Analysis of correlation patterns in hyperfine sublevel correlation spectroscopy of<Emphasis Type="Italic">S</Emphasis>=1/2,<Emphasis Type="Italic">I</Emphasis>=3/2 systems

The properties of two-dimensional four-pulse electron spin echo envelope modulation spectra of nuclei with nuclear spinI=3/2 in disordered systems are discussed by means of calculated lithium hyperfine sublevel correlation spectra. Since a case of small coupling is treated, the multifrequency transition components appear only in the first quadrant of the two-dimensional spectra. The influence of the hyperfine and nuclear quadrupole interaction parameters on the cross peak ridges in the two-dimensional powder pattern is analyzed. Furthermore the possibility to determine the magnitude of the dipolar hyperfine and the nuclear quadrupole coupling from the cross peak line shapes is investigated. In this context the potential of those spectral features representing the so-called multiquantum transitions for the interpretation of the spectra is demonstrated.     

Field-dependent nuclear relaxation of spins 1/2 induced by dipole-dipole couplings to quadrupole spins: LaF3 crystals as an example

A general theory of spin-lattice nuclear relaxation of spins I=1/2 caused by dipole-dipole couplings to quadrupole spins S1, characterized by a non-zero averaged (static) quadrupole coupling, is presented. In multispin systems containing quadrupolar and dipolar nuclei, transitions of spins 1/2 leading to their relaxation are associated through dipole-dipole couplings with certain transitions of quadrupole spins. The averaged quadrupole coupling attributes to the energy level structure of the quadrupole spin and influences in this manner relaxation processes of the spin 1/2. Typically, quadrupole spins exhibit also a complex multiexponential relaxation sensed by the dipolar spin as an additional modulation of the mutual dipole-dipole coupling. The proposed model includes both effects and is valid for an arbitrary magnetic field and an arbitrary quadrupole spin quantum number. The theory is applied to interpret fluorine relaxation profiles in LaF3 ionic crystals. The obtained results are compared with predictions of the 'classical' Solomon relaxation theory.     

Two-dimensional hyperfine sublevel correlation spectroscopy: powder features for S=1/2, I=1

The lineshapes of two-dimensional magnetic resonance spectra of disordered or partially ordered solids are dominated by ridges of singularities in the frequency plane. The positions of these ridges are described by a branch of mathematics known as catastrophe theory concerning the mapping of one 2D surface onto another. We systematically consider the characteristics of HYSCORE spectra for paramagnetic centers having electron spin S=1/2 and nuclear spin I=1 in terms of singularities using an exact solution of the nuclear spin Hamiltonian. The lineshape characteristics are considered for several general cases: zero nuclear quadrupole coupling; isotropic hyperfine but arbitrary nuclear quadrupole couplings; coincident principal axes for the nuclear hyperfine and quadrupole tensors; and the general case of arbitrary nuclear quadrupole and hyperfine tensors. The patterns of singularities in the HYSCORE spectra are described for each case.     

Comparison of electron spin relaxation times measured by Carr-Purcell-Meiboom-Gill and two-pulse spin-echo sequences

Electron spin relaxation times obtained by two-pulse spin-echo and Carr-Purcell-Meiboom-Gill (CPMG) experiments were compared for samples with: (i) low concentrations of nuclear spins, (ii) higher concentrations of nuclear spins and low concentrations of unpaired electrons, (iii) higher concentrations of nuclear spins and of electron spins, and (iv) dynamic averaging of inequivalent hyperfine couplings on the EPR timescale. In each case, the CPMG time constant decreased as the time between the refocusing pulses increased. For the samples with low concentrations of nuclear spins (the E' center in irradiated amorphous SiO2) the limiting value of the CPMG time constant at short interpulse spacings was similar to the Tm obtained by two-pulse spin echo at small turning angle. For the other samples, the time constants obtained by CPMG at short interpulse spacings were systematically longer than Tm obtained by two-pulse spin echo. For most of the samples, the CPMG time constant decreased with increasing electron spin concentration, which is consistent with the expectation that the CPMG sequence does not refocus dephasing due to electron-electron dipolar interaction between resonant spins. Dynamic processes that average inequivalent hyperfine couplings contributed less to the CPMG time constant than to the spin-echo decay time constant. The impact of nuclear echo envelope modulation on CPMG time constants also was examined. For a Nycomed trityl radical in glassy D2O:glycerol-d8 solution, the CPMG time constant was up to 20 times longer when the time between pulses was approximately equal to integer multiples of the reciprocal of the deuterium Larmor frequency than when the time between pulses was an odd multiple of half the reciprocal of the deuterium Larmor frequency.     

