Elliptical orbits

We shall concenPate on the potential energy term of the nuclear Hamiltonian and adopt a sPategy similar to the one used in simplifying the equation of an ellipse in Chapter 2. There we found that an arbiPary elliptical orbit can be described with an arbiParily oriented pair of coordinates (for two degrees of freedom) but that we must expect cross terms like 8xy in Eq. (2-40)... 

I should also mention Sommerfeld, who extended Bohr s theory to try and account for the extra quantum numbers observed experimentally. Sommerfeld allowed the electrons to have an elliptic orbit rather than a circular one. 

EUipsenbohn, /. elliptic orbit, ellipsoidfdrmig) a. ellipsoidal, elliptisch) a. elliptic, elliptical. 

Now consider the hypothetical problem of trying to teach the physics of space flight during the period in time between the formulation of Kepler s laws and the publication of Newton s laws. Such a course would introduce Kepler s laws to explain why all spacecraft proceed on elliptical orbits around a nearby heavenly body with the center of mass of that heavenly body in one of the focal points. It would further introduce a second principle to describe course corrections, and define the orbital jump to go from one ellipse to another. It would present a table for each type of known spacecraft with the bum time for its rockets to go from one tabulated course to another reachable tabulated course. Students completing this course could run mission control, but they would be confused about what is going on during the orbital jump and how it follows from Kepler s laws. 

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

Here A is the distance of the foci, which are found on the. s 12-axis. For = 0we have plane polar coordinates. Varying v e [0,2n at constant u describes an elliptical orbit with a = yjA2 + u2 and b = u its semimajor and semiminor axis, respectively. 

Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics. Its initial applications to physics were in celestial mechanics, and its goal was to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets. Today, there is hardly a field of theoretical physics and chemistry in which perturbation theory is not used. Many beautiful, fundamental results have been obtained using this approach. Perturbation techniques are also used with great success in other fields of science, such as mathematics, engineering, and economics. 

We find Saurer 1 to be a member due to its Vgsr [2] did not correct this RV to Vgsr before incorrectly excluding Saurer 1 from the GASS member clusters. 2 It is noted from the spatial distribution of the clusters that GASS should have an elliptical orbit. If this change is made,... 

In general, if a particle is bound (E < 0) it will oscillate (classically) between some limits r = a, and r = b. For example, in an elliptic orbit of a hydrogen atom, the radius oscillates periodically between inner and outer limits. Only for a circular orbit is there no oscillation. Among the eigenvalues which have the same n, the one with lowest l has the largest amplitude in the vicinity of the nucleus. 

This is obviously positive for C positive. It has also the interesting property that, on the edge of the cluster where <1> tends to zero, the diffusion becomes more effective, because D becomes very large. This is a reasonable property of this model, since one expects elliptical orbits close to parabolic and thus large diffusion. 

The orbital or azimuthal quantum number (/) defines form (i.e., eccentricity of elliptical orbit cf Pauling, 1948) and indicates which sub-level is occupied by the electron. It assumes integer values between 0 and n —. ... 

To a first approximation, we may assume that the electrons follow elliptical orbits with the nucleus at one focus of the ellipse. This model of the atom thus resembles the Solar System, with the electrons in the role of planets and the nucleus standing in for the Sun. 

Which is all very well, but the Bohr atom is wrong. The picture of a dense nucleus surrounded by electrons is accurate enough, but they do not follow nice elliptical orbits like those of the planets. Venus and Mars follow Newton s laws, but electrons are governed by the... 

While the theory of Bohr was a major step forward, and it helped to rmderstand the observed hydrogen spectrum, it left many other observations in the dark. New light was shed on the subject of atomic structure and the line spectra by Arnold Sormnerfeld (1868-1951) (27). He elaborated the basic theory of Bohr by observing that the orbits eould also be elliptical, and that for each principal energy level, there eotrld be a specific number of elliptical orbits of different... 

Atomic. From spectroscopic studies, it is known lliat when an electron is bound to a positively charged nucleus only certain fixed energy levels are accessible to the electron. Before 1926, the old quantum theory considered that the motion of the electrons could be described by classical Newtonian mechanics in which the electrons move in well defined circular or elliptical orbits around the nucleus. However, the theory encountered numerous difficulties and in many instances there arose serious discrepancies between its predictions and experimental fact. 

Fig. 4.2 Classical orbits for planets or electrons. One circular orbit and two elliptical orbits are shown.

Bohr s theory was extended in various ways, especially by Somerfeld, who showed how to deal with elliptical orbits. There was a certain amount of qualitative success in applying the theory to atoms with several electrons. These developments in what is now called the old quantum theory were important as they laid much of the groundwork necessary for a correct theory. Ultimately, they were unsuccessful. Bohr s theory did not really explain what is going on why should only some orbits be allowed Where does the quantization condition (eqn 4.12) come from Following the developments of... 

Figure 19. It is assumed here that the concept of the ellipsoids of observation apply to all fields moving with the velocity of light, such as electric or gravitational fields. Thus, precession of the perihelion of Mercury (the rotation of its elliptic orbit) can be explained by the asymmetry of the gravitational forces as the planet advances toward (resp. retreats from) the sun.

There were lines in the spectra corresponding to transitions other than simply between two n values (cf. Eq. 4.14). This was rationalized by Sommerfeld in 1915, by the hypothesis of elliptical rather than circular orbits, which essentially introduced a new quantum number k, a measure of the eccentricity of the elliptical orbit. Electrons could have the same n but different k s, increasing the variety of possible electronic transitions k is related to what we now call the azimuthal quantum number, Z l = k — 1). 

The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n the azimuthal quantum number Z (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n Why should electrons spin, with only two values for this spin Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. 

FIGURE 5.6 For elliptical orbits, the sum of kinetic and potential energy is conserved, as is the quantity /. =mv1r (angular momentum). The orbiting body moves fastest at the minimum separation. 

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 radial quantum number (n ) was introduced to specify the eccentricity of elliptic orbits and an azimuthal quantum number (k) to specify the orientation of orbits in space. The three quantum numbers are related by... 

The Sommerfeld model for Ne is shown in figure 2.8. The He atom presented a special problem as the quantum numbers restrict the two electrons to the same circular orbit, on a collision course. One way to overcome this dilemma was by assuming an azimuthal quantum number k = for each electron, confining them to coplanar elliptic orbits with a common focal point. To avoid interference they need to stay precisely out of phase. This postulate, which antedates the discovery of electron spin was never seen as an acceptable solution to the problem which eventually led to the demise of the Sommerfeld model. 

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. 

The remarkable accord between the postulates of van t Hoff and Sommer-feld s elliptic orbits must, no doubt have convinced many sceptics of a more fundamental basis of both phenomena to be found in atomic shape. The new quantum theory that developed in the late 1920 s seemed to define such a basis in terms of the magnetic quantum number mi. 

---