Diamagnetic Term Research Articles

The effects of external magnetic field on the surface charge distribution of spherical nanoparticles

The finite element method has been implemented in order to investigate the electronic structure of spherical quantum dots (QDs) in an external magnetic field. The Schrödinger equation has been discretized by means of the Galerkin's weighted residue method with a nonuniform mesh of triangular elements. Unlike other approaches, the computational effort required to obtain converged results is independent of the strength of the magnetic field. The effects of the diamagnetic term on the energy levels and surface charge density distribution for semiconductor metal oxide QDs in alkaline aqueous colloids have been discussed.

DFT study of the NMR properties of xenon in covalent compounds and van der waals complexes-implications for the use of 129Xe as a molecular probe.

The NMR properties (chemical shift and spin-spin coupling constants) of (129)Xe in covalent compounds and weakly bound complexes have been investigated by DFT methods including relativistic effects. For covalent species, a good agreement between experimental and calculated results is achieved without scalar relativistic effects, but their inclusion (with a triple-zeta, double-polarization basis set) leads to some improvement in the quality of the correlation. The spin-orbit coupling term has a significant effect on the shielding constant, but makes a small contribution to the chemical shift. Coupling constants contain substantial contributions from the Fermi contact and paramagnetic spin-orbit terms; unlike light nuclei the spin-dipole term is also large, whereas the diamagnetic spin-orbit term is negligible. For van der Waals dimers, the dependence of the xenon chemical shift and anisotropy is calculated as a function of the distance. Small (<1 Hz) but non-negligible through-space coupling constants between (129)Xe and (13)C or (1)H are predicted. Much larger couplings, of the order of few Hz, are calculated between xenon and (17)O in a model silicate residue.

Magnetoexcitons from spatially separated electron–hole systems in semiconductor heterostructures with disorder

The effect of the diamagnetic shift due to a magnetic field on the formation of excitons is investigated for double layer semiconductor systems in which electrons and holes are spatially separated. The effect of disorder due to interface roughness of the double layer structures is also included in the calculation of exciton density. Numerical calculations are performed for type-II AlAs/GaAs by using an effective lattice-gas model where the electron–hole system is divided into unit cells. It is found that exciton density decreases under the influence of the diamagnetic term.

Landau diamagnetism of a weakly bound muonium atom

Abstract The ionization with temperature of weakly bound muonium atoms in undoped CdS has been studied using the technique of muon spin relaxation in a transverse magnetic field of 10 mT. For this atom the Coulomb binding energy between the muon and the electron is sufficiently small that the Landau diamagnetic term determines the magnetic behavior of the system: due to the diamagnetic interaction the muon precession in a transverse magnetic field exhibits a frequency shift of approximately 0.5% around the ionization temperature.

The computation of Karplus equation coefficients and their components using self-consistent field and second-order polarization propagator methods

The Karplus equation has been investigated by ab initio computation of the spin-spin coupling constants for a series of rotated ethane geometries. The couplings have been calculated at the self-consistent field (SCF) level as well as using the second-order polarization propagator approximation (SOPPA) and the second-order polarization propagator approximation with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)) and have been compared with results of previous calculations. The four principal components of the coupling constants rather than just the Fermi-contact have been calculated, and the common supposition that the Fermi-contact term is totally dominant has been confirmed. The derivatives of the orbital paramagnetic and orbital diamagnetic terms are significant but opposite in sign for the case of this rotation in ethane. It is found that the coefficients in the Karplus equation are largely overestimated at the SCF level, whereas the SOPPA(CCSD) results are in good agreement with coefficients derived from experimental coupling constant data or the results of multi-configurational self-consistent field (MCSCF) calculations. It is further observed that extending the Fourier series in the Karplus equation to include cos(3θ) and cos(4θ) terms neither significantly improves the quality of the fit nor significantly changes the values of the other coefficients. In order to simulate the Abraham and Pachler equation, calculations varying the nuclear charge on hydrogen have been performed. These will allow an abstract but flexible prediction of the effect of electronegative substituents.

Fine structure of excitons in a quantum well in the presence of a nonhomogeneous magnetic field

The trapping of excitons in a semiconductor quantum well due to a circular symmetric nonhomogeneous magnetic field is studied. The effect of the spin state of the exciton on its trapping energy is analyzed, and the importance of the interaction of the orbital and spin Zeeman effect as compared to the diamagnetic term in the exciton Hamiltonian is emphasized. Magnetic field profiles are considered, which can experimentally be created through the deposition of ferromagnetic disks on top of a semiconductor heterostructure. This setup gives rise to a magnetic dipole type of profile in the $xy$ plane of the exciton motion. We find that the spin direction of the exciton influences its localization by changing the confinement region in the effective potential. The exciton confinement increases with magnetic field intensity, and this is more pronounced when the exciton g-factor is different from zero. The numerical calculations are performed for GaAs/Al$_{x}$Ga$_{1-x}$As quantum wells and we show that it open up a new realistic path for experiments designed to probe exciton trapping in semiconductors.

