Dale W. Mitchell

Excitation electric field inhomogeneities in a cubic ion cyclotron resonance cell: ion motion for away from the cyclotron frequency

A theory of ion motion in a cubic ICR cell is presented for differential sinusoidal excitation that explains the observed stability, orders of magnitude, and resonance positions for excitation frequencies away from the cyclotron frequency. This is accomplished by transforming the equations of motion to an amplitude-phase representation where resonance is identified as terms with slowly varying phase. The assumption of isolated resonance reduces the analysis to a case-by-case study of individual resonances. Two classes of nonlinear resonances are identified: those dependent only on the excitation field inhomogeneities and those dependent also on the linear excitation. The latter resonances are negligible compared to the first class. For the strong resonances in the first class, resonance frequencies are predicted at 2ωz ± ω−, ω+ ± 2ωz, ω+ ± ω−, 2ω+ ± ω−, 3ω+, and 3ω−, where ω+, ωz, and ω− are the cyclotron, z-axis, and magnetron frequencies characteristic of the unperturbed motion, respectively. The amplitude—phase equations show that the z-node coupling coefficients are proportional to (mass)−12 while the radial modes are mass-independent in the low mass limit. Therefore, if the initial mode amplitudes are mass independent, mass discriminatory effects related to excitation inhomogeneities in quadrupolar traps are attributable to the z-mode. The coupling coefficient for the z-mode equation of motion has a relative order of magnitude approximately (ω+ − ω−)/ωz greater than the corresponding radial mode coefficients, hence the resonances at 2ωz ± ω− and ω+ ± 2ωz hve the strongest effect on ion motion. The stability of ion motion in the isolated resonance approximation is governed by a constant of motion relating the mode-amplitudes for the mode coupling resonances. The observed stability at ω+ — 2ωz and instability at w+ + 2ωz are explained by this invariant. Finally, FT-ICR double resonance experiments are used to show changes in the ICR signal for excitation near ω+ ± 2ωz, 2ωz, 2ω+, and 3ω+. Experimental results agree qualitatively with theoretical predictions. Excitation at ω+ − 2ωz shows a small increase in the FT-ICR signal due to reduction in the z-mode amplitudes. The resonances at ω+ + 2ωz and near 2ωz show a decrease in the signal due to z-axis ejection. Excitation at 3ω+ and near 2ω+ gave changes in the FT-ICR signal consistent with radial mode excitation.

z-axis oscillation sidebands in FT/ICR mass spectra

Abstract Modulation of the cyclotron and trapping frequencies arising from ion motions in a dual cubic-cell instrument produces sidebands in FT/ICR mass spectra that may be up to a few percent of the amplitude of the associated mass peak. One or two pairs of sidebands occur symmetrically disposed about the peak, with their frequency offsets directly proportional to the square root of the trapping voltage, as expected from theory. Two lines of evidence indicate that the pairs represent first- and second-order modulations: (1) comparison of the theoretical correlation between frequency offset and trapping voltage for the single-ion approximation with the correlation observed in experimental data over a large range of trapping voltages (0.25–10.0 V); and (2) narrow-pulse ejection experiments that identify twice the trapping frequency as the excitation signal that results in maximum energy absorption in the z direction. Under typical operating conditions, a sideband of one mass may occur at the same frequency as the peak of another mass, thereby giving a potential source of error in isotope-ratio measurements.

Averaging methods applied to quadrupolar FT-ICR perturbation problems

Abstract Many FT-ICR systems are approximately described by single ion motion in a constant magnetic field and a quadratic electrostatic potential. In this study, the quadrupole approximation—a magnetic field combined with a quadratic trapping potential—is considered the unperturbed problem while all other forces are treated as perturbations to this motion. Transformations are derived for the equations of motion describing the time variation of the constants of motion of the quadrupole problem. Reformulation of the quadrupolar problem in these variables can offer significant analytical and numerical advantages. In this form approximate analytical solutions to the perturbed problem are obtainable by averaging methods. Numerical integration of the transformed equations can give directly the time-dependence of the mode amplitudes and instantaneous frequencies. Applications of the theory include the derivation of frequency shifts, approximate invariants and the equations of motion describing non-linear resonance.

Theory of trapped ion motion in the non-quadrupolar electrostatic potential of a cubic ion cyclotron resonance cell

Abstract Perturbation theory is employed to analytically investigate the dynamics of a single ion trapped in a cubic ion cyclotron resonance cell. The trapping potential is expanded in a Taylor series with terms which are quadratic, powers of four and powers of six in coordinates treated as zero-, first-, and second-order perturbations, respectively. Frequencies are calculated to second order while the mode amplitudes and coordinates are solved to first order. The frequency shifts derived by this method agree very well with numerically evaluated frequency shifts obtained using the exact cubic cell trapping potential. For low m/z, the cyclotron frequency shift is shown to be due to just the cylindrically symmetric part of the cubic cell potential. An internal resonance involving energy exchange between the mode amplitudes is predicted to occur for an ion in a cubic cell when m ≈ mc/3, where mc is the high mass limit.

Electronic structure of Zn in oxide spinels

Using67Zn Mössbauer absorption and emission spectroscopy, we have investigated the electronic structure at the A and B sites in the normal spinels (Zn)[Al2]O4, (Zn)[Fe2]O4 and (Zn)[Ga2]O4. Within each system, the center shift Sc at the A site is more positive. In all systems investigated, the electric field gradientVu at the B site is negative. The values for SC andVU scale with oxygen nearest-neighbour distance to Zn. In the Fe spinel, a transferred magnetic hyperfine field is observed at the Zn site below the antiferromagnetic ordering temperatureTN=10 K. For a more detailed discussion of the chemical bond, we have performed ab initio Hartree-Fock cluster calculations for the Al and Fe spinels. Our experimental and theoretical results show that all hyperfine parameters are essentially determined by covalency effects. Our data on the Ga spinel raise the question of a partially inverse structure.

