Bohr model#Refinements

The Bohr - Sommerfeld model of the atom, Sommerfeld model of the atom or the Sommerfeld expansion is a physical description of the electron orbits in an atom. It has been proposed 1915/16 by Arnold Sommerfeld and represents a refinement of the Bohr model of the atom

Overview

The Bohr - Sommerfeld atomic model of 1916 is based on the Bohr model of 1913, making it one of the older quantum theories prior to the development of quantum mechanics. It is assumed that the electrons move around the nucleus in well defined paths, which initially result in classical mechanics from the equations of motion, ie the ellipses known from planetary motion. Quantum theoretical principles are introduced through additional quantization conditions ( Bohr -Sommerfeld quantization). This lead to only a small selection of tracks is allowed, which would be possible according to classical mechanics. As a consequence of this, associated with the orbital motion conserved quantities (energy, angular momentum ) do not accept any more, but only certain discrete values ​​, so they are " quantized ".

The progress of the Sommerfeld atomic model over its predecessor is, above all, that the fine structure of the hydrogen spectrum, ie the small splittings of the classically calculated energies, predictable makes ( fine structure constant ) by taking into account the equation of motion of special relativity. The fine structure is attributed to the increase in the inertial mass, which the spec. RT predicts for increased speed. At the same principal quantum number n, this effect is stronger the closer the electron is at perihelion flies past at the core, so the greater the eccentricity of the ellipse, or the smaller the orbital angular momentum. The different paths to a principal quantum number no longer have exactly the same energy level, but the energy is also on the orbital angular momentum dependent.

Another advance of the Sommerfeld model of the atom is that it can explain the normal Zeeman effect and the Stark effect.

The Bohr - Sommerfeld atomic model has high explanatory value because of its clarity; instead of the previously in Bohr's model justified single quantum number of the electron states is delivered correctly all three spatial quantum numbers and allowed for the first time at least a qualitative physical explanation of the periodic table of chemical elements.

However, the drilling - Sommerfeld model failed, as have been the Bohr model in all calculations of atoms with more than one electron. The fact that this failure of the mistaken belief stemmed defined, classical particle trajectories, was from 1925 clear when the new quantum mechanics could explain a lot more observations and make predictions, and these usually even quantitatively correct. In it there is no defined paths more, eg how to be seen from the Heisenberg uncertainty principle, but only probability distributions.

Geometry of the electron orbits

While in the model of Niels Bohr, the possible orbits of the electron are circles around the atomic nucleus, Sommerfeld introduced a more general elliptical orbits, as the special case of the circle still occurs. The core is based on this model in one of the two foci of an elliptical orbit, thus resulting in a geometrical configuration as the planet orbits by Kepler's laws. On these tracks should - move the electron without emission of electromagnetic radiation, such as would occur after the classical electrodynamics - as in the Bohr model.

Thus, the Bohr - Sommerfeld model represents a kepler cal planetary system in miniature, while the Bohr model of the older Copernican idea corresponds. This analogy is obvious, since the force fields of the Coulomb force of the nucleus and the gravitation of the sun have the same shape:

The consideration of the special theory of relativity resulting tracks approximately in the shape of an ellipse whose major axis is rotating slowly ( perihelion ).

Bohr-Sommerfeld quantization

An ellipse can not be more like a circle by a single parameter (radius ) are described, but which one requires two ( eg, large and small semi-axis ). Therefore, two quantum numbers are necessary in elliptical orbits to define the shape. A third is required for the orientation of the plane of the web in space. A quantum number of these, but only one can contribute the Bohr quantization of the orbital motion, because in the spherically symmetric potential of the nucleus all the tracks have a specific angular momentum. Sommerfeld generalized Bohr quantization effect that each coordinate must meet its own quantum condition:

It is

• A coordinate
• Their canonically associated pulse
• The quantum number
• Planck's constant.

The line integral is the area within the respective track in the plane. It is referred to in the mechanism as the effect that is associated with this movement.

In the 1 -dimensional case can easily be the x coordinate and the normal pulse. Then results from the quantum condition, for example, immediately the quantization of the harmonic oscillator with energy levels. By canonical transformation can, however, come to other variables, then automatically satisfy the same condition.

When moving in two or three dimensions one can eg choose a rotation angle, to which the canonical momentum then part of the angular momentum of coordinates. The action integral for a full orbit is then

And it results in the Drehimpulsquantisierung as in Bohr

Quantum numbers

Sommerfeld considers the system in the three spherical coordinates ( r distance and two angles ), and subjecting each of the new quantization. So he gets three quantum numbers: the radial, azimuthal and magnetic.

Principal quantum number

The quantum number n, which is determined here as in the Bohr model and the Rydberg formula, the energy is now called the principal quantum number and proves to be

Or actually as

Quantum number

The azimuthal quantum number, now called quantum number, are the (path ) of angular momentum ( is Planck's constant divided by. ) For a given n, the quantum number can take as values ​​the natural numbers from 1 to:

Where the maximum angular momentum ( ) belongs to the Bohr orbit. The value is explicitly excluded, because in this case the electron oscillates back and forth on a straight line that goes through the core.

After the quantum-mechanical calculation, the Bohr -Sommerfeld model replaced in 1925, the angular momentum is, however, by exactly one unit less and the proper value range thus

(also see adjacent figure). For this part of a spherically symmetric orbital.

Magnetic quantum number

The magnetic quantum number indicates the angle of inclination of the angular momentum to the z-axis, or to be precise, the size of the projection of the angular momentum in the z-axis:

The value range of this quantum number

Total different values. Thus, the directional quantization is predicted, because there are exactly parallel to exactly antiparallel only these finite number of possible settings. ( Quantum mechanics are thus the two extreme settings but not quite coincide with the z- axis of the angular momentum vector instead of the length. )

The moving around the nucleus electron forms a magnetic dipole whose direction is perpendicular to the orbital ellipse, ie parallel to the vector of the orbital angular momentum. One brings the atom in an external magnetic field ( the z- axis is defined ), then its energy depends also on the setting angle. Because of the directional quantization splits the energy depending on the value of the magnetic quantum number ( hence their name ) in different values ​​( Zeeman effect).

Spin quantum number

In addition to these introduced in Bohr-Sommerfeld 's atomic model spatial quantum numbers for each electron there is also the spin quantum number for the always exactly two settings of its intrinsic angular momentum (spin ). It is with the values ​​of ½ or - ½ specified or the symbols ↑ or ↓. This quantum number is not due to Sommerfeld quantization conditions, but was inserted into the model later due to otherwise unexplained experimental findings (eg even- splitting in the Stern - Gerlach experiment and the anomalous Zeeman effect). The energy of each of the previously mentioned tracks can be split into two energies.

Pauli exclusion principle

Due to the Sommerfeld by the model made ​​it possible order in the understanding of atomic structure was Wolfgang Pauli in 1925 to discover the Pauli exclusion principle: each of the determined by the three spatial quantum numbers car can seat a maximum of two electrons must have opposite spin quantum number.

Swell

• Arnold Sommerfeld: For the quantum theory of spectral lines (I II). In: Annals of Physics. 51, 1916, pp. 1-94. (instead of vol 51 is true at Wiley Online: Vol 356)
• Helmut Rechenberg Quanta and Quantum Mechanics: Laurie M Brown et al. (Ed.) Twentieth Century Physics vol. I, IOP Publishing Ltd.. AIP Press. Inc. 1995, ISBN 0750303530
• Friedrich Hund: History of quantum theory, BI university paperbacks Vol 200/200a, Bibliographical Institute Mannheim 1967