On the contribution of thediamagnetic coupling term to the two-body retarded dispersioninteraction

The contribution of the leading diamagnetic coupling term tothe retarded dispersion interaction between a pair of neutralatoms or randomly oriented molecules in the ground electronicstate, is investigated using the multipolar formalism ofnon-relativistic quantum electrodynamics. Within thisframework the interaction energy is viewed as arising from theexchange of two virtual photons and is calculated usingconventional time-dependent perturbation theory. Explicitexpressions are presented for energy shifts involving theinteraction of a diamagnetic species with (a) anelectric-dipole polarizable body, (b) a magnetic-dipolesusceptible body, (c) an electric-quadrupole polarizable bodyand (d) another diamagnetic species. The resulting potentialsobtained are applicable for all separation distances beyondthe electron overlap region. Also given are the limiting formsof the energy shift at large and small inter-atomic/molecularseparations.

Some mathematical properties of gauge transformations with respect to the Coulomb gauge: Variational analysis of an energy functional

As gauge invariance of computed magnetic properties, usually partitioned into diamagnetic and paramagnetic terms, is not achieved within the algebraic approximation, unless ad hoc techniques are adopted, a general variational treatment is analyzed, attempting to minimize the term more difficult to evaluate accurately, i.e., the paramagnetic contribution to magnetic susceptibility, by means of a gauge transformation. It is shown that an absolute minimum in a variational sense cannot be determined a priori. However, a “local” minimum of the paramagnetic contribution to magnetic susceptibility can be arrived at by employing general gauge transformations of polynomial form. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 599–606, 2000

Empirical calculations of 29Si NMR chemical shielding tensors: A partial charge model investigation of hydrolysis in organically modified alkoxy silanes

The 29Si NMR chemical shifts for a series of organically modified silane compounds, MexSi(OR)y(OH)4 − x − y (OR = OMe, OEt), were evaluated using a partial charge model (PCM) approach to describe the chemical bonding. These PCM results allow the relative contributions of the diamagnetic and paramagnetic terms in the 29Si NMR chemical shielding constant to be discussed. An explanation for the unique variations in the 29Si chemical shifts during for hydrolyzed organically modified silanes is presented.

High-resolution laser spectroscopy of diamagnetic helium in the chaotic regime

The diamagnetic Rydberg atom, a quantum-mechanical system with chaotic dynamics in the semiclassical limit, is a prototype system to investigate ‘‘quantum chaos’’ @1,2#. Classically, irregular motion arises when the spherical Coulomb term in the Hamiltonian becomes of the same order of magnitude as the cylindrical diamagnetic term, a situation which can easily be created in an experiment with Rydberg atoms. The scaling properties of Rydberg atoms can be used to perform experiments under fixed classical conditions. In such experiments the field strength g (g5B/2.35310 5 T) is adapted to the laser frequency exciting the Rydberg atom ~excitation to energy E) in order to keep the scaled energy «5Eg 22/3 constant @3#. In the field-free case («52‘),

Beryllium-9 NMR Study of Solid Bis(2,4-pentanedionato-O,O‘)beryllium and Theoretical Studies of 9Be Electric Field Gradient and Chemical Shielding Tensors. First Evidence for Anisotropic Beryllium Shielding

