Electron Probability Density Research Articles

Estimate of the escape time of resonant tunneling electrons from a quantum well in double-barrier heterostructures

An estimate of the escape time of a resonant tunneling wave packet from a quantum well (QW) of double-barrier heterostructures (DBH) is presented by using the properties of the propagator of the system. For large times, regardless of the detailed structure and composition of the system, the probability density of finding electrons inside a QW decays exponentially with time. The escape time is proportional to the reciprocal of energy width Γn of the quasibound states at resonance. The full width at half maximum of the transmission peak is just equal to the energy width of the quasibound states. However, the dwell time of electrons depends on the electron incident direction on the DBH for an asymmetric system. In the symmetric potential the dwell time is equal to twice the escape time.

Sub-band populations and the spatial distribution of electrons in GaAs/(Al,Ga)As modulation-doped quantum wells

High-field Hall and Shubnikov-de Haas (SDH) measurements on modulation-doped GaAs/(Al,Ga)As quantum well (QW) structures have been interpreted on a model in which an asymmetric potential distribution across the QW causes a well-width-dependent spatial separation of the probability densities of electrons in different sub-bands. In narrow wells, <150 AA, this effect is small, and the electron distribution can be considered as a single two-dimensional electron gas (2DEG) extending throughout the well, with a low mobility characteristic of the inverted (GaAs on (Al,Ga)As) interface. In the wide well samples, >or approximately=450 AA, the electron density in the lowest sub-band is spatially concentrated near to the inverted interface, while the second sub-band has a peak density close to the normal, high mobility, interface; the electrons in these sub-bands thus have different mobilities, and magnetoresistance effects characteristic of parallel conduction in two distinct 2DEG channels are observed. For intermediate thickness wells, approximately=250-300 AA, the two sub-band populations are close to those in the corresponding wide well samples, but because of the increased spatial overlap of the wavefunctions, the mobilities in the two subbands are similar. This results in a characteristic 'beating' phenomenon in the SDH curves. This model is shown to accord with published self-consistent calculations for modulation-doped quantum wells.

Optimization of a distributed Gaussian basis set using simulated annealing: Application to the adiabatic dynamics of the solvated electron

A molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many-body system. The instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions, each with equal amplitude. We present generalized equations of motion for the coupled system, which optimize the positions and widths of the Gaussians by simulated annealing. The condition of equal amplitude ensures the aggregation of the Gaussians in regions of finite electron probability density and hence yields a particularly efficient representation of localized ground states. The method is applied to an electron solvated in liquid ammonia and results for equilibrium properties are compared to quantum path integral calculations. New results for the dynamics are discussed in the light of mobility measurements.

Localization and ALCHEMI for zone axis orientations

Under zone axis diffraction conditions, small changes in orientation cause large changes in fast electron probability density ψψ ∗ on specific crystallographic sites. X-ray fluorescence spectra for GaAs near the 〈111〉 zone axis orientation were obtained to provide a simple yet severe test as to whether differences in localization are measurable when Ga and As project onto equivalent sites for beams in the zeroth-order Laue zone. Differences in the K- to L-shell fluorescence ratio occur for both Ga and As when ψψ ∗ on the atoms is varied by an order of magnitude, as do the respective As/Ga ratios for K- and L-shell excitation. Experiment is found to compare favourably with calculations based on a fully dynamical 49-beam (e, 2e) model for ionization. The “interaction potential” is plotted as a function of distance, and an estimate of impact parameter obtained from the (e, 2e) model. The effect of HOLZ beams is also discussed.

Non-zeroth-order Laue zone effects and alchemi

The depth-dependence of fast electron probability density is calculated for the 〈111〉 zone axis orientation of GaAs, and the effects of slight tilt from this orientation. Although Ga and As project onto equivalent sites for beams in the zeroth-order Laue zone, differences in mean probability density on Ga and As occur, due to higher-order Laue zone effects. Whether preferential peaking takes place on Ga or As depends upon which atom is assumed to be in the top surface layer. 64×64 beam multislice calculations are compared with 49-beam Bloch wave computations, in which the fast oscillatory depth-dependence of probability density is not accounted for. These results have implications for the interpretation of dynamical X-ray fluorescence effects, and particularly for site determination under zone axis diffraction conditions.

Evolution of electronic probability density in alpha -H collisions.