Initial relative ion abundances and relaxation times from apodized, segmented FT/ICR time domain signals

Abstract A Fourier transform ion cyclotron resonance (FT/ICR) time domain signal (TDS) can be simply described by a superposition of exponentially decaying sinusoids. If the mass components decay at different rates, however, then the FT magnitude peak heights give erroneous ratios of actual ion abundances. A method is presented to circumvent this difficulty that divides the original TDS into shorter (overlapping) segments as a function of the time of the first point, T i , in each segment. For a particular mass component, the peak height maximum, M , of the FT magnitude spectrum of each segment is related to T i such that a plot of ln [ M ] versus T i is linear, with slope and intercept proportional to the decay constant (inverse of relaxation time) and initial amplitude (TDS amplitude at beginning of detection), respectively. Inclusion of an arbitrary apodization function in the derivation of the equation for M leaves the slope of the above plot invariant but introduces a decay constant-dependent shift in the intercept. Numerical simulations using this scheme generally predict initial amplitudes and damping constants to better than 0.1%. Electron-beam ionization experiments on Kr ( P = 3.3 × 10 −7 torr) are compatible with exponential damping and suggest that the dominant isotopic fractionation mechanism in FT/ICR is dependent on ion number rather than mass. Application of the segmenting procedure to the Kr experimental data yields isotope ratios with greatly reduced scatter and apparently improved accuracy.

Dynamic short-range order observed in the (Zn)[Fe2]O4 spinel by neutron diffraction, μSR, and Mössbauer experiments

Using neutron diffraction (ND), muon-spin rotation/relaxation (μSR), and57Fe-Mössbauer spectroscopy (MS) we have investigated magnetic properties of the normal spinel (Zn)[Fe2]O4. In compounds which are slowly cooled from 1200°C to room temperature inversion is below detection limits. AtTN = 10.5 K the spinel exhibits long-range antiferromagnetic order (LRO). The transition as seen in thermal-scan spectra by MS is very sharp. However, ND andμSR experiments show that already at temperatures of ∼ 10TN a short-range antiferromagnetic ordering (SRO) develops which extends through ∼70% of the sample volume just aboveTN. BelowTN SRO and LRO coexist. At 4.2 K still ∼25% of the sample is short-range ordered. The regions over which the SRO extends have a size of ∼ 3 nm. Their fluctuation rates are in the GHz range. Modern ab initio cluster calculations successfully describe the magnetic hyperfine fields as well as the electric field gradient (EFG) tensor at the Fe sites. Covalency of the Fe-O and Zn-O bonds is important. The physical origin of the regions exhibiting SRO, however, remains unresolved at this point.

67Zn-Mössbauer study of nanostructured ZnO

Using the 93.31 keV Mössbauer transition in67Zn we have investigated hyperfine interactions and lattice-dynamic effects of nanostructured hexagonal (wurtzite) ZnO. The nanocrystals with particle sizes between 3 and 30 ran were produced by a reactive-gas sublimation method and were subsequently compacted into solid bodies under external pressure. The nanocrystalline materials were used as sources in Mössbauer emission experiments. A drastic increase of the asymmetry parameter was found fromη=0 (single crystal) toη≈0.5 (nanostructured material). Our theoretical Hartree-Fock cluster calculations show that displacements of the Zn atoms off the hexagonal symmetry axis as small as 0.005 Å can lead to such enhancedη parameters. The Lamb-Mössbauer factorf drastically drops when the particle size is reduced below ∼ 10 nm. Two lattice-dynamic models are discussed which successfully describe this behavior.

High-Pressure phase transition in67ZnO

Using a high-pressure Mössbauer spectrometer for the 93.3-keV resonance in67Zn and an X-ray diffractometer of the Guinier type67ZnO powder was investigated at pressures up to ≈19% at the phase transition. Surprisingly, the center shift (CS) of the cubic phase is by ≈20μm/s more negative than CS of the wurtzite structure. The Mössbauer data as well as theoretical cluster calculations show that the volume collapse is mainly due to the increase in coordination number of nearest-neighbour (nn) atoms. Covalency of the Zn−O bond, however, is reduced due to a≈7% increase of nn distance.

High-pressure phase transition in ZnSe

A high-pressure Mössbauer spectrometer for the 93.3-keV resonance in67Zn and an X-ray diffractometer of the Guinier type are used to investigate67ZnSe powder at high external pressures. At ∼ 13 GPa, a phase transition from the zinc blende to the fcc (NaCl) structure is observed. The transition is accompanied by a 15.2% decrease of the volume of the unit cell. The Lamb-Mössbauer factor (LMF) increases fromf=0.53% at ambient pressure tof=1.07% at 6.1 GPa. It thendecreases tof=0.91% as the pressure is further increased to 8.2 GPa. This behavior is investigated by lattice-dynamical calculations using an 11-parameter rigid-ion model. The decrease of the LMF at high pressures is caused by the softening of the TA phonon modes, which occurs already far below the phase transition. The s-electron density ρ(0) at the67Zn nucleus increases with reduced volume. The change of ρ(0) as well as theoretical Hartree-Fock cluster calculations show that covalency of bonding 4s/4p states of Zn and Se plays an essential role and determines the isomer shift.