Despite the favorable NMR properties of 9Be (I = 3/2), NMR spectroscopy of this nucleus in the solid state remains comparatively unexplored, perhaps owing to the extreme toxicity of beryllium and its compounds. We present here an integrated experimental and theoretical study of the Be chemical shielding (CS) and electric field gradient (EFG) tensors in bis(2,4-pentanedionato-O,O‘)beryllium [Be(acac)2]. Interpretation of the 9Be NMR data was facilitated by crystal X-ray diffraction results, which indicate two crystallographically unique sites (Onuma, S.; Shibata, S. Acta Crystallogr. 1985, C41, 1181). Beryllium-9 NMR spectra acquired at 4.7 and 9.4 T for magic-angle spinning (MAS) and stationary samples have been fitted in order to extract the nuclear quadrupole coupling constant (CQ), asymmetry parameter (η), and isotropic chemical shift (δiso). The best-fit nuclear quadrupole parameters for the two sites were determined to be CQ(1) = −294 ± 4 kHz, η(1) = 0.11 ± 0.04; CQ(2) = −300 ± 4 kHz, η(2) = 0.15 ± 0.02. Our analyses of the stationary samples also reveal a definite anisotropy in the beryllium CS tensor and allow us to place upper and lower limits on the spans of 7 and 3 ppm. This is the first evidence for anisotropic shielding in beryllium. Ab initio calculations of the beryllium CS tensors in Be(acac)2 at the RHF level indicate spans ranging from 7 to 9 ppm; this represents a substantial fraction of the total known chemical shift range for Be (<50 ppm). The calculated CQs are also in good agreement with the experimental results. To put the Be(acac)2 results in context, calculations of the beryllium CS tensors for a series of compounds encompassing the known range of 9Be chemical shifts are also presented. The calculations are in outstanding accord with experimental data from the literature. On the basis of calculations for linear molecules, it is shown that the assumption that the 9Be chemical shift is governed essentially by the diamagnetic term is erroneous. For some of these molecules, the calculated Be CS tensor spans are greater than the total known chemical shift range.

Relativistic theory of nuclear shielding in one-electron atoms 1. Theoretical foundations and first-order terms

The nuclear shielding in NMR spectroscopy is given in relativistic theory (Pyper, N. C., 1983, Chem. Phys. Lett., 96, 204) as the sum of the exact relativistic analogues of the non-relativistic diamagnetic and paramagnetic terms augmented by a further purely relativistic term. In this paper this theory, based on the Gordon decomposition of the Dirac current, is extended to encompass the point charge—point dipole model of the nucleus, the previous formulation (Pyper, 1983) being restricted to spatially extended descriptions of the nuclear charge and magnetization. For the point nucleus case, analytical expressions for both the purely relativistic shielding contribution and diamagnetic term are derived for all orbitals. The result for the diamagnetic shielding is derived by invoking the relativistic virial theorem, and thus depends only on the orbital energy and magnetic quantum number m. Therefore it shows, for given m, the same ‘accidental’ degeneracy between orbitals differing in the sign of the quantum number κ as the energy. The purely relativistic contribution to the shielding is shown to differ from the electronic contribution to the hyperfine energy by a factor of only m multiplied by fundamental constants. The present purely relativistic term is shown to equal the previous result (Pyper, 1983) even though appearing in a different form. For the interaction with a uniform magnetic field the relativistically exact Hamiltonian obtained from the Gordon decomposition is used with the virial theorem to derive a new expression for the g factor for all orbitals. The resulting simple expression depends only on the orbital energy in addition to the quantum numbers κ and m.

NMR Chemical Shifts. 3. A Comparison of Acetylene, Allene, and the Higher Cumulenes

The 13C NMR shieldings for acetylene, allene, and some higher cumulenes have been calculated at the MPW1PW91/6-311+G(2d,p) level using both the GIAO and IGAIM procedures. The calculations satisfactorily reproduce the tensor components of the shielding for both acetylene and allene, including the remarkably large difference in the components about the long axes of these molecules. A linear molecule such as acetylene has no paramagnetic shielding about the symmetry axis. The strong diamagnetic shielding about this axis for acetylene results in part from the energetically and spatially degenerate π-orbitals. With allene, the π-orbitals are energetically degenerate, but are not spatially degenerate, and as a result they cannot be mixed and cannot give a shielding diamagnetic term. Instead, in the presence of a magnetic field, they mix with virtual orbitals, leading to a downfield paramagnetic term. The shieldings for all the compounds are analyzed on an MO basis.

On the origin and contribution of the diamagnetic term in four-component relativistic calculations of magnetic properties

The relativistic Dirac Hamiltonian that describes the motion of electrons in a magnetic field contains only paramagnetic terms (i.e., terms linear in the vector potential A) while the corresponding nonrelativistic Schrödinger Hamiltonian also contains diamagnetic terms (i.e., those from an A2 operator). We demonstrate that all diamagnetic terms relativistically arise from second-order perturbation theory and that they correspond to a “redressing” of the electrons by the magnetic field. If the nonrelativistic limit is taken with a fixed no-pair Hamiltonian (no redressing), the diamagnetic term is missing. The Schrödinger equation is normally obtained by taking the nonrelativistic limit of the Dirac one-electron equation, we show why nonrelativistic use of the A2 operator is also correct in the many-electron case. In nonrelativistic approaches, diamagnetic terms are usually considered in first-order perturbation theory because they can be evaluated as an expectation value over the ground state wave function. The possibility of also using an expectation value expression, instead of a second-order expression, in the relativistic case is investigated. We also introduce and discuss the concept of “magnetically balanced” basis sets in relativistic calculations.

