Directed elliptic orbits

The Sommerfeld extension of the Bohr model was based on more general quantization rules and, although more successful at the time, is demonstrated to have introduced the red herring of tetrahedrally directed elliptic orbits, which still haunts most models of chemical bonding. 

Newton s laws can also be used to describe elliptical orbits, and it is then found that a vector quantity called angular momentum is conserved (always stays constant), just as the total momentum p stayed constant in a closed system. Angular momentum is conventionally denoted L. The vector L points perpendicular to the orbit (in the z-direction by our definitions) and has length... 

The vehement opposition generated by van t Hoff s proposals was evidently directed against the notion that atoms possessed non-spherical three-dimensional structures. The equivalent, more careful, formulation of Le Bel referred explicitly to the symmetry of molecules, such as methane, and is free of this criticism. It is unfortunate that it was the van t Hoff picture which became established, first in terms of Sommerfeld s elliptic orbits and later on in Pauling s hybrid orbitals. 

For propagation in an isotropic medium or along a pure-mode direction of a crystal (e.g., a plane of symmetry). Equation 3.38 reduces to a Rayleigh wave, characterized by having no transverse component Ux = 0. Since Uy and Uz are 90° out of phase, the particles move in an elliptical orbit in the sagittal plane die surface motion resembles that of the ocean under the influence of a passing wave. 

In wave mechanics the quantum numbers / and m, are obtained directly from the theory. In the Bohr treatment, on the other hand, the value of / was associated with the magnitude of the minor axis of the elliptical orbit, and hence the lowest possible value of I was i, since when / is zero the ellipse degenerates into a straight line and the electron during its motion would pass through the nucleus. Before the introduction of wave mechanics and the theoretical proof that / may have the value zero, it had been necessary to modify the Bohr theory, by putting / = o in order to conform with spectroscopic evidence. 

The physical interpretation of the quantum numbers / and m, whereby they are connected with the axes of the elliptical orbits and their orientation in space, is now no longer tenable and it is only possible to say that the principal quantum number n is related according to equation i. 8 with the total energy of the atom in a particular state, and the quantum numbers / and m are related to the angular momentum. The angular momentum is given by / (/ + 0 component in the direction of the z... 

The elliptical orbits can take on different orientations with respect to an external electric or magnetic field and the projections on the direction of the field also are quantitized and given by ... 

Outside the terrestrial atmosphere, the total flux of solar energy is approximately 1.4 kW/m2 in a plane perpendicular on the direction to the Sun. This value oscillates a few percent for various reasons. Because of the elliptic orbit excentricity of the Earth, the distance to the Sun varies by 3.3% and consequently, the insolation is 6.6% larger at perhelion than at aphelion. Besides the well-known 11 year periodicity of sun-spots, it appears that the Sun is an irregular variable star with a small luminosity amplitude. 

In space, to change the plane or direction of an orbit, a spacecraft must add a velocity component sideways, enough to make the resultant velocity vector point along the new direction. This is best done at the apogee of an elliptical orbit, where the orbital speed is lowest. 

Based on Sommerfeld s atomic model [11] of elliptic orbits, directed towards the corners of a cube, a number of chemists, including Kossel, Lewis, Langmuir and Bury, developed an electronic theory to account for atomic structure and valency at... 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

Circular orbits are defined by n = 0. The principal quantum number specifies energy shells. For n = 1 the only solution is n = 0, k = 1, which specifies two orbits with angular momentum vectors in opposite directions. The solutions n = 0, k = 2 and n = 1, k = 1 define 8 possible orbits, 4 circular and 4 elliptic. The angular momentum vectors of each set are directed in four tetrahedral directions to define zero angular momentum when fully occupied. Taken together, these tetrahedra define a cubic arrangement, closely related to the Lewis model for the Ne atom. 

But the history of the Milky Way may not be so simple. Detailed studies of the globular clusters in the halo and the different ages of stars in the halo and disk reveal some anomalies. For example, the contents of the halo do not always orbit in the same direction that the disk does. In addition, portions of the halo have very different ages. From this evidence, astronomers have concluded that the Milky Way may have formed as the result of a merger of smaller systems such as globular clusters or dwarf elliptical galaxies. 

In fact, the elliptical-cone coordinate dependent part of the Laplacian is identified as the square of the orbital angular momentum, via a direct comparison of Equations (28) and (47) ... 

The electron density can also be characterized by its ellipticity, the extent to which it deviates from cylindrical symmetry, reflecting the contribution of rr orbitals. While the C=C bond in ethyne is cylindrically symmetrical, the C-C bonds in ethene and benzene have greater extension in the direction of the rr component. Ellipticity is defined by... 

Polarization. The emitted synchrotron radiation is also highly polarized. For those x-rays emitted in the plane of the electron orbit, that is, in the direction = 0, the radiation is completely linearly polarized, with the electric vector of the radiation parallel to the electron orbit. For the x-rays emerging in the direction not exactly on the orbital plane, that is, j/ 0, the radiation is elliptically polarized, with a small vertically polarized component present. The polarization factor given in Equation (1.35) for the scattering of x-rays from an electron is valid only for unpolarized incident x-rays and has to be suitably modified for measurements with synchrotron radiation. 

Cyclotron radiation is emitted in all directions and has a frequency equal to the gyrofrequency. The radiation is polarized with the polarization depending on the direction of propagation. The polarization is circular when viewed along the direction of the magnetic field and linear when viewed in the plane of the orbit. At intermediate angles the polarization is elliptical. 

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