and pi^ are functions of the particular state that the spin bath is in. Now what we are going to do now is to replace p2^ with its average value; that is, we take P\^\>Z l7i1)|2sin22^ ) (5.83) kit — 1 \ / Since ^^^m-'M^ (5.84) the average value of is simply P?, = %^E I^Psin^^ (5.85) The approximation of neglecting the width of the distribution of numbers {piM} is justified whenever this width is much less than TM/i (which is certainly the case here). With this approximation the sum in question becomes —e"^ £ (5.86) In order to clarify the procedure that we shall adopt in what follows, let us imagine how we would proceed in a specific case. As a typical example, let us consider how to deal with the protons in Fes-In this situation, there are an enormous number of possible states because NH (and therefore CM[]) is large. In Fe8 there are 120 protons per molecule. This means that there are 2120 ~ 1036 states in 2NH + 1 = 241 polarization groups. The Mjj polarization group contains NH choose \MH — NH\/2 states, each of which contribute to the sum (5.86). In Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 173 addition, we note that the values that e% can take are bounded. So the situation that we must deal with is one where we have an extraordinary number of states squashed into a bounded energy range. It is natural in this situation to convert the sum over initial states into an integral over e,. Now because of the exceedingly large number of states available to the spin bath, we may invoke the central limit theorem in order to supply an appropriate weighting function for the integration. That is, as we increase Ng, the numbers that we get for for each particular state of the spin bath will begin to approximate a gaussian distribution centered at some value ef^ with some width W(M/i). In this limit we find that ^^£l#W& (5.87) Now the width W(MM) is a little trickier to dealt with. In general it is apparent that this quantity has its maximum for MM = 0 and monotonically decreases to zero for \M^\ = Ng. How it does this will in general be a function of how the fields |T!^| are distributed. It turns out, however, that to a good approximation we can consider this width to be independent of M^. There are two factors that make this approximation reasonable. The first is that the number of states available in a given polarization group (which, for spin 1/2 nuclei, is simply N choose |JV - M|/2), fall off sharply for lAf^j >~ vW- This means that the behaviour of W(MA1) for |MM| > VNf will be irrelevant, simply because there aren't enough states in these higher polarization groups to make any difference in the final relaxation rate. The second factor is that for \M^\ < y/~Ng, the width W(M/i) will be roughly of the form W(MM) ~ (l - J^lj W((VJ , |M,| < yfw < Ng (5.88) for Ng large. The zero polarization group width we get from the central limit theorem; Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 174 it is simply 2 N' -+ -?= £ 1^ | cos2 A (5-89) Note that to be consistent here we must also take -> r0/J for the same reason that we took W(Mfl) —»• W(O^). We shall hold off on using these approximations for the time being. We can make excellent progress in evaluating our expressions with unspecified rM(i, ef^ and W(M/t). Until we reach the point where we need to have explicit forms for these we shall just leave them as functions of the polarization group. Using the gaussian weighting allows us to rewrite our sum in the form l ^ l r°° , „ -tjl^Hl 'fx X flu. 1 /-oo tg1 e1 > tlfX y el^r = e~^y-=— -/ de^e ™2W e{^r «M W2(MM)/2+P2 = elA^"e ^ x (5.90) Substitution of (5.90) into the expression for the transition probability (5.82) yields dY r dxA^^e^e2 J-IT/2 J-OO (5.91) pM^ = A^OS$|2 r/2 ^ r a vA^XY+Lx] ^'Z.- w2^ +f^W x, tl CO 2 J-K/2 Evaluation of the Ensemble Average Over Polarization Groups Examination of P^ reveals the following useful fact. Because the energy spreads in the different polarization groups overlap considerably, ie. « ^(M^ + ^ + r^ (5.92) we may approximate the ensemble average over all polarization groups by integrations. Formally this is done by taking APG fi \i = r dM,e-MV2N° KN? J-oo " (5.93) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 175 where we have extended the integration limits (the gaussian cuts off the large M contri butions). From (5.28) we have APG n A^|cos$|2 r/2 J-ix/2 J-OO n M„ W(MM) J= f00 dM,e-Mll^eV< » X A: w2(M^)/2+^M+r2 X2 (5.94) where we have written ef^ explicitly as a function of using (5.87). In order to perform the integrals over the set {M^} we now invoke our W(M/1) —» W(O^) approximation. The integrals over the set {M^} are then easily performed, giving A2|cos$|2 W2 T/2 •Pf-I- ~ f'2 dY r dxA%XY+Me-$x' J-n/2 J-OO (5.95) where we have defined the full energy width (5.96) where all these quantities are evaluated for the zero polarization group spin set for each species. Evaluation of the X and Y Integrals The X integral can now be performed, yielding PU -Changing variables to A21 cos $ I2 r/2 yfUloW J-n/2 dY exp m2 (2AY t f' '4W2 \ TTLU LU (5.97) LU (2AY C Z = + -2W V vro; LU (5.98) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 176 allows us to write our relaxation rate in the final form , o> An| cos $|2 rz+ ,„ 72 , where ^ = ^ ("<») Evaluation of the Relaxation Rate in Various Interesting Limits In the limit where \A\ 3> y/^2+~W2 we recover our old large A result, which is of course independent of , A^|cos$|2 r"1 = —^— L 5.101 A v ' If \A\ <C \/£2 + W2 then the relaxation rate becomes A2|cos$|2 2 414/2 (5.102) Discussion of Results: I. Pure Orthogonality Blocking The general result (5.99) produces two limiting cases (5.101, 5.102) that are both quite interesting. The first of these (5.101), valid for large A, tells us that in this limit orthog onality blocking effects do not affect the relaxation rate at all. How can we explain this physically? In order to understand the result, we shall present a graphical depiction of how orthogonality blocking works in a molecular magnet. The central spin complex, a two-level system, begins its evolution in some fz eigenstate which we choose without loss of generality to be | y>. Now for all the time that it remains in this state the nuclear spins Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 177 feel its dipolar field. The k nuclear spin initially is exposed to this field which we have called 7^. We saw previously that because of the nature of pure orthogonality blocking we could choose the axis of quantization of the kth nuclear spin freely and for convenience chose the z axis to be parallel to the direction of 7^. Physically what this means is that for this choice of basis the kih nuclear spin feels only a longitudinal field as long as the central spin complex remains in its initial state | y>. Now when the central spin complex tunnels to the other fz eigenstate | i>, the kth nuclear spin feels a different field 7^ which contains transverse components. This means, of course, that the kth nuclear spin will precess in the new field 7^. The diagram that we are evaluating in order to solve for the transition probability is the one shown in figure 5.5. We see that this process contains a length of time tx — i2 where the central spin complex is in the state | \>. During this length of time the kth nuclear spin precesses in the field 7^. Now what does this have to do with our large A result (5.101)? Let us go back to an earlier expression (5.95) and look closely at the integrals involved. In particular, let us examine the Y integration. Notice that there is only one term under the integration that is a function of Y. This term may be isolated and is Now if A/to is "large enough", we see that this expression will begin to approximate a delta function in A'. How large is large enough? Looking back to the expression (5.95) we see that the X integral is now of the form (5.103) (5.104) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 17& t=0 Figure 5.5: Depiction of the precession of the kth nuclear spin during a "blip". The central spin is shown in black, with a schematic nuclear spin underneath. This nuclear spin feels a field y± for times t < ti, which we choose to be the axis of quantization. After the central spin flips at ti the nuclear spin feels a different field 7^ which contains in general transverse components. This causes a precession of the nuclear spin. After the central spin flips back, the nuclear spin will be in a state that has less than full overlap with its original state. Changing variables to Z = AX/LJ allows us to write Now here is the crux of the