Rescaling of 2D ESEEM Data as a Tool for Inverse Problem Solving

A rescaling procedure is proposed for electron spin echo envelope modulation spectra observed at several electron paramagnetic resonance transitions. Analytical expressions describing the relations between the rescaled frequencies and hyperfine and quadrupole parameters of the remote nucleus are obtained. The dependences of the rescaled data on the external magnetic field and spin projections of the ion nucleus and the remote nuclei are used to derive the parameters of the nuclear state in the crystals Y₂SiO₅ and YVO₄ doped by ion Nd³⁺.     

Pulsed EPR studies of polycrystalline cesium hexamethyl hexacyclen sodide

The electron spin echo envelope modulation (ESEEM) technique of pulsed EPR spectroscopy has been used to measure weak¹³³Cs hyperfine couplings to trapped electrons in polycrystalline cesium hexamethyl hexacyclen sodide. Two magnetically distinct groups of weakly coupled¹³³Cs nuclei were found — one with an isotropic electron-nuclear hyperfine (Fermi contact) coupling of 0.34 MHz and the other with a contact coupling less than 0.1 MHz. Analysis of these data, by computer simulation, shows that the number of Cs nuclei that give rise to these interactions and their distance from the paramagnetic center is consistent with the hypothesis that the trapped electrons occupy vacant anion sites. The results indicate that the spread of unpaired electron spin density, along the axis formed by the “contact ion pair,” may be greater than that in the plane perpendicular to this axis.     

The glories of ESEEM: Measuring electron-nuclear dipolar couplings in orientationally disordered solids

In orientationally disordered solids, the anisotropy of spin interactions can cause a spectral line-broadening that severely complicates the acquisition and analysis of ESEEM (electron spin echo envelope modulation) spectra. The reduction or removal of powderbroadening is accomplished in ESEEM spectroscopy through the cancellation of antagonistic, static interactions. Three, distinct examples of this approach to spectral line-narrowing in nuclear spin one-half systems are described, together with their implications regarding the determination of electron-nuclear dipolar couplings. Applications involving¹⁵N and¹H nuclei in protein and model systems are discussed.     

Electron spin-echo envelope modulation at S-band. 2: Hyperfine and weak nuclear quadrupole interaction effects forI>1/2

The analysis of the nuclear modulation often observed in the electron spin-echo signals gives information on the structural arrangement around the paramagnetic center and on the interactions undergone by the spin system. Most applications to date have been carried out at X-band (≈9 GHz). However, to run experiments at different operating frequencies yields more accurate results and increases the number of experimental situations that can be investigated. Here we analyze the two- and three-pulse modulation patterns simulated at S-band (≈3 GHz) where the modulation is due to two different nuclei of experimental interest at different electron-nucleus distances, and comparison is done with the corresponding X-band patterns.²H (I=1) and⁷Li (I=3/2) nuclei are chosen due to their low quadrupole moment, so that comparison can be done between the effects due to weak nuclear quadrupole interaction at the two bands. It is shown that at short distances the complicated patterns due to hyperfine couplings are successfully interpreted by Fourier transform in the frequency domain, and that the use of S-band is really of some advantage with respect to X-band in detecting the presence of a weak nuclear quadrupole interaction.     

