Transmission coefficients for the exact triangular barrier: an exact general analytical theory that can replace Fowler & Nordheim's 1928 theory

In field electron emission theory, evaluating the transmission coefficient D ET for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for D ET , by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electron's outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and ‘flyover’ (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler–Nordheim 1928 formula for D ET ) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler–Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of D ET are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.

AN AB INITIO STUDY OF PROTON-TRANSFER REACTION IN Be2+(H2O)n AND THE SPATIAL CHANGING FEATURE IN THE FORMATION PROCESS OF HYDROXIDE

The proton-transfer reaction in Be 2+( H 2 O )n is investigated by an ab initio calculation. With an increasing number of water molecules, there are different formation processes of hydroxide, and the reaction barrier is dependent on the cluster size n. By MELD ab initio program and own-coding programs, we have calculated the potential acting on an electron within a molecule, and have investigated the changing of spatial appearance for the formation process of hydroxide, by the molecular intrinsic characteristic contour defined in terms of the classical turning point of electron movement.

THE CHARACTERISTIC BOUNDARY RADII OF ATOMS

The characteristic boundary radius of an atom has been defined as the distance from the classical turning point of electronic motion to the nucleus of the atom. With the ab initio method, the atomic boundary radii for elements from H through Xe are calculated. For the atoms in the same group, the radii defined in this way are of good linear relationship with the empirical radii commonly accepted, such as the van der Waals and covalent atomic radii determined by experimental data.

TUNNELING OF MACROSCOPIC UNIVERSES

The meaning of 'tunneling' in a timeless theory such as quantum cosmology is discussed. A recent suggestion of 'tunneling' of the macroscopic universe at the classical turning point is analyzed in an anisotropic and inhomogeneous toy model. This 'inhomogeneous tunneling' is a local process which cannot be interpreted as a tunneling of the universe.

THE INFLUENCE OF THE INERTIAL DRAGGING EFFECT ON GRAVITATIONAL RADIATION EMITTED FROM A PARTICLE MOVING IN EQUATORIAL GEODESIC CIRCULAR ORBIT ABOUT A KERR’S BLACK HOLE

Using Chranowski and Misner’s equations,1 gravitational radiation emitted from a particle moving in an equatorial geodesic circular orbit about a Kerr’s black hole is calculated. Outside the classical turning point, the radiation energy can be represented as a continuous function of orbital radius of the particle, and thus a corresponding curve is obtained. Using Wilkins’ approach,2 two inertial dragging related functions are obtained by restricting the orbit of the particle to the equatorial plane of a Kerr’s black hole. By comparing the curve of the gravitational radiation and the curves of the simulating functions (consisting of the angular frequency and the drag related function), we come to the conclusion that inertial dragging effect on a particle is one of the main factors that influences the gravitational radiation.

Vibrational relaxation with an anisotropic intermolecular potential

The quantum mechanical theory of vibrational excitation of a diatomic molecule by collision with an atom is developed so as to allow the molecular rotation to relax under the perturba­tion due to the angular dependence of the internal potential. In the impact region near the classical turning point the perturbed rotation is better regarded as pseudo-vibration, centred on the energetically favoured broadside configuration. For a thermal distribution over initial rotational states, the average excitation cross-section is shown to include a novel factor, f ‡ vib / f rot , where f rot is the initial rotational partition function and f ‡ vib is a partition function for the pseudo-vibration, with each term modified by a factor which represents the mean steric and anharmonicity correction for the state in question. Calculations on the H 2 + He system indicate that, particularly for heavy diatomic molecules, with a strongly anisotropic potential, the magnitude of the cross-section may be reduced by one or two orders of magni­tude by this modification of the theory, but that its temperature dependence is relatively unaffected.