matter. If A is much bigger than £ and W, then the terms in for easy solution of (5.105). Moreover, we see that because these quantities drop out of the expression all references to the nuclear spins disappear! What has happened? We see that this expression is dominated by small values of Z. But Z is nothing but Z = AX/LO — A(ti — t2). Therefore contributions only come from t\ — t2 <~ ^. We see then that large A reduces the length of time that the central spin stays in the "blip" (5.105) the exponentials grow slowly compared to the term and we can neglect them, allowing Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 179 and therefore the length of time that the nuclear spins can precess in the field jk . This is the physical reason why orthogonality blocking does not affect our large A relaxation rate-the processes involved are completely dominated by short blips and therefore the nuclear spins don't have time to precess out of their initial states. Let us now turn our attention to the other limiting case that we have worked out, that for small A (5.102). What we have found is that the relaxation rate falls off like a gaussian with applied bias £. This seemingly contradicts results obtained for A = 0 where the relaxation rate was found to be exponential with bias [20] where £0 is an energy scale in the Prokofi'ev and Stamp theory [20] such that r0 < £0 < W (W is the full energy width of the nuclear spin distribution-in our case we shall find that it is W ~ 100 — 400MHz depending on the choice of isotopes in the Fe$). Why it is that our result seemingly disagrees with this earlier result can be traced to a subtle difference in the way that our expression (5.102) and the expression (5.106) were calculated. When A = 0 there is an additional constraint when we calculate the transition prob ability for the central spin, and that is that when the central spin complex tunnels I t, M >—>• I I, M' > it is necessary in order to conserve energy that M' = — M. That is, the total energy of the central spin plus the environmental spins has to be the same before and after a flip and therefore the environmental spins cannot go into any M' they want. Now in the calculation of (5.106) this constraint, which is explicitly used in the calculation of the transition probability, is responsible for the exponential dependence on external bias. In our case we have not explicitly put in any such constraint. Instead we have al lowed in our expression for the transition probability | t, M >—> | I, M' > for any M'. The energy constraint here of course still exists (it is implicit in the expression for the T-i = ±Le-|£l/to £0 (5.106) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 180 transition probability) but it is of a slightly different nature than when A = 0. This is because when A/0 the Hamiltonian is explicitly time-dependent and therefore the system of central spin plus nuclear spins does not in general conserve energy-energy may be exchanged with the external field. Moreover, our expression (5.102) is not valid for A = 0 because we are in the fast passage limit AQ <§C Au. So it appears that the exponential dependence on bias (5.106) arises solely because of the explicit addition of an energy constraint. Now in a molecular magnet, one can ask if this constraint is realistic. The answer here is that it is not. In these materials, there are dynamic external dipolar fields coming from all the other molecules in the crystal (in addition to any externally applied fields of the type that we are discussing). In terms of the structure of the transition probabilities we see that each individual molecular magnet will have an explicitly time-dependent field acting on it from a external source. This means that in calculating the one-molecule relaxation rates it is crucial to not put in any artificial energy constraints on the central spin plus nuclear spin system. Let us now summarize our point of view on the discrepancy between our result and the A = 0 result. The exponential dependence on bias found previously [74, 20] is a limiting case that is not relevant for a real molecular magnet, arising because of the addition of an explicit energy constraint. Our result, arising from an explicitly time-dependent Hamiltonian, is the description that is relevant for molecular magnets even in the absence of an externally applied time-varying field because of the presence of time-varying internal dipolar fields which can exchange energy with the one-molecule system. 5.5.2 The General Case; Inclusion of Topological Decoherence We now turn our attention to the more general case, including the contributions due to the topological decoherence terms AkND. The inclusion of these terms increases the technical difficulties involved in evaluating the trace over the spin bath for the following Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 181 reason. The basis in which the AkN D are calculated is that of the spin Hamiltonian for the problem. That is, the axes of quantization of the nuclear spins cannot be chosen arbitrarily without properly rotating the AkNj) into this new basis. In order to get a concrete feel for why this presents a problem, let us consider the specific case of the easy-axis easy-plane spin Hamiltonian discussed at length in chapter 2. For molecules with this symmetry we showed that (2.73) nS 9nkh'n Lpe{pt} pe{pj.} ' |_pe{pi-} Tk N,D £ Mfk - E MJS-4 D — E 2E E M$- £ Mt (5.107) for ligand nuclear spins k = 1..N and xk _ g7r4|c y — %\ ID- E 2E -x (5.108) for any 57Fe in the material. By splitting the real and imaginary parts of these we can write ~(l)~ k . -~(2) ~k AN D — ak hl + iak h2 (5.109) where a B-eAkN D\ 42) = \ImAk\ (5.110) and ReA%,D -K ImAkN D n2 = • (D.lll) lk ak for all nuclear spins k = 1..N + 8. Now as we did previously, we choose the axes of quantization to be such that 7^ = z and 7^ = —cos2(lkz + sin2/?^x for all k. Then we see that since these new axes do not in general correspond to those of the spin Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 182 Hamiltonian, it is necessary to rotate the topological decoherence terms into these new bases. These rotations yield in general new topological decoherence terms r,N,D —  ak  nl ^  lak  n2 (5.112) in terms of which we may write n/^A^c^ £ e ,, J—Tr/2w J—n/2u u L1 n rL,(ti-«2)2 _± CM, i{h$—m<&*) /i,m=±l E n ih^\l{kX)\{ti-t2) e "f-< Mpk\emakL>n^Ik^~imakZ>A2-h^e-i(h-t2)1^ \j^pk > (5.113) Note that now that we have topological decoherence effects it is necessary to include all the nuclei (and therefore the product over species a) here because the sum over m, h doesn't commute with the product over nuclei. Changing variables to A (ix +12) x = u(h -12) Y = 2ir (5.114) allows us to write TTpff- = fA,2uJ dY r dXeAXY+hx] y ei(W-m** a  U J-A/2. 7-00 ,t±l n C M, E II el-^r-h^x < Mpk\emakl)^'rk*~imak)h*'Tk»e~l1^ \Mpk > (5.115) where we have taken the limits on the X integral to infinity as before. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 183 Evaluation of the Inner Product over Spin Bath States As we saw previously in the case of pure orthogonality blocking, because the gaussian term (in X) coming from the average over the nuclear-nuclear bias field cuts off large X contributions to the expression (5.115), we may expand the orthogonality blocking term inside the spin inner product to quadratic order in X. As well, we shall restrict our attention to the case when all the |o4 ' 1^1, which is again the physical limit for the materials in which we are interested. This allows us to expand the topological (1 2) decoherence terms inside the inner product to quadratic order in ' as well. Keeping (1 2) only up to quadratic in X, 04 ' or products thereof allows us to write the environmental spin inner product in the form _(2) < M£fc|ema*1}"i'^-ima*2^ |Mf > -(2) X2 l-iX< Mf |^ • /fejMf > -— < Mf\ '-(2) UJ |Mf > + < Mf | (m£$tD • Ik + hAkr^D • Ik) |Mf > +- < Mf | ([%$tD •Ik}\[Akr^D • fk]2) |Mf > +mh < Mf I Ak/1* • f, •"•r.N.D 1k Akr:N,D • ik] \Mf > -iX < Mf | m 4^* • J, r,N,D 1k -(2) UJ + h -(2) UJ •™r,N,D ik Mf > (5.116) We shall present results for these terms one by one and then combine the results after we are finished. We have that < Mf | (ml^n • Ik + hAhry!D • h) |Mf > =< Mf | (m + h) a^n^ • 4 - »(m - h)a{k2)h2k^ • /fcjMf > = (m + h) a{k^hlkiiZIkiiZ - i(m - hja^^Jk^ (5.117) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 184 The next term is 2 < Mf,k 4f K • 4)2 - «lf fa*, • 4)21 wf > a (2/,2 -^ \kpZ kfj,Z -a {2Y (1 - n\kJ (5.118) Next we have < > --< Mf a , (2)2 •vp. (or2 - T2 ) . T (1) (2) /- ~ . . - \ • 4)2 + 4f fa*, • 4)2 + ictfta^ [nlkfl • Ik^n2k, • 4]] \Mf > (oT2 - T2 \ (5.119) And finally <Mf\ (m [Al^-h] v l7£)|cos2/3^ -(2) h LO LO l7g|sin2/?fc)i LU T2 1kuz ai^nik,z(m + h) - ia{k/jn2kfiZ(m - h) (2)„ hi2 -12 z) " 4 + h)hlkpiX - ia[kJ{m - h)n2kiiX) {ak»(m ~ h)hik»y - ictklim + h)h2kiiy) (5.120) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 185 We now substitute these results into the expression for the transition probability, giving M = 2A|r MA- dy roc i[xy+ix] i(ft#_m#., 11 t; V-yl/aj ./-co _f^,, n A J-A/2U) J-oo «" X2o? m,h=±\ 1 e  p X 2 " ^^g(^+^)[P31^-i^p4iM]-l(m-/l)[p32M-iXp42M]+(l-t-m/l)/921M-(l-m/l)p22/i PGM? (5.121) where we have defined P21/x P32/x P42fi and T „ /or2_r2 \ , AT]1 (1) 2/i 2 \VZV J^2' , 1 T (!) (2)/- - \ (! -  nik,z) j-^- + 9 £ hazOi\la\l[n2kiiXnXKy - n2Mnlfc(iX) JV," JV" E4f(i--U (242-4lJ . l JV," 4 + 2 £ ^%al("°t|.