Relaxation times and line widths of isotopically-substituted nitroxides in aqueous solution at X-band

Optimization of nitroxides as probes for EPR imaging requires detailed understanding of spectral properties. Spin lattice relaxation times, spin packet line widths, nuclear hyperfine splitting, and overall lineshapes were characterized for six low molecular weight nitroxides in dilute deoxygenated aqueous solution at X-band. The nitroxides included 6-member, unsaturated 5-member, or saturated 5-member rings, most of which were isotopically labeled. The spectra are near the fast tumbling limit with T₁∼T₂ in the range of 0.50–1.1 μs at ambient temperature. Both spin–lattice relaxation T₁ and spin–spin relaxation T₂ are longer for ¹⁵N- than for ¹⁴N-nitroxides. The dominant contributions to T₁ are modulation of nitrogen hyperfine anisotropy and spin rotation. Dependence of T₁ on nitrogen nuclear spin state m_I was observed for both ¹⁴N and ¹⁵N. Unresolved hydrogen/deuterium hyperfine couplings dominate overall line widths. Lineshapes were simulated by including all nuclear hyperfine couplings and spin packet line widths that agreed with values obtained by electron spin echo. Line widths and relaxation times are predicted to be about the same at 250 MHz as at X-band.     

Three-pulse spin echo signals from quadrupolar nuclei in magnetic materials

The time evolutions of the three-pulse spin echo signals from quadrupolar nuclei ⁶³Cu and ⁵³Cr in ferromagnetic CuCr₂S₄:Sb have been investigated at the temperature T=77 K. The experimental results were well explained by the developed theory of the time evolutions of the three-pulse echoes. The main assumption of this theory is the assumption that the temporal fluctuations in the electron magnetization lead to the fluctuations in the hyperfine and quadrupole interaction Hamiltonians.     

Observation of coherent oscillations in a single electron spin

Rabi nutations and Hahn echo modulation of a single electron spin in a single defect center have been observed. The coherent evolution of the spin quantum state is followed via optical detection of the spin state. Coherence times up to several microseconds at room temperature have been measured. Optical excitation of the spin states leads to decoherence. Quantum beats between electron spin transitions in a single spin Hahn echo experiment are observed. A closer analysis reveals that beats also result from the hyperfine coupling of the electron spin to a single 14N nuclear spin. The results are analyzed in terms of a density matrix approach of an electron spin interacting with two oscillating fields.     

Influence of the anisotropic hyperfine interaction on the 14N ENDOR and the ESEEM orientation-disordered spectra

The influence of the anisotropic hyperfine interaction on the 14N electron-nuclear double resonance/electron spin echo envelope modulation spectra is studied by approximate analytical and graphical methods for the case of the isotropic g-factor. The suggested determination of the modified characteristic directions of the magnetic field due to anisotropy enhances the insight in the structural details of the system and analytical solutions of the secular equation for these conditions are derived. The graphical method, previously used for the analysis of the orientation dependence of the 14N nuclear-transition frequencies in orientation-disordered samples for isotropic hyperfine interaction is extended to the case of arbitrary anisotropic hyperfine tensor. The above analytical and graphical methods are illustrated and tested against exact simulations in two practically important cases: (i) isotropic hyperfine interaction (hfi) exceeding other nuclear interactions in nuclear spin Hamiltonian. (ii) Cancellation of the isotropic part of the hfi.     

Multiple-pulse NMR of oriented radioactive nuclei under electric-quadrupolar interactions

The formation of the nuclear spin echo in the Angular Distribution of Nuclear Radiations (ADNR) in the presence of combined magnetic-dipole and electricquadrupole interactions (EQI) is theoretically considered. The case of a perturbative EQI in addition to the dominant magnetic hyperfine interaction is analysed. When both interactions are extremely large, with the EQI much greater than the magnetic inhomogeniety, the excitation of spin-echo in ADNR by pulsed rf field, including transitions between a definite pair of quadrupole shifted Zeeman levels is considered.     