The evolving electronic probability density in 1-8-keV $\ensuremath{\alpha}$-H collisions has been calculated and graphed. This density is the modulus squared of the time-dependent electronic wave function in the previous coupled-molecular-state calculations of Winter, Hatton, and Lane without and with plane-wave translational factors. New, larger-basis results are also presented. The mechanism for electron transfer and the extent of sensitivity to the choice and size of the molecular basis are clearly seen in these graphs of the electron cloud.

Adiabatic molecular properties beyond the Born–Oppenheimer approximation. Complete adiabatic wave functions and vibrationally induced electronic current density

The conceptual and mathematical basis for describing adiabatic molecular behavior beyond the Born–Oppenheimer (BO) approximation is presented. The traditional nonadiabatic corrections to the BO approximation are shown to contain a kinetic energy induced adiabatic part and a complete nonadiabatic, or diabatic, part. The lowest order contributions to the new adiabatic terms can be described as momentum adiabatic and are complementary to the traditional BO position adiabatic terms. A complete adiabatic wave function comprised of contributions involving position and momentum is obtained that provides a complete description of the response of the electron probability density to the dynamics of the nuclear motion. This response is comprised of nuclear momentum induced electronic current density and nuclear displacement induced electronic charge flux density for all points in the Cartesian coordinate space of the molecule. These nuclear induced charge fluxes and currents are shown to obey an equation of continuity for the conservation of electronic charge. Relations between the position and velocity formulations of infrared absorption intensity and the charge fluxes and currents are provided.

ESR Studies on Platinum Uracil Blue

Abstract ESR spectra of platinum uracil “blue” are discussed. Two main types o f paramagnetic species have been found, one of which can be explained by assuming an unpaired electron spin delocalized over dz²-orbitals of two equivalent Pt centers, the other one is attributed to a more localized d7 Pt(III) center. In both cases the hyperfine structure suggests an additional smaller probability density of the unpaired electron at one adjacent platinum center. The smallest possible oligomers to realize this facts are tetramers in the first case and dimers in the latter one.

Quantum chemical approach to electron transfer in molecular systems

A theoretical treatment of electron transfer in and between molecules is presented. The nuclei are assumed to move classically. A time-dependent electron probability density is calculated using general equations given by Nikitin. Variational methods of different kinds may be employed in these equations. In the present paper some examples are studied at the extended Huckel level for the systems H2-H 2 + , H2-O2−-H 2 + , H2-S2−-H 2 + and H2-H2-H 2 + .

A statistical test of Anderson localization

Numerical demonstrations of localization in random systems are difficult to obtain and interpret because of statistical fluctuations in the electron probability density. This difficulty can be avoided through the use of correlation functions defined in terms of the electron probability density. The fluctuations can then be eliminated by averaging over a large number of Anderson Hamiltonians. The resulting averaged correlation functions clearly show that electrons are exponentially localized. The localization demonstrated here is sufficient to insure a zero dc conductivity in the limit of large systems.

Nuclear-Magnetic-Resonance Measurement of the Conduction-ElectrongFactor in CdTe

The effective $g$ factor ${g}^{*}$ of conduction electrons in degenerate CdTe has been determined by using measurements of the ${\mathrm{Cd}}^{113}$ nuclear spin-lattice relaxation time ${T}_{1}$ and the Knight shift $K$. It is shown that the magnitude of ${g}^{*}$ is given by the Korringa product ${T}_{1}T{K}^{2}=C{({g}^{*})}^{2}$, where $C$ is a known constant and $T$ is the absolute temperature, and that the sign of ${g}^{*}$ is given by the sign of $K$ for a spherically symmetric conduction band. The measured value, ${g}^{*}=\ensuremath{-}1.1\ifmmode\pm\else\textpm\fi{}0.1$, is within the range allowed by effective-mass theory. Also, the electronic probability density at the nucleus, normalized to unity in an atomic volume, is calculated to be ${|{\ensuremath{\psi}}_{F}(0)|}^{2}\ensuremath{\simeq}6.5\ifmmode\times\else\texttimes\fi{}{10}^{25}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, about 70% of that found for the free Cd ion in a $5s^{2}S_{\frac{1}{2}}$ state.