«Hv _ ^2/cM2/nifcMX) = £ J (i). E ^4?^* E — *in N'. ku-l (2Il tk^zJ (2). E ^^2^ v-;- , (1)- (5.122) P<*P = E A (5.123) In other words, if there is a /J subscript on one of these quantities then it refers to the species \i. For example, p21/t is the quantity defined above for the specific subset of nuclear spins in species ji. If this subscript is not there then it refers to the sum over all species-for example, p2i is the sum over the contributions from all the species P21 = E/i P2Ui-Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 186 The Large A Limit Now there is a limit here where this simplifies considerably, and that is when A is much larger than all the other energy scales in this expression. In this case, the integral over Y gives a delta function in X, for reasons identical to those discussed in the pure orthogonality blocking case. This allows us to write 2A27T m,h=±l 11 T; LOA ^ 11 1_ ^ \e{rn+h)pzipt-i{m-h)pz2^+{l+mh)p2\^-{l-mh)p22A (5.124) The Sum Over States in the M*h Polarization Group All four quantities p2itl, P22p,, P3ip and P32p. depend explicitly on the state of the spin bath via their dependence on the set {h^z}- At this stage of the calculation we wish to perform the sum over the states in the Mjj1 polarization group. In order to do this we define the quantities V" (!)2 ft 2 ^ZV 1kP.z) , IMj, v-- I (1) (2)/. „ P2\p = Z, akJ \l-nik,z) T^ + 2iVf ^ \a^akJ(n2^xnik,y - n2k,yn1^x)\ hp = 1 fcfi — 1 I^Ip - 1) ffi (1)2, 2 \ 1 Mp, I (1) (2),„ . \ I = TT, Z, < U - nu„J + ^ Z, 1%<.K*nit,» - n2kiiynlkiiX)\ 11 fc„=i ^7Vs fc„=i \^ (2)2/. 2 ^Il~lt»z) , r I (1) (2)/- - - - M p22^ = Z. (1 - n2kiiZ) j-^— T-^^ \oi\ykl{n2kiiXnxktiy - n2k^ynlkfiX)\ kfi=l k^ — l ^(5/^ - 1) ffi (2)2 2 1M^ (1) (2) „ . . = T~2 Z, al (X ~ n2^z) + 2 iVf Z, K>1("2*^nlfc)1j, - n2fcM?/rilfcMX)| fcp. — 1 kfi — 1 _ •"'•X/I \~~* I (1) ~ I - _ I (2) ~ P3lp - TmL lafcMnl^l > P32/X - -T7JI 2_, |ttfcMN2AM2 s fcB=l YVs fc„=l Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 187 w3lli = E l (1)-JV," P £ l42W. (5.125) For reasons identical to those given in the section on pure orthogonality blocking we may now change the sum over states to an integration; explicitly we take 1 (m+/i)p3i/i —i{m-h)p32^ +(l+mh)p2ill-(l-mh)p22n 1 2irW-31p -(P31u.-P31ii) dp3i^e 2W^ 27rW32J-oodP3211 ~(P32^~P32/i) e (p21M~p21M) 2W"2 e 2M ~(p22M~P22^) 2W2 e 2^ exp [(m + /i)p3iM - «(m - /i)/o32/i + (1 + mh)p21fl - (1 - mh)p22)M} (5.126) Note that the reason that these integrations can be taken to be independent is because we are only keeping to quadratic order in |a^|. These gaussian integrations are easily performed, giving 1 CM, E 3(m+/i)p3iM-i(m-/i)p32M+(l+m/i)p2iM-(l-m/i)p22,, exp (1 + mh) [W32ltl + p2iJ - (1 - mh) [W22/1 + p22J + (m + h)p3lll - i(m - h)p32lx (5.127) Inserting these results into (5.124) gives us 2A2,7T IP1 M, \i _ 0" gi(/i*-m**) exp I (1 + m/i) \W'£X + P211 - (1 - mh) \ W%2 + p221 + (m + /i)An - «(m - /i)p32 M ~" m,/i=±l ^2,-1 /1 ITT/2 (5.128) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 188 Evaluation of the Ensemble Average Over Polarization Groups Our expression for Yl^P^ depends explicitly on the set {M^} via the quantities p2i, P22, P31 and p32. In order to perform the ensemble average over the set {M^} we perform the same calculation as was introduced in the section on pure orthogonality blocking; namely we take En%->n APG A* A» L= r dMtie~MH2N' Substitution of this into the expression for the relaxation rate (5.129) (5.130) APG Ii gives 9A2 r-l _ ZAA0 ei(/i*-m**) A m,/i=±l exp [(1 + mh) [w3\ + p2i] - (1 - mh) [\W22 + p22] + (m + h)p31 - i(m - /i)p32] (5.131) Performing the {M^} integrals gives r-i = 2A| ^ ei(h*-m*.)exp A m,/i=±l [1 + m/i) [1 — mh) W,29 + 4(5^-1) 12 12 exp 1 [(m + /i)W3i -i(m-/i)W32] (5.132) We can now perform the {m, h} sums. The result can be written T~l = [eAl cosh 2cp! + e~x'2 cos 2<p0 (5.133) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 189 where we have defined 6 Al = < + W-1) £ £ 4f(l - n?M) (5-134) A« fc„=i A2 = 4iy322 + ^(5Jg " 1} £ £ 4f (1 - (5.135) fcM=i Defining new phases (/>o = ^ cos-1 (e~A2 cos2(/?0) and (f>i = - cosh-1 (eAl cosh 2ipx) we write $ = </>o + t(f>i (5.136) in terms of which the expression (5.133) can be written •, An|cOS$|2 r"1 = 01 A 1 (5.137) which is identical to the result without topological decoherence except that the phase $ has been renormalized by the interaction with the nuclei. This is exactly what we would expect to have happen here. Note that the inclusion of topological decoherence does not change the result that in the large A limit orthogonality blocking effects disappear. General Solution for Arbitrary A Now let us attack the arbitrary A case, starting with (5.121). The strategy that we will use in this most general case is as follows. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 190 1. First, we perform the sum over states in Mp, by converting this sum to a set of integrals over quantities that are functions of the set {p G Mp}; explicitly we take 1 V-„2 (P31,-P31,)2 -) rOO -(P32,-P32,)2 2Vf2 e 32, dpzi„e 2MW / dp32/, -oo \/ ZlTvVii,. J —CO (P21,-^21,) 1 ^ -^22,-P22,) 2W.2 1 f°° 2w* 1 Z"00 \ roo ^ /-oo ^ roo 7=1 dp41fl5(p41lJ, - p41^)—= / dp42^(P42n ~ PA2U)-7== / dpi^ipi^ - plfl) Z7T -'-oo VZ7T J-oo X/Z7T J-oo \/27r ^-oo " ' V27T ./-oo ' — " ™" • V27T 1 r00 1 1 -==— / de^e 2w2W (5.138V^rW(M^) J-oo 1 ^ ; As we have seen previously, taking all these integrations to be independent is justified as long as we only keep to quadratic order in our small parameters. Note that the widths of the three distributions {pi^}, {pAip\ and {pA2ii\ are all quadratic in small parameters and therefore have been dropped (because these appear in quartic order in small parameters after the integrations), effectively replacing integrations over gaussians with integrations over delta functions. 2. Next we perform the ensemble average over polarization groups M^. Explicitly we take APG V V• L L= r dNLe-MH2N° \irN? J-OO " 3. Next we integrate over the variable X, which represents the "blip length" 4- We then perform the sums over {m, h}. 5. Finally, we change variables from Y —> Z. This gives us our final answer. Let us perform all five of these steps explicitly, starting with (5.121); 2A2TT rAI* (5.139) TTA^ = I dY T dXeiXY+^ V e*(**-™**) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 191 n 1 < x2"l ^2 elX" jr£e(m+ft)[P3iM-^/'4iM]-i(m-/i)[p32M-iXp42M]+(l+m/i)p2iM-(l-m/i)/922M (5.140) In order to complete step 1. we must calculate the following; 1 S X2p\ lXu J2^g(m+ft)[P31M-J^/541M]-J(m-/l)[p32M-iA'p42M]+(l+m/l)p21^-(l-m'l)P22Al UMf PGM? —>• (P31H~P31M) e 31f -(P32M~/'32M) e 32M -^21^-^21^) 2f 2nW< 1 -(P|2^-P22M) \ roo 1^ roo ]_ /•oo ?= / dp4iltS(p4ill - P41/i)-^= / dp42^S(p42^ ~ p42n)-7= ^Pl/APl/i - plp) Z7T J—oo V^TT •/-co v27T ./-co /oo -co (tf-e,")2 v2„2 -g 2VV2(Mjl) g V2TTW(MA y-oo exp [(m + h) [p31/i - iATp4i^] - z(m - /i) [/932Ai - iXpi2fi] + (1 + mh)p2iIJL - (1 - mh)p22ll] (5.141) Evaluating these integrals and inserting the results into (5.140) gives t4- //i/zuj roo r * i -A/2u J-oo ... ,-, n exp X2 f^2 2 W' M- ) + i—-ef^ — iXp4ill(m + h) — (m — h)Xp, 2 / LO 2 '42/x exp [(1 + mh)(Wz\ + p2l) - (1 - m/i)(W|2 + p22) + (m + h)p3l - i(m - h)p32 (5.142) We now move on to step 2. The relaxation rate is written (5.28) T~L= V APG I* n [WM»TMI Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 192 2A2, rA^ / / dY dxAXY+^ y e<("*-™*-) A J-A/2u J-oo .. ,, , m,/i=±l n exp 1= r dM,e~MV2N° KM J-OO " X2 ,,,2 + ^ LO —^ ) + z—ef" - iXp41ll(m + h) - (m - h)Xp42» Z I LO exp [(1 + m/i)^ + p2x) - (1 - m/i)(Ty322 + p22) + (ra + /*)p31 - i(m - /i)p32]] (5.143) Performing the integrations over {M^} gives exp _! _ 2A2 A 7-/l/2i j dY T dxAXY+»x\ y e^~m^exp J-Al2w J-oo ... L , , m,/i=±l (m + h) h z(ra — h) LO LU (l + mh)^-(l-mh)^ (5.144) where W, \\ and A2 are as previously and A - WO»W A - Wo^F 4 ~ ~WW32 (5.145) (5.146) We now perform step 3., the integration over the blip length X. This gives 2A27r1/2u; -A/2w .