Large-magnitude high-spin nuclear parameters in a Ti3+ center from X-band EPR measurements at 10 K

Precise 10 K X-band EPR measurements and subsequent spin-Hamiltonian analysis by direct matrix diagonalization methods are reported for a Ti3+ (S = 1/2) center in tetragonal zircon (zirconium silicate, ZrSiO4). A special, and previously unobserved, feature of the supposedly uniaxial spectrum is the marked angular dependence of the titanium hyperfine lines in the perpendicular, ab crystal plane. As discussed, this can only arise from the presence of high-spin nuclear terms of dimension BIk, SIk (k = 3, 5) in the spin Hamiltonian. Parameters arising from these terms were determined to have magnitudes very much larger than observed previously in first-row transition ions. The consequences of precise determination of these high-spin parameters are significant and several: a precise determination of the nuclear quadrupole tensor leading to a ratio 47P/49P in excellent agreement with the ratio derived from the corresponding nuclear quadrupole moments; an apparent anisotropy in the nuclear Zeeman interaction which can be identified with anisotropy in the chemical shielding "tensor"; a marked hyperfine anomaly. The origin and significance of these observations are discussed.     

Calculation of Double-Quantum-Coherence Two-dimensional Spectra: Distance Measurements and Orientational Correlations

The double-quantum-coherence (DQC) echo signal for two coupled nitroxides separated by distances ?10 Å, is calculated rigorously for the six-pulse sequence. Successive application of six pulses on the initial density matrix, with appropriate inter-pulse time evolution and coherence pathway selection leaves only the coherent pathways of interest. The amplitude of the echo signal following the last π pulse can be used to obtain a one-dimensional (1D) dipolar spectrum (Pake doublet), and the echo envelope can be used to construct the 2D DQC spectrum. The calculations are carried out using the product space spanned by the two electron-spin magnetic quantum numbers m ₁, m ₂ and the two nuclear-spin magnetic quantum numbers M ₁, M ₂, describing, e.g. two coupled nitroxides in bilabeled proteins. The density matrix is subjected to a cascade of unitary transformations taking into account dipolar and electron exchange interactions during each pulse and during the evolution in the absence of a pulse. The unitary transformations use the eigensystem of the effective spin Hamiltonians obtained by numerical matrix diagonalization. Simulations are carried out for a range of dipolar interactions, D, and microwave magnetic field strength B ₁ for both fixed and random orientations of the two ¹⁴N (and ¹⁵N) nitroxides. Relaxation effects were not included. Several examples of 1D and 2D Fourier transforms of the time-domain signals versus dipolar evolution and spin-echo envelope time variables are shown for illustration. Comparisons are made between 1D rigorous simulations and analytical approximations. The rigorous simulations presented here provide insights into DQC electron spin resonance spectroscopy, they serve as a standard to evaluate the results of approximate theories, and they can be employed to plan future DQC experiments.     

Characteristics of ESEEM and HYSCORE spectra of S>1/2 centers in orientationally disordered systems

One- and two-dimensional electron-spin echo envelope modulation (ESEEM) spectra of Kramers’ multiplets in orientationally disordered systems are simulated using a simple mathematical model. A fairly general high-field spin Hamiltonian is considered with a general g-tensor and arbitrary relative orientations between all tensors involving the electron-spin S, the nuclear spin I, and their interaction. The zero field splitting (ZFS) and the nuclear quadrupole interactions are, however, approximated by their respective secular part in a way that retains all orientation dependencies and it is assumed that the nuclear quadrupole interaction is smaller than the hyperfine interaction. These approximations yield an effective sublevel nuclear Hamiltonian for each EPR transition and are sufficient to account for the most important characteristics of the ESEEM spectra of high electronic multiplets in orientationally disordered systems. Moreover, they allow to obtain some analytical expressions that for I=1/2 illuminate important aspects of 2D hyperfine sublevel correlation (HYSCORE) experiments in S=3/2, 5/2 systems. The pulses are considered as ideal and selective with respect to the different EPR transitions. The contributions of the latter to the echo intensity are weighed according to their different nutation angles and equilibrium Boltzmann populations. For simple axial cases with I=1/2, analytical expressions, analogous to the S=1/2 case, were derived for: (i) the modulation depth, (ii) the lineshapes of the HYSCORE cross-correlation ridges, and (iii) ENDOR powder pattern. Experimental results obtained from Mn(D₂O)₆²⁺ and VO(D₂O)₅²⁺ in frozen solutions are presented, compared, and analyzed in light of the theoretical part.     