Influence of [formula omitted] and ∇ B drift terms in 2-D edge/SOL transport simulations

Classical particle drifts across the magnetic field can play an important role in tokamak edge-plasma transport. The relative influence of these terms is studied for self-consistent simulations by including them, together with anomalous diffusion transport, in a 2-D fluid model of edge-plasma transport for the DIII-D tokamak geometry. The drifts cause asymmetries in the plasma parameters between the inner and outer divertor regions which depend on the direction of the magnetic field, B. The basic results can be understood by dividing the drifts into three categories: diamagnetic, E×B , and ∇ B. The dominant effect near the divertor plates is from the E×B drifts, while the weaker ∇ B drifts cause an increase in the magnitude of the radial electric field inside the magnetic separatrix. The diamagnetic terms, defined here as divergence free, do not contribute to transport.

The high-frequency constitutive relation of axisymmetric toroidal plasmas

The response of a hot axisymmetric toroidal plasma to oscillatory fields is derived by integrating the linearized Vlasov equation along the unperturbed orbits written in the drift approximation. The details of the equilibrium configuration are taken into account with the same degree of accuracy as in the gyrokinetic approach; instead of approximating the kinetic equation, however, we approximate its solution. The integral constitutive relation obtained is valid to all orders in the Larmor radius, and for arbitrary frequency, except for the omission of relativistic effects. Finite Larmor radius wave equations adequate for the numerical simulation of ion cyclotron heating and current drive in tokamaks can be obtained from the general results by expanding to second order in the Larmor radius, while neglecting the perpendicular particle drifts and the diamagnetic terms in the equilibrium distribution functions. In the low-frequency domain, in which these effects are important, our wave equations are essentially equivalent to those obtained from a gyrokinetic equation. Some additional approximations, not explicitly stated in the gyrokinetic derivation although usually physically justified, are, however, required to prove this equivalence.

Anomalous photon energy and field dependencies of the Faraday rotation in ZnMnSe under high magnetic fields

Pulsed-field measurements of the Faraday rotation (FR) spectra over a wide photon energy range for (x = 0.015) were made up to 150 T at low temperatures by means of a streak spectrometer. We observed a reversal of the direction of the FR with increasing field. Dividing the Verdet constant into a paramagnetic term in the low-field range and a diamagnetic term in the high-field part, we found that the latter exhibits anomalies at around 2.4 eV.

The effect of static applied magnetic and electric fields on the dielectronic recombination of

A new calculation was performed of the dielectronic recombination rate coefficient for the ion, in cases for which static magnetic and electric fields are applied simultaneously in the collision region. The intermediate-state Hamiltonian was diagonalized in a basis of states of fixed n (principal quantum number of the high Rydberg state electron) and included the paramagnetic, diamagnetic and Stark terms. Our results showed enhancements of , when both magnetic and electric fields were present, beyond what occurred for an applied electric field alone. Generally, these additional enhancements depended nontrivially upon the angle between the two applied fields. Also, in a region of elevated magnetic field strengths , the diamagnetic term was found to play an important role, when .

Magnetic-field-induced linear Stark effect for doubly excited and states of

We have carried out a theoretical study for the lowest doubly excited Feshbach and resonance states of below the threshold under the influence of external parallel electric and magnetic fields. The method of complex-coordinate rotation is used. At first, when only the magnetic field is turned on, both the and states are shifted upward by the diamagnetic term, with the latter state being shifted faster than the former, resulting in the accidental degeneracy for these two states at a magnetic-field strength of about . With such a magnetic-field strength being fixed, and with the external electric field then turned on, the linear Stark effect for the and states is observed. Field effects on the resonance positions and widths are discussed.

Different scenarios of transitions into improved confinement modes

This review addresses transitions into regimes with improved confinement. The paradigm is tested on the example of L - H transitions. It is demonstrated that L - H transitions may emerge either due to an amplification of the diamagnetic drift term caused by enhanced pressure gradient or due to an increased poloidal rotation velocity at the separatrix. In general, it is asserted that the emergence and dynamics of transitions are very sensitive to fine details of prelude plasma profiles. The L - H transition occurs provided the radial electric field changes dramatically within a few poloidal gyro-radii from the separatrix. The adoption of a significant power flux to the separatrix as the trigger of the L - H transition appears from the theoretical point of view to be a simplistic `brute force' solution to the issue of confinement control. The propagation of the front of the barrier is very fast. The dynamics of interest consists of threshold conditions for barrier formation, barrier propagation speed and profile steepening rates, barrier limits and mechanisms for their relaxation and termination.