-1 exp LO 1jy WA 2 rAI2w J-AI2w ... , e»(h*-m**)exp m,/i=±l 1 + mh) — — (1 — mh) — 2 2 H h (m + h) i(m — h) LU LO LO We next perform step 4-, the sums over {ra, /i}. This gives ,2 2A2TT1/2O; M/2w T = r— / dY WA J-A/2u exp cosh (5.147) + exp -A2 + A2 AW' LO COS 2** + ^(y + £) W LO (5.148) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 193 Finally, step 5. is a change of variables which allows us to write this in the form A2 r~l = o ° / dZe~z2 [eAl_A2- cosh [2(p, + 2\SZ] + e"A2+A2 cos [2<p0 + 2A4Z]1 2y7rA Jz~ L J (5.150) where * - (^) <"«> Defining new phases ^(Z) = ^ cos"1 [e"A2+A2 cos [2<p0 + 2X4Z}] (5.152) and ^(Z) = ]- cosh-1 [eAl_A3 cosh [2<pi + 2A3Z]] (5.153) we write $(Z) = 0o(Z) + ^i(Z) (5.154) in terms of which our final relaxation rate may be written T-i= Ao ^+ / +rfZe_z2 |cos$(zT)|2 (5.155) y/nA Jz-Evaluation of the General Relaxation Rate in the Small A Limit When A is large the general relaxation rate reduces to that for pure topological decoher ence (5.137). When A becomes small, however, we expect to see some interplay between Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 194 orthogonality blocking and topological decoherence effects. In this limit we may rewrite the relaxation rate in the form (5.156) A2 exp _e_ 4W2 cos$ ' ^ 2WJ We see from (5.156) that there is a seemingly unusual dependence upon the external DC bias field when both orthogonality blocking and topological decoherence effects are present. However the reason for this form is evident. As in the case of pure orthogonality blocking, we see that there exists a gaussian profile with external bias. As well, the topological phase term has acquired a longitudinal bias dependence. This last occurs because the axes of quantization of each of the k nuclear spins that we have chosen, that is zk — and xk = **k ^^p^ ' ^° not m general correspond with the axes of the central spin Hamiltonian, which are also the axes with which the topological decoherence terms are described. When we calculate the values for the topological decoherence terms AkN D, these are by definition off-diagonal in the axes of the spin Hamiltonian. But when these are rotated into the axes defined by the orthogonality blocking parameters they acquire, in general, longitudinal components. It is these longitudinal components, coming from this mismatch of preferred axes, that gives rise to the presence of a longitudinal bias term in the topological phase in (5.155) and (5.156). When orthogonality blocking effects disappear, such as we found happens in the large A limit, this rotation of axes is not necessary and this effect is not present (5.137). 5.6 The General Single Molecule Relaxation Rate in Fe8 We now wish to evaluate the quantities entering into (5.155) for the specific case of the Fe$ molecular magnet. We will begin by evaluating the contributions that arise from Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 195 Quantity Definition LOQ W2(0) w2 1 viY+8-,ll 1 v7V+8 (, .11 , , \2 Ef=>* k=1 \ k °> r2 + p2 + w2(o) Table 5.1: Quantities coming from orthogonality and degeneracy blocking. orthogonality and degeneracy blocking effects, namely p2, T2, and W(0). How these quantities are defined, in terms of the more fundamental quantities in the theory, is repeated for convenience in table 5.1. In table 5.2 we present zero field values for T0 and LO0 for the three species -Fe8*, Fe8D and 57Fes- The quantities p\ and W(0) will be functions of any external bias field present, both because of the field magnitudes \jkJ and their changing directions, imbodied in the orthogonality blocking parameters . Shown in figures 5.6 through 5.11 are pi, W(0) and T0 as functions of an external static transverse magnetic field oriented along the x direction (as defined by the central spin Hamiltonian) in the Fes molecular magnet, for various species. In figure 5.12 is shown the full width W for the three varieties Fe$*, Fe^o and 57Fe8-Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 196 Fe8 Fe8D 57Fe8 r0 10.64 3.22 12.96 UJ0 8.32 2.93 10.04 Table 5.2: Zero external field values for T0 and u>0 for the three species shown. Units are in MHz. We now consider the terms owing their existence only to topological decoherence effects. These terms, unlike the ones dealt with in the preceding, are not strong functions of an externally applied transverse field (although there is a small dependence, since the application of an external field shifts the minima of the central spin object, thereby changing the phase that it accumulates in tunneling from | y> to | i> or viceversa). Therefore it is enough to calculate their values in zero transverse field. The definitions of Ai and A2 are repeated in table 5.3. Contributions to these from different nuclear species are shown in table 5.4. Values for Ai and A2 for the three varieties Fe8*, Fe8D and 57Fe8 are shown in table 5.5. Quantity Definition Ai A2 ^ \4 \rN° \nWf, , ll2 1 '*(5/*.-l) v-Atf (l)2n ~2 J ^ [4 \TN' la(2)n t ll2 1 W-1)?"' o(2)2fl n2 )} t—'l1 Nt [^k^l n2ku.z\\ "I" 6 Lku. = laku \l n2kaz) Table 5.3: Quantities coming solely from topological decoherence effects. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 197 400.0 300.0 I 200.0 100.0 0.0 0.0 0.1 0.2 External Field Hx[T] Figure 5.6: pi^ (dashed) and W(0M) (solid) for p =l H. V = 79 Br i47v 13^ 170 57Fe Ai^ 1.45 • 1(T6 0.06 • 10"6 0.03 • 10"6 0.66 • 10~6 0.11 • 10"6 0.18 • 10~6 12.8 • 10"6 0.13 • 10"6 0.13- 10"6 0.61 • lO"6 0.31 • 10"6 0.60 • lO"6 Table 5.4: Zero field values of the topological decoherence terms for species in Fe%. Finally, we consider terms that arise from the interplay of orthogonality blocking, degeneracy blocking and topological decoherence. These quantities are listed in table 5.6. These of course will be functions of any external transverse field because of the presence of the orthogonality blocking terms. Shown in figure 5.13 are A3 and A4 as functions of Hx for Fe8*, Fe8D and 57Fe8. The "bare" phase $ = op0 + i<pi is uniquely defined by the choice of a central spin Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 198 Figure 5.8: p1(l (dashed) and W(O^) (solid) for p. =14 N. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 199 300.0 200.0 100.0 0.1 External Field Hx [ T ] Figure 5.9: plfl (dashed) and W(O^) (solid) for p =57 Fe. Figure 5.10: plfl (dashed) and W(O^) (solid) for p =17 O. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 200 200.0 0.1 External Field Hx [ T ] 0.2 Figure 5.11: plfl (dashed) and W(0M) (solid) for p. =13 C. 0.1 External Field Hx [ T ] 0.2 Figure 5.12: Full width W for Fe8* (dotted), Fe8D (dashed) and 57Fe8 (solid). Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 201 Fe8* Fe8D 57Fe8 Ai 1.54 • lO-6 0.53 • 10"6 1.72-10"6 A2 13.06- 10"6 4.16 • 10"6 13.66 • 10"6 Table 5.5: Topological decoherence terms for three varieties of Fe8. Quantity Definition A3 A4 v VJVS" W0, i (1)- | v Wo, . (2)- , Z^ti Z^/cM = l y/~^w 1"*:, n^z\ Table 5.6: Quantities that come about due to interplay between orthogonality blocking, degeneracy blocking and topological decoherence effects. Hamiltonian. In our case we have chosen to use the spin Hamiltonian H = -DS2Z + E{S% + S2_) + C(S4+ + St) (5.157) where D = 0.292A", E = 0.046AT and C = -2.9 • 10"5A". The bare tunneling splitting between nearly degenerate ground states | ± 10 > can be found via exact diagonalization (see chapter 2) and is A0 « 3.9 • \Q~*K (5.158) Truncating the quartic spin term gives us an "easy-axis easy-plane" spin Hamiltonian Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 202 0.0005 0.0000 0.0 0.1 Hx[T] 0.001 0.000 0.2 0.0 0.2 Figure 5.13: A3 (left) and A4 (right) for the three varieties Fes* (dotted), Fe8D (dashed) and 57Fe8 (solid). which gives bare phases cpo = TTS TrSgnBHy S2ivgiJ,BHx (5.159) ' ' ~ 2E Looking back at our general expression for the single molecule relaxation rate (5.155), we see that we have now calculated all of the parameters in this expression. Fixing the spin Hamiltonian gives a single molecule AC relaxation rate with no free parameters. 