Fine and hyperfine structure of the 22 P term of7Li; determination of the nuclear quadrupole moment

Microwave transitions between the Zeeman split doublet states 2² P _1/2 and 2² P _3/2 of⁷Li have been measured using the optical double resonance method. The fine structure separation was determined to bev _fs=10.053.184(58) MHz. The hyperfine structure was well resolved and led to considerably improved precision of the diagonal coupling constantsa _1/2=45.914(25) MHz,a _3/2=?3.055(14) MHz,b=?0.221(29) MHz. Furthermore, the contact, spin dipolar and orbital magnetic hyperfine constants are evaluated using additional results from level crossing experiments, which are more sensitive on the non-diagonal magnetic hyperfine constanta _3/2,1/2. The results are in good agreement with recent theoretical calculations. The nuclear quadrupole moment is derived fromb to beQ(⁷Li)=?41(6) mbarn.     

Frequency (250 MHz to 9.2 GHz) and viscosity dependence of electron spin relaxation of triarylmethyl radicals at room temperature

Electron spin relaxation times for four triarylmethyl (trityl) radicals at room temperature were measured by long-pulse saturation recovery, inversion recovery, and electron spin echo at 250 MHz, 1.5, 3.1, and 9.2 GHz in mixtures of water and glycerol. At 250 MHz T(1) is shorter than at X-band and more strongly dependent on viscosity. The enhanced relaxation at 250 MHz is attributed to modulation of electron-proton dipolar coupling by tumbling of the trityl radicals at rates that are comparable to the reciprocal of the resonance frequency. Deuteration of the solvent was used to distinguish relaxation due to solvent protons from the relaxation due to intra-molecular electron-proton interactions at 250 MHz. For trityl-CD(3), which contains no protons, modulation of dipolar interaction with solvent protons dominates T(1). For proton-containing radicals the relative importance of modulation of intra- and inter-molecular proton interactions varies with solution viscosity. The viscosity and frequency dependence of T(1) was modeled based on dipolar interaction with a defined number of protons at specified distances from the unpaired electron. At each of the frequencies examined T(2) decreases with increasing viscosity consistent with contributions from T(1) and from incomplete motional averaging of anisotropic hyperfine interaction.     

Superposition of scalar and residual dipolar couplings: analytical transfer functions for three spins 1/2 under cylindrical mixing conditions

The superposition of scalar and residual dipolar couplings gives rise to so-called cylindrical mixing Hamiltonians in dipolar coupling spectroscopy. General analytical polarization and coherence transfer functions are presented for three cylindrically coupled spins 12 under energy-matched conditions. In addition, the transfer efficiency is analyzed as a function of the relative coupling constants for characteristic special cases.     

Pulsed ESR investigations of anisotropic interactions in M@C82 (M=Sc,Y,La)

82 , Y@C₈₂, and La@C₈₂ in frozen solutions. We were able to determine the g tensors of these molecules by analysing magnetic field spectra at X-band (9.5 GHz) and W-band (94 GHz) frequencies. Moreover, in Y@C₈₂ we have investigated the hyperfine interaction of the ⁸⁹Y nuclear spin (I=1/2) with the electron spin on the C₈₂ cage. The principal values of the hyperfine tensor A and the relative orientation of g and A tensors were determined by applying three- and four-pulse electron spin echo envelope modulation techniques (ESEEM). Received: 3 September 1997/Accepted: 10 November 1997