Nuclear magnetic resonance in liquid noble metal-tin alloys

Measurements have been made as a function of composition of the Knight shift K of the 119Sn resonance in the liquid alloy systems SnCu, SnAg and SnAu, and of the 63Cu resonance in SnCu. K( 119 Sn) increases monotonically with increasing noble metal concentration in all three alloys, the fractional change of shift increasing with the atomic number of the noble metal impurity. In SnCu the fractional changes of K( 63 Cu) and K( 119 Sn) with composition are equal in magnitude and of opposite signs. The results are interpreted in terms of the variation with composition of the conduction electron probability density at the nucleus. This has been calculated in two ways. The first method was a partial wave analysis of the impurity scattering while in the second method the probability density is obtained at different compositions by explicitly orthogonalizing the conduction electron pseudo wavefunctions to the ionic core states. Both methods give qualitative agreement between theory and experiment.

Aspects of the electron distribution in adenine, thymine and cytosine as given by probability density curves from nonempirical calculations

The electron probability density curves of adenine, thymine and cytosine are given and discussed.

Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermal-motion effects

The usual crystallographic structure-factor equation, with three positional and six anisotropic-temperature-factor coefficients, assumes that the thermal-motion probability density function is centrosymmetric. However, phenomena such as libration and anharmonic vibration can cause skewness. In this study, ten more coefficients per atom representing the third cumulant of the probability density function for thermal-motion are added to the structure-factor equation to permit a determination of the nature of the skewness. The Edgeworth series expansion based on the normal probability density function is used to analyze the results. The equations are generalized to include also the fourth cumulant, which describes kurtosis. The `cumulant-expansion model' for thermal motion is a statistical model without kinematic constraints and provides an unbiased estimate for the skewness of the density function for thermal motion. Application of the model to neutron diffraction data from crystals containing methyl groups (which are undergoing torsional oscillation) confirms the assumption that the density functions for the hydrogen atoms of a methyl group are skewed as an arc about the axis of torsional oscillation. The model has not been applied with X-ray diffraction data; if it were, the resulting parameters would describe the skewness of the combined electron and thermal-motion probability density functions.

Nuclear Magnetic Dipole Moment of ^165Ho

The first energy interval of the 4f116s configuration of Ho ii was recently found by Spector and Sugar to be 637.8 cm−1. The hfs of four of the lines defining this interval have been measured with an HYPEAC spectrometer at a resolution of 600 000. Seven independent equations relating the splitting factors a6s, a4f, and b4f to the measured hfs were derived. Known values of a4f and b4f from Ho i (deduced in an atomic-beam magnetic-resonance experiment) were inserted in these equations, giving a6s = 0.4437 cm−1 (corrected for core polarization) with a probable error of 0.0004 cm−1. From a6s, a value for the nuclear magnetic dipole moment μI = (3.94±0.05)μN was derived. The uncertainty of μI is the rms deviation due to uncertainties of a6s, and ψ6s2(0), the value of the probability density of the 6s electron at the nucleus.

Fading channel simulators

Summary form only given, as follows. In the light of a recent letter to the Editor (ibid., vol. 54. p. 1072, August 1966) it may be of interest to readers to know that water tank fading channel simulators of the type described in this letter have been in operation in the graduate electrical engineering laboratories both of the Polytechnic Institute of Brooklyn and of the Massachusetts Institute of Technology since 1964. The MIT unit was developed under the guidance of Prof. Jacobs and uses recirculated water jets as a source of disturbance. The PIB unit operates from 200 kHz to 4 MHz and uses both mechanical movement and/or air bubbles as sources of disturbance. The PIB unit has been described in the literature. Additional results from this unit are presented in a recent paper (An electronic probability density machine, IEEE Trans. on Insrrumention and Measurement, vol. 15, pp. 25-29, March-June 1966) describing a probability density machine that provides a useful adjunct for such a simulator. Several additional papers describing work done on this unit are now in preparation. The PIB simulator is used regularly both by graduate students taking our Graduate Communications Laboratory course and as a research tool by M.Sc. and Ph.D. candidates.

Isotope Shift in the Spectrum of Mercury Hydride*

The isotope shift in the HgH spectrum for Hg198H–Hg200H has been measured for 698 lines in the A2Π−X2∑+ system. The variation of the isotope shift with rotational frequency of the line is different for the various branches of a band, and greatly exceeds the variation predicted by the usual rotational and vibrational mass effects alone. This phenomenon is a consequence of the different nuclear volumes of the mercury isotopes. The results are evidence that the total electron probability density at the mercury nucleus increases with rotational frequency at a faster rate for the ground state than for the excited state.