5.6.1 Effect of the Nuclear Spin Environment on the Large A Single Molecule Relaxation Rate in Fe8 Shown in figure (5.14) are plots of two quantities. The first is the quantity A = V'AT-1 calculated without the addition of any nuclear spins; that is, A = An I cos $1 (5.160) The second is the quantity A, with the addition of nuclei to the effective Hamiltonian; that is, A = An|cos<l>| (5.161) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 203 We see that the phase renormalization $ —>• $ coming from the nuclei causes a tiny smoothing of the "nodes" seen in A calculated using the bare spin Hamiltonian. Fur thermore, because we know how this renormalization depends on the nuclear spins we can predict the shape of this smoothing effect. This being said, it is obvious in this particular material that this effect is extremely weak, because of the small values of the topological decoherence parameters. One should note that what we are calculating here is the single molecule relaxation rate. If we were to include many molecules, as is the case in a real crystal, we would have to include the effects of the transverse dipolar fields in $ (ie. Hx and Hy don't only come from the external field; there are also internal dipolar fields of this kind). This is the basic reason why the data shown in figures 1.20 and 1.21 is so "smoothed"-this effect comes about because of non-zero Hy in the phase $ coming from internal dipolar fields (we shall treat the multi-molecule case in the next chapter). 5.7 Summary and Discussion of Results In this chapter we began with the full effective Hamiltonian (5.6) for a general molecular magnet in an external time dependent field. We derived a formal expression for the one-molecule relaxation rate in such a system, and solved it for the special case of a sawtooth external field in the absence of quadrupolar couplings to the nuclei. The general one-molecule relaxation rate in this case was found to be where needed definitions may be found in (5.151) and the equations that directly follow, (5.134), (5.135), (5.145) and (5.146). This form for the relaxation rate simplifies in the limit that the sweeping amplitude is greater than both the energy width W and any external bias £. In this limit the (5.162) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 204 -7.0 -11.0 Figure 5.14: Presented here are (1) the bare result A0|cos$| (the lower curve) and (2) the large A result with nuclear spins A0|cos$| (the middle curve) plotted in units of Kelvin. Note the logarithmic vertical scale. The horizontal axis is Hx in Tesla-here we have (p = 0(Hy = 0). relaxation rate can be written A2,| cos$|2 A (5.163) where the nuclear spins have caused a renormalization of the Berry phase $ —> $ in a way that we can calculate with no free parameters (once the positions of the nuclei are fixed). Note that in this case orthogonality blocking does not affect the relaxation rate at all. We saw in the case of the Fe$ system that this phase renormalization coming from the nuclear spins was slight. However this need not be the case in general. We may draw some far reaching conclusions from this result- basically that the presence of spin environments that couple to magnetic degrees of freedom "randomize" the central Berry phase term, destroying Aharonov-Bohm type oscillations. This is hardly surprising. This Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 205 being said, we have shown quantitatively herein just how it happens and how to calculate the effects of a spin environment on this kind of topological decoherence. When the amplitude of the sweeping field is much smaller than the total energy spread + we find that There are two things to note here. One is that the form for the one-molecule relaxation rate is gaussian, not exponential as is reported in the literature. We have explained the reasons behind this discrepancy and the reasons why we believe the gaussian form is more appropriate to the physics of molecular magnets. The second thing here is that there is an explicit interplay between topological decoherence and orthogonality blocking effects here which manifests itself in a £ dependent Berry phase. However, we have shown that for the specific case of Fe8, the oscillations in £ coming from this term are far too small to be seen. Again, we do not expect this to hold in the general case-it is quite possible that in systems that have large amounts of topological decoherence coming from a spin bath the oscillations in T_1(£) will be apparent (these could be measured in a similar manner to recent experiments performed on Fe8 which extract just this quantity). We compared our general large A relaxation rate with that of the simple no-environment spin Hamiltonian (5.157). We found that the nuclear spins smooth out the cusps near the nodes in the relaxation rate. The effect of the nuclei considered is rather small in Fe8; however, this need not be the case in general. (5.164) Chapter 6 AC Relaxation in a Crystal of Molecular Magnets In this chapter we use the results obtained in the previous chapter to investigate what form the relaxation of the magnetization in a crystal of molecular magnets takes. In essence what we shall do is insert the one-molecule relaxation rate calculated in chapter 5 (5.155) into a master equation [15, 167] and solve it in various limits. In particular we shall be concerned with short-time relaxation in the presence of an external sawtooth field of arbitrary amplitude. 6.1 Preamble The results derived in the previous section are single-molecule relaxation rates. In other words, in their derivation we have omitted completely from consideration all effects com ing from intra-molecular terms in the full crystal Hamiltonian. The full Hamiltonian of a crystal of molecular magnets may be written in the form ff = £tti + £Vy (6.1) i i<j where Hi is the single-molecule Hamiltonian for the ith molecule (including external fields) and Vij gives the total interaction between the ith and jth molecules. This interaction will be dominated by magnetic dipolar interactions between off-site magnetic atoms. Now when we attempt to calculate the relaxation characteristics of the entire crystal, we see that it is not enough to know the single-molecule relaxation rates-there are other important terms in the crystal Hamiltonian. 206 Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 207 In order to get a feel for the physics going on inside one of these crystals during a relaxation experiment, let's attempt to qualitatively describe this relaxation first before dealing directly with (6.1). This qualitative treatment will follow the course of the standard experiments of this type that are currently being performed on these systems (see chapter 1). There are two standard ways of initially preparing these materials before their relax ation characteristics are measured. The first of these involves placing the sample in a large (5-8 T) DC magnetic field aligned along the z axis of the crystal at some high tem perature, and then cooling the sample down slowly to the mK regime. This technique has the effect of preparing the material in an initially polarized state-all the magnetic ions will be such that each molecule will be in the "up" state at the beginning of the ex periment. The second way to prepare the sample is to place the crystal in a smaller (0-5 T) z axis magnetic field at a high temperature, and then quench the system's tempera ture very quickly down to the mK regime. This has the effect of giving the crystal some less than full initial magnetization (which is a function of the strength of the original DC field) while making sure that there are no significant off-site correlations at the beginning of the experiment (why this consideration is important will soon become apparent!). Let's begin our qualitative ponderings with some generally applicable remarks. In either of the above cases, each molecule in the sample will feel some bias field coming from the sum over the contributions of all the other molecules. These bias fields will have some distribution over the crystal, whose details depend on the sample shape. Now if the longitudinal (ie., along the S-axis) bias on any particular molecule is much bigger than the "resonance window" given by our single-molecule relaxation rate (5.155), this molecule will be frozen in its original state. It is completely unable to relax! We note in passing that if we remove the effects of nuclear spins from our single-molecule relaxation rate, the resonance window shrinks down to encompass only external biases £ < AQ. Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 208 Because the values of the bare tunneling matrix elements in these materials are small (typically < 10~eK), any significant longitudinal bias distribution will freeze the entire crystal. Typically the initial bias distribution in these materials varies over a range of ~ 0.1 — 1 K, orders of magnitude larger than A0! Because significant short-time relaxation is seen experimentally in both Fe8 and Mn^, we infer that the nuclear spins are actually crucial to the physics here [20, 51, 49]. Now let's imagine what will happen as the crystal starts to relax. We now have to specify the type of field applied during the relaxation process. There are two main possibilities for which experimental results have been obtained; either we apply a DC field in any direction, or we apply a longitudinal AC field. Now regardless of the form of the applied field, the following considerations apply. We have imagined that there exists a distribution of longitudinal biases in the crystal. Now there will be some fraction of the crystal for whom the bias is smaller than their resonance window (which is a function of the applied field through (5.155)). This fraction is able to relax. When one of these molecules does relax (by tunneling to the "down" state) it rearranges the distribution of bias fields over the crystal. This is because instead of contributing an "up" to the total internal field it now contributes a "down". We see therefore that the internal field distribution will evolve in time in a peculiar way-all molecules that are inside the bias window will begin to flip, and will "dig a hole" in the bias distribution near zero bias, sending their weight out into the distribution's wings. This will slow down the relaxation, as there are less and less molecules available in the resonance window as time progresses. As we have seen in chapter 1, this effect has recently been observed in Fe8 [51]. Now in terms of a quantitative theory, what does this mean? Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 209 6.2 The Generalized Master Equation There is one characteristic here that is apparent, and that is that as the crystal starts to relax, correlations between different sites begin to develop. That is, the relaxation of the crystal is strongly dependent on its history. It has been suggested [15] that these correlations may be dealt with similarly to the way that correlations in spin glasses are treated, namely by the definition of a series of many-molecule distribution functions which are related via an expression of the BBGKY type. We shall in what follows use this technique and see what results we can obtain and which remain elusive. We begin by writing the magnetization of a crystal of molecular magnets in the form M(t) = £ j dHM{r, n,t) = yj a"H{2Pt{r, H, t) - 1) (6.2) r r where Pf(r, ri, t) is the normalized probability of the central spin complex at site f to be "up" (ie. in state \SZ = +S)) and in a DC bias field ri at time t. To obtain a solution for Pf (f, ri, t) we shall proceed in the following manner. We noted in the previous chapter that for frequencies lower than the nuclear T2 energy scale >~ T0 each pass of the AC field through resonance is decorrelated from all other such passes. This means that we may calculate the transition probability for a single pass and by simply dividing this by the period of the ac field obtain a single-molecule relaxation rate T~1(A,UJ;'H). Knowledge of this rate allows us to write a kinetic or "master" equation of the form [15, 20] Pa(r,il,t) = -T-\A,u-%){Pa{r,n,t) - P_a(?,H,t)} - £ /dH'T-\A,ufH')[P™,(?tT 4& -J i ^ r ,a (6. where P^ is a 2-molecule distribution function, ie. the joint probability of two molecules Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 210 having the characteristics implied by their arguments. Vi)(r — r) represents the DC bias at f coming from the molecule at site r , which as we have mentioned is due to magnetic dipolar interactions. We have not explicitly written out the higher order multimolecular terms P^Z\P^A\ etc., representing these by an ellipsis (...). As the crystal relaxes, the influence of these higher order terms will begin to be significant. However, in the initial stages of relaxation, we shall assume that these can be neglected. This approximation will only hold for some limited time from the preparation of the initial state and it remains to be shown that this time is long enough to be significant. 6.3 Short Time Dynamics In the early stages of the relaxation of a crystal of molecular magnets, there will be two dominant contributions to the physics. The first of these is given by the first term on the right-hand side of (6.3), and corresponds to "local" relaxation. If we were to neglect all terms but this first one we would find that the relaxation would be linear in time. Another way of saying this is that if all of the molecules relaxed independently of each other, then the total relaxation of the crystal has to be exponential (and therefore linear at short times). Defining the quantities M(f, H, t) = Pt(f, H, t) - P^f, %, t) , M(H, t) = £ M(f, H, t) , M(t) = J dHM(H, t) f allows us to write PQ(f, li, t) = -T~\A, to; U){Pa(r, U, t) - P.a(r, H, t)} (6.4) in the form M(H,t) = -r-1(A,w;H)M(??,t) (6.5) Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 211 Now if this is the whole story (ie. we neglect all corrections to this) this is easily solved, giving M{H,t) = M(ft, 0)e-t/r(A'^) (6.6) and therefore M(t) = J driM(n, o)e-2/T{A'UJ'il) (6.7) which has short time behaviour M(t) ~\-tj dfi M(H, 0) T~1 (A, LO; it) (6.8) which, as we have claimed, is linear in time. Now we turn to the leading sub-dominant term in (6.3). How do we incorporate the leading corrections to the local term, and what effect will this have on the relaxation characteristics? As the crystal begins to relax, the internal bias distribution (and therefore M{ri,t)) will change in time in a way that one can calculate analytically in an ellipsoidal crystal (see Appendix A). This redistribution of biases is exactly the effect that the leading correction term in (6.3) has on the relaxation. We can therefore take into account the leading corrections by simply inserting the form derived in Appendix A for M(ri,t) into the local term (6.5). This gives us the following equation for the relaxation rate, incorporating both the local relaxation and the first corrections to it; M{t) = - j dH r"1 (A, LO] it) M(H, t) (6.9) where M(it, t) is the (evolving) distribution of biases derived in Appendix A and r~l (A, LO; it) is the general single-molecule relaxation rate derived in the previous chapter (5.155). Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 212 6.3.1 Strongly Annealed Samples and the Large A Limit As we have seen earlier when the external sweeping amplitude is large enough to sweep all molecules through resonance, ie. when A > Hz + WD, the relaxation rate simplifies considerably. We have also seen that in the Fe8 material topological decoherence effects are minimal. If we ignore these we find that the relaxation rate can be written in the form A20\cos^{Hx,Hy)\2 r-1(A,Hx,Hy) = A (6.10) Let us now consider the experimentally relevant case of strong annealing, where the initial magnetization M(t0) <C Mo, the saturated magnetization. In this situation the initial magnetization distribution over the applied field is gaussian (see [200] and refs. therein) 3/2 M(H,t0) = M(t0) exp £ (Hi - H?) (6.11) nW2(M)/ The half-width WD(M) is directly measured in experiments [] and is much larger than E0 for Fe$. Note that the observation that Wr> is a function of M and is directly extractable from experiment is due to work by Stamp and Tupitsyn [198]. Plugging (6.10) and (6.11) into (6.9) allows us to solve for M(t). We find that M(t) = M(t0) exp [-TAC(H, M)t (6.12) where the relaxation rate is 2A§ T,c(H,M) = ^wh{M)A exp J dHx j dHy |cos $(HX, Hy Wl(M) {Hx - H°x)2 + (Hy - H\ 0\2 (6.13) Now as we have stressed throughout this document, our analysis depends on the choice of a particular spin Hamiltonian (for Fe$ this was the easy-axis easy-plane Hamiltonian). Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 213 The choice of the parameters in the spin Hamiltonian strongly affects the dependence of the tunneling matrix element on external transverse fields. This is relevant here as the dependence on Hx and Hy in the relaxation rate changes if we use a different spin Hamiltonian as we saw in chapter 2. In addition, in deriving our relaxation rate (6.9), we have assumed the instanton calculation which of course is not exact. In order to extract the correct tunneling amplitude as a function of external field, we have to exactly diagonalize the spin Hamiltonian plus the external field. That is, the correct expression takes our A0| cos$(Hx, Hy)\ to \A(Hx,Hy)\ extracted by exact diagonalization. There is a specific case where we can get around the first difficulty, and that is near the nodes of the tunneling amplitude-this case has been treated independently by Stamp and Tupitsyn [198, 200] and we will review their results in the following section. In this case the tunneling amplitude will be extracted by exact diagonalization and therefore comparing our results to theirs allows us to gauge the accuracy of our instanton approximation. In the case where the spin Hamiltonian is the easy-axis easy-plane model we have seen that, in the limit being considered, cos ${Hx,Hy)\ cos 2TTS 2irSgrtB(Hy + H°y) On + cosh S2KgnB(Hx + Hl) E (6.14) where we have split the transverse fields into two contributions, one from externally applied fields Hxy and one from internal transverse fields Hx>y. Integrating over the internal fields Hxy gives us our internal dipolar field-averaged relaxation rate TAC(H,M) = ^\cos$(H°x,H°y) where cos 2n2 S2g2n2Bwl(M) COS 2irS-2irSg^BH0y O0 + (6.15) Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 214 s47r2g2M2 w2 (M) 2£2 cosh E (6.16) The relaxation rates (6.15) are shown in figure 6.1 for several values of WD(M). 6.3.2 General Solution Near the Nodes Stamp and Tupitsyn [200, 199] have demonstrated that near the nodes of \A(HX, Hy)\2 one can expand \A(HX,Hy)\ around its zeroes, as \A(HX,Hy)\ = A0coshZ, where for the Fe8 biaxial symmetry Z = 4X+ilby= [(Hy/Fy) + I {Hx ~ H^)/FX (6.17) (see [200] for complete details of this calculation). The "periods" Fx and Fy vary slowly with Hx and Hy, and so we expand as Fi{Ht) — F;(0) + Fi(0)"Hf/2 + .... We can thus solve for the AC relaxation rate around the nih "nodal field" H^; r^(Hl,M) = ^ Cn(M) + (rft + tfj (6.18) Cn(M) = *-Wl{M) \F-\HX^) + F-2(0) (6.19) This is a parabolic increase around minima with values exactly proportional to the inde pendently measurable Wp(M). Choosing a specific form for the Fe8 spin Hamiltonian (such as the easy-axis easy-plane model we have been considering) allows us to use (6.13) to plot r^c(-^j°\ M) over the whole range of Hj_ for different values of M (see figure 6.2). By varying M, the field angle 9 = sin_1(i7°/77°) and H0 , AC relaxation measurements ought to obtain information about the giant spin Hamiltonian inaccessible to ESR experiments [200]. Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 215 Figure 6.1: Top figure: Logw A2 vs. Hy, for 6 = sin"1 (Hll/Hy) = 0. Curves are, from bottom to top, WD == 0,10, 20, 30 and 50 mT. Bottom figure: Same, but with 9 = 1°. Note that these results are obtained from our relaxation rate which was derived assuming the instanton approximation for the tunneling amplitude. ERROR^undefined 4 ^ Relaxation in a Crystal of Molecular Magnets OFFENDING COMMAND: STACK: Chapter 7 Summary and Outlook In this thesis we have presented a quantitative theory of the physics of magnetic relaxation in the presence of time-dependent external fields in crystals of molecular magnets. In so doing we have achieved considerable insight into the effect of spin environments on the quantum dynamics of mesoscale magnetic moments. This research produced several interesting results, which we reproduce here. We used the standard Prokof'ev and Stamp theory [15, 20, 122], together with the crystallographic structure of the Fe8 molecular magnet, to extract quantitative values for the topological decoherence {ajj.1'2^} and orthogonality blocking {j3k} parameters for this material. We found that topological decoherence effects were minimal (lajj.1'2^ ~ 10~3) but that orthogonality blocking was significant. We found that the primary source of the orthogonality blocking came from the large number of protons in the organic matter in the Fe8 molecule. In terms of the value K = — In [T[k cos (3).] m the Prokof'ev and Stamp theory, we find that K quickly rises with transverse field, reaching K ~ 1 for transverse field magnitude of \H\ ~ 110 gauss in Fe8* (see figures 3.14 and 3.15). All quantities that are functions of the hyperfine couplings were found to exhibit a significant isotope effect. In particular, A, K, and the nuclear spin linewidth W all depend on the isotopic concentration in an Fe8 sample. Fitting the shape of oscillations in tunneling amplitude to our derived relaxation rate, we found that internal dipolar transverse fields in Fes are °n the order of yjH2 + H2 ~ 0.03^ ~ 30 gauss. This means that even in the absence of any external transverse fields 217 Chapter 7. Summary and Outlook 218 K ~ 0.09 in Fes*, which is just large enough to cause significant decoherence [15, 20, 122], erasing any hope of seeing macroscopic quantum coherence in Fe8*. We derived an expression for the single-molecule relaxation rate in any molecular magnetic substance with an effective Hamiltonian of the form (5.6) being acted upon by a sawtooth AC field, which is given by where 2W (7.2) with £ the longitudinal bias, W the total energy spread available to the molecule, and A the amplitude of the field. A0 is the bare tunneling matrix element of the spin Hamilto nian part of (5.6) and <&(Z) is the Berry phase of the central spin complex, renormalized by the interaction with the nuclear spins. This expression may be interpreted in the following way. The effect of the nuclear spins is both to broaden the linewidth W of the molecule and to renormalize the Berry phase $ —>• $(Z). The first of these effects comes about from orthogonality blocking and degeneracy blocking. The second comes about primarily from topological decoherence, although the Z dependence of $ is due to an interplay between topological decoherence and orthogonality blocking. In the limit where A is much bigger than \W + £|, the linewidth W becomes irrelevant, and all orthogonality blocking effects disappear, giving , A^|cos$|2 , N r"1 = 01 A 1 (7.3) In the opposite limit A <C \W + £| we find that -i A2 2 AW2 cos$(c;/2W0|2 (7.4) Chapter 7. Summary and Outlook 219 ie. the relaxation is gaussian, with an unusual bias dependent Berry phase. We showed that in Fe8, because the topological decoherence effects are so small, this peculiar feature is overpowered by the gaussian and it is permissible to take <5>(£/2W) —>• $. This gaussian dependence disagrees with previously quoted results which find the relaxation to be exponential in bias [15, 20, 122]. We believe that the reason for this disparity is that in the calculation leading to the exponential bias dependence the authors added an energy constraint which is unphysical for these systems (see chapter 6 for a discussion of this point). We used our single molecule relaxation rates as input into a master equation in order to take into account the crystalline nature of the real material [15, 167]. We studied several specific cases and found the following general observations. If the sweeping amplitude of the field is larger than the total longitudinal bias available to the molecules A > \W + £ + ED\ the relaxation is linear at short times. When the sweeping amplitude A <C \W + £ + ED\ the relaxation is initially linear, turning over to square root at later times. This occurs for both Lorentzian and gaussian initial longitudinal bias profiles. We used these general results in the specific case of AC measurements performed on Fe8 [39]. In this case we are in the large A limit, giving linear relaxation A?I ros <I>I2 M(t)~M(0)(l-^&t) (7.5) where <§ is the renormalized Berry phase, averaged over the transverse internal dipolar fields (7.11). We then fit this relaxation rate to the experimental data. We found the following interesting results. Our derived expression for the tunneling amplitude produces results which quantita tively match those fund using exact diagonalization techniques for transverse fields less than ~ 0.5 T. This provides evidence that the instanton calculation is valid here. Our expression also qualitatively agrees with experimental results. Chapter 7. Summary and Outlook 220 In addition to these results, our research produced several new results related to the central problem investigated here. In chapter 2 we showed how the WKB, perturbation theory and instanton methods for calculating tunneling splittings in spin Hamiltonians for selected Bravais lattice symmetries compare to exact results obtained via diagonalization. We found that the standard WKB result's prefactor [25, 26] (corresponding to the first correction ~ h in the WKB expansion) disagrees with the exact result in each case studied. In chapter 3 we extended the work of Tupitsyn, Prokofiev and Stamp on the effective Hamiltonian of a central spin system [74, 75, 15, 20, 122] to allow for the calculation of the parameters {ak ' '} and {(5k) in real systems. As well, we included in our derivations the effect of electric quadrupolar terms coming from nuclei with spins greater than 1/2. In chapter 4 we proposed a classification scheme for all exactly solvable time depen dent generalized Landau-Zener Hamiltonians. Our proposal is that if the equation for the wavefunction can be mapped to Riemann's equation then the problem is solvable. We presented two such mappings, along with exact results for the transition probabilities in each case. This concludes the presentation of our results. In conclusion, we would like to point out some avenues for future research related to this work. It will be quite straightforward at this point to perform investigations on molecular magnets other than the seminal Fe8, using the methods developed herein. This will give us a new quantitative tool with which to study decoherence and relaxation in these materials. In terms of the mathematical physics side of this work, there were many tantalizing relations between the theory of special functions and integral and differential equations glimpsed which may be of interest; we have noted these in the text whenever one occured. The problem of how to treat multiple crossings coherently in the presence of a spin bath Chapter 7. Summary and Outlook 221 (ie. when our coarse graining approximation breaks down), or the related problem of how to go beyond the fast-passage approximation, remain works in progress. The resolution of the conflicting results obtained with the inclusion of an oscillator bath, necessary in the 0(DS2) effective description, have to be addressed in order to understand the thermal/quantum crossover. The inclusion of the quadrupolar terms in the transition probability may provide many hours of gruesome enjoyment. Appendix A Bias Distribution in a Dilute Solution of Dipoles In this appendix, we derive a general expression for the bias distribution inside a crystal of molecular magnets. The general arguments that we will use have been published previously [123]. It is however worthwhile to understand how the formulae quoted in the text arise, and that shall be the purpose of this exposition. We begin by presenting the following question. Given an ensemble of dipoles in a crystal of arbitrary shape, what will be the probability that the total dipolar field at some site r has the value ril Basically we can just write down the general solution to this problem; it is a sum over all possible ways of obtaining this value with the spins we've got, Ti Here each site fi ^ f in the crystal can either have a spin pointing "up" or a spin pointing "down", ie. in the ±z directions. The dipolar field is taken to be H(?{ - f) = pr^Ta (4 - 3f(4 • r)) (A.2) where v is a unit cell volume, ED is the dipolar field scale, and d^ is the direction that the ith spin is pointing. We now specialize our treatment to the case of an ellipsoidal sample. In this case the f dependence drops out of P(f,rl), and we can write n 222 Appendix A. Bias Distribution in a Dilute Solution of Dipoles 223 Writing the delta function out as an exponential gives pM = <k>f^ff*"%'-U**>f (A'4) This can be rewritten as We note that the contribution from H(fi) flips sign depending on the sign of . There fore we can write the preceding as pM = ^ / *w* n / f 9 n / g^«>* ( A.6) where we have subdivided the total number of spins TV into two subsets, those that are up ({t}) and those that are down ({!}). We now note that because we are only allowing the spins in our solution to point in the ±z directions we have a symmetry here that we can exploit; the solution for P^H) has to be of the form P(H) = 6{HX = 0)5(Hy = 0)P(Hz) (A.7) That is, the final solution in this idealized case has the probability for the transverse fields to be zero. We therefore recast our expression (A.6) into an expression for P(HZ), ie. the probability of being in a longitudinal bias Hz; (.4.8) We can perform these integrations; /rw - '"/?(',y) \l Jo JO ' Appendix A. Bias Distribution in a Dilute Solution of Dipoles 224 Changing variables to x = 1 /r3 gives 27r r ,„ . „ f°° dx 1 _ |L rd0sme f° ^ (l - e±iYE^l-z™2e>) (A.10) This can be broken up into two parts, like this 27T r ,n . „ r°° dx 1 30 y dBsindJ ^[{l-cosYEDv{l-3cos29)x)±i[s\nYEDv{l-3cos29)x)) (A.ll) The leftmost integral is easy to do. We find that 27T r .„ . „ r°° dx ZTT fn r00 cLx / \ — d9 sin 9 — (l-cosYEDv(l-3cos29)x) 30 Jo Jo XZ V ' r d9 sin 0|1- 3 cos2 9\ HI Jo (A.12) ir2EDv\Y\ r 30 8TT2EDV\Y 9V3Q The second one is a little trickier. We write it as follows; —— / d9sm9 — sinYEDv(l - 3cos29)x 30 Vo 7o x2 = ^7^" I d6sind I — (sin fy/Wl - 3cos2 0)z) - YEDv(l - 3cos2 6)x) oil Jo Jo xl v v ' ' (A.13) The reason we can do this is that the term we have added is equal to zero, as d9sm6(l-3cos2 9) = 0 (A.14) J 0 and the apparently divergent integral over x contains a cutoff which we've suppressed here (besides, the divergence of this term is only logarithmic). With this new expression we find, defining b = YEDv(l-3cos29) (A.15) Appendix A. Bias Distribution in a Dilute Solution of Dipoles 225 that rA dx [ (sin (bx) — bx) Jo xz = 6(Ci(A6)-ln(A) + l-ln(6)-7) (A.16) and the whole expression is ±iYEDv2n „ ^ / d9 sin 9(1 - 3 cos2 9) oil Jo (Ci(AYEDv(l - 3 cos2 9)) - ln(A) + 1 - ln YEDv(l - 3 cos2 9) - 7) ±iYEDv2ir [V d9 sin 9(1 - 3 cos2 9) Jo 3Q (Ci(AYEDv(l -3cos29)) - ln (YEDv(l - 3 cos2 9))) (A.17) The leftmost term is ±iYEDv27r r* 3Q j* d9 sin 0(1 - 3 cos2 9) (Ci(AYEDv(l - 3 cos2 9)) I\Y\EDVTT28 /A The rightmost term is ±iYEDv27r r« 3ft Collecting our results, we find that dr 3Q lo d° Sln 9iyl ~ 3 C°s2 ^ {1u(AYEdV(1 ~ 3 cos2 e)) ±iYEDv2ir (A 8 1 , T 1—. 4 1— \ V3 ~ 9V^i?e(tanh~ ^) + 9V^TriJ (A.19) J n = 1 - pY [1 ± 1 ± i =F iK] ,{Y>0} = l + pY[l^l±i±iK] , {Y < 0} (A.20) Appendix A. Bias Distribution in a Dilute Solution of Dipoles 226 where we have defined Going back to our original expression for P(£) we see that 1 r°° dYeinzl m m P(HZ) = — / dYeiH*Y II [1 " PY H + iK]} JJ [1 - pY [2 + i - iK}] + -!- [° dYelH*Y Y[[i + pY[2-i- iK]} [1 + pY [+i + iK]] 27r J~°° m m (A.22) This can be written P(HZ) = — f°° dYeiY^Hz+{1~{N^~N^}p{K~l)IN)e~2^[N^~~Ni}YplN z 2ix Jo 1 roo _|_ _ / dye-iy(^+{l-(iVt-iV1)}p(A--l)/iV)e-2T(l-{iVt-iVA}yp/JV 27T 7o (A.23) where iVf, iVj. are the numbers of up/down spins respectively. As well, we have assumed that Q is large enough so that we can rewrite the sums as exponentials. Performing the integrals gives TM 1 F™ = lT(g, + g)» + P (A'24) where we have defined E(t) = p(K - 1)JV(1 - M) , T(t) = 2npN(l - M) (A.25) where M = ^Njt~N^ is the magnetization of the sample. Now in order to write our full P{ri), we have to include the Hx and Hy dependence which we've seen is in the ideal case very simple. In a real material, the delta functions in this dependence will be spread somewhat. For this reason we are going to replace the Appendix A. Bias Distribution in a Dilute Solution of Dipoles 227 delta functions with gaussians, whose width will be much smaller than the energy scale ED of the longitudinal distribution. Because we have a x -H- y symmetry here, we shall choose the transverse widths to be the same. This gives, for our final result, TM 1 T2 eXP Hi + Hi ™2t (A.26) 7T (Hz + E)2 + T2 2irWjt where, as we have mentioned, W^t <C ED is the width of the transverse field distribu tion. Appendix B Time Evolution of Nuclear Spin States In this appendix we shall show how the nuclear spin states evolve over time due to their coupling with the central spin complex in a molecular magnet. We shall assume the same conditions as were presented in chapter 6, and furthermore we shall assume that we are in the large A limit. We may write down a master equation for the flow of the one-molecule system in the space \S,M > where \S > is the two-level state of the central spin and \M > is the state of the spin environment of the form Ps,M = -PS,M £ TJ-is',M->M' + £ Ts'\S,M'^MPS',M' (B-1) s' ,M' S' ,M' where PS,M is the normalized probability of being in state \S,M > and T~^S, M M> is the transition rate from state \S, M > to state \S', M' >. Now as we explicitly showed in chapter 6, these relaxation rates do not select for any particular state of the central spin system. That is, the rate of flow | t>—> \ I> is identical to the rate of flow | J,>—>• | y> in the central spin space. This means that we can take Ts->s',M^>M' ~ TS'->S,M->M' — TM->M' (B-2) and write our master equation in the form PM = -2PM £ r-^M, + 2 £ r-)^MPM, (B.3) M' M' where we have defined PM = Y,pSM (B.4) s 